EPA-600/2-77-179d
August 1977
Environmental Protection Technology Series
PREDICTION OF MINERAL QUALITY OF
IRRIGATION RETURN FLOW
Volume IV. Data Analysis
Utility Programs
Robert S. Kerr Environmental Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Ada, Oklahoma 74820
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7. Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL PROTECTION TECH-
NOLOGY series. This series describes research performed to develop and dem-
onstrate instrumentation, equipment, and methodology to repair or prevent en-
vironmental degradation from point and non-point sources of pollution. This work
provides the new or improved technology required for the control and treatment
of pollution sources to meet environmental quality standards.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/2-77-179d
August 1977
PREDICTION OF MINERAL QUALITY
OF IRRIGATION RETURN FLOW
VOLUME IV
DATA ANALYSIS UTILITY PROGRAMS
by
Bureau of Reclamation
Engineering and Research Center
Denver, Colorado 80225
EPA-IAG-D4-0371
Project Officer
Arthur G. Hornsby
Source Management Branch
Robert S. Kerr Environmental Research Laboratory
Ada, Oklahoma 74820
ROBERT S. KERR ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
ADA, OKLAHOMA 74820
-------
DISCLAIMER
This report has been reviewed by the Robert S. Kerr
Environmental Research Laboratory, U.S. Environmental Protection
Agency, and approved for publication. Approval does not signify
that the contents necessarily reflect the views and policies of
the U.S. Environmental Protection Agency, nor does mention of
trade names or commercial products constitute endorsement or
recommendation for use.
-------
FOREWORD
The Environmental Protection Agency was established to coordinate
administration of the major Federal programs designed to protect the
quality of our environment.
An important part of the Agency's effort involves the search for
information about environmental problems, management techniques and
new technologies through which optimum use of the Nation's land and
water resources can be assured and the threat pollution poses to the
welfare of the American people can be minimized.
EPA's Office of Research and Development conducts this search
through a nationwide network of research facilities.
As one of these facilities, the Robert S. Kerr Environmental
Research Laboratory is responsible for the management of programs to:
(a) investigate the nature, transport, fate and management of pollutants
in groundwater; (b) develop and demonstrate methods for treating waste-
waters with soil and other natural systems; (c) develop and demonstrate
pollution control technologies for irrigation return flows; (d) develop
and demonstrate pollution control technologies for animal production
wastes; (e) develop and demonstrate technologies to prevent, control
or abate pollution from the petroleum refining and petrochemical
industries; and (f) develop and demonstrate technologies to manage
pollution resulting from combinations of industrial wastewaters or
industrial/municipal wastewaters.
This report contributes to the knowledge essential if the EPA is
to meet the requirements of environmental laws that it establish and
enforce pollution control standards which are reasonable, cost effective
and provide adequate protection for the American public.
William C. Galegar
Director
Robert S. Kerr Environmental
Research Laboratory
111
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PREFACE
This report is one of a set which documents the development and
verification of a digital computer modeling effort to predict the
mineral quality changes in return flows occurring as a result of
irrigating agricultural lands. The set consists of five separate
volumes under one general title as follows:
"Prediction of Mineral Quality of Irrigation Return Flow"
Volume I. Summary Report and Verification
Volume II. Vernal Field Study
Volume III. Simulation Model of Conjunctive Use and
Water Quality for a River System or Basin
Volume IV. Data Analysis Utility Programs
Volume V. Detailed Return Flow Salinity and Nutrient
Simulation Model
This set of reports represents the culmination of an effort started
in May 1969 by an interagency agreement between the U.S. Bureau of
Reclamation and the Federal Water Pollution Control Administration
on a joint research proposal on the "Prediction of Mineral Quality
of Return Flow Water from Irrigated Land." This research project
has had three different project identification numbers during the
project period. These numbers (13030 EH, EPA-IAG-048-(D) , and
EPA-IAG-D4-0371) are given to avoid confusion on the part of indi-
viduals who have previously tried to acquire project reports for
the earlier project numbers.
IV
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ABSTRACT
This volume of the report contains a description of the data
analysis subroutines developed to support the modeling effort
described in Volume III. The subroutines were used to evaluate
and condition data used in the conjunctive use model. The sub-
routines include (1) regression analysis, (2) Gaussian probability
function, (3) Beta distribution, and (4) Pearson's incomplete
gamma function. For each of these subroutines, a brief theory is
given plus a program listing and sample problem.
This report was submitted in fulfillment of EPA-IAG-D4-0371 by
the Bureau of Reclamation, Engineering and Research Center, under
the sponsorship of the Environmental Protection Agency.
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CONTENTS
Page
Introduction 1
Regression Analysis 2
Program Listing Regression Analysis 20
Sample Problem Regression Analysis (Ralston and Wilf). ... 47
Sample Regression Analysis (Hold Data) 56
Normal or Gaussian Probability Function 69
Program Listing Normal Probability Function 73
Sample Problem Normal Probability Function 77
Beta Distribution 85
Program Listing Beta Distribution 96
Sample Problem Beta Distribution 105
Pearson's Incomplete Gamma Function 177
Program Listing Incomplete Gamma 186
Sample Problem Incomplete Gamma 201
VII
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INTRODUCTION
Hydrology and related fields have a continued requirement for various
tools of modern statistical theory. Most of the tools are needed for
the examination, extension, and analyses (time series) of different
forms of basic or observed data.
Prior to the advent of the electronic computer the extensive use of the
techniques of modern statistics was prohibitive because of the many
laborious and tedious hand computations. However, as computer technology
advanced, resulting in greater computation speeds and larger rapid access
storage facilities, the usage of these statistical techniques grew as
various forms of computer applications.
It is obvious that no one tool will solve all problems encountered in
the analysis of hydrologic data. Further, a comprehensive computer
package that was portable and not exclusively propertied was beyond
the scope of a reasonable effort both in time and money. Within the
limits of a reasonable effort it was determined that a computer package
composed of computer applications of regression analyses, tools for
the solutions of the normal, beta, and gamma distributions would be a
good initial step. Many of the other tools of modern statistical
theory such as chi-square tests, Markovian analyses, and pseudo-distribution
functions, to name a few, are extensions of the tools contained in this
computer package.
In the subsequent narrative a brief description is given of some of the
applications of these statistical tools in the field of hydrology.
1
-------
REGRESSION ANALYSIS
The computer application of this regression analysis has Been designed
to be interactive with a terminal. The interactive feature provides
easy access to the computer and reduces the narrative required in terms
of a user's guide.
Regression analyses are used extensively in the field of hydrology in
the manipulation and extension of data. In this respect, the computer
application was adapted to fit the more common needs of the practiced
hydrologist with some fundamental skill in statistics. This adaptation
does not, however, preclude other uses of the regression analysis for
problem-solving in areas not related to hydrology.
The subsequent discussion will concern the various features of the
computer application and.where necessary.a brief explanation of statistical
theory will be included.
Data Input
The input data are limited to a maximum of 11 variables, one of which
is the dependent variable and the remaining 10 are the independent
variables. The maximum number of observations (data points) is 500.
The two maxima were chosen to fit the majority of hydrology applications.
Each variable must have a title and must also have a reference index of
at least one and not more than two digits.
-------
The entry of the above data can be accomplished by any one of two
choices. When the sample size is relatively small and the number of
variables are few, entry of the data via the terminal is more expedient.
Data entered in this manner will be formatted internally and written
on an output file labeled OUTO. The internal format used in this case
is compatible to that required when data are entered via a read device.
The remaining means of data entry is via a read device. The input file
name for the read device is labeled OUTA. Entry of data via the read
device is more expedient when the sample size (number of data points)
is large, the number of variables many, and where variables are trans-
formed prior to entry into the regression analysis. Surface responses
are cases in particular where simple stand-alone mechanisms would
be a judicious choice of transformation of variables prior to data
entry.
The format required for entry of data concerning the variables is as
follows:
Variable title (first line) - 1X,7A10,A9
Variable index and number of data points (second line) - 9X,I3,6X,I3
Data points for variable (third, ....lines) - 1X,6F13.0/(1X,6F13.Q)
Data points must include decimal point in field, if applicable.
The title of the variable is stored in a two-dimensional array labeled
LTITLE, the index of the variable is stored in a single-dimensional
-------
array labeled MVAR, the data points are stored in a two-dimensional
array titled XOB, and the number of data points are stored in a single-
dimensional array labeled NPX. A check is made to determine if all
variables entered have the same number of data points and when this
check fails, a program-controlled error message is displayed at the
terminal, otherwise the number of data points entering the regression
analysis is stored with the label NP.
All other information pertinent to the regression analysis is entered
via the terminal as responses to queries displayed at the terminal.
Whenever the query requires a "yes" or "no" response, any single
character will suffice for the "no" response; however, the "yes"
response must be the three alpha characters YES.
The query format as displayed at the terminal is self-explanatory and
the response required is direct, hence any discussion in this respect
would be redundant.
Output Files
The computer application for the regression analysis has provisions for
three output files. The file assigned the label OUTPUT is used for all
queries displayed at the terminal as well as the display of all error
messages displayed at the terminal that are program generated.
The output file assigned the label OUTA is used to store the formatted
data, i.e., variable name, index, and data points as accepted via terminal
input.
-------
The output file assigned the label OUTB is used to store the results
of regression analysis, the intermediate matrices and the residual
lists. Options are provided to write the intermediate matrices and
residual lists. The intermediate matrices are the (1) raw sums of
squares and products (A. matrix), (2) covariances (C matrix), and
(3) simple coefficients of regression.
The residuals can be listed at each, step of the regression analysis.
A list of variables from the input set will always be included as part
of this output file.
Program Structure
The program (electronic computer application) structure is composed
of the main program and four subroutines and requires approximately
115,000,, words to load. The program can be used as a stand-alone
application or can be used as five subroutines where four remaining
subroutines are monitored by the main subroutine or program.
Main Program - REGRES
The main program monitors all subroutines and nearly all of the
terminal queries are contained in the main program.
Problem Title
The problem title can be represented with at least one line (79 or
less characters per line) or with as many as five lines. The
number of lines are stored under the label NTTTLE and the lines of
-------
title information are stored in a two-dimensional array labeled
KITTLE. The lines are entered left justified and will Be
centered internally on the basis of an 80-character line.
Variable Sequence
The variable sequence is controlled by the order in which each
variable is entered at time of input either via terminal or read
device. The variable sequence is stored in a single-dimensional
array labeled KRANK. The variable sequence can be changed and only
selected variables within the input data set will be allowed to
enter the regression analysis. When selected variables are not a part
of the input data set, a program-controlled error message will be
displayed at the terminal. The index of the dependent variable is
stored under the label KVAR.
Variable Transformation
The purpose of variable transformation is to use a simple linear
regression model in terms of the transformed variables rather than a
more complicated model in terms of the original variables. When a
nonlinear model, defined as nonlinear in parameters to be estimated,
can be expressed by adequate transformation of variables in the form
of a linear model, it is referred to as being intrinsically linear,
and such is the limitation of suitable transforms. This particular
computer application of a regression analysis has purposely been
further constrained by allowing only six rather simple transformations
-------
to be accomplished internally. These six transforms are as
follows:
Form Alpha configuration
X'±. - log X.. LOG10
X' = In X±. LOGE
X1 . = /1T7~ SQRT
x'ij = Xi1 EXP
X1 = sin X.. SIN
X' .. = cos X. . COS
13 13
The above transforms are those most commonly used in hydrology where
the sine and cosine transforms are a bit far out and the log and In
transforms seem redundant where one can be expressed in terms of the
other. In any event, the computer application can be easily modi-
fied to discard and replace some of the six transforms listed
above or number of transforms performed internally can be expanded
to as many as 10.
The alpha configurations of the transforms are stored in a single
dimensional array labeled ITMNS.
When it is desirable to represent a "kthu order model - single
independent variable, another alpha configuration is used. This
alpha configuration is the three-character signal POW. The
response to this signal will be a query concerning the degree
(order) of the polynomial.
-------
The exponents or order of polynomial are stored in the single
dimensional array EXPO.
The use of transformed variables should Be investigated with an
examination of the residuals as obtained as a result of the
regression equation. An aid in this examination has been provided
by listing the residuals when the dependent variable has been
transformed in terms of the original independent variable (exception
sine and cosine).
F Levels
Each of the observations of the dependent variable are random
variables, hence all variables that are functions of this set
are also random variables. These two functions are the mean square
due to the regression and the mean square of the residuals. Both
functions have particular distributions, means, variance, and
moments. Assume that the errors (deviations) e. are independent,
i.e., N(0,a2) variables, and also that it can be demonstrated that
mean square due to the regression multiplied by its degrees of
freedom (regression) and the mean square of the residuals multi-
plied by its degrees of freedom (residual), both will follow a
chi-square distribution. It can also be shown that the two variables
are independent. Without further discussion it can be said that:
_ mean square due to regression
mean square due to residuals
represents the F ratio and said ratio follows an F distribution.
-------
Nearly all statistical texts or handbooks have tables for the
1 percent (99 percent) and 5 percent (95 percent) F levels.
The regression analysis as represented by this computer application
has provision for the entry of F levels. The F level to enter a
variable in the regression is stored under the label FLEVEN and
the F level to exclude a variable from the regression is stored
under the label FLEVRE.
When no F level is given the variables enter the regression in the
sequence stored in the array KRANK.
The F level to exclude a variable from the regression should always
be less than or equal to the F level to enter a variable in the
regression.
It is suggested that one assume two degrees of freedom for the regres-
sion and twenty degrees (>20 data points) to estimate F levels.
Under such an assumption use F level ^3.5 for 5 percent (95 percent)
and F level ^6.0 for 1 percent (99 percent).
-------
Ranking of Data Points
This application has the provision to rank the data points (observa-
tions) by magnitude. In some hydrology studies this sort of license
is taken without statistical justification. One would have to
possess a keen knowledge of the observed data to use this facet with
a linear model. However, the mechanism to do the ranking is a good
tool in setting up distribution studies.
The alpha signal to rank in ascending order of magnitude is LOW and
to rank in descending order of magnitude the alpha signal is HIGH.
These alpha signals are stored in a single dimensional array labeled
IRANK,
Miscellanea
The number of variables entered during input is stored under the
label KCONT. The number of variables in regression analysis
is stored under the label KNUM. Intermediate storage of ranked
data points is provided by the single dimensional array labeled
XSORT.
Subroutine VARTRN
This subroutine is a simple setup mechanism for the six transformations
The argument list is as follows:
KA = Order of variable to be transformed
NUM = Number of data points
KTRAN = Order in the transform list
EXPP = Exponent, if any
10
-------
All variables are shoved through this subroutine even though no
transform is desired. The data points for each variable (transformed
or raw) are stored in a two dimensional array labeled XRT. All data
ranking, if required, is completed before entry to this subroutine.
Subroutine PRESET
This subroutine is used for the sole purpose of setting the indices JB
and KB in preparation of the quadratic sort mechanism used in ranking
the data points by magnitude.
Subroutine SORT
This subroutine is used to rank the data points in terms of magnitude.
The subroutine uses a quadratic sort mechanism. Detailed discussion
of this subroutine is not considered pertinent with respect to the
regression analysis.
Subroutine REGCOR
This subroutine contains the actual computational sequence of the
regression model. Before discussion of the details of this subroutine
it might be wise to begin with a little of the philosophy of the linear
model where the model has the following form:
y = a0 + axx + a2X2 + . . . + s^x^ + e (0 < n < 11 )
where: e represents the residual and every other
term on the right side of the equation
represents the regression.
11
-------
i'he fundamental notion is to complete the regression of all the
variables as a series of straight line regressions where each of
the straight line regressions represents a step,or as some say, stage.
The first computational effort is to determine the previously mentioned
A matrix which contains the sums and cross products of all the variables
in the regression, including the dependent variable.
The second computational effort is to determine the covariances of all
the variables, which is the previously mentioned C matrix.
The third computational effort is to determine the simple correlation
coefficients of all the variables, which is the previously mentioned
R matrix.
The next and final computational effort prior to the initiation of the
straight line regressions or steps is to augment the R matrix in the
following manner (n = number of independent variables):
B =
R(n x
T( 1 V
\ -L A
-I(n x
—
n) !
n) .
n) .
T'(n
S(l x
0(n x
x 1) !
1) .
1) .
I(n x n)
0(1 x n)
0(n x n)
12
-------
where: R(n x n) = matrix of simple correlation coefficients
T(l x n) = correlation vector with response to dependent
variable
-I(n x n)
T' (n x 1)
S(l x 1)
I(n x n) = identity matrix
negative identity matrix
transpose of T
correlation of response with itself (R ~= 1)
F yy
As an example, consider the following model:
y = aQ + alXl
a, x.
4 4
then:
Rll
R21
R
31
R
4 1
R .
7l
-1
0
0
0
R12
R22
R
32
R, .
42
R
72
0
-1
0
0
R13
R23
R
33
R
4 3
R
73
0
0
-1
0
R14
R24
R
R
|L |l
_
R
74
0
0
0
-1
R17
R2y
R
37
R
4y
L _
= 1
0
0
o
0
1
o
0
0
0
Q
0
0
0
0
1
Q
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
the B matrix is used to determine the F level of each of the independent
variables and choosing that independent variable with the highest F
level to enter the regression, if this F level is equal to or greater
than the critical F level for entry. The computation of the F level
is as follows:
13
-------
Let : V be a measure of variance
DEGF - degrees of freedom of the residual
then: V. = (Bi>n+1> (B±jn+i)/B. )± i - i, .... n
F.=v. (DEGF)/(Bn+1>n+1- V.) 1-1, .... n
Initially the degrees of freedom for the regression equals zero and
the degrees of freedom for the residual = NP-1, where NP = number of
data points (observations) .
The mechanism used in updating the B matrix (augmented R matrix) as each
variable enters the regression and the updating of the B matrix after
a variable has been excluded from the regression will be discussed in
a subsequent portion of this narrative.
Continuing with the philosophy, it might be well to examine a few
transformations of variables and the resulting models.
No Transformations (1)
y=a + a x + e - one independent variable
y = a + a Xi + a->xo + e ~ two independent variables
y = a + a x + a x +...+ ax + e - n independent variables
Logarithmic Transformations (2)
Iny = Ina + a Inx + e "1
01 >• one independent variable
logy = log aQ + a logx + e J
y = a + a Inx + a Inx + e - two independent variables
14
-------
Square Root Transformations (3)
1/2 1/2
y = a + a,x + ax + e - two independent variables
One Independent Variable(4)
o
y=a + a,x + ax+e- second-order models
y = a + ax + ax2+...+a_xk+e- kth order models
J O I 2 K.
Two Independent Variables (5)
2 2
y - aQ + a1x1 + a2x2 + a^ +a4x2 + a^x^ + e ~ second-order
models - surfaces
y = ao + axx + a^ + a^ + a^Xlx2 + a^ +
aexf + a_xfx0 + aDx x. + ax + e - third-order models -
bl /iz o i ^ yz
surfaces
The above examples of transformations are but a few of the many that are
possible. However, the examples should display the general idea of
transformations.
Transformations (1), (2), (3), and (4) are handled by the computer
application. Transformation (5) can be accomplished by manipulation
of the input data file with a stand-alone technique. It can be seen
that a third-order model of a surface would be near the limitation of
10 independent variables.
15
-------
Algorithms - Updating B Matrix
Let k = index of variable entering regression
n = number of independent variables
i = 1,2, ..., 2n-l
j = 1,2, ..., 2n-l
B.. =Dkj/Dkk *
Dij -
Let k = index of variables excluded from regression
n = number of independent variables
I = k + n+1
m = 2n-l
i = 1,2, .. . , 2n-l
j = 1,2, ..., 2n-l
Update ktn vector
i = 1,2, . .., 2n-l
B., = Dik - D. ID
ik 1K im £m
Note: The D matrix is an exact image of the B matrix from
the previous step (stage)
16
-------
Labels
Item Array
A matrix A(ll,ll)
B matrix B(21,21)
C matrix C(ll,ll)
D matrix 0(21,21)
R matrix RC11,11)
Independent variables X(11,500)
Dependent variables Y(500)
Students T value STUDT(ll)
Standard deviation STDEV(ll)
Partial F values PARFL(ll)
Regression coefficients REGCOF(IO)
Variables in regression KIN(11)
F levels FLEV(ll)
Variables not in regression KOUT(ll)
Original variable index KSETS(ll)
Variances V4(ll)
Degrees of freedom - regression NSTEP
Degrees of freedom - residual IDEG.DEGF
Step or stage of regression NSTEP
All other labels are self-explanatory.
17
-------
Statistics
Let n = number of independent variables
m = n+1
I, C =
Original variance (ORGVAR) independent variable = C
mm
Percent explained variance (PEREXP) = (1 - B )(100)
mm
Percent unexplained variance (PERUNE) = B (100)
Multiple correlation coefficient (RMULT) = / 1.0 -
Residual sum of squares (RESQR) = C
Standard error of residuals (STDRES) = VRESQR/DEGF
Let k = index of variable
£ = k + m
KJI = number of variables in regression
i = 1,2, ..., KJI
Regression coefficient (REGCOF(i)) = B. ., /t;
im V ii
•n
Standard error (STDREG(i)) = STDRES ..
2
Partial F value (PARFL(i)) = DEGF (B. ) /B .B
im mm ££
The partial F value is tested against critical F level to exclude
a variable from the regression because of a nonsignificant contribu-
tion to the response of the dependent variable.
18
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Regression Analysis References
1. N. R. Draper, H. Smith, Applied Regression Analysis,
Second Printing, 1967, John Wiley and Sons, Inc.,
New York, London, Sydney.
2. R. S. Burington, D.C. May, Jr., Handbook of Probability
and Statistics with Tables, 1953, Handbook Publishers, Inc.,
Sandusky, Ohio
3. M. A. Efroymson, "Multiple regression analysis," Chapter 17,
Mathematical Methods for Digital Computers, Ralston and
Wilf, Second Printing, 1962.
19
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PROGRAM REGRES 7i./7<. OPT = 1 FTN 1..2+P380 75/02/28. 13.13.13. PAGE
PROGRAM REGRES(INPUT,OUTPUT,OUT A,OUTB,OUTD,TAP£5=OUTA,TAPEG=OUTB,
1TAPE7 = OUTD)
COMMON/VARBLS/ XOB(11,500), XRTfll,500)
COMMON/TITLES/KTITLE(5,8>, LTITLEtll,8),NTITLE
5 COMMON/INDICE/KCONT,KNUM,IDY,FLEVEL,NP,CIT,COT,IXR
COMMON/KE-VS/IRANK(il> ,ITRANS(11),EXPC(11) ,KSORT (11) ,KRANK(11)
DIMENSION MTEST(IO) , NPX(11>,IFQRM(2>,ITEM(8>,KTEM(8),XSORT(500)
DIMENSION MVARU1) ,NFCRM<2>
INTEGER CIT.COT
10 DATA((MTEST(K),K=1,10)=5RLOG10,5RLOGE ,5RSQRT .5REXP , 5RSIN ,
1 5RCOS , <.*5R )
DATA(KTEST=310523B)
CIT=5 $ COT=f> $ IXR=7 t ISH=0
PRINT 10
15 10 FORMAT(2X»THIS IS A REGRESSION ANALYSIS PACKAGE WITH*
1 /2X'A MAXIMUM NUMBER OF ELEVEN VARIABLES*)
ISH=0
KCQNT=C
00 20 K=l,ll
20 MVAR(K)=0
NPXtK)=0
IRANK(K)=0
ITRANS(K)=0
EXPO(K)=0.0
25 KSORT(K>=0
KR«NK!K)=0
20 CONTINUE
30 FORMAT(R3)
35 PRINT 37
30 37 FORMAT(2X»00 YOU WANT TO ENTER ANOTHER VARIABLE - YES OR NO*)
READ 30.ITEST
IF(ITEST.NE.KTEST) GO TO 100
PRINT 39
39 FORMAT(2X*00 YOU WANT TO ENTER VARIABLE VIA TERMINAL-YES OR NO*)
35 READ 30.ITEST
IFUTEST.EQ.KTEST) GO TO 59
= IF(ISW.NE.l) GO TO *t3
PRINT *
(,5 2 /2X»DECIM.«L POINT MUST BE IN FIELD*)
<*6 READ(CIT,<«9iITEM
1.9 FORMAT (IX,7A10.A9)
IF(EOFICIT)) 35.51
51 READICIT,53)INDEX,NA
50 53 FORMAT(9X,I3,6X,I3>
KA=KCONT=KCONT*1 $ MPX(KA)=NA t MVAR(KA)=INDEX
IF(KCONT.GT.ll) GO TO 1020
DO 55 K=l,«
LTITLE(KA,K)=ITEM(K)
55 55 CONTINUE
REAO(CIT,57)(XOB(KA,K),K=1,NA)
57 FORMAT!1X,6F13.0/1IX,6F13.0))
-------
PROGRAM REGRES 7<»/7l» OPT*1 FTN
60 CO TO 1(6
59 PRINT 60
60 FORMAT(2X*TYPE IN INDEX OF VARIA8LE-(1-11)*>
KCONT=KA = KCONTU t IF(KCONT.GT.11) GO TO 1020
READ*,INDEX
65 PRINT 65
65 FORMAT(ZX*TYPE IN TITLE OF VARIABLE*)
READ 67,
67 FORMAT<7A10,ft2>
70 MVAR 95,80
80 JX=JXH
XOB(KA,JX)=TEMPX
GO TO 75
85 FORMAT(1X*INDEX = »I3,6X,I3)
80 90 FORMAT(1X,6F13.6/(1X,6F13.6))
95 NPX(KA)=JX
HRITE(IXR,85)INDEX, JX
HRITEUXR,90> (XOBCKA.K) ,K=i,JX)
GO TO 35
100 PRINT 102
102 FORMAT(2X*ARE ALL VARIABLES ENTERED TO BE IN ANALYSIS WITHOUT*
1 /2X»TRANSFORMS OR RANKING-YES OR NO*)
READ 30,ITEST
IF(ITEST.NE.KTEST) G.O TO 10<(
90 DO 103 K=1,KCONT
KRANK(K)=MVAR(K>
ITRANS(K)=5ROUT
103 CONTINUE
KNUM=KCONT
95 GO TO 160
10
-------
PROGRAM REGRES 7I./71. OPT = 1 FTN i».3*P3BO 75/02/2B. 13.13.13. PAGE
115 135 KRANK(KNUM)*KVAR(KA)
PRINT lf.0
lltO FORMAT (2X»ENTER TRANSFORM OR (CR>»)
READ l
GO TO 153
11.9 IFCJTEST. NE.378) GO TO 153
130 PRINT 151
151 FORMAT(2X»ENTER DEGREE OF POLYNOMIAL*)
READ'.IDEG
EXPO(KA)=IDEG
153 PRINT 155
135 155 FORMAT(?X'ENTER RANKING INDEX(HIGH,LOH OR »)
READ 157,IRANK(KA)
IF(EOF(5LINPUT))156,158
156 IRANK(KA) = GO TO 110
160 PRINT 165
165 FCRMAT(ZX»ENTER INDEX OF DEPENDENT VARIABLE*)
1<,5 READ*,ITEST
DC 167 K=l,ll
IFdTEST.NE.HVAR(K) ) GO TO 167
KVAR=K
GO TO 1B1
150 167 CONTINUE
PRINT 169
169 FORMAT<2X* DEPENDENT VARIABLE NOT IN DATA SET--ABORTED—*)
CALL EXIT
181 DO 185 K=1,KCONT
155 IFCKSORT(K).EQ.O) GO TO 185
DO 182 L=l,500
XSORT(L)=0,0
182 CONTINUE
NS=NPX1K>
160 DO 183 L=1,NS
XSORT(L)=XOe(K,L)
183 CONTINUE
CALL PRESRT)
DO 181. L=1,NS
165 XOB(K,L)=XSORT(L)
18k CONTINUE
185 CONTINUE
NP=NPX<1)
DO 190 K=3,KCONT
170 KB=K t IFtNPX(K).NE.NP) GO TO 200
190 CONTINUE
-------
PROGRAM REGRES 7
210 FORMAT(2X*NO. OF POINTS ARE BAD*/
175 1 'VARIABLE INDEX = *I1»,2X*NO. OF POINTS = »!<»)
CALL EXIT
220 HIN=1 t MAX=KCONT $ MSW=0 $ LINE=0
IFCMAX.LT.7) GO TO 221
MSW=1 $ MAX=6
180 221 KOT=MAX-MIN*1 S IFORM(1)=20H
ENCODE(19,223,IFORM)KOT
NFORM(1)=IFORM(1) $ NFORM(2)=IFORM(2>
00 226 J=1,NP
IFILINE.NE.O) GO TO .22
222 FORMAT(1H1,///1X*LIST OF INPUT VARIABLES*//)
WRITE(COT,NFORM)(MVAR(JA),JA=MIN,MAX)
223 FORMAT (lfH(lX,Il,mH(8X,*X(*I2»>*>)
22t, HRITE(COT,225) (X08( JB,J) ,JB = MIN,HAX)
190 225 FORM«T(1X,6F13.6)
LINE=LINE4l $ IFfLINE.GT.50) LINE=0
226 CONTINUE
IF(MSW.EQ.O) GO TO 2<»8
MIN=7 J MAX=KCONT S LINE=0 S GO TO 221
195 2li8 PRINT 2c,9
2"»9 FORMAT (ZX'OIO YOU ENTER VARIABLES IN THE ORDER*/
1 2X»YOU WOULD LIKE THEM TO ENTER ANALYSIS*/2X»TYPE YES OR NO*)
READ 30.ITEST
FLEVEL = 99-0
200 IF(ITEST.EQ.KTEST) GO TO 270
250 PRINT Z6Q
260 FORMAT(2X*TYPE F LEVEL TO ENTER(COMHA) AND F LEVEL TO REMOVE*)
REAQ*,FLE\jEN,FLEVRE
270 PRINT 280
205 280 FORMAT(2X*ENTER NUMBER OF TITLE CARDS*)
REAO*,NTITLE
NT = 0
290 PRINT 300
300 FORMAT(ZX'TYPE IN A TITLE CARD*)
210 NT = NT+1
READ 310, ITEM
310 FORMAT(SAIO)
IFORH(1)=20H
LA=fl $ LB=0 $ LC=0
215 320 ITEST=ITEK(LA)
330 JTEST=0 $ JT£ST=ITEST.AND.77B
IF(JTEST.NE.55B) GO TO 350
ITEST=SHIFTfITEST.-&) S LC = LC* 1
IF(LC.EQ.IO) GO TO 3 GO TO 370
225 00 360 K=l,8
KTEM(K)=ITEM(K)
360 CONTINUE
GO TO 395
-------
PROGRAM REGRES
7W71, OPT = 1
FTN lU2*P380
75/02/28. 13.13.13.
PAGE
230
235
21,0
21,5
250
255
260
265
270
275
280
370 IB=LB/2 S tBB=LB-IB
IC=80-L8 S ICC=IC/10 $ ICO=IC- (ICC»10 >
IF(ICO.GT.O) GO TO 380
ICD=1 $ IBB=IBB-1
380 ENCOOE(20,390,IFORM) IB, ICC, ICD.IBB
390 FORMAT {•(•^•X," 1 2»A10,A»I2»,»I2»X)»)
ENCODE(80iIFORM,KTEM) ITEM
395 DO 1,00 K=l,8
KTITLE) GO TO <»20
CALL VARTRN(KA,NP,KB,EXPO(J»
GO TO <(30
",20 CONTINUE
KB=0 S CALL VARTRN(KA,NP,KB,EXPO( J) )
<»30 CONTINUE
WRITE(COT,<»35)
1,35 FORMAT(1H1,///1X»PROBLEM INFORMATION'/)
DO !»50 J=1,NTITLE
WRITE (COT, 1,1,0) (KTITLEU.IO ,K=1,8)
(,(,0 FORHAT(1X,8A10)
<»50 CONTINUE
HRITc(COT,<,«,5)
1,1,5 FORMAT I///8X)
00 1,60 K=1,KNUM
IOV=KRANK(KI
IF(IOV.EQ.O) GO TO 460
WRITE (COT, 1,55 ) IDV, (LTITLE (IOV.L) ,L=1, 8)
1,55 FORMAT(IX»X(»I2») = *7A10,A2)
<»60 CONTINUE
WRITE ( COT, <<70)NP,KNUM,MVAR(KVAR)
1,70 FORMAT (// IX'NUMBER OF EVENTS = *!<,, 10X»NUMBER OF VARIABLES = »
i II»///IX»INOEX OF INDEPENDENT VARIABLE « *i«»)
IF(FLEVEN.EQ.O.O) GO TO 1,90
HRITE(COT,lt80)FLEVEN,FLEVRE
J.80 FORMAT (//1X»F LEVEL TO ENTER = »F8.J,,5X»F LEVEL TO EXCLUDE = »
1 FS.i,)
GO TO 510
<»90 WRITE(COT,500)
500 FORMAT (1X*F LEVELS HERE NOT PART OF INPUT AND HILL NOT BE USED*)
510 CALL REGCOR(KVAR,FLEVEN,FLEVRE,MVAR)
CALL EXIT
1000 PRINT 1010
1010 FORMAT (2X»YOU HAVE READ AN EOF IN ERROR")
GO TO 2000
1020 PRINT 1030
1030 FORMAT 12X»NUMBER OF VARIABLES GREATER THAN 11")
2000 CALL EXIT
END
-------
PROGRAM REGRES
OPT = 1
FTN (..2+P380
75/02/38. 13.13.13.
PAGE
SYMBOLIC REFERENCE MAP (R=l)
ENTRY POINTS
12257 REGRES
VARIABLES SN TYPE
5 CIT
26 EXPO
11,321 FLEVEN
11,327 IB
11,331 1C
11*333 ICO
li»33i* IOV
11,362 IFORM
0 IRANK
11,361, ITEM
13 ITRANS
11,316 J
11,320 JB
11,300 JX
11,277 KA
0 KCONT
11,315 KOT
«il KSORT
13233 KTEST
11*302 KTYPE
11,306 L
11,325 LB
11,311, LINE
11,312 MAX
11,313 MSH
15370 MVAR
151,03 NFORM
11,31,7 NPX
11,323 NT
11*301 TEMPX
12571, XRT
FILE NAMES
0 INPUT
201,1 OUTPUT
EXTERNALS
EOF
PRESRT
VARTRN
INTEGER
REAL
REAL
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
REAL
REAL
HODE
MIXED
FMT
TYPE
REAL
INLINE FUNCTIONS TYPE
SHIFT
NO TYPE
RELOCATION
INOICE
ARRAY KEYS
ARRAY
ARRAY KEYS
ARRAY
ARRAY KEYS
INDICE
ARRAY KEYS
ARRAt
ARRAY
ARRAY
ARRAY VARBLS
1,102 OUTA
1,102 TAPES
ARCS
1
3
it
ARGS
2 INTRIN
6
3
11,322
11,330
11,332
li«30i,
2
11,275
11,272
11,271,
7
11,317
11,303
11,273
11,310
1
51*
li»37i»
0
11,305
11,321*
11,326
50
11,311
11,335
11,276
i*
11,307
200
0
11,1,01,
COT
FLEVEL
FLEVRE
IBB
ICC
IOEG
IDY
INDEX
ISH
I TEST
IXR
JA
JTEST
K
KB
KNUM
KRANK
KTEM
KTITLE
KVAR
LA
LC
LTITLE
MIN
MTEST
NA
NP
NS
NTITLE
XOB
XSORT
611,3
611,3
EXIT
REGCOR
INTEGER
REAL
REAL
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
REAL
REAL
OUTB
TAPE6
STATEMENT LABELS
13237 10
12277 35
13315 1,1
12323 <>6
13365 53
131,07 58
FMT
FMT
FMT
FMT
0
13257
12320
13351,
0
12363
20
37
1,3
1,9
55
59
FMT
FMT
ARRAY
ARRAY
ARRAY
ARRAY
ARRAY
ARRAY
ARRAY
INOICE
INOICE
INDICE
INDICE
INDICE
KEYS
TITLES
TITLES
INDICE
TITLES
VARBLS
10.201, OUTD
10201, TAPE7
13252 3D
13276 39
13327 1,5
0 51
13377 57
131,15 60
FMT
FMT
FMT
FMT
FMT
INACTIVE
-------
PROGRAM REGRES
0°T =
FTN 1..2+P380
75/02/28. 13.13.13.
STATEMENT LABELS
PAGE
131,31
121*30
13502
1352t»
13547
13577
12533
13636
12563
13671.
12606
12626
0
12657
1371,5
13766
1(*Q07
1«,01«*
12776
11,073
13016
0
Ii»ll6
13102
UU7
11.165
11.220
13173
13201
65
75
90
102
105
120
135
11.5
lt.9
155
158
167
182
185
210
222
225
249
270
300
330
360
390
1*20
1.1.0
i*55
1.80
510
1020
FMT
FHT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
COMMON BLOCKS LENGTH
VARBLS 11000
TITLES 129
INDICE 8
KEYS 55
STATISTICS
PROGRAM LENGTH
BUFFER LENGTH
CM LABELED COMMON LENGTH
13«.53
0
12i»t,2
0
12505
12527
13621.
1251.&
13660
0
12613
13731
0
0
12703
13777
0
0
11.057
lijlOt.
13026
1301,0
13057
13111
liflSii
13156
13171
0
1«*255
67
80
95
103
110
125
li»0
146
151
156
160
169
183
190
220
223
226
250
280
310
31.0
370
395
430
«.i*5
«»60
%90
1000
1030
3H.6B 1638
1221.5B 5285
25670B 11192
FMT
FMT
FMT
FMT
FMT
INACTIVE
INACTIVE
INACTIVE
FMT
FMT
FMt
INACTIVE
FMT
131*60
131*76
121,63
12502
13560
13613
0
1361,3
12573
13707
13711,
12632
0
12671.
12712
12736
12763
11.01,0
13003
13011.
13032
13053
0
11.131*
0
11*177
11*233
li»2«»5
13203
70
85
100
1 Qi*
115
130
11.1
li*7
153
157
165
181
18i»
200
221
221*
21,8
260
290
320
350
380
1,00
1*35
450
1,70
500
1010
2QOO
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
FMT
INACTIVE
-------
SUBROUTINE VARTRN 7.GT.a.O) XRT =AUOG10 (XOS (KA.K) >
35 CONTINUE
GO TO 2000
««0 00 <*5 K=1»NUM
15 XRT(KA,K)=0.0 S IF (XOBtKA.K) .GT.O .0 ) XRT (KA,K) =AUOG (XOB (KA,K) )
itS CONTINUE
GO TO 2000
50 00 60 K=1,NUM
XRT{KA,K) =0.0
20 IF(XOB(KA,K).GT.O.O) XRT (KA,K> =SQRT {XOB (KA,K) >
60 CONTINUE
GO TO 2000
70 00 80 K=1,NUK
XRT(KA,K) =0.0
25 IF(XOQ(KA,K) .GT.0.0) XRT (KAfK)=XOB(KA,K) **EXPP
80 CONTINUE
GO TO 2000
90 00 100 K=lfNUM
XRTtKA,K)=SIN(XOB(KA,K)>
30 100 CONTINUE
GO TO 2000
110 DO 120 K=1,NUH
XRT(KA,K)=CQStXOB(KA,K))
120 CONTINUE
35 2000 RETURN
END
-------
SUBROUTINE VARTRN
7
-------
SUBROUTINE PRESRT 7^/7>t OPT=1 FTN e».2*P380 75/02/28. 13.13.20.
PAGE t
SUBROUTINE PRESRT (YB.N.KEY)
DIMENSION YB (N)
COMPA = 0.0
COMPB = 0.0
5 COMPC = 0.0
COMPA = N
COMPB = COMPA * 5.E-1
IF ( COMPB . LE. 0.0 ) GO TO 100
COMPC = SORT ( COMPB >
10 JB = COMPC
KB = N / JB
IF •( MOO (N, KB ) . EQ. 0 ) GO TO 10
KB = KB •«• 1
NTOT - KB * JB
15 NP1 = N * 1
C"*» SET KEY FOR RANKING
RP=1.£18
00 5 K = NP1 * NTOT
YB IK) = RP
20 5 CONTINUE
10 CfiLL SORT ( YB, JBt KB, YB, N.KEY )
RETURN
to 100 PRINT 110
<° 110 FORMAT I 8X, 39HSORRY NO SORT DUE TO BAO INDICE BREAKUP >
25 CALL EXIT
END
-------
SUBROUTINE PRESRT
SYMBOLIC REFERENCE MAP
-------
SUBROUTINE SORT rif/n. OPT=I FTN ^.Z+PSSO 75/02/29. 13.13.1.1. PAGE
SUBROUTINE SORT (ARRAY ,JB,KBf Y,LLIM,KEY)
DIMENSION ARRAY(JB,KB)
DIMENSION YMINI200) ,KPOS(200>
DIMENSION Y(LLIM) ,Z<500>
5 C»»»» SET KEY FOR RftNKING
ISW=0 $ IFtKEY.EQ. 1D110710B) ISH=1
C
c
C FILL YHIN KITH THE MINIMUM VALUE FROM EACH COLUMN. SAVE THE ROW POSIT
10 DO 60 K=1,K8
COLMIN = SRRAYU.K
JPOS = 1
DO 50 J=liJB
IF (ARRAY(J.K) ,G£. COLMIN) GO TO 50
15 COLMIN = »RRAY(J,K>
JPOS = J
50 CONTINUE
YMIN(K) = COLMIN
KPOS
-------
SUBROUTINE SORT
7<»/7J» OPT =
FTN &.2+P380
75/02/28. 13,13.1*1.
PAGE
60
65
70
75
80
85
90
95
YMIN(LCOL) = COLMIN
KPOS(LCOL) = JPOS
C
C
C IF THE LAST MINIMUM CAME FROM THE COLUMN INTO WHICH IT HAS SORTED, OR
C IF THE LAST SORTED VALUE REPLACED THE MINIMUM OF THE COLUMN IN WHICH
C PLACED, FOR THE BALANCE (IF ANY) OF THAT COLUMN FIND THE NEW MINIMUM,
C AND PLACE IT IN THE CORRESPONDING POSITION IN YMIN.
200 IF {KROW .EQ. J3) GO TO 300
IF (LCOL .EQ. KCOL) GO TO 250
IF (KROW .NE. KPOS(KCOD) GO TO 300
250 Jl = KROW * 1
COLMIN = ARRAY(J1,KCOL>
JPOS = Jl
00 260 J=J1,JB
IF (ARRAY(J,KCOL) .GE. COLMIN) GO TO 260
COLMIN = ARRAY(J.KCOL)
JPOS = J
260 CONTINUE
YMIN(KCOL)
KPOS(KCOL)
300 CONTINUE
350
l»00
<»50
C»»»*
500
2000
= COLMIN
= JPOS
IF(ISH.EQ.O) GO TO <»50
RESET ARRAY FOR DESCENDING ORDER
NG=LLIK
DO 350 K=1,LLIM
Z(NG)=Y(K>
NG=NG-1
CONTINUE
DO kOO K=i,LLIM
Y (K)=Z(K)
CONTINUE
IF(MRS.NE.O) GO TO 2000
LIST THE SORTED ARRAY
PRINT 500, Y
FORMAT (IX ,5F12.'»/ ( IX, 5F12. 4) >
RETURN
END
-------
SUBROUTINE SORT
SYMBOLIC REFERENCE HAP fR=i)
OPT
FTN
ENTRY POINTS
3 SORT
VARIABLES SN
0 ARRAY
217 ISH
0 JB
231 Jl
0 KB
0 KEY
225 KROH
0 LLIW
232 MRS
0 Y
23«* YMIN
FILE NAMES
OUTPUT
TYPE
REAL
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
REAL
REAL
MODE
FMT
STATEMENT LABELS
".1 50
110 150
1«»1 260
0 «|00
202 2000
STATISTICS
PROGRAM LENGTH
RELOCATION
ARRAY F.P.
P.P.
F.P.
F.P.
F.P.
ARRAY F.P.
ARRAY
221
223
222
220
22*»
5<»<»
227
230
233
226
105*»
COtMIN
J
JPOS
K
KCOL
KPOS
LCOL
LROH
NG
YLIT
Z
RtAL
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
REAL
REAL
60
200
300
173 1.50
0
115
75/02/28. 13.13.<»1.
PAGE 3
ARRAY
ARRAY
6«» 100
123 250
0 350
212 500
FMT
2070B
1080
-------
SUBROUTINE REGCOR 7W7* OPT = 1 FTN l».2tP380 75/02/28. 134l3»<»$4 PAGE
SUBROUTINE REGCOR(KVAR.FLEVEN.FLEVRE,MVAR)
DIMENSION Y1500),SX (11),SY(11),A(11,11), STDERC(ll)
COMMON/VflRBLS/X09tl1,500),XRT<11,500)
COMHON/TITLES/KTITLE<5,8!.LTITLEt11,3),NTITLE
5 COMMON/INCICE/KCONT,KNUM,IOY,FLEV£L,NP,CIT,COT,IXR
COMMON/KEYS/IRANKdl) ,1 TRANS (11) , EXPO (11) ,KSORT(11),KRANK(11)
DIMENSION Clllt 11), CY(ll), Rdl, 11), RY(ll), KFAdl), STUDT(ll)
A ,FLEV(11),STOEV(11),KOUT(11),PflRFL(11)
DIMENSION SUMS(11),MEANS(11),X(11,500),KSETS(11),IFORM(2),
10 1 TEMP (10 ,10) .TEMPA(ll) ,V!»(11> ,KIN(11),B(21.21) ,0(21,21) ,
2 R£GCOFdl),S10REG(ll),MVAR(ll>,KEX(ll)
REAL MEANS
INTEGER CIT, COT
OATA(IFORM(1)=20H
15 00 1 J=l,ll
KSETS(JI=0
KEX(J)=0
DO 1 K=l,500
X(J,K)=0.0
20 1 CONTINUE
DO 2 K=l,500
Y(K)=0.0
2 CONTINUE
JAY=MVAR(KVAR)
25 ISH=0
FMIN=0
KMIN=Q
KAX=0
IMH = 0
30 NSTEP=0
PRINT 73
READ 7<(,ITEST
IF(ITFST.EQ.3RYES) ISW=1
DO 3 K=1,NP
35 Y(K)=XRT(JAY.K)
3 CONTINUF.
NA=0
YEXP=EXPO (JAY) J IFtYEXP.EQ.0.0) YEXP=1.0 S KYTRAN=ITRANS(JAY)
DO i. K = l,ll
1,0 IF(ITRANS(K).NE.5RPOH ) GO TO l»
KA=K $ GO TO 7
J» CONTINUE
DO 6 1=1,11
IF(KRANKd).EQ.O) GO TO 6
(.5 IF(KRANKd).EQ.JAY) GO TO 6
J=KRANK(I) $ NA=NAtl t KSETS(NA)=J
DO 5 K=1,NP
X(NA,K) = XRT(J,K)
5 CONTINUE
50 6 CONTINUE
GO TO 9
7 MftX=EXPO(KA) J KAX=MVAR(KA>
DO 8 JA=1,MAX
NA=JA S KSETS(JA)=NA
55 DO 8 J8=1,NP
X(JA,JB)=XOB(KA,JB)»»JA
8 CONTINUE
-------
SUBROUTINE REGCOR 7l»/7it OPT = 1 FTN <».Z+P380 75/02/28. 13.13. = 0.0
00 11 K = 1, 11
A(J, KJ = 0.0
75 C
-------
SUBROUTINE REGCOR 7<./7it OPT = 1 FTN <»;2+P380 75/02/28. 13.13.<»3. PAGE
115 DO 16 J8=1,MAX
B(JA,JB) = 0.0 I D(JA,JB)=0.0
16 CONTINUE
C»"* IN MATRIX OPERATION JA=ROH AND J8=COLUHN
MIN=M1*1
120 00 17 JA=MIN,MAX
DO 17 jn=l,MT
JC=JB+M1
IF(JA.NE.JC) GO TO 17
B(JA,J8)=-1.0
125 17 CONTINUE
MIN=M1*1
00 18 JA=1,MT
JC=JA*M1
DO 18 JB=MIN,MAX
130 IF(JC.NE.JB) GO TO 18
B(JA,JB)=1.0
18 CONTINUE
C
C »*»*«» FORM *A* MATRIX FOR INDEPENDENT VARIABLE AND *SY» COLUMN
135 C »»»»»» MATRIX FOR DEPENDENT VARIABLE
C »*»»*» LET K = ROW AND L - COLUMN
C
DO 35 J = 1, Nf>
LA = 1
1<*0 MIN = 0
CONC = 1.0
CONB = 1.0
IF (M.EQ.O) CONB = X(LA, J)
SY2 = SY2 t Y(J) * Y(J)
1<,5 DO 3? K = 1, Ml
SY(K) = SYtK) * Y(J) » CONC
MIN = MIN *• 1
DO 25 L = MIN, Ml
A(K, L) = A(K, L) + CONB » CONC
150 A(L, K) = A(K, L)
IF (M.EQ.O) GO TO 20
CONC = CONC » X
-------
SUBROUTINE REGCOR
71t/7l» OPT = 1
FTN <(.2»P380
75/02/28. 13.13.<»3.
PAGE
175
130
185
190
195
200
205
210
215
220
225
STOY = SQRTICYY / OENOM / (DENOM - 1.0)>
STOEVY = SQRTtCYY / BIAS )
FORM *C* MATRIX , COVARIANCE » AND *CY* COLUMN MATRIX
LET K = ROW AND L = COLUMN
=1.0
C2 = CtK, X) » CYY
IF CC2.LE.O.O) GO TO 1060
»»** STANDARD DEVIATION OF COMPUTED INDEPENDENT VARIABLE
AND SET B MATRIX ON THE FIRST PASS ONLY
STDEVIK) = SQRTtClK, K)/ BIAS )
RY(K)=CYtK)/SQRTIC2) t IF (MARK.EQ.O) B /SQRTtC2) $ IF tMARK.EQ. 0) B IL,K)=B (K,L)=R !K»L>
CONTINUE
KZ = K
CONTINUE
B(M1,M1) = 1.0
DO 58 K=1,MT
8tMl,K)=BtK,Ml>
CONTINUE
FORMAT I2X*00 YOU HANT TO LIST INTERMEDIATE MATRICES*
1 /2X»TYPE YES OR N0»>
FORMAT IR3)
IFtISW.EQ.0) GO TO l»00
LIST A MATRIX - RAH SUM OF SQUARES AND CROSS PRODUCTS
HRITEICOT,86)
FORMAT (1H1,///1X»RAH SUMS OF SQUARES AND PROOUCTS-OR tA) MATRIX*)
DO 90 LA=1,M1
TEMPAtLA)=SYtLA)
DO 90 LB=1,M1
TEMP(LA,L8)=AtLA,LB)
CONTINUE
MATE=0
-------
SUBROUTINE REGCOR 71./7I. OPT = 1 FTN = CU,K>
92 CONTINUE
235 C»*»* LIST COVARIANCE MATRIX
WRITE!COT,93)
93 FORMAT! ///1X*COVARIANCE MATRIX OR (C) MATRIX')
MATE=1 $ ASSIGN 9 $ GO TO 3«tO
260 320 IF(AMAX.GT.1.0E<»> GO TO 330
IFO = 6RF10.3) S GO TO 3 +1 t IXY = MAX-MIN+2
ENCOOE<13,350,IFORM)IXX,IXY,IFO
350 FORMAT(*(*I2»X,*I2,R6>
270 IF(MATE.EQ.l) GO TO 355
WRITE (COT, I FORM) (TEMP(JA,J8) , J8=MIN,MAX) , TEMPA (JA)
GO TO 358
355 WRITE(COT,IFORM)
-------
SUBROUTINE REGCOR 7t./7(, OPT*1 FTN 1..2+P380 75/02/28. 13.13./.3. PAGE
415 CONTINUE
1.20 KHAX = Q 1! VMAX = -1.0E18 $ ITOEG=NP-1-NSTEP * OEGF=IOEG= ITDEG-i
DO <»30 JA=1,MT
KN=KOUT(JA> $ IF(KN.LT.O) GO TO 1.30
290 Vi,(KN>=BtKN,Ml>*B(KN,Ml)/B(KN,KN)
FLEV(KN) = (Vi.*OEGF)/(B(Ml,Ml)-Vi. I KMAX = KN
i»30 CONTINUE
295 FMAX=FLEV(KMAX>
IFUMW.NE.O) GO TO Ski
WRITE,FLEV,STUDT(K)iSTDEV{K>,MEANS{ZX,F13.6) 1)
WRITE(COT,i.70)STOEVY, YMEAN.SUMY
1.70 FORMAT (//IX'STANDARO ERROR OF Y = *F13.6, lOX'MEAN OF Y = »F13.6
310 1 //1X*SUM OF Y = *F13.6)
IF(FLEVEN.NE.O.O) GO TO 1.75
KIN(1)=KN = KFA CD SFMAX=FLEV(KN) $ KMAX=KN $ GO TO 5i»l
"»75 KIN(1)=KN = KMAX
IF(FHAX.GE. ELEVEN) GO TO 51.1
315 WRITE(COT,500) FMAX
500 FORMATCIX'ALL INDEPENDENT VARIABLES FAIL F LEVEL TEST T0«
1 »ENTER REGRESSION F LEVEL MAX = *F13.6/2X» JOB IS ABORTED*)
CALL EXIT
541 IFCIMW.NE.O) GO TO 51.2
320 IMK = 1 $ GO TO 1.00
51.2 NSTEP = NSTFP*1 S IF (NSTEP. GT. MT) GO TO 150
KJO=0 S KJI=NSTEP S IFtFLEVEN.EQ. 0. 0 ) KMAX=KFA ( NSTEP)
FMAX=FLEV(KMAX) $ KIN (NS TEP) =KMAX $ IF ! KMAX.EQ.KMIN) GO TO 1100
IF(FLEVEN.EQ.O.O) GO TO 5<.3
335 IFIFKAX.LT. ELEVEN) GO TO 1080
51.3 DO 51.5 JA=1,KJ[
DO 51.1. JO = 1,MT
IF(KOUT(JE).LT.O) GO TO 51.1.
330 IF(KN.EQ.JB) KOUT(JB)=-1
51.1. CONTINUE
51.5 CONTINUE
DO 5i»7 JA = 1,MT
51,7 CONTINUE
335 KJI=NSTtP
KN=1JA=MJS=KIN[NSTEP) S KZ=KN
C»»»» COMPUTE B MATRIX FOR THIS STEP
548 MAX = MH-MT
DO 555 JA=1,MAX
31,0 DO 553 JB = 1,MAX
IF(0(MJA,!^JS) .EQ.0.0) GO TO 101.0
IF(JA.NE.KN) GO TO 551
-------
SUBROUTINE REGCOR Tt/Tt OPT = 1 FTN <>.2 + P380 75/02/29. 13.13.1»3. PAGE
B(JA,JB)=D(JA,J8)/0(MJA,MJB)
GO TO 553
3<>5 551 B-(D(JA,KZ)*D(KZ, J8)/0(MJA,MJB»
553 CONTINUE
555 CONTINUE
IF(MJB.NE.MAX) GO TO 558
DO 5S& JA=1,MAX
350 IFUA.EQ.KN) GO TO 556
B/D(MJA,MJB»
556 CONTINUE
558 ORGVAR = C(M1,H1) S PEREXP=11.0-B(Ml,HI)) *1. OE2
PERUNE=8(M1,M1)*1.0E2 S RESQR=C*8(Ml,M1>
355 STDRES = SQRT(RESQR/DEGF) $ CONST = 0.0 $ RMULT = SQRT< l.-B (M1,M1»
DO 570 JA=1,KJI
KN=KIN(JA> $ REGCOF(KN)=8(KN,M1>*SQRT(C(M1,M1>/C(KN,KN))
MZ=M1+KN
STDREG(KN)=STDRES»SQRT(B(MZ,MZ)/C(KN,KN) 1
360 CONST=CONSTtREGCOF(KN>*MEANS(KN>
PARFL(KN)=
570 CONTINUE
CONST=YMEAN-CONST
IF(ISW.EQ.O) GO TO 700
365 700 WRITE(COT,710)NSTEP
710 FORMAT (1H1,////1X*REGRESSION INFORMATION FOR STEP NO. *!((//)
WRITE(COT,715)KSETS(KMAX),FMAX
715 FORMAT(1X»VARIABLE ENTERING REGRESSION THIS STEP IS X(*
K i I?*)<15X»F LEV;L TO ENTER =»Fi3.&/>
O 370 HRITE(COT,720)ORGVAR,PEREXP,PERUNE,RMULT,RESQR,STORES,NSTEP,ICEG,
1 CONST
720 FORHAT (IX'ORIGINAL VARIANCE*21X ,F 12 . J,
1 /1X»PERCENT EXPLAINED VARI ANCE» 15X ,F9.i»
2 /1X»PERCENT UNEXPLAINED VARIANCE*13X,F9.<»
375 A /IX'MULTIPLE CORRELATION COEFF» 12X.F12. it
3 /1X*RESIOUAL SUM OF SUUARES'15X,F12.4
>t /IX'STANDARO ERROR OF RESIDUALS'llX ,F12.I»
5 /IX'DEGREES OF FREEDOM OF REGRE SSION»1I,X,!(,
6 /IX'OEGREES OF FREEDOM OF RES IDUALS'ISX ,11.
380 7 /IX'CONSTANT'SOXtFl?.^/)
WRITE(COT,725)
725 FORMAT (1 X *VAR I ABLE"tX'REGRESS ION* 7X*STANDARO»6X*PARTI AL*
1 /12X*COEFFICIENTS*3X*ERROR*9X»F VALUE*/)
DO 727 JA=1,KJI
385 KN=KIN(JA) $ KO=KSETS(KN)
WRITE(COT,726>KO.REGCOF(KN>,STDREG(KN>,PARFL(KN>
726 FORMAT ( 3X ,I3,i»XtF13 .6 ,2 (2X.F13. &»
727 CONTINUE
DIFF2=0.0 $ AEXP=1.0/YEXP
390 PRINT 200,NSTEP
READ 7
DO 880 JA=1,NTITLE
WRITE(COT,870)(KTITLE(JA,JB),JB=1,8)
-------
SUBROUTINE REGCOR 7<(/7J( OPT = 1 FTN it.2*P380 75/02/28. 13.13.1.3. PAGE
"•00 870 FORMAT(1X,7A10,A9>
880 CONTINUE
WRITECCOT,890>
890 FORMAT
<»05 900 FOFX=0.0
DO 903 JB=1,KJI
KN=KINCJB> t FOFX = FOFXKREGCOF(KN)»X GO TO 905
IF(ITEST.EQ.5RCOS ) GO TO 905
YOBS=XOB(JAY,J)
IFUTEST.EQ.5RLOG10 ) YCOMP=10.0••YCOMP
<»15 IF(ITEST.EQ.5RLOGE ) YCOMP = EXP (YCOHP)
IFHTEST.EQ.SRSaRT 1 YCOMP = YCOMP*YCOMP
IFIITEST.EQ.5REXP > YCOMP=YCOHP*»6EXP
905 OIFF=Y03S-YCOMP I OIFF2=DIFF2*(DIFF»DIFF)
WRITE(COT,910)J,YOBS,YCQMP,DIFF
<>20 910 FORHAT(lX,I3,5X,F13»6,«tX,Fi3.6,2X,Fi3.65
UINE=LINE+l S IF1LINE.GT.50! LINE=D
930 CONTINUE
WRITE(COT,930)OtFF2
930 FOR^AT(//1X»SUM OF DEVIATIONS SQUARED = *F20.6>
«5 932 IFtFLEVRE.EQ.0.0) GO TO 400
DO 935 JA = 1,KJI
KN=KIN(JA)
IF(PARFL(KN).LT.FLEVRE> GO TO 9iiO
935 CONTINUE
l»30 GO TO UOO
91(0 KMIN = KN $ FMIN=PARFL(KMIN) $ NSTEP=NSTEP-1
HRIT E ( COT, 9<»5)KS£TS(K MINI ,FMIN,FUEVRE
91(5 FORMAT(//1X»VARIABLE WITH INDEX = "lit* HAS BEEN *
1 'EXCLUDED FROM REGRESSION WITH F LEVEL = »F15.6
V35 2 /IX'HHERE F LEVEL TO EXCLUDE = »F15.6)
JB=0
DO 950 JA=1,KJI
KN=KIN(JA)
IFCKN.EQ.KMINt GO TO 950
i«t(0 ja=JB+l S KEX(J8)=KN
950 CONTINUE
KJI=KJI-1
DO 955 JA--1,KJI
KIN(JA)=KEX(JA)
1,1(5 955 CONTINUE
DO 960 JA=1,MT
KN=KOUT(JA)
IF(KN.GE.O) GO TO 960
KOUT(JA)=KMIN
j,50 GO TO 965
960 CONTINUE
965 KN=KMIN t MJA=KN+M1 $ HJB=M1+HT J KZ=MJB
ITDEG=NP-1-NSTEP t DEGF=IOEG=ITOEG
00 970 JA=1,MJB
1,55 00 970 JB = 1,MJB
D(JA,JB)=B(JA,JB)
-------
SUBROUTINE REGCOR
OPT =
FTN
465
«»70
<»75
970 CONTINUE
GO TO 5i»8
1000 PRINT 1010
1010 FORMATC2X*ERROR IN NUMBER OF INDEPENDENT VARIABLES*)
CALL EXIT
101)0 PRINT 1050
1050 FORMAT(IX'DIAGONAL ELEMENT IS ZERO-JOB ABORTED')
CALL EXIT
1060 PRINT 1070
1070 FORMAT<2X*C2 IS LESS THAN OR EQUAL ZERO—ABORTED')
CALL EXIT
1080 HRITE(COT,1090)KSETS 14.0, are considered not significantly different than zero or one.
69
-------
The power series as shown below was used in the computer application
to approximate the solution of the aforementioned integral for the
limits -4.0 f_ x ^ 4.0. The accuracy, or maximum error with this
i i -8
series, is | e(x)| < 1.0 x 10 .
n=50
2n+1
j^
n=0
x
Power series: P(x) = 1/2 + Z(x) ^ j^f - (2n+l)~ + e(x)
-9
The series was truncated when |P(x)n - p(x)n+;jj < 1.0 x 10 .
Another mechanism that is required when the normal probability function
is used is that of obtaining the inverse of integral. A popular
polynomial approximation to this inverse is as follows :
a + a t
x = f (t) = t -- 2 - 1 - + e(p)
p z
where: 0 < p < 0.5 and t = fcn 2
~ V P
and: aQ = 2.30753 &l = 0.2760
b = 0.99229 b = 0.0448
and: | e(p) | < 3.0 x 10~3
The above polynomial does not provide the same maximum error term as
that in the approximation of the integral in the range, -4.0 <_ x <_ 4.0.
A Taylor series expansion of the higher derivatives was examined
as a method of obtaining a better approximation to the inverse.
70
-------
2
1 ~x /2
Let: Z(x) = - £
then: Z(1) (x) = - xZ(x)
Z(2) (x) = (x2 - 1) Z(x)
(3) 3
v ' (x) = (3x - x ) Z(x)
(4) 4 2
Zv (x) = (x - 6x +3) Z(x)
Note: d(u,v)/dx = u — + v ^ was the general form used in
obtaining the higher derivatives.
This approach to the approximation became messy in solving for the required
root in terms of the fourth degree polynomial resulting from the Taylor
series expansion. The technique as used in the computer application for
the solution of inverse of the integral combines the polynomial
approximation as described earlier and Newton's method of successive
approximations.
Let: xn = x = f(t) + e(p) (initial guess)
o p
then: x, = x _ f(x )/f'(x )
1 o o o
or: x .. = x - f(x )/f'(x_), n-1, 2, 3, ..., k
n+1 n n
71
-------
where: f(x)=P(x)-C
f'(xn)=Z(xn)
C = True value of the integral
i
The successive approximations were truncated when f(x )| < 1.0 x 10
— ft
The maximum error is e(p) | < 1.0 x 10 for the range —4.0 <_ x <^ 4.0.
The electronic computer application has attached a main program called
DRIVER. The main program was used to test the maximum error and can be
discarded.
The subroutine used to approximate the solution of the integral and its
inverse is titled NORM, where the entry for the inverse is titled NORM1.
The subroutine is entered with three arguments labeled XR, PROB, XC.
The label XR is the argument entered for the solution of the integral.
The label PROB is the value of the integral. When entry is made via
NORM1 (inverse of integral), the label PROB is the true value of the
integral and the label XC is the approximation of the argument.
The label SEED is the first guess (x ) and uses the polynomial approxi-
mation shown earlier.
All program-controlled error messages are displayed at the terminal.
72
-------
PROGRAM DRIVER 7t*/7t, OPT = 1 FTN ^»2*P380 75/03/03. 15-11.<»3.
PAGE 1
PROGRAM DRIVER(INPUT,OUTPUT,OOT21,TAPE5=OUT21}
XT = -<».Q20 $ LINE=0 S IH = 5
10 XT=XT*C.QZ $ IF(XT.GT.%.3) CALL EXIT
CALL NORM (XT,PROB,XC>
5 CftLL NORM1(XT,PR03,XC)
DIFF=XT-XC
IF(LINE.NE.O) GO TO 35
WRITE(IW,30)
30 FORMAT (lHl»////<3X»P(X)*9X»X*12X*XC*liX*DIFF»)
10 35 WRITE{IH,
-------
PROGRAM DRIVER
OPT =
FTN (*.2+P380
75/03/03. 15.11.«»3.
PAGE 2
SYMBOLIC REFERENCE MAP
ENTRY POINTS
6151 DRIVER
VARIABLES SN TYPE RELOCATION
62l»3 DIFF REAL 62<*0
6237 LINE INTEGER 62^1
62<»2 XC REAL 6236
FILE NAMES MODE
0 INPUT 20^1 OUTPUT
EXTERNALS TYPE ARCS
EXIT 0
NORM1 3
STATEMENT LABELS
6156 10 6212 30 FMT
6227 t»0 FMT
STATISTICS
PROGRAM LENGTH 1018 65
BUFFER LENGTH 61«»3B 3171
IW
PROB
XT
INTEGER
REAL
REAL
<»102 OUT21
i»102 TAPE5
NORM
617*. 35
-------
SUBROUTINE NORM
SYMBOLIC REFERENCE MAP (R=i)
OPT=1
FTN
-------
SUBROUTINE NORM 7<»/7«» OPT = 1 FTN <».2+P380 75/03/03. 15-11.«*5-
PftGE 1
SUBROUTINE NORM{XR,PR08,XC>
0 ATA (PI = 3. li»l 5 9265359)
CONST=2.0*PI $ CONST=1.0/SQRTCCONST)
X = ABS(XR) $ IF(X.LE.*ZX
35 DENOM=1.0 $ POFX=0.5 $ PPOFX=0.0
DO (»0 K = l ,50
15 N=K-1 $ KN=2*N+1 S ZAP=KN $ DENOM=DENOM*ZAP
POFX=POFX*ZX»X**KN/DFNOM
ERROR=ABS (POFX-PPOFX) S IF(ERROR.LT.1.OE-9 ) GO TO 50
PPOFX=POFX
kQ CONTINUE
20 PRINT itStN, ERROR
tt5 FORHAT(2X*N = *I<», 5X*ERROR = *F20.9)
CALL EXIT
50 PROB=POFX t IF(XR.LT.O.O) PROB=1.0-PROB
GO TO ISX (150,100)
25 ENTRY NORM1
P=PROD=FR08 $ ISIGN=0
IF(PROD.GE.0.53 GO TO 60
ISIGN=1 $ PROO=1.0-PROD
60 IF(P.GT.0.5) P=1.0-P
30 IF(P.LE.O.O) P=1.0E-5
TEE=SQRT GO TO 90
35 PRINT 80
60 FORMAT(1X*TRUE VALUE OF INTEGRAL IS GREATER THAN*
1 /1X*LIMIT ARGUMENT IS SET TO SEED*)
XC=SEEO'$ GO TO 120
90 ASSIGN 100 TO ISX $ GO TO 25
itO 100 DIFF = POFX-PRCO $ IF {ABS (OIFF) . LT. 1. OE-9) GO TO 120
X1=X-(DIFF/ZX) $ NC=NC*1
X=X1 $ IF(NC.LT.30) GO TO 90
PRINT 110,X
110 FORMAT(2X*NO CONVERGENCE FOUND IN X ERROR = *F15.9)
i»5 CALL EXIT
120 XC=X $ IF(ISIGN.EQ.l) XC=-XC
150 RETURN
END
-------
P(X)
.00003167
.00003446
.00003748
.00004074
.00004427
.00004810
.00005223
.00005669
.00006152
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cOT J21577
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X
-4.00000000
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xc
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-3.00000000
OIFF
-.oooooooo
-.00000000
-.00000000
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.00000000
-.oooooooo
-.oooooooo
-.00000000
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.00000000
-.oooooooo
-.oooooooo
-.oooooooo
-.00000000
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.00000000
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.00000000
-.oooooooo
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-.oooooooo
-.oooooooo
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-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.00000000
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.oooooooo
-.00000000
77
-------
P(X)
. 00144124
.00153820
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.00175016
. 001865 81
.00198838
.00211821
. 00225558
.00240118
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X
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XC
-2.93000000
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-------
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-------
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81
-------
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83
-------
P(X)
.99915526
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-------
BETA DISTRIBUTION
A random variable is said to have a beta distribution if its
probability density is:
k.X3"1 (1 - X)b-1 for 0 < X < 1
f(x) =
i
elsewhere
T(a + b)
where k =
r(a) r(b)
It is obvious from the above that the beta distribution is a two-
parameter function and the distribution is related to the gamma
distributions.
The beta distribution is fundamental in the solution of the I and J
integrals used in total sediment in transport analysis. This
distribution is also fundamental in the determination of the dis-
tribution of sediment deposited in a reservoir. This distribution
is used extensively when the argument "X" (or its transform) is in
the range zero to one.
The integral of f (x) is evaluated from "0" to "x" with the use of
the following derivation:
85
-------
Let the beta density function be defined as:
a-1 b-1
f (x) = X— (1-X) Where 0 < X < 1 and
f3(a, b) - -
a > 0, b > 0
and the beta distribution function be defined as:
1 px a-1 b-1
F(x) = fj(a, b) J t (1 - t dt
o
where
3(a,b) = J t: (1 -t)D-xdt = ^a ^ (1)
o
See 6.2.1, page 258, Handbook of Mathematical Functions, U. S.
Department of Commerce, National Bureau of Standards, Applied
Mathematics Series 55, June 1964. On page 263, 6.6.1, of this same
handbook the incomplete beta function is:
6x(a, b) - ta (1 - t)13" dt
and
F(x) = Ix (a, b) = gx(a, b)/g(a, b) (2)
and .
(a, b) = a~ Xa F(a, 1-b; a+1; X)
where F(a, 1-b; a+1; X) is the hypergeometric series.
86
-------
The circle of convergence of the Gauss hypergeometric series
is |Z| < 1. Within this circle F is defined as shown:
F(c) V r(a+n) F(b+n) Zn
F(a, b; c; z) = r(a) r(b) ^ r(c4tl) n.
r(c) r(a) r(b) z" . r(a+i) r(b+i) z ,
r(a) r(b) I r(c) 0! rr
r(a+2) F(b+2) Z2 + F(a+3) T(b+3) Z3
F(c+2) 2! F(c+3) 3!
Using recurrence formulas:
T(z+l) = ZF(z) and
r(z+n) = (n-1 + z) (n - 2 + z) ... (1 + Z) F(1 + z)
then:
r(c) [ r(a) r(b) ar(a) br(b) z
F(a, b; c; z) = r(a) r(b) I r(c) cr(c)
T(a) b (b+1) F(b) Z2 +
T(c) 2!
3
a(a+l)(a+2) T(a) b (b+1) (b+2) T(b) Z
c(c+l)(c+2) T(c) 3!
and upon further reduction:
87
-------
r(c) [ r(a) r(b) 4- r(a) r(b) abz
F(a, b; c; z) = r(a) r(b) r(c) r(c) c
F(b) a(a+l) b (b+1) Z
2
r(c) c(c+l) 2!
F(b) a(a+l)(a+2) b(b+l) (b+2)Z3
T(c) c(c+l)(c+2) 3!
and:
T(c) . F(a) T(b) [" abz a(a+l) b(b+l) Z2
F(a, b; c; z) = r(a)F(b) r(c) L T(c) c(c+l) 2!
a(a+l)(a+2)b(b+l)(b+2) Z3
(c+2) 3!
Let: a=a, b = 1 - b, c=a+l, and Z = X
then:
F(a, 1-b; a+1; x) = 1 + a+1+ (a+l)(a +1+1) 2!
a(a+1) (a+2) (1-b) (1-b+l) (l-b+2) X3 ,
(a+1) (a+1+1) (a+1+2) 3!
and:
a(1-b) (2-b) (3-b) X3
(a+3) 3!
88
-------
then:
(n-b) 0.50. By taking advantage of the symmetry which is expressed
as follows :
I (a, b) = 1 - 1^ (b, a) (5)
a good comparison was obtained. In the data analysis model the
integral was solved using the series of equation (4) and the
symmetry of equation (5) where the series (4) was truncated
when the last term was equal to 1'10~7 or less. In the model
"a" and "b" were allowed to assume any value greater than zero without
89
-------
any apparent discontinuity. Solving the inverse of the integral can
be accomplished using Newton's method and will be described in a
later portion of this narrative.
It is left to the experience of the user to use this technique with
good judgment.
Solving for mean and variance of the beta function given "a" and
"b", where a > 0 and b > 0
The general term for the derivation of moments is as follows:
= T(a+b) T(a+r)
yr T(a) T(a + b + r)
where r = 1, 2, 3, ... or the first, second, third ... moments.
When r = 1 the first moment or mean is:
,. _ r(a+b)
1 T(a) T(a + b + 1)
Let: a + b = c
and the recurrence formula:
T(z + 1) = Z T(z)
then
F(c) a T(a) . a
r(a) c r(c) = a/c = ~ = mean'
90
-------
the second moment =• u_ and:
Ha+2)
* T(a) T(a + b + 2)
Let: a + b = c
and using the recurrence formulas
T(z + 1) = ZF(z)
T(z + n) = (n - 1 + z) (n - 2 + z) ... (1 + z) r(z + 1)
then
r(c) a (a+1) T(a) = a(a+1)
r(a) c (c+1) F(c) c
or
(a+1)
M2 " (a+b) (a + b + 1)
Variance = a2 = E(x2) - y2 and E(x2) = y
1 2
Then:
2
a (a+1)
0 - (a+b) (a + b + 1) (a+b)2
and:
a2 = a(a+b) (a+1) - a2(a + b + 1)
(a+b)2 (a + b + 1)
which reduces to:
ab
n2 _ o = variance.
0 = (a+b)2 (a + b + 1)
91
-------
Solving for "a" and "b" in terms of mean and variance
Solve first for "b" in terms of "a" and V'
b =
y
i
Solve for "a" by substitution of "b" in equation for variance
a- 2
92
-------
Inverse of Integral
The inverse of the integral of the beta distribution function was
first attempted using Newton's method. The computation of the
inverse in this manner proved to be very sensitive at the end points
and also lacked continuity for large values of the parameters "a"
and "b". Tne mechanism used was as follows:
Let: ZX = X S~ (1 - Xt)
FN = value of integral for X
FN = true value of integral
Then:
FN - FN
Xt - —^-^—£ (Newton's method)
The final attempt for the solution of inverse of the aforementioned
integral made use of the following recursion formulas:
.an ,-
x (1-x)
_
Ix(a,b) - Ix(a,b-l) + (a+b-l)B(a,b)
and: Ix(a,a) = 1/2 [1 + Ix,(l/2,a)]
=1-1/2 I , (a, 1/2)
JL""*X
where: x' - 4(x - 1/2)2 or x = 1/2(1 + 41
The technique used to estimate the value of the inverse then became:
93
-------
Let: FN = true value of integral
FN' = X - X ,
n n-I
FN" = FNX - FN (values of Integral for Xfl and
n n-1
Then: X =X - FN'ClN - FN )/FN"
n
The subsequent narrative includes the computer generated tables for
a range of distribution parameters "a" and "b". The table is self-
explanatory and the values shown therein were compared with those
included in K. Pearson's Tables of the Incomplete Beta Function,
Cambridge (1956). This comparison indicated the computer generated
table was within all significant digits of those of Pearson's Tables.
SUBROUTINE BETAX
This subroutine is the electronic computer application for the solution
of the beta distribution function and its inverse.
94
-------
The entry for the solution of the inverse is labeled BETAX1.
The arguments for entry are labeled as follows:
A = value of a
B = value of b
XR = value of x
BETA = value of distribution function
and true value for the inverse
computation
XC = value of the inverse
The label CONGAM is the title of the computer function for the solution
of the complete gamma function. This function (subroutine type) will
be discussed in a subsequent portion of this narrative which concerns
the subroutines required for the electronic computer application for
the solution of the incomplete gamma function.
All other labels included in the BETAX subroutine are self-explanatory.
95
-------
PROGRAM QPIVTR 7U/71, 0°T = 1 FTN
CALL 9ETOX1 ( A , 8, X , BFT A, XC)
DIFF=X-XC
IFtitC.EQ.S9.» GO TO 21
WRITE (IH,?0)X,3FT41,XC,DIFF
20 FORMAT I20X.F6.2, 3(2X,FIO. 7) )
GO TO 25
25 21 HPITE(IW,22)X,BETA1
22 FORMAT(?OX,F6.2,2X,F10.7,2(1X»NOT DEFINED'))
25 COIITIHUE
LINE=LINF«1 S IF tLINE.EQ.2)LINE=0
GO TO 10
30 101 IF I rj . GT. 5. ) GO TO 200
8=8*0.5
A = B
GO TO 1?
201 CALL FXIT
35 E«D
OPIV^R 7<«/7
-------
DRIVER
71./7". OPT-l
FTN i,.2tP3SO
75/03/25. 09.12.28.
PAGE
SYMBOLIC REFERENCE MSP IR =
ENT»Y POINTS
1C21U DRIVER
VARIABLES
SN TYPE
ft
BETS
OIFF
FX
IK
10366
10573
1Q376
0
0
10370'
1
FILE NAMES
1.102 FIN
TSPE5
REAL
REAL
INTEGER
INTEGER
MODE
RELOCATION
VARBLS
UNITS
V4RBLS
6li»3 FOU
61
-------
SUBROUTINE 3fTSX 1<>/1<, 0"T=1 FIN "..3«P3SO 75/03/35. 09.13.Z9. PACE
sun=>ouirnF BETS* ia,e,*«,nFTa,xci
COH-10N/IJNITS/IH, IR
COMMON/VAP8LS/FX,XAB,FOFX
X = XR
5 ASSIGN eooo TO isx
10 IOH=0 1 XA = X t IF(A.NE.B) GO TO 1?
IOW=1 t 87=0.5 1 TZ=BZ t MAX = 0
XA=XA-0.5 $ XA = GT TO 30
AR=A J 1C=OZ » ISH=0
5 GO TO 50
30 «a=OZ J BC=A t XA=1.D-XA t ISH=1
50 GX=ar? J FACTN=1.D t FOFX = 1.0 « BETAM=1.0
00 liO K = l ,50
BK=K J FACTN=FACTN'BK J BD=BK-BC
3 GX=GX«XA"ED J FX=GX/(
FOFX = FOFXtFX H I F ( ABS (FX) . L T . 1. OE-1 0) GO TO 90
60 CONTINUE
PRINT 70
70 FOP'-tAT IIX'NO COIIVERGFNCE FOUND FOR FX - ABORTED*)
5 CALL FXIT
90 XM=»BS(XA> | BETAH = XM«*al/AB J ARGA = AB<-BC
XAIliCOIir.ACtABOA) /(CONCAH(HB) »COW, AM I BCI I
BKTAH^xaO'GETAM'FOFX j IF(ISW.Ea.11 BETAM=1.0-BETAM
IFC3ETAH.LK.1.0E-1"!) BET»H = 0.0
1 IF(MAX.HE.0) GO TO 100
GO TO 150
100 XBP*X»M I CONA = CONGAM(a)
00 110 K=1,HAX
CK=K r CKK=CKtT7 t APGA=A*CKK $ COND=CONGAM?ZX*XAP = *FK.6
1 /IX'XAn = 'F15.6>
110 COHTIMUF
>1 l?o fOPIBT II X'MAX = "Iit,SX*X = •F15.7,5X»n? = »F15.7
t /1X«A = »F15.?,2X'n = 'F15.7i2X«aETA = *F15.7)
6TAH t IFdOH.NE.ll GO TO 160
a = t .0- (O^'B^TAH) t IF ( X.LT. 0. 5) BETA=1,0-Bf TA
lf,Q GO TO I".X (?000,250,300,31Q)
1.5 ENT»Y BStaxl
X=1.0 t IF (BFTA.EO.1.1) GO TO "t50
X=t.O-1.0E-5 ! UC=0 t TEST=BETA t SAVE=X
IF(3ETA.GT.I.nE-7) GO TO 200
X=91. I GO TO i»50
50 ?00 ASStHH 750 TO ISX « GO TO 10
?50 IFINC.GT.OI GO TO 300
IFCDETA.LT.TESTI GO TO Z10
IFOFT4.fO.TFSn GO TO (.50
SAVF=X * x=X-1.0E-l t IF(X.LE.O.O» GO TO 2*0
55 SAVE=X t GO TO »00
?BO X=X1=SA«E * ASSIGN 300 TO ISX « GO TO 10
300 FX1=BETA » X=X3=Xl-l.OE-7
SUBP1UTH.E BFTAX 7W71. OPT=l FTN l,.3»P3'iO 75/03/35. 09.13.39.
105 ASSIGN ?10 TO ITX « GO TCI 10
i.n r*7-oFTA » nx=x?-Xl t OXDP=FX2-FX1
-------
SUBROUTINE BF.TSX 7t,/?tt OPT=1 FTN I..2+P390 75/03/25- 09.13.29. PAGE
305 ASSIGN 510 TO ISX t GO TO 10
310 FX2 = 8fFT« « OX=X2-X1 1 OXOP=FX2-FX1
60 IF(OXOO.EO.O.O) GO TO <*50
X3=X?-(OX*(FX2-TEST(/DXDP) $ ERROR = X3-X2
IFIUBS(ERROR) .LT.1.0F.-7) GO TO 1(50
X1=X2 $ X=X2=X3 t NC=NCtl $ FX1=FXZ
IF(XZ.GT.X1»3.0)X=X2=3.0»X1
65 IFINC.LE.100) GO TO 305
i.50 XC=X I GO TO 2000
2000 RFTURN
END
-------
SU1RCUTHE BETiX
74/74 OPT = 1
FIN 4.2*P380
75/03/Z5. 09.1Z.Z9.
P«GE
SY190LIC PEFEPE'tCE MAP
O
o
ENTRY POINT"
3 BETAX
ve»le=iLES SN
o a
414 A PGA
403 PC
0 ': £ T a
411 8K
375. 3Z
420 CK<
421 COM3
434 CXQP
406 FACTN
0 FX
43? FX2
373 IGW
404 ISH
o Iw
377 MAX
400 NP
425 TEST
371 X
1 XA9
415 XAP
0 XC
1 XR
431 X2
FILE 'JAPES
OUTPUT
EXTEP'IALS
CONGAS
IMLIN-: FUNCTIONS
ans
"TATEHFNt LAf-ELC
11 10
45 30
206 BETAX1
TYPE RELOCSTION
°E4L F.P.
REAL
R^ AL
°EaL F.P.
REAL
REAL
R EaL
REaL
RF8L
REaL
RFAL VAROLS
"EBL
INTEGE0
INTEGER
U.TEGrP UNITS
IMEG-R
INTEG'P
REAL
REAL
REAL VSRflLS
0 r AL
?-tl F.P.
°EAL F.P.
PFAL
MCD^
FnT
TYPE APGS
OFAL 1
TYPE APGS
REAL 1 INTRIN
25 12
53 50
402
0
412
407
401
417
416
433
436
2
430
405
1
372
410
424
426
376
374
423
422
413
427
435
OR
8
in
SET AM
RP
CK
CONA
OX
ERROR
FOFX
FX1
GX
IR
ISX
K
NC
savE
TZ
xa
XAC
XAQ
XM
XI
X3
EXIT
REAL
REAL
REAL
REAL
REAL
REAL
REAL
REAL
REAL
PEAL
PEAL
PEAL
INTEGER
INTEGER
INTEGER
INTEGER
PEAL
PC AL
RFAL
REAL
REAL
RFAL
REAL
REAL
324 70 FHT 103 90
134 100
173 150
231 250
253 305
!04 2000
COMMCN BLOCKS
UNITS
VAPBLS
STATISTICS
»ROG»AH LENGTH
0 110
203 160
243 260
255 310
LENGTH
2
3
4378 217
CM LaPELEO CCfMON LENGTH 58 5
F.P.
VAR6LS
37 15
0 60
332 95
342 120
227 200
247 300
302 450
FMT NO REFS
FMT NO REFS
74/74 OPT=1
75/03/75. 09.12.31.
FUNCTION CONGAM (SLF )
COMMON/UNITS/IH,IR
rtTwenrrnw or rut
-------
FUNCTION CQNGAM
7W7I.
OPT = 1
FTN <».2+P390
75/03/25. 09.12.31.
PAGE
15
O
35
(.0
1.5
50
FUNCTION CONGAM (ALF )
COMMOII/UNITS/IH.IR
DIMENSION 8C (8)
DATAUBC (K> , K = 1,8 )
I -0.397056937
2 -0.193527818 ,
PI = 3.111159265358979
KSH - 1
CONGAM = 0.0
-0.577191652 , 0.988205691 ,
0.918P06857 , -
0.03556831,3
0.75670<.078
)
0. <(8219939<«t
IF ( ALF . GT. 1. E-10 )
IF ( ALF . LT. 1. E-10 )
CONGAM = 1.0
GO TO 100
NCP - ALF
SAVE = ALF
ALFl = A3S (NCP)
COMP = 0.0
COMP = ALF » ALFI
GO TO 1
GO TO 5
IF ( ABS (COMP) .GT. l.E-10 ) GO TO 15
PRINT 10
10 FORMAT ( 8X, 28HALPHA IS A NEGATIVE INTEGER
GO TO 100
15 KSW = 2
ALF = i.o - ALF
IF ( ALF . GT. l.E-10 ) GO TO 1
PRINT 11
H FORMAT ( «X, 30H9AO EQUATION IN TERMS OF ALPHA
CALL EXIT
19 ALF = SAVE
CONGAM = PI / ( CONGAM » SIN ( PI » ALF) )
GO TO 100
1 MARK - 1
FR = 1.0
N = ALF
COM" = 0.0
8X, E19.1? )
N = COMP
CK OT(" Cl 8 FORMAT ( 8X, lit ,
KN = N - 1
IF (KN.GE.l) GO TO 20
IF ( ALF .LT. 1.0 ) GO TO 9
X = ALF - 1.0
GO TO a
= 1, KN
- FK
FA
GO TO "tO
-------
FUNCTION CONGAM
7"./7<» OPT = 1
FTN . SEVERITY GETULS
7
-------
FUNCTION COSGAM 7i*/7<« OPT = l FTN i».Z»P380 75/03/25. 09.12.31. PAGE
CARC US. SEVERITY GETMLS DIAGNOSIS OF PROBLFM
16 I BSSIC EXTFRMAL OR INTRINSIC FUNCTION CALLED WITH ViRONG TYPE ARGUMENT.
FUNCTION COMSM 7I./71. OPT = I FTN
-------
FUNCTION COS01M
FTN
+ P380
75/03/25. 09.12.31.
PAGE
SYMBOLIC REFERENCE MSP
ENTRY
<.
PC IHTS
CONGA*
VARIABLES SN
0
175
153
163
173
0
167
155
16".
15!.
172
FILE
EXTE=>
INLI'I
ALF
GC
CONGAM
F8
FOFX
IH
' K
KSH
N
PI
TEST
NAHES
OUTPUT
NJLS
FXIT
E FijNCTIC'IS
ABS
TYPE
RFAL
REAL
REAL
REAL
PFAL
INTEGER
INTEGER
INTEGER
INTEGER
REAL
REAL
MOCE
F«T
TYPE
TYPE
REAL
RELOCATION
F.p.
ARRAY
UNITS
APGS
0
ARGS
1 INTRIN
160
161
171
170
1
17".
165
162
156
157
166
ALF1
COMP
FA
FK
IR
JB
KM
HARK
NCP
SAVE
X
SIN
REAL
REAL
REAL
REAL
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
RFAL
REAL
RFAL
UNITS
1 LIBRARY
STATElEf.T
132
37
105
126
10
19
1.0
100
C01MON OLCCKS
UMTS
LENGTH
16
30
65
107
5
IS
20
50
6? 9
li»2 18
0 30
116 60
FMT
STBTISTICI
PROGRAM LFNGTh 2058 133
CH LABELED CCKM3N LENGTH 28 Z
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
,60
.64
.68
.72
.76
.30
.34
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
,52
.56
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
.5 AND E
BETA (X)
.12B1884
.1825549
.2251989
.2619798
.2951673
.3259320
.3549785
.3827767
.4096655
.4359058
.4617105
.4872642
.5127358
,5382895
.5640942
.5903345
.6172233
.6450215
.6740660
.704332?
.7330202
.7748011
.8174451
.8718116
1.0000000
1.0 AND E
BETA (XJ
.0202041
.0408337
.0619163
.0834849
.1055728
.1232202
.1514719
,1753789
.2000000
.2254033
.2516685
,,2738897
.3071797
.3366750
,3675445
,4000000
.4343146
.4708497
.5101021
.5527864
.6000000
.6535698
.7171573
.soooooo
1,0000000
= .5
INVERSE
.0400000
. 0800000
. 1200000
.1600000
.2000000
.2400000
.2800000
. 3200000
. 3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
,6400000
.6300000
.7200000
.7600000
. 8000000
. 3400000
. 8800000
,9200000
.9600000
1.0000000
.5
INVERSE
.0400000
.0300000
.1200000
. 1600000
.2000000
.2^00000
.2800000
.3200000
.3600000
,4000000
.4400000
.4300000
.5200000
,5599999
,6000000
.6400000
.5300000
.7200000
. 7600000
.8000000
. 3400000
88800000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
. 0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000 000
.0000000
-.0000000
-.0000000
-.0000000
-.0000000
-.0000000
.0000000
0.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
-.OOCQOOO
.0000000
-.0000000
. 0000000
.0000001
-.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
o.ooooooo
105
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.2k
.28
.32
.36
• 40
.44
.43
.52
.56
.60
.54
.53
.72
• 76
.80
.34
.83
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
• 52
.56
.60
• 54
.68
.72
.76
.80
.34
.38
.92
.96
1.00
1.5 AND B
BETA (X)
.0034369
.0098443
.0 183220
.0285911
.0405193
.0540424
.0691369
.0858087
.1040 880
.1240270
.1457008
.1692090
.1946808
.2222798
.2522156
.28^7571
.3202555
.3591301
.4021786
.4501849
.5046317
.5679242
.6447346
.7470601
1.0000000
2.0 AND 8
BETA (X)
.0006082
.0024670
.0056319
.0101636
.0161301
.0236066
,0326779
.0434395
.0560000
.0704840
.0870356
.1053233
.1270464
.1509441
.1773078
.2030000
.2419815
.2303556
.323940S
.3739010
.4320000
.5011694
.5870496
.7040CQO
1.0000000
= .5
INVERSE
.0400000
. 0800000
. 1200000
.1600000
.2000000
.2400000
.2800000
,3200000
.3600000
.4000000
.4400000"
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.3400000
.3800000
.9200000
.9599999
1, 0000000
= .5
INVERSE
. 0400000
,0300000
.1200000
,1600000
.2000000
.2400000
.2800000
.3199999
.3600000
.4000000
.4400000
.4300001
.5200000
.5600000
.6000000
.6400000
.6800000
. 7200000
.7600000
.8000000
.3400000
.8300000
.9200000
.9600000
1.0000000
106
XC ERROR TERM
-. 0000000
-.0000000
-.0000000
-.0000000
-.0000300
-.0000000
-.0000000
. ooooooo
-.0000000
-.ooooooo
-.0000000
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.0000001
o.ooooooo
XC ERROR TERM
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.0000001
-.ooooooo
-.0000000
-.ooooooo
-•OQUOOQl
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
-.0000000
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
,0000000
-.ooooooo
.ooooooo
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
• 64
.63
.72
.76
.30
.8**
.38
.92
.96
1.00
A =
X
.Oc*
.08
.12
• 16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.34
.83
.92
.96
1.00
K
2.5 AND E
BETA (X)
.0001102
.0006330
.0017718
.003&963
.0065663
.0105400
.0157798
.0224556
.0307494
.0408594
.Q530046
.0674314
.0844217
.1043029
.1274641
.1543774
.1856300
.2219762
.2644212
.3143727
.3739340
.4465564
.5388054
.6672192
1.0000000
3.0 AND 8
BETA (X)
.0000203
.0001650
.0005662
.0013651
.0027137
.0047762
.0077312
.0117740
.0171200
.0240082
.0327067
.0435194
.0567944
.0729370
.0924263
.1158400
.1438917
.1774888
.2178289
.2665697
.3261600
.4005719
.4972754
.6343800
1.0000000
= ,5
INVERSE
.0400000
. 0800001
.1200000
.1600000
.2000000
.2400001
.2800000
. 3200000
.3600000
.4000000
.4400000
.4800000
.5199999
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
.8800000
.9200000
.9600000
1.0000000
.5
INVERSE
.0400000
.0300000
.1200000
.1599999
.2000000
.2400000
.2300000
.3200000
,3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7199999
.7600001
.8000000
.3400000
.8800000
.9200000
.9600000
1. 0000000
107
XC ERROR TERM
-. 0000000
-.0000001
-.0000000
.0000000
-.0000000
-.0000001
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000001
-.0000000
-.0000000
-.0000000
.0000 000
. ooooooo
-.0000000
-.0000000
-. ooooooo
.ooooooo
-.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
-.ooooooo
-.ooooooo
.ooooooo
.0000001
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.0000 000
.0000001
-.0000001
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A -
X
.04
.09
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.34
.33
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.55
.60
.64
.63
.72
.76
.30
.34
.83
.92
.96
1.00
3.5 AND B
BETA (X)
.0000038
.0000435
.0001330
.0005098
.0011338
.0021676
.0038273
.0062372
.00-56279
,0l42««58
.0203755
.0233483
.0385539
.0514492
.0675834
.0876230
.1123938
.1429467
.1806648
.2274529
.2861052
.3611135
.4608415
.6059013
1.0000000
4.0 AND B
flETA(X)
.0000007
.0000116
.0000596
.0001920
.0004776
.0010101
.0019103
.0033299
.0054560
.0085163
.0127S61
.0185978
.0263519
.0365338
.0497356
.0666880
.0883074
.1157637
.1506247
.1950155
.2520720
.3268003
.4284484
.5795840
1.0000000
.5
INVERSE
. 0400000
. 0300000
.1200000
.1600000
.2000000
.2400000
.2800000
. 3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.5300000
.7200000
.7600000
.3000000
. 3400000
.8800000
.9200000
.9600000
1. 0000000
.5
INVERSE
. 0400000
. 0800000
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3599999
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
, 3400000
.3799999
.9200000
.9600000
1.0000000
108
XC ERROR TERH
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000 000
.0000 000
-.0000000
. ooooooo
-.0000000
-.0000000
.ooooooo
-.0000000
.ooooooo
o.ooooooo
XC ERROR TERM
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.0000001
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.0000 000
.ooooooo
-.ooooooo
.0000000
-.ooooooo
-.ooooooo
.0000001
-.ooooooo
.ooooooo
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
• 16
.20
.24
.28
.32
.36
.'+0
.44
.43
• 52
.56
.60
• 64
.68
.72
.76
.80
.8%
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
• 2k
.23
.32
.36
.40
.44
.43
.52
.56
.60
• 64
.63
.72
• 76
.80
.84
.88
.92
.96
1.00
4.5 AND B
BETA (X)
.0000001
.0000031
.0000196
.0000728
.0002025
.0001+691*
.0009593
.0017887
.0031104
.0051211
.0030697
.0122691
.0181100
.0260794
.0367877
.0510034
.0697075
.0941742
.1261034
.1678508
.2228685
.2966651
.3993615
.5554455
1.0000000
5.0 AND B
8ETA(X)
.0000000
.0000003
.0000065
.0000277
.0000863
.0002192
.0004842
.0009656
.0017813
.0030941
.0051167
.0031307
.0125005
.0186962
.0273229
.0391629
.0552348
.0768851
.1059339
.1449276
.1976173
.2699963
.3730427
.5331354
1.0000000
.5
INVERSE
.0400000
. 0800001
.1199999
.1599999
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400001
.6800000
.7200000
.7600000
.3000000
. 8400001
.8800000
.9200000
.9600000
1.0000000
= .5
INVERSE
NOT DEFINED
.0800000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.5000000
.6400000
.6800000
.7200000
.7600000
.8000000
. 8400000
.3800000
.9200000
.9600000
1.0000000
109
XC ERROR TERM
-.0000000
-.0000001
.0000001
.0000001
-.0000000
•0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000001
.0000000
-.0000000
.0000000
-.0000000
-.0000001
.0000000
-.0000000
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.34
.83
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.43
.52
.56
.60
.64
.63
.72
.76
.80
.34
.88
.92
.96
1.00
5.5 AND B
BETA (X)
.0000000
.0000002
.0000021
.0000106
.0000369
.0001023
.0002454
.0005234
.0010243
.0018767
.0032568
.0054094
.0086603
.0134509
.0203632
.0301709
.0439060
.0629593
.0892442
.1254670
.1756517
.2462522
.3*490844
.5123899
1.0000000
6.0 AND B
BETA (X)
.0000000
.0000001
.0000007
.0000041
.0000159
.0000484
.0001248
.0002846
.0005914
.0011421
.0020796
.0036090
.0060131
.0097060
.0152201
.0233084
. 0 349S44
.0516885
.0753654
.1088643
.1564496
.2250075
.3271668
.49300-37
1.0000000
- .5
INVERSE
NOT DEFINED
.0800000
. 1200000
. 1600000
.2000000
.2400000
.2300000
. 3200001
,3600000
.4000000
.4400000
.4799999
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.3000000
.3400000
.8800000
.9200000
,9600000
1.0000000
.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1500000
.2000000
.2399999
.2300000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.8000000
.8400000
.8800000
.9200000
.9599999
1.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000001
.0000000
-.0000000
.OOOOflOO
.0000001
-.0000000
.0000000
-.0000000
.0000 000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.0000000
-.0000000
-.0000000
.0000001
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000 000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000001
0.0000000
110
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0^
.03
.12
.16
.20
. 24*
.23
.32
.36
.4*0
.4*4*
.4*3
.52
.56
.60
»6*f
.68
.72
.76
.30
. 34*
.83
.92
.96
1.00
A =
X
.0/4
.03
.12
.16
.20
.2**
-23
.32
.36
.4*0
.4*4*
.4*3
.52
.56
.60
• 6
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A -
X
.04
.08
.12
.16
.20
.2k
.23
.32
.36
.40
.44
.43
• 52
.56
.60
.64
.68
.72
.76
.80
.34
.S3
.92
.96
1.00
A =
X
.04
.08
.12
.15
.20
.24
.23
.32
.36
.40
.44
.43
.52
.56
.60
• 54
.69
.72
.76
.80
.84
.88
.92
.96
1.00
7.5 ANC 6
BETA (X)
.0000000
.0000000
.0000000
.0000002
.0000013
.0000051
.0000167
.0000465
.0001155
.0002614
.0005497
.0010882
.0020490
,0036986
.0064427
.0108397
.0179443
.0289497
.0459122
.0718797
.1116377
.1731612
.2713245
.4415161
1.0000000
3.0 ANC 8
OETA (X)
.0000000
.0000000
.0000000
.0000001
.0000006
.0000024
.0000086
.0000255
.0000672
.0001605
.0003542
.0007326
.0014361
.0026913
.0048549
.0084795
.0144114
.0239405
.0390405
.0627720
.1000251
.1590616
.2554269
.4262064
1.0000000
.5
INVERSE
NOT DEFINED
NOT DEFINED
NOT DEFINED
.1600000
.2000000
.2400001
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.560000,0
.6000000
.6400000
.6300000
.7200000
. 7600000
.8000000
. 8400000
.8300000
.9200000
.9600000
1. 0000000
.5
INVERSE
NOT DEFINED
NOT DEFINED
NOT DEFINED
NOT DEFINED
.2000000
.2400000
.2800000
. 3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.5400000
.6800000
.7200000
, 7500000
.8000000
. 3400000
. 8800000
.9200000
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
NOT DEFINED
-.0000000
-.0000000
-.0000001
.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000 000
-.0000000
-.0000 000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
NOT DEFINED
NOT DEFINED
-.0000000
.0000000
.0000000
,0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
o.ooooooo
112
-------
INCONPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.75
.80
.34
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
• 6*f
.68
.72
.76
.80
.84
.88
.92
.96
1.00
3.5 AND B
BETA(X)
.0000000
.0000000
.0000000
.0000000
.0000002
.0000012
. 0000 Gki*
.0000140
.0000392
.QQOOC88
.0002286
.0004940
.0010084
.0019616
.0036641
.0066125
.0115903
.0193247
.0332395
.0548836
.0897189
.1462538
.2406614
.4116811
1.0000000
9.0 AND 8
BETA (X)
.0000000
.0000000
.0000000
.0000000
.0000001
.0000006
.0000023
.0000077
.0000229
.0000608
.0001477
.0003335
.0007088
.0014314
.0027688
.0051629
.009332^
.0164358
.0283323
.0480375
.0805539
.1345^63
.2269203
.397873-0
1.0000000
.5
INVERSE
NOT DEFINED
NOT DEFINED
NOT DEFINED
NOT DEFINED
.2000000
.2400000
.2800000
. 3200000
.3600000
.4000000
.4400000
.4300000
.5200001
.5599999
.6000000
.5400000
.5800000
. 7200000
.7600000
.8000000
. 8400000
.8800000
.9200000
.9600000
1.0000000
.5
INVERSE
NOT DEFINED
NOT DEFINED
NOT DEFINED
NOT DEFINED
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.8000000
.8400000
.8800000
.9200000
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
NOT DEFINED
NOT DEFINED
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000001
.0000001
-.0000000
.0000 000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
NOT DEFINED
NOT DEFINED
-.0000000
.0000000
.0000000
,0000000
.0000000
-.0000000
.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
o.ooooooo
113
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.30
.34
.83
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.23
.32
.35
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.30
.34
.88
.92
.96
1.00
1.0 AND 8 -
BETA (X)
.0400000
.0800000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.3400000
.8300000
.9200000
.9600000
1.0000000
1.5 AND 8 =
BETA (X)
.0030000
.0226274
.0415692
.0640000
.0394427
.1175755
.1481621
.1810193
.2160000
.2529822
.2918630
.3325538
.3749773
.4190656
.4647530
.5120000
.5607424
.6109403
.6625526
.7155418
.7693727
.8255132
.8824330
.9406041-
1.0000000
1.0
INVERSE
. 0400000
.0800000
.1200000
. 1600000
.2000000
.2400000
.2800000
. 3200000
.360000-0
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
.8300000
.9200000
.9600000
1.0000000
1.0
INVERSE
. 0400000
. 0800000
.1199999
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6799999
.7200000
.7600000
.8000000
.3400000
.3800000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.OOOOOOO
. ooooooo
.OOOOOOO
.ooooooo
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000 000
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
-.ooooooo
-.ooooooo
.0000001
-.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
. ooooooo
-.ooooooo
.ooooooo
.0000001
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
114
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.2k
.28
.32
.36
.40
.44
.<+3
.52
.55
.60
.64
.68
.72
.76
.30
.64
.88
.92
.96
1.00
A =
X
.0^
.03
.12
.16
.20
.24
.28
.32
.36
.40
.*»4
.43
.52
.56
.60
.64
.68
.72
.76
.30
.Bit
.88
.92
.96
1.00
2,0 AND B
BETA (X)
.0016000
.0061+000
.0144000
.0256000
.0400000
.0576000
.0784000
.1021+000
.1296000
.1600000
.1936000
.2304000
.2704000
.3136000
.3600000
.4096000
.4624000
.5134000
.5776000
.6400000
.7056000
.7744000
.8464000
.9216000
1.0000000
2.5 AND B
BETA(X)
.0003200
.0018102
.0049833
.0102400
.0178385
.0232181
.04lt»854
.0579262
.0777600
.1011929
.1284197
.1596258
.1949882
.2346768
.2783548
.3276800
.38130^8
.4398770
.5035400
.572«*334
.6466931
.7264516
.811833^
.9029799
1.0000000
1.0
INVERSE
.0400000
. 0800000
.1200000
.1600000
.2000000
.2^00000
.2300000
.3200000
.3600000
.4000000
.«+400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7500000
.8000000
. 3400000
.8800000
.9200000
.9599999
1.0000000
= 1.0
INVERSE
. 0400000
. 0800001
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600001
.4000000
.4400000
.4800001
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
. 8000000
. 3399999
.8300000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
-.OQOOOOO
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000001
0.0000000
XC ERROR TERM
-.0000000
-.0000001
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000001
-.0000000
-.0000000
-.0000001
.0000000
-.0000000
-.0000000
-.0000 000
-.0000000
.0000000
-.0000000
-.0000000
.0000001
-.0000000
.0000000
-.0000000
o.ooooooo
115
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04*
.03
.12
.16
.20
.24*
.23
.32
.36
.1*0
.4*4*
.4*3
.52
.56
.60
.64*
.68
.72
.76
.80
.84*
.88
.92
.96
1.00
A =
X
.04*
.03
.12
.16
.20
.2k
.23
.32
.36
.4*0
.4*4*
.4*3
.52
.56
.60
.64*
.63
.72
.76
.80
.34*
.88
.92
.96
1.00
3.0 AND B
BETA (X)
.000064*0
.0005120
.0017230
.004*0960
.0080000
.013824*0
.0219520
.0327680
.04*66560
.064*0000
.035184*0
.1105920
.14*06080
.1756160
.2160000
.26214*%Q
.314*4*330
.3732t*SO
.4*389760
.5120000
.592704*0
.63H*720
.7786880
.834*7360
1.0000000
3.5 AND 8
BETA (X)
.0000128
.0001t*«*8
.0005986
.0016384*
.0035777
.0067723
.0116159
.0135364*
.0279936
.04,04*772
.056504*7
.0766204*
.1013939
.1314*190
.1673129
.2097152
.2592873
.3167114*
.3826904*
.4*5794*67
.54*32222
.6392774*
.74*68913
.8663607-
1.0000000
1.0
INVERSE
. Ot*00000
. 0800000
. 1200000
.1600000
.2000000
.24*00000
.2800000
.3200000
.360000-0
.t*000000
.4*4*00000
.4*800000
.5200000
.5600000
.6000000
.64,00000
.6800000
.7200000
. 7600000
.8000000
.34*00000
.8800000
.9200000
.9600000
1.0000000
1.0
INVERSE
.04*00000
.0800000
.1200000
.1600000
.2000000
.24*00000
.^SOOOOO
. 3200000
.3600000
.4*000000
.4*4*00000
.4*800000
.5200000
.5600001
.6000000
.64*00000
.6300000
.7200000
.7600000
.8000000
. 34*00000
.3800000
.9200000
.9600000
1.0000000
116
XC ERROR TERM
-.0000000
-.0000000
,0000000
.0000000
-.0000000
.0000000
,0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
-. 0000 000
-.0000000
.0000000
-.0000000
-.0000000
-.0000 000
-.0000000
.0000000
-.0000000
o.ooooooo
XC ERROR TERM
-.0000000
-.0000000
. ooooooo
-.0000000
-.ooooooo
.0000000
. 0000000
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.0000001
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.0000000
-.ooooooo
.ooooooo
-.ooooooo
0.0000000
-------
INCOMPLETE 9ETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
• 64
.68
.72
.76
.80
.Stt
.88
.92
.96
1.00
A =
X
.o«f
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
4.0 AND B
BETA (X)
.0000026
.0000410
.0002074
•0006554
.0016000
.0033178
.0061466
.0104858
.0167962
.0256000
.0374810
.0530842
.0731162
.0983450
.1296000
.1677722
.2138138
.2687386
.3336218
.4096000
.4978714
.5996954
.7163930
.8493466
1.0000000
4.5 AND B
BETA (X)
.0000005
.0000116
.0000713
.0002621
.0007155
.0016254
.0032525
.0059316
.0100777
.0161909
.0248621
.0367773
.0527243
.0735946
.1003877
.1342177
.1763153
.2230322
.2908447
.3663574
.4563056
.5625641
.6871400
.8321863
1.0000000
1.0
INVERSE
.0400000
. 0800000
. 1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400001
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
. 7199999
.7600000
.3000000
.3400000
.8800000
.9200000
.9600000
1.0000000
= 1.0
INVERSE
. 0400000
.0800000
.1199999
.1600000
,2000000
.2400000
,2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
,6400000
.6800000
.7200000
.7600000
. 8000000
.8400000
.8800000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
. 0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000001
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000001
-.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
o.ooooooo
XC ERROR TERM
-.0000000
-.0000000
.0000001
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
o.ooooooo
117
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.Oft
.03
.12
.16
.20
.2k
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.30
.84
.83
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.2k
.23
.32
.36
,40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.30
.84
.88
.92
.96
1.00
5.0 AND B
BETA (X)
.0000001
.0000033
.0000249
.0001049
.0003200
.0007963
.0017210
.0033554
.0060466
.0102400
.0164C16
.0254904
.0380204
.0550732
.0777600
.1073742
.1453934
.1934S18
.2535525
.3276800
.4132119
.5277319
.6590815
.8153727
1.0000000
5.5 AND B
BETA (X)
.0000000
.0000009
.0000086
.0000/419
.0001431
.0003901
.0009107
.0018981
.0036280
.0064763
.0109393
.0176533
.0274169
.0412130
.0602326
.0858993
.1198944
.1641832
.2210420
.2930859
.3832976
.4950564
.6321688
.7988988
1.0000000
1.0
INVERSE
. 0400000
. 0800000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.36 00 '000
.4000000
.4400000
.4300000
.5200000
.5600000
.5000000
,6400000
.6800000
.7200000
.7600000
, 8000000
.3400000
.8800000
,9199999
.9600000
1,0000000
= 1.0
INVERSE
NOT DEFINED
.0800000
.1200000
.1600000
.2000000
.2400000
.2800000
,3200000
.3600000
.4000000
,4400000
.4800000
.5200000
.5600000
.6000000
.6400001
.6800000
.7200000
.7600000
.3000000
.3400000
.3800000
.9200000
.9600000
1. 0000000
XC
-
-
-
-
-
-
-
-
-
-
-
-
0
XC
NOT
-
—
-
-
_
_
-
_
-
-
-
-
0
ERROR TERM
0000000
0000000
0000 000
0000000
ooooooo
0000000
ooooooo
ooooooo
ooooooo
ooooooo
013 0 0 0 0 0
ooooooo
ooooooo
ooooooo
ooooooo
0000 000
ooooooo
ooooooo
ooooooo
ooooooo
ooooooo
ooooooo
0000001
ooooooo
0.0000000
ERROR TERM
DEFINED
.0000000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000 001
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000000
.ooooooo
.ooooooo
118
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A s
X
.0**
.03
.12
.16
.20
• 2it
.28
.32
.36
.40
.44
,48
.52
.56
.60
.64
.68
.72
.76
.30
.34
.88
.92
.96
1.00
A =
X
.04
.08
.12
.15
.20
.24
.23
.32
.36
.kQ
.1*1*
.43
.52
.56
.60
.64
.63
.72
.76
.80
.Bit
.88
.92
.96
1.00
6.0 AND 8
BETA (X)
.0000000
.0000003
.0000030
.0000168
.0000640
.0001911
.0004819
.0010737
.0021768
.0040960
.0072563
.0122306
.0197706
.0303410
.0456560
.0687195
.0938675
,1393141
.1926999
.2621440
.3512930
.4644041
.6063550
.7827578
1.0000000
6.5 AND 6
•BET A(X)
.0000000
.0000001
.0000010
.0000067
.0000236
.0000936
.0002550
.0006074
.0013061
.0025905
.0043133
.003(4736
.0142563
.0230793
.0351396
.0549756
.0 315282
.1132119
.1679919
.2344637
.3219700
.4356496
.5315953
.7669429
1.0000000
1.0
INVERSE
NOT DEFINED
. 0300000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200001
.3599999
.4000000
.4400000
.4799999
.5200000
,5600000
.5000000
.6400000
.6300000
.7200000
.7600000
.3000000
.3400000
.3300000
.9200000
.9600000
1.0000000
= 1.0
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
. 1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
. 7600000
. 3000000
.3400000
.8800000
.9200000
.9600000
1.0000000
119
XC ERROR TERM
NOT DEFINED
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000001
.0000001
-.0000000
.0000000
.0000001
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
-.0000000
-.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000 000
-.0000000
.0000000
-. ooooooo
-.0000000
.ooooooo
-.ooooooo
.ooooooo
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
,12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.84
.88
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.84
.83
.92
.96
1.00
7.0 AND 8
BETA(X)
.0000000
.0000000
.0000004
.0000027
.0000123
.0000459
.0001349
.0003436
,0007836
.0016384
.0031923
.0058707
.0102807
.0172709
.0279936
.0439805
.0672299
.1003061
.1464519
.2097152
.2950903
.4086756
.5578466
.7514475
1.0000000
7.5 AND 8
BETA (X)
.0000000
.0000000
.0000001
.0000011
.0000057
.0000225
.0000714
.0001944
.0004702
.0010362
.0021178
.0040673
.0074135
.0129244
.0216837
.0351844
.0554392
.0851126
.1276738
.1875750
.270t»548
.3833717
.5350677
.7362652
1.0000000
1.0
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
. 1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.3000000
.8400001
.8799999
.9200000
.9600000
1.0000000
= 1.0
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600001
.4000000
.4399999
.4800001
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.8000000
.8400000
.8800000
.9200000
.9600000
1. 0000000
120
XC ERROR TERM
NOT DEFINED
NOT DEFINED
-.0000000
-.0000000
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.9000000
-.0000000
.0000000
-.0000000
.0000 000
.0000 000
-.0000000
.0000000
-.0000000
-.0000001
.0000001
-.QQOOOOO
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
-.0000000
-.0000000
-.0000000
-.0000000
-.0000000
.0000000
-.0000001
-.0000000
.0000001
-.0000001
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
,1+k
.48
.52
.56
.60
.64
.68
.72
.76
.30
.84
.83
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.2k
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.84
.83
.92
.96
1.00
8.0 AND B
BETA (X)
.0000000
.0000000
.0000000
.0000004
.0000026
.0000110
.0000378
.0001100
.0002821
.0006554
.0014048
.0028179
.0053*«60
.0096717
.0167962
.0281475
.0457163
.0722204
.1113035
.1677722
,21*78759
.3596345
.5132189
.7213896
l.OOOOOOQ
3.5 AND B
BETA (X)
.0000000
.0000000
.0000000
.0000002
.0000011
.000005
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A -
X
.04*
.08
.12
.16
.20
.2k
.23
.32
.36
.4*0
.4*t*
.4*3
.52
• 56
.60
.64*
.68
.72
.76
.30
.84*
.33
.92
.96
1.00
A =
X
.04*
.03
.12
.16
.20
,2k
.23
.32
.35
.
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
,2k
.23
.32
.36
.40
• 44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
2.0 AND B
BETA (X)
.0029597
.0116750
.0258945
.0453578
.0697957
.0989284
.1324648
.1701013
.2115200
.2563872
.3043511
.3550390
.4080543
.4629721
.5193338
.5766400
.6343i,Q<3
.6918229
.7483884
.80322&0
.8553600
.9035594
.9461467
.9804800
1.0000000
2.5 AND B
BETA(X)
.0006425
.0035806
.0097159
.0196290
.0337287
.0523023
.0755397
.1035476
.1363570
.1739276
.2161499
.2623445
.3137606
.3685713
.4268677
.4881494
.5518112
.6171235
.6832036
.7489718
.8130781
.8737710
.9286247
.9738084
1.0000000
1.5
INVERSE
. 0400000
.0800000
.1200000
.1600000
.2000000
.2400000
.2800001
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
,7200000
.7600000
.8000000
.8400000
.8300000
.9199999
.9600000
1.0000000
1.5
INVERSE
. 0400000
.0800000
.1200000
,1600000
.2000000
.2400000
.2800000
.3199999
.3600000
.^000000
,4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
.8800000
.9200000
.9600000
1.0000000
123
XC ERROR TERM
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000001
.0000000
-.000,0000
-.0000000
.0000000
-.0000000
.0000000
. 0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000001
.0000000
o.ooooooo
XC ERROR TERN
-.0000000
-. 0000000
-.0000000
. 0000000
-.0000000
-.0000000
.0000000
.0000001
-.0000000
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000 000
.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.2k
.28
.32
.36
.1+0
.44
.43
.52
.56
.60
.64
.63
.72
• 76
.80
.Sit
.88
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
,2k
.28
.32
.36
.40
.44
.43
.52
.56
.60
• 64
.63
.72
.76
.30
.34
.38
.92
.96
1.00
3.0 AND B
BETA (X)
.0001379
.0010858
.0036056
,0084039
.0161301
.0273727
.0^26566
.0624388
.0871040
.1169598
.1522302
.1930488
.2394496
.2913567
.3435708
.4107520
.4773972
.5473094
.6210541
.6958948
.7706880
.8432009
.9102370
.9666560
1.0000000
3.5 AND B
BETA (X)
.0000293
.0003265
.0013269
.0035688
.0076528
.0142151
.0239071
.0373765
.0552504
.0781185
.1065162
.1409067
.1816614
.2290376
.2331540
.3439599
.4111977
.4343540
.5625943
.6446680
.7237624
.8122521
.8912020
.9590921
1.0000000
= 1.5
INVERSE
.0400001
. 0800000
.1200000
.1500000
.2000000
.2400000
.2799999
. 3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6399999
.6800000
.7200000
.7600000
.8000000
.3400000
.8800000
.9200000
.9600000
1.0000000
1.5
INVERSE
.0400000
.0800000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
. 3400000
.8800000
,9200000
.9600000
1.0000000
XC ERROR TERM
-.0000001
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000001
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000 001
-.0000 000
.0000000
. ooooooo
.0000000
.0000000
,0000000
.ooooooo
.0000000
0.0000000
XC ERROR TERM
-.ooooooo
-.ooooooo
.ooooooo
-.0000000
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.0000 000
-.0000 000
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
124
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0'*
.08
.12
.16
.20
.2*»
.28
.32
.36
.<*Q
.M*
.^8
.52
.56
.60
• 61*
.68
.72
.76
.30
.8*t
.88
.92
.96
1.00
A =
X
.0^
.08
.12
.16
.20
• 2*
.28
.32
.36
.tfO
• kk
.1*8
.52
.56
.60
.6*
.68
.72
.76
.30
• 8*
.88
.92
.96
1.00
**.0 AND B =
BETA (X) °
.0000062
.0000975
.000^852
.0015059
.0036081
.0073371
.0133193
.0222^i»8
.Q3tt3^93
.0518937
.Q7MM5
.10233*t3
.1371627
.179231*6
.2290367
.2868890
.3528885
.1*268380
.5081511
.5957189
.6877091*
.7812328
.8716939
.9511731
1.0000000
tf.5 AND 8 =
BETA(X)
.0000013
.0000290
.0001765
.0006321
.0016926
.Q03768£»
.00738*7
.0131768
.0218808
.031*3200
.0513862
.07**0152
.1031567
.1397360
.18/4607i»
.238^956
.3019208
.3751037
.^578365
.5^*930^6
.6f»7819t«
.7503817
.85135it3
.9f,29«462
1.0000000
1.5
INVERSE
.OffOOOOO
. 0800000
.1200000
.1600000
.2000000
.2^+00000
.2799999
.3200000
.3600000
.JfOOOOOO
.'t^OOOOO
.^800000
.5200000
.5600001
.6000000
,5^00000
,6800001
.7200000
.7600000
.8000000
.HttOOOOO
.8800000
.9200000
.9599999
1.0000000
1.5
INVERSE
.0^00000
.0800000
.1200000
.1600000
.2000000
.2399999
.2800000
.3200000
.3599999
,<*000000
.^00001
,^800000
.5200000
.5600000
.6000000
.6^00000
.6300000
,7200000
.7600000
.8000000
.8'* 00 000
.8800001
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.0000000
•0000000
.0000000
-.0000000
.0000000
.0000001
-.0000000
•0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000001
-.0000000
-.0000000
-.0000001
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000001
o.ooooooo
XC ERROR TERM
-.0000000
-.0000000
. 0000000
.0000000
-.0000000
.0000001
-.0000000
-.0000000
.0000001
-.0000000
-.0000001
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000 000
.0000000
-.0000000
-.0000000
.0000000
-.0000001
.0000000
.0000000
0.0000000
125
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.84
.83
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.2k
.23
.32
.36
.40
• 44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.84
.33
.92
.96
1.00
5.0 AND B
BETA (X)
.0000003
.0000086
.0000639
.0002642
.0007907
.0019275
.0040780
.0077749
.0136861
.0226139
.0354376
.0533484
.0773249
.1035977
.1483512
.1977076
.2576393
.3288512
.4116190
.5055606
.6092947
.7198845
.8318013
.9344516
1.0000000
5.5 AND B
BETA(X)
.0000001
.0000025
.0000231
.0001101
.0003681
.0009825
.0022**^
.00if5?25
.0085330
.Olt»8541
.02^337
.0383397
.0577935
.08^1706
.1139098
.163^987
.2193561
.287703*4
.36937ft3
.i»6«f537l
.5722726
.6898862
.8116332
.92572^0
1.0000000
1.5
' INVERSE
.0^00000
.0300000
.1200000
.1600000
.2000000
.2^00000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
. 8800000
.9200000
.9600000
1.0000000
= 1.5
INVERSE
NOT DEFINED
.0800000
.1200000
.1599999
.2000000
.2400000
.2799999
.3200000
.3500000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7199999
.7600000
.8000000
.8400000
.3800000
.9200000
.9600000
1.0000000
XC ERROR TERM
-. 0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
. 0000 000
. ooooooo
-. ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
NOT DEFINED
-.OOOOOOO
.ooooooo
.0000001
-.ooooooo
.ooooooo
.0000001
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.0000001
-.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
0.0000000
126
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0**
.08
.12
.16
.20
.2<+
.28
.32
.36
.<»0
.*+**
,tf3
.52
.56
.60
,6
.0000000
.0000002
.0000030
.0000189
.0000791
.0002532
.0006743
.0015688
.0032909
.0063598
.0114965
.0196537
.0320680
,0502251
.0759076
.1111373
.1531081
.2190545
.2960312
.3905581
.5030442
.631810*4
.7712653
.907687-6
1.0000000
1.5
INVERSE
NOT DEFINED
. 0800000
. 1200000
. 1600000
.2000000
.2«+00000
.2799999
. 3200001
. 3600000
.ffOOOOOO
.
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.2k
.28
.32
.36
.40
.4 4
.43
.52
.56
.60
.64
.63
.72
.76
.80
.34
.83
.92
.96
1.00
A -
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.54
.63
.72
.76
.30
.84
.38
.92
-96
1.00
7.0 AND 8
BETA (X)
.0000000
.0000001
.0000011
.0000073
.0000365
.0001281
.0003683
.0009158
.0020371
.0041482
.0078614
.0140344
.0238163
,0386879
.0604847
.0913938
.1339081
.1907133
.2644744
.3574607
.4709052
.6038853
.7511948
.8984295
1.0000000
7.5 AND 8
BETAfX)
.0000000
.0000000
.0000004
.0000032
.0000163
.0000647
.0002008
.0005336
.0012586
.0027008
.0053663
.0100020
.0176589
.0297534
.0481217
.0750481
.1132557
.1658251
.2360026
.3268225
.4404191
.5767725
.7312700
.8890407
1.0000000
= 1.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2400000
.2799999
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.3400000
.8800000
.9200000
.9600000
1.0000000
1.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2400000
.2800000
. 3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.8000000
.8400000
. 8800000
.9200000
.9600000
1. 0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
-.0000000
.0000000
-.0000000
-.0000000
.0000001
. ooooooo
-.0000000
-.ooooooo
.0000000
-.0000000
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.0000000
-.ooooooo
-.ooooooo
,0000000
.ooooooo
.ooooooo
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
-.OOOOOOO
.OOOOOOO
-.ooooooo
-.OOOOOOO
.ooooooo
.ooooooo
-.ooooooo
-.0000000
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.0000000
.0000 000
-.ooooooo
.0000000
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
0.0000000
128
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0**
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.6«*
*68
.72
.76
.30
.84
.88
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
,2k
.28
.32
.36
.40
.44
.1+8
.52
• 56
.60
.64
.68
.72
.76
.80
,84
.83
.92
.96
1.00
3.0 AND B
BETA (X)
.0000000
.0000000
.0000001
.0000013
.0000077
.0000326
.0001093
.0003104
.000776**
.0017556
.003657**
.0071174
.0130738
.0228494
.0382328
.0615447
.0956691
.1440169
.2103703
.2985229
•4115644
.550506*4
.711533**
.8795405
1.0000000
3.5 AMD B
-BETA(X)
.0000000
.0000000
.0000000
.0000005
.0000036
.000016*t
.000059*+
.0001803
.0004782
.0011396
.0021*892
.0050578
.0096666
.0175250
.0303386
.0504113
.0807227
.1249450
.1873394
.2724347
.3843071
.5251103
.6920213
.8699463
1.0000000
1.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2^00000
.2800000
.3200000
.36000.00
.4000000
.4400000
.4800000
.5200000
.5600000
.&OOOQOO
.6400000
.6800000
.7200000
.7600000
.3000000
.8400000
.8800000
.9200000
.9600000
1.0000000
- 1.5
INVERSE
NOT DEFINED
NOT DEFINED
NOT DEFINED
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
. 8800000
.9200000
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.0000000
.0000000
-•OQOOOOO
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000 000
.0000 000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
NOT DEFINED
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
. ooooooo
-.0000000
o.ooooooo
129
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0«»
.08
.12
.16
.20
.2
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
• 40
.44
.43
.52
.58
.60
.64
.68
.72
-76
.80
.3%
.88
.92
.96
1.00
A =
X
• 04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.84
.88
.92
.96
1.00
2.5 AND 8
BETA (X)
.0010880
.0059736
.0159626
.0317440
.0536656
.0 818326
.1161591
.1564007
.2021760
.2529822
.3082073
.3671393
.4289741
.1*928212
.5577096
.6225920
.6863437
.7477909
.8056640
.8586501
.9053703
. 9443871
.9742060
.9932779
1.0000000
3.0 AND B
BETA (X)
.0002*483
.0019251
.0062899
.0144179
.0272000
.Ott53t«27
.0693683
.09961£t7
.1362355
.1792000
.2282931
.2831155
.3^30835
.^07^291
.tf752000
.5452595
.6162867
.6867763
.7550387
.3192000
.8772019
.9268019
.9655731
.99090«»3
1.0000000
2,0
INVERSE
.0^00000
.0800000
.1200000
.1599999
.2000000
.2^00000
.2300000
.3200000
.3600'OQO
.ffOOOOOO
.^399999
.ttSOOOOO
.5200000
.5600000
.6000000
.&i»00000
.6800000
.7200000
.7600000
.8000000
.aifooooo
,8800000
.9200000
.9600000
i.ooqoooo
2.0
INVERSE
.0400001
. 0800000
.1200000
.1500000
.2000000
.2^00000
.2800000
.3200000
.3600001
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7599999
. 3000000
.8400000
.8800000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.0000000
-.QOOOOOO
.0000001
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000001
-.0000000
.0000000
.0000000
.0000000
.0000 000
.0000 000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
0.0000000
XC ERROR TERM
-.0000001
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000001
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
iQOOOOOO
-.0000 000
.0000000
.0000001
•ooooooo
.QOOOOOO
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
131
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.63
.72
.76
.80
.84
.88
.92
.96
1.00
A =
X
• 04
.03
.12
.16
.20
.2k
.23
.32
.36
.40
• 44
.43
.52
.56
.60
.6*+
.68
.72
.76
.80
.84
.83
.92
.96
1.00
3.5 AND B
BETA(X)
.0000558
.0006111
.0024423
.0064553
.0135953
.0247868
.0408880
.0626530
.0906993
.1254792
.1672533
.2160695
.2717356
.3333042
.4015509
.4739564
.5496890
.6270886
.7041503
.7785094
.8474266
.9077739
.9560209
.9882212
1.0000000
4.0 AND B
BETA (X)
.0000124
.0001917
.0009373
.0028574
.0067200
.0134038
.0233487
.0390070
.0597943
.0870400
.1214383
.1634992
.2134S92
,2714321
.3369600
.4093641
.4874954
.5697257
.6538986
.7372800
.8165090
.3375491
.9456387
.9352420
1.0000000
2.0
INVERSE
.0400000
.0300000
.1200000
.1600000
,2000000
.2400000
.2800000
.3200000
. 3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.3000000
.8400000
.8799999
.9200000
.9600000
1.0000000
2.0
INVERSE
.0400000
.0800000
. 1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
. 8399999
.3800000
.9200000
.9600000
1 .0000000
132
XC ERROR TERM
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
,0000000
. 0000000
.0000001
.0000000
.0000000
o.ooooooo
XC ERROR TERM
-.0000000
-.0000000
. 0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
-.0000000
-.0000 000
.0000000
-.0000000
.0000000
.0000001
.0000000
.0000000
.0000000
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04*
.08
.12
.16
.20
.21*
.28
.32
.36
.4*0
.i*t»
.4+8
.52
.56
.60
.6*+
.63
.72
.76
.30
.8^*
.33
.92
.96
1.00
A =
X
.04*
.08
.12
.16
.20
,2k
.23
.32
.36
.4*0
.<+«»
.^ 3
.52
.56
.60
• 64*
.63
.72
.76
.80
.34*
.33
.92
.96
1.00
<+.5 AND 6
BETA (X)
.0000027
.0000595
.0003563
.0012530
.0032915
.007184+1
.0137901*
.024*0825
.0391015
.0599062
.087514*4*
.1228378
.1666104*
.2193120
.2810856
.3516504+
.4+302094*
.5153528
.604*9570
.6960790
.734*84,74+
.86634+37
.934»51Q(*
.9819798
1.0000000
5.0 AND 8
BETA (X)
.0000006
.0000182*
.00013«t4«
.0005^53
.0016000
.0033221
.0079168
.014+764*0
.0253953
.04*09600
.0626682
.0917294*
.129269'*
.176234*2
.2332500
,3006«i77
.3730227
.4*64*3302
.5573156
.6553600
.7527G15
,9i*i*37ll
.922714*1
.9734*4*72
i.ooooooo
2.0
INVERSE
. 04*00000
.0800000
.1200000
.1600000
.2000000
.24*00000
.2300000
.3200000
,3600000
.4*000000
.4*4*00000
. 4*800000
.5200000
,5600001
.6000000
.64*00000
.6800000
.7200000
.7600000
,8000000
.84*00000
.3300000
.9200000
.9600000
1.0000000
= 2.0
INVERSE
. 04*00000
.0300000
.1200000
.1600000
.2000000
.24*00000
.2800001
.3200000
.3600000
.4*000000
.4* <*00000
.4*799999
.5200000
.5600000
.6000000
.64*00000
.6300001
.7200000
.7600000
.3000000
. 34*00000
. 8800000
.9200000
.9599999
1.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000001
-.0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
o.ooooooo
XC ERROR TERM
-.0000000
-.0000000
. ooooooo
-.0000000
-.ooooooo
.0000000
-.0000001
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.0000001
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.0000001
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000001
o.ooooooo
133
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0^
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.63
.72
.76
.80
.34
.38
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
• 72
.76
.30
.84
.33
.92
.96
1.00
5.5 AND B
BETA (X)
.0000001
.0000056
.0000503
.0002357
.0007728
.0020207
.00^5170
.0089971
.0163984
.0278^83
.0446324
.0631419
.0997975
.1409484
.1927444
.2559801
.3309036
.4170253
.5128174
.6154804
.7205994
.8217937
.9103231
.9746566
1.0000000
6.0 AND B
BETA (X)
.0000000
.0000017
.0000188
.0001013
.0003712
.0010625
.0025637
.0054546
.0105356
.0188416
.0316375
.0503900
.0767100
.1122612
.1586304
.2171535
.2836930
.3733617
.4701878
.5767168
.6835441
.7937750
.8974054
.9706197
1.0000000
2.0
INVERSE
. 0400000
. 0800000
.1200000
.1600000
.2000000
.2400000
.2800000
. 3200000
.3600000
.4000000
.4400000
,4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
. 7600000
.3000000
. 3400000
.8800000
.9200000
.9600000
1.0000000
2.0
INVERSE
NOT DEFINED
.0800000
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3&00000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.3000000
.8400000
. 8800000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.0000000
. 0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
•ooooooo
.0000000
.0000000
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.ooooooo
.0000000
-.ooooooo
.ooooooo
.ooooooo
-.0000000
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
134
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0**
.08
.12
.16
.20
«2£f
.28
.32
.36
.z*0
»i*k
»k9
,52
.56
.60
,6<+
.6S
.72
.76
.30
.8%
.88
.92
.96
1.00
A =
X
.Ok
.03
.12
.16
.20
.2k
.28
.32
.35
.<*0
»kk
.k6
.52
.56
.60
,6*f
.68
.72
.76
.80
.8^
.83
.92
.96
1.00
6.5 AND B
BETA (X)
.0000000
.0000005
.0000070
«,0000«»3«4
.0001775
.0005561
.001^8*4
.0032921
.0067393
.0126936
.0223337
• 03711** l»
.0587380
.0890860
.1301025
.183618**
.2511069
.3333576
.2*300593
.5392781
.6568187
.775i»56*»
«33**Q2%8
»96&3t»80
1,0000000
7.0 AND B
.BETA(X)
.0000000
-.0000002
.0000026
.0000185
.00008^5
.0002899
,,0008150
.0019791
.Q0*t29*»%
.0035197
.0157085
.0272^00
.0^«f8239
.07014655
,1063757
e15^8112
92l782ft8
^2969061
8392^S12
^5033165
,6255915
.7519631
,8702«,Q7
.9613528
1.0000000
2,0
INVERSE
NOT DEFINED
• 0300000
,1199999
. 1600000
,2000000
«2399999
92800QOO
.3200001
B3599999
.^000000
»%4QOOQQ
.JfSOOOOO
85200000
.5600000
,6000000
S6^00000
^6300000
,7199999
.7600000
. 3000000
, SJiOOOOO
. 8800000
.9200000
®9&OOOOQ
1* 0000000
= 2.0
INVERSE
NOT DEFINED
S0800000
^1200000
.1600000
82000000
a2«400000
S2800000
83200000
,,3600000
s^000000
.itffOOOOQ
e^800000
.5200000
B5&QOQOO
.6000000
86^00001
s&800000
,7200000
,7600000
,8000000
. 8^00000
S8300000
.9199999
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
~»0000000
.0000001
.0000000
-.0000000
sQOOOQOl
-«ooooooo
-eOOQOQQi
^0000001
-«ooooooo
.0000000
.= 0000000
-,,0000000
.0000000
~«0000000
-^ooooooo
*ooooooo
.0000001
-*QQOOOOO
-4,0000000
SQOOOGOO
-.0000000
.0000000
BOOOOOOQ
o.ooooooo
XC ERROR TERM
NOT DEFINED
-sQQOOQOQ
-.OOOOOOO
-.0000000
-9OQOOOQQ
-»OQOOOOO
.0000000
^ooooooo
-.0000000
-,0000000
*OQOQOQQ
-.0000000
-.0000000
.0000000
-aOOOOOOO
-.0000001
^ooooooo
-.0000000
-.0000000
-. ooooooo
.0000000
-.ooooooo
.0000001
.0000000
o.ooooooo
135
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.2k
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.80
.84
.33
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.fit*
.63
.72
.76
.30
.34
.33
.9?
.96
1.00
7.5 AND E
SETA (X)
.0000000
.0000000
.0000009
.0000073
.0000401
.0001505
.0004569
.0011856
.0027271
.0056992
.0110128
.0199299
.0341022
.0555749
.0857350
.1301822
.183/4932
.2638490
.3574863
.4689374
.5950005
.7284062
.8561083
,9571447
1.0000000
8.0 AND a
BETA (X)
.0000000
.0000000
.0000003
.0000033
.0000189
.0000779
.0002554
.0007081
.0017265
.0038011
.0076984
.0145405
.0258745
.0437162
.0705439
.1092123
.1627501
.2339941
.3250062
.4362076
.5651570
.7048837
.8416790
.9522342
1.0000000
2.0
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5599999
.6000000
.6400000
.6799999
.7200000
, 7600000
.8000000
.8400000
.8300000
.9199999
.9600000
1.0000000
= 2.0
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2400000
.2800000
. 3200000
.3600001
.4000000
.4399999
.4300000
.5200000
.5600000
.6000000
.6400000
.5800000
.7200000
.7600000
.3000000
, 3400000
.8800000
.9200000
.9600000
1.0000000
136
XC ERROR TERM
NOT DEFINED
NOT DEFINED
-.0000000
.0000000
-. ooooooo
-.0000000
.0000000
.ooooooo
-.0000000
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.0000001
-.ooooooo
.0000 000
.0000001
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.0000001
.ooooooo
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.OOOOOOO
.OOOOOOO
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
-.0000001
-.ooooooo
.0000001
-.ooooooo
- .ooooooo
.ooooooo
-.ooooooo
.ooooooo
.0000 000
-.ooooooo
.ooooooo
-.ooooooo
-.0000000
-.ooooooo
. 0. 0 0 0 0 0 0
.noooooo
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.2k
.28
.32
.36
.4*0
.1*1*
.43
.52
.56
.60
.64
.63
.72
.76
.30
.84
.88
,92
.96
1.00
A =
X
• 04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.75
.30
.84
.38
.92
.96
1.00
8.5 AND 6
BETA (X)
.0000000
.0000000
.0000001
.0000014
.0000089
.0000402
.00011*23
.0094217
.0010901
.0025284
.0053675
.0105816
.0195836
.0343065
.0572451
.091t,231
.1402390
.2071300
.2949777
.4051620
.5361495
.6814815
,8270006
.9471315
1,0000000
9.0 AND 8
8ETA(X)
,0000000
.0000000
.0000000
.0000006
.00000^2
.0000207
.0000791
.0002505
.0006865
.0016777
.0037335
.0076828
.01^7891
.02636^2
.Oi»6357^
.0763810
.1206179
.1830354
.2673064
.3753096
.5080464
.6582750
.3121175
.9413462
1.0000000
2.0
INVERSE
NOT DEFINED
NOT DEFINED
. 1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
,6800000
.7200000
.7600000
,3000000
.3400000
,8300000
.9200000
.9600000
1.0000000
2.0
INVERSE
NOT DEFINED
NOT DEFINED
NOT DEFINED
. 1600000
,2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4800000
,5200000
.5600000
.5000000
,6400000
.6300000
.7200000
.7600000
.3000000
.8400000
. 3800000
.9200000
.9600000
1.0000000
137
XC ERROR TERM
NOT DEFINED
NOT DEFINED
. ooooooo
-.0000000
-.ooooooo
.0000000
.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.0000000
-.ooooooo
.ooooooo
-.ooooooo
• 0000 000
.0000000
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
NOT DEFINED
-.0000000
-.ooooooo
.ooooooo
.0000000
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.0000000
.ooooooo
-.0000000
. ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooono
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.6ft
.63
.72
.76
.30
.34
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
,2k
.23
.32
.36
.40
• 44
.43
.52
.56
.60
.6**
.68
.72
.76
.80
.34
.33
.92
.95
1.00
2.5 AND B
BETA (X)
.0016645
.0090042
.0236^74
.0463959
.0771837
.1157809
.1615941
.2138329
.271531,8
.3336096
.3938727
.4660741
.5339259
.6011273
.6663904
.7284652
.7861671
.8384059
.8842191
.9223113
.9536041
.9763026
.9909953
.9933355
1.0000000
3.0 AND 0
BETA (X)
.0004013
.0030625
.0098465
.0222001
.0411741
.0674439
.1013313
• lt»28268
.191613t«
.?if70<=20
.303ff076
.37tf%778
.fff»tt0233
.3156008
.5876390
.653f+781
.726£»1«»6
.7897521
.3£*63603
.8962£»6(+
.9366««51
.9671370
.9873232
.9976218
1.0000000
2.5
INVERSE
.OitOOOOO
. 0300000
.1200000
. 1600000
.2000000
.2^00000
.2800000
.3200000
.3600000
.ffOOOOOO
.^00000
, ^ 8 0 0 0 0 0
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.3000000
.8t,00000
.8300000
.9200000
.9600000
1.0000000
= 2.5
INVERSE
. 0400000
. 0800000
.1200001
.1600000
.2000000
.2400001
.2300000
.3199999
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
. 3300000
.9200000
.9600000
1. 0000000
138
XC ERROR TERM
-. 0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
. 0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
. 0000 000
.0000 000
. ooooooo
.0000000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
-.ooooooo
-.0000000
-.0000001
.0000000
-.ooooooo
-.0000001
.ooooooo
.0000001
-.ooooooo
-.ooooooo
-.ooooooo
-iOOOOOOO
.0000000
.ooooooo
.ooooooo
. ooooooo
.ooooooo
.0000000
.0000000
.ooooooo
.0000000
.0000000
.ooooooo
.ooooooo
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A -
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
• 44
.48
.52
.56
.50
.64
.68
.72
.76
.30
.84
.83
.92
.96
l.QO
A =
X
.0**
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
• 76
.80
.34
.33
.92
.96
1.00
3.5 AND B
BETA (X)
.0000947
.001020?
.0040114
.0104212
.0215599
.0335909
.0624595
.0938425
.1331127
.1303149
.2351530
.2969370
.3643390
.4374079
.5130959
.5900434
.6661770
.7392714
.3070292
.8671827
.9176295
.9566165
.9830123
.9967658
1.0000000
4.0 AND B
-RET A {X)
.0000220
.0003347
.0016085
.0048169
.0111213
.0217623
.0379626
.0608310
.0912844
.1299730
.1772186
.2329632
.2967302
.3675997
.4441981
.5247060
.6068862
.6331361
.7655701
.3361409
.3963154
.9448235
.9780729
.9957633
1.0000000
2.5
INVERSE
. 0400000
. 0800000
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400001
.6800000
.7200000
.7600000
.8000000
.8400000
. 8300000
.9200000
.9599999
1.0000000
2.5
INVERSE
. 0400000
.0800000
. 1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.3000000
. B400000
. 8800000
.9200000
.9599999
1. 0000000
139
XC ERROR TERM
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000001
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000001
0.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000 000
.0000000
. ooooooo
.0000000
.ooooooo
.0000000
.ooooooo
.0000001
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.1*8
.52
.56
.60
.64
.68
.72
.76
.80
.34
.83
.92
.96
1.00
A =
X
.0^
.08
.12
.16
.20
.24
.23
.32
.36
.40
.44
.43
.52
.56
.60
.64
.63
.72
.75
.30
.34
.33
.92
.96
1.00
ft. 5 AND 8
BETA (X)
.0000050
.0001083
.0006366
.0021984
.0056660
.0121258
.0228056
.0389898
.0619243
.0927180
.1322436
. 18101*17
.2392315
.3064323
.3317005
.4634862
.5496150
.6373044
.7232229
.8036071
.374459S
.9313683
.9725197
.9946130
1.0000000
5.0 AND 8
8ETA (X)
.0000011
.0000347
.0002493
.0009927
.0028563
.0066880
.0135657
.0247529
.0416215
.0655576
.0973493
.1395635
,1914156
.2536333
.3253593
.4069866
.4951272
.5875363
.6306213
.7700249
.3503121
.9173352
.9663712
.9933113
1.0000000
2,5
INVERSE
.0400001
. 0300000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.5800000
.7200000
. 7600000
.3000000
. 3400000
.8800000
.9200000
.9600000
1.0000000
2.5
INVERSE
.0400000
.0300000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.359999-9
.4000000
.4400000
,4300000
.3200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
. 3000000
.8400000
. 3300000
.9200000
.9600000
1 . 0000000
140
XC ERROR TERM
-.0000001
-.0000000
. ooooooo
.0000000
-.ooooooo
.0000000
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
. ooooooo
-.ooooooo
-.ooooooo
.0000 000
-.ooooooo
. ooooooo
. ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
-. ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.0000-001
-.ooooooo
-.ooooooo
.ooooooo
-. ooooooo
-.ooooooo
-.ooooooo
-.0000 000
-.0000 000
.OOQOOOO
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000000
O.QOOOOOO
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.01*
.03
.12
.16
.20
• 2i»
.23
.32
.36
.1*0
.1*1*
.1*8
.52
.56
.60
.61*
.63
.72
.76
.30
.81%
.83
.92
.96
1.00
A =
X
.0**
.08
.12
.16
.20
.2**
.23
.32
.36
.1*0
«ifl*
.1*8
.52
.56
.60
.61*
.63
.72
.76
.80
.88
.92
.96
1.00
5.5 AND B
BETA (X)
.0000003
.0000110
.0000967
.000i*i«i*2
.0011+277
•0036569
.0080015
.0155860
.0277539
.01*59996
.0718701
.1068366
.15211*36
.2086371
.276581*2
.3551*907
.1*1*39321
.5391*162
.6382993
.7357931
.8261099
.9028305
.95961*88
.9918571*
1.0000000
6.0 AND 8
BETA{X}
.0000001
.0000035
.0000372
.0001972
.0007080
.001981*6
.001*6852
.0 0971*1*1*
.0183793
.0320615
.0521*500
.0812821
.1202230
.1706753
.2335551
.30901*1*0
.3963322
,1*93371*9
.5966929
.701261*2
,8005765
.886951*9
.9523760
.9902506
1.0000000
= 2.5
INVERSE
.01*00000
. 0300000
.1200000
.1600000
.2000000
.21*00000
.2800000
.3200000
.3600000
.<*OOOGOO
. 1*1*00 ooo
.1*799999
.5200000
.5600000
.6000000
.61*00000
.6800000
.7200000
.7600000
.3000000
.81*00000
.8800000
.9200000
.9600000
1.0000000
= 2.5
INVERSE
NOT DEFINED
.0800000
.1200000
.1600000
.2000000
.21*00000
.2300000
.3200000
.3600000
.1*000000
. 1*1*00 000
.1*800000
.5200000
.5600000
.6000000
.61*00000
.6800001
.7200000
.7600000
.8000000
.81*00000
.3300000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-•ooooooo
.0000000
.0000000
-.ooooooo
.0000000
-.0000000
.ooooooo
.0000001
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.QOOOOOO
.ooooooo
.ooooooo
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
-.OOOOOOO
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.0000000
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.0000 001
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
141
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
,2k
.23
.32
.36
.1*0
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.38
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.2k
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.34
.88
.92
.96
1.00
6.^ AND 8
BETA (X)
.0000000
.0000011
.0000142
.0000870
.0003488
.0010701
.0027259
.0060543
.0120976
.0222157
.0380609
.0615041
.0945073
.1389358
.1963136
.267523**
.3524684
.4497149
.5561549
•6667461
.7744194
.3703031
.9445781
,9834907
1.0000000
7.0 AND 8
.-SET A (X)
.0000000
.0000003
.OOQ005<*
.0000381
.0001708
.0005736
.0015770
.0037409
.0079203
.0153135
.0274804
.0463133
.0739478
.1126008
.1643269
.2306899
.3123565
.4036297
.5169605
.6325035
.7478291
.8529866
.9362815
.9365778
1.0000000
2.5
INVERSE
NOT DEFINED
. 0800000
.1199999
. 1600000
.2000000
.2400000
.2800000
.3200001
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8399999
.8300000
.9200000
.9600000
1.0000000
2.5
INVERSE
NOT DEFINED
. 0800000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
,4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.3400000
.8800000
.9200000
.9600000
1.0000000
XC ERROR TERH
NOT DEFINED
-.0000000
, 0000001
.0000000
-.0000000
.0000000
-.0000000
-.0000001
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000 000
. ooooooo
.0000000
-.0000000
.0000000
.0000001
.ooooooo
.0000000
.ooooooo
0.0000000
XC ERROR TERM
NOT DEFINED
-.OOOOOOO
-.ooooooo
,0000000
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.0000000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
142
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A -
X
. 0'+
.08
.12
.16
.20
»2k
.28
.32
.36
.**0
.****
.*»3
.52
.56
.60
»6«f
.68
.72
-76
.80
.3**
.88
.92
.96
1.00
A =
X
• Q*f
.08
.12
.16
.20
.2k
.28
.32
.36
.**0
.<4*+
.i+8
.52
.55
.60
.S*f
.63
.72
.76
.30
.3*+
.33
.92
.96
1.00
7,5 AND E
BETA (X)
.0000000
.0000001
.0000020
.0000166
.0000832
.0003060
.0009078
.0023002
.0051606
.0105068
.0197519
.03*47230
.0576201
.0908956
.1370368
.1982323
.275920**
.3702257
.*t79318it
,598761*+
.7209793
.8350838
.927513*+
.981*5127
1.0000000
8.0 AND B
BETA (X)
.0000000
.0000000
.0000008
.0000072
.GOQO*tO**
.OQOl62i»
.0005203
.001*4082
.0033*482
.0071788
. Q1M395
.G2593U
.0*4^7233
.0731102
.1133393
.1697976
,2*»30165
.33^5381
.*4*433788
.565707*4
. 69i+0267
.8166839
.9133016
.9822963
1.0000000
2.5
INVERSE
NOT DEFINED
. 0800001
. 1200000
.1600000
.2000000
,2*+00000
.2800000
. 3200000
.3600000
.*tOOOOOO
.*t*400000
.*t300000
.5200000
.5600000
.6000000
,6*+00000
.6300000
.7199999
.7600000
.8000000
. 3*400000
.8300000
.9200000
.9600000
1.0000000
2.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200001
.1600000
.2000000
.2t»00001
.2800000
.3200000
.360000-0
,*4000000
.*4*400000
.^300000
.520,0000
.5599999
.6000000
.6*t00001
.6300000
.7199999
.7600000
. 3000000
.8'+00000
. 8300000
.9200000
.9600000
1.0000030
XC ERROR TERM
NOT DEFINED
-.0000001
-.0000000
-.0000000
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000 000
.0000001
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
-.0000001
. 0000000
-.0000000
-.0000001
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000001
-.0000000
-.0000 001
.0000 000
.0000001
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
o.ooooooo
143
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.30
.84
.88
.92
.96
1.00
A =
X
.Off
.03
.12
.16
.20
.21*
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.8^
.83
.92
.96
1.00
3.5 AND B
BETA (X)
.0000000
.0000000
.0000003
.0000031
.0000195
.0000859
.0002970
.0008587
.0021639
.0048864
.0100848
.0192971
.0346024
.0536125
.0943575
.1450168
.2134571
.3015471
.4092434
.5334960
.6671118
.7978870
.9086741
.9799300
1.0000000
9.0 AND B
BETA (X)
.0000000
.0000000
.0000001
.0000013
.0000094
.0000453
.0001689
.0005218
.0013936
.0033147
.0071639
.0143133
.0266652
.0468439
.0779524
.1235195
.1870245
.2711898
.3769714
.5022511
.6403589
.7737579
.3936537
.9774154
1.0000000
2.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2400000
.2300000
. 3200000
.3600000
.4000000
.4399999
.4300000
.5200000
.5600000
.6000000
.6400000
.6799999
.7200000
.7600001
.8000000
.3400000
.3800000
.9200000
.9600000
1. 0000000
2.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
. 1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6300000
,7200000
.7600000
. 3000000
.8400000
.3800000
.9200000
.9600000
1. 0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.0000000
.0000000
-.0000000
.0000000
.0000000
. ooooooo
.0000000
-.ooooooo
.0000001
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.0000 000
. 0000 001
-.ooooooo
-.0000001
-.0000000
-.ooooooo
-.ooooooo
. ooooooo
.ooooooo
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.OOOOOOO
.OOOOOOO
-.ooooooo
.OOOOOOO
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
. ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.0000000
.ooooooo
.ooooooo
0.0000000
144
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A ~
X
.04
.03
.12
.16
.20
• 2k
.28
.32
.36
.40
.44
.48
.52
.56
.50
.64
.68
.72
.76
.30
.84
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
• 2k
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
3.0 AND B
BETA (X)
.0006022
.0045253
.0143189
.031758?
.0579200
.0932512
.13764*78
.1905263
.2508973
.3174*400
.3835753
.4625400
.537*4600
. 611i*2i»7
.6825600
.7491027
.8094737
.8623522
.9067488
,9<*2Q80G
.9682A13
.9856811
.9951*71+7
.9993C78
1.0000000
3.5 AND 8
BETA (X)
.0001487
.0015764
.0060928
.0155592
.0316269
.0555915
.0883088
.1301513
.1809954
.2402319
.3067978
.3792249
.4557046
.5341656
.6123651
.6879917
.7587733
.8226263
.8777337
.9227626
.9569t»02
.9802680
.9936642
.9991436
1.0000000
3.0
INVERSE
. 0400000
.0300000
.1200001
.1600000
.2000000
.2<*00000
.2800000
.3200000
.3600000
.4000000
.4400000
.4799999
.5199999
.5600000
.6000000
.6400000
.6300000
.7199999
.7599999
.3000000
.8400000
.3800000
.9200000
.9600000
1,0000000
3.0
INVERSE
.0400000
.0800000
,1200000
.1600000
.2000000
.2400000
.2799999
.3200000
.3600001
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.3000000
.8399999
. 3800000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-«ooooooo
-.0000001
.0000000
-•ooooooo
-.0000000
.0000000
.0000000
-.ooooooo
-.0000000
.ooooooo
.0000001
.0000001
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000001
.0000001
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
-.ooooooo
-.0000000
.0000000
-.ooooooo
-.ooooooo
.ooooooo
.0000001
-.0000000
-.0000001
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000001
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
145
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.83
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.55
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
4.0 AND B =
BETA (X)
.0000360
.0005384
.0025431
.0074816
.0169600
.0325671
.0557124
.0874932
.128591**
.1792000
.2389786
.3070383
.3819583
.4618279
.5443200
.6267968
.7064407
.7804168
.8460648
.9011200
.9439641
.9739053
.9914879
.9938316
1.0000000
4.5 AND B =
BETA (X)
.0000086
.0001809
.0010447
.0035420
.0089536
.0138019
.0346555
.0580245
.0901833
.1320365
.1839991
.2459036
.3169394
,3956300
.4798533
.5669088
.6536362
.7365897
.8122711
.8774259
.9294054
.9655977
.9389319
.9934574
l.OOOOOQO
146
3.0
INVERSE
.0400000
.0800000
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.3000000
.8399999
.3300000
.9200000
.9500000
1. 0000000
3.0
INVERSE
.0400001
.0300000
.1200000
.1600000
.2000000
.2400000
.2799999
. 3200000
.3600000
.4000000
.4400000
.4300000
,5200000
.5600000
' .6000000
.6400000
.6800000
.7200000
. 7600000
.3000000
. 8400000
.8799999
.9200000
.9500000
1. 0000000
XC ERROR TERM
-.0000000
-.0000000
.0000 000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.ooooooo
.0000 000
.ooooooo
.ooooooo
.ooooooo
.0000001
.ooooooo
,0000 000
.ooooooo
o.oqooooo
XC ERROR TERM
-.9000001
-..ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.0000001
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
. oooonoo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.0000000
.ooooooo
.ooooooo
.0000001
.ooooono
.ooooooo
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.Of*
.08
.12
.16
.20
.24
.28
.32
.36
.40
.kk
.t*Q
.52
.56
.60
.6%
.68
.72
• 76
.30
.84
.83
.92
.96
1.00
A =
X
.0%
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.43
.52
.56
.60
.64
.63
.72
.76
.30
.84
.38
.92
.96
1.00
5.0 AND B
BETA{X)
.0000020
.0000600
.0004234
.0016551
.0046720
.0107209
.0212996
.0380373
.0625^62
.0962560
.1402448
.1950 779
.2606679
.3361667
.4199040
.5093831
.6013469
.6919265
.7768850
.8519680
.913374S
.9583612
.9859860
.9980162
1.0000000
5.5 AND 8
BETA (X)
.0000005
.0000196
.0001697
.0007647
.0024099
.0060481
.0129558
.0246859
.0429610
.0695236
.1059537
.1534675
.2127113
.2335701
.3650098
.(+5^97^5
.550363*+
.6<*71117
.7^0^022
.8250365
.895956^
.9£»92212
.9326^32
.9975051
1.0000000
= 3.0
INVERSE
. OitOOOOO
.0800000
.1200000
.1600000
.2000000
.2399999
.2800000
.3200000
.3600000
.UOOQOOO
.^00000
.^300000
.5200000
.5600000
.6000000
.o*»QQOOO
.5300000
.7200000
.7600000
.8000000
.Sit 00 000
.8300000
,9199999
.9600000
1. 0000000
= 3.0
INVERSE
.0^+00000
. 0300000
.1200000
.1599999
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400001
. Jf 8 0 0 0 0 0
.5200000
.5600000
.£000000
.6399999
.6300000
.7200000
.7600000
.3000000
.8400000
. 3300000
.9199999
.9600000
1. 0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000001
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
. 0000000
-.0000000
-.0000000
. 0000 000
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000001
.0000000
o.aoooooo
XC ERROR TERM
-.0000.000
-.0000000
.0000000
.0000001
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000001
.9000000
-.0000000
--OOOOQOO
-.0000000
. ooooooi
-.QQGQOOQ
. ooooooo
.0000000
.0000000
.0000000
.ooooooo
.0000011
.ooooooo
o.oocoooo
147
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.84
.83
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.83
.92
.96
1.00
6.0 AND 8 =
BETA (X)
.0000001
.0000064
.0000673
.0003499
.0012314
.0033805
.0078097
.0158811
.0292594
.0498074
.0794247
.1193402
.1723681
.2376483
• 3153^6
.4041805
.5012977
.6027284
.7032777
.7969178
.8774020
.9392108
.9788995
.9969203
1,0000000 1
6.5 AND e =
BETA (X)
.0000000 NOT
.0000020
.0000265
.0001583
.000&240
.0018742
.0046704
.0101381
.0197791
.0354256
.0591265
.0929639
.1333041
.1979971
.2710469
.3572563
.4546013
.5592605
.6659199
.7678851
.8577280
.9233694
.9747537
.9962588
1.0000000 1
148
3.0
INVERSE
, 0400001
. 0800000
.1200000
.1600000
.2000000
.2400000
.2799999
. 3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
. 7600000
.8000000
.8400000
.8800000
.9199999
.9600000
.0000000
3.0
INVERSE
DEFINED
. 0300000
.1199999
.1600000
.2000000
.2400000
.2300000
.3200000
. 3500000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
.3300000
.9200000
.9600000
.0000000
XC ERROR TERM
-.0000001
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000001
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000 000
-.0000 000
. 0000000
.0000000
. ooooooo
.0000000
.ooooooo
.0000001
,0000000
0.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
.0000001
-.0000000
-.ooooooo
.0000000
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.0000000
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.0000 000
.ooooooo
. ooooooo
. ooooooo
.ooooooo
. ooooooo
,0000000
.ooooooo
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
• 2k
.23
.32
.36
.40
• 44
.48
.52
.56
.60
.64
.68
.72
.76
.30
.84
.33
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.43
• 52
.56
.60
• 54
.63
.72
.76
.30
.34
.88
.92
.96
1.00
7.0 AND B
8ETA(X)
.0000000
.0000007
.0000103
.0000715
.0003139
.0010316
.0027735
.0064277
.0132818
.0250348
.0437436
.0716881
.1111469
.1640878
.2317870
.3144075
.4105864
.5170982
.6286889
.7381975
.8371123
.9167411
.97020&8
.9955176
1.0000000
7.5 AND B
8ETA(X)
.0000000
.0000002
.0000040
.0000320
.0001568
.0005642
.0016367
.0040504
.0088658
.0175898
.0321828
.0549862
.0885472
.1353313
.1973221
.2755288
.3694467
.4765453
.5918960
.7080955
,315o916
.9043738
.9652621
.9946942
1.0000000
= 3.0
INVERSE
NOT DEFINED
.0800000
.1200000
. 1600000
.2000000
.2400000
.2300000
.3200001
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800001
.7200000
.7600000
.8000000
. 8400000
.3800000
.9200000
.9600000
1.0000000
3.0
INVERSE
NOT DEFINED
.0800001
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.JfOOOOOO
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.3000000
.3400000
.8800000
.9200000
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000001
. ooooooo
-.0000000
.ooooooo
.0000000
-.ooooooo
. ooooooo
-.ooooooo
-.ooooooo
-.0000001
.ooooooo
-.ooooooo
.0000000
.0000000
.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
NOT DEFINED
-.0000001
-.ooooooo
.ooooooo
-.ooooooo
-.0000000
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.0000000
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000000
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
149
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.i+O
.44
.43
.52
.56
.60
.64
.58
.72
.76
.80
.34
.88
.92
.96
1.00
A =
X
.04
,08
.12
.16
.20
.24
.28
.32
.36
.40
.«t 4
.43
.52
.56
.60
.64
.63
.72
.75
.30
.34
.88
.92
.96
L.OO
8.0 AND 6
BETA (X)
.0000000
.0000001
.0000015
.00001*42
.0000779
.0003063
.0009605
.0025384
.0053364
.0122946
.0235583
.0419713
.0702161
.11112*43
.1672898
.2405373
.3312783
.4378290
.5558051
.6777995
.7935995
.8913182
.9599246
.9937863
1.0000000
3.5 AND e
8ETA(X)
.0000000
.0000000
.0000006
.0000063
.0000385
.0001660
.0005608
.0015829
.0038893
.0085529
.0171662
.0318958
.0554447
.0903304
.1412913
.2092507
.2961002
.4011090
.5206356
,6
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.ttk
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.83
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.43
.52
.56
.60
.6**
.63
.72
.76
.80
.34
. 33
.92
.96
1.00
9.0 AND 8
BETA(X)
.0000000
.0000000
.0000002
.0000028
.0000189
.000089**
.0003259
.0009326
.0025585
.0059245
.0124564
.0241413
.0436112
.0740499
.1139168
.1814410
.2638673
.3664868
.4865654
.6174015
.7479110
.8633530
,9431000
.9917087
1.0000000
3.5 AND B
8ETA(X)
.0002203
.0022981
.0037361
.0219231
.0438115
.0756422
.1179749
.1706364
.2327766
.3029507
.3792263
.4593106
.5406394
.6207737
.6970493
.7672234
.8293636
.3820251
.9243573
.9561835
.9780669
.9912639
.9977019
.9997797
l.OOOOOOQ
= 3.0
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4399^99
.4300000
.5200000
.5600000
.6000000
,6400001
.6300000
.7199999
.7600000
.3000000
. 8400000
„ 8799999
.9200000
.9600000
l.OOQOOOO
3.5
INVERSE
.0400000
. 0800000
.1.200000
.1599999
.2000000
.2400000
.2300000
.3199999
.3600000
.4000000
.4400900
.4300000
.5199999
.5600000
.6000000
.6400000
.6800000
.7200000
. 7600090
,8000000
.3400000
,3300000
.9200000
,9600000
1,0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.0000000
.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
-.0000000
.0000001
-.0000000
-.0000000
.0000000
-.0000000
-.0000 001
.0000000
.0000001
-.0000000
-.0000000
.0000000
.0000001
.0000000
.0000000
0.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000001
-.0000000
.0000000
.0000000
.9000001
-.0000000
-.0000000
.0000000
-.0000000
.0000001
.0000000
.0000000
.0000 noo
.0000000
.noooooo
.0000001
. 0 0 0 0 0 0 0
.0000000
.0000009
.0000090
.0000000
0.0000000
151
-------
INCOMPLETE BF.TA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.54
.63
.72
.76
.80
.84
.33
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.34
.88
.92
.96
1.00
4.0 AND B
BETA (X)
.0000554
.0008148
.0037833
.0109353
.02i»3445
.0458824
. 0769977
.1185560
.1707450
.2330378
.3042096
.3824026
.4652335
.5499371
.6335401
.7130571
.7857007
.8490958
.9014873
.9419266
.9704207
.9330178
.9967956
.9996878
1.0000000
4.5 AND B
8ETA(X)
.0000137
.0002835
.0016036
.0053559
.0132951
.0273697
.0494530
.0811167
.1234311
.1768110
.2409160
.3146107
.3959896
.4824635
.5709099
.6578780
.7398441
.3135034
.8760855
.9256723
.9614393
.9841365
.9956874
.9995730
1.0000000
3.5
INVERSE
. 0400000
. 0300000
. 1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6399999
.6800000
.7199999
.7600000
.8000000
,8400000
.8300000
.9200000
.9600000
1. 0000000
3.5
INVERSE
. 0400000
. 0300000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4800001
.5200000
.5599999
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
. 3400000
.8300000
.9200000
.9600000
1 .0000000
XC ERROR TERM
-. 0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-. 0000000
.0000000
-.0000000
-.0000000
-.0000000
. ooooooo
-.0000000
.0000000
.0000 001
.0000 000
. 0000001
.ooooooo
.0000000
.ooooooo
.ooooooo
.0000000
.ooooooo
O.OOOOQOO
XC ERROR TERM
-.0000000
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.0000001
.ooooooo
.0000001
.ooooooo
.0000 000
.ooooooo
. ooooooo
. ooooooo
.ooooooo
.ooooooo
.ooooooo
. ooooooo
.ooooooo
o.ooooooo
152
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
• 40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.34
.88
.92
.96
1.00
A =
X
.04
.03
.12
.15
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.84
.83
.92
.96
1.00
5.0 AND 8
B£TA(X)
.0000033
.0000971
.0006733
.0025836
.0071543
.0160946
.0313267
.0547699
.0331060
.1325496
.1886478
.2561263
.3338009
.4195659
.5104673
.6028717
.6927172
.7753592
.8434796
.9075463
.9512833
.9795959
.9943616
.9994327
1.0000000
5.5 AND 8
BETA(X)
.0000008
.0000328
.0002781
.0012301
.0033011
.0093480
.0196079
.0365558
.0621977
.0983242
.1462503
.2065682
.2789436
.3619799
.4531796
.5490186
.6451523
.7367590
.8190168
.8876964
.9398290
.9743798
.9923043
.9992644
1.0000000
- 3.5
INVERSE
. 0400000
. 0800000
.1199999
.1599999
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
.8800000
.9200000
.9600000
1.0000000
3.5
INVERSE
.0400000
.0300000
.1200000
,1600000
.2000000
.2400000
.2800001
.3200000
.3600000
.4000000
.4400000
.£^800000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.3000000
, 3400000
. 8800000
,9200000
.9600000
1. 0000000
XC ERROR TERM
-.0000000
-.0000000
.0000001
.0000001
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
0.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000001
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000 000
.0000 000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
o.ooooooo
153
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.80
.84
.88
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.38
.92
.96
1.00
6.0 AND B
BETA (X)
.0000002
.0000109
.0001135
.0005790
.0019973
.0053710
.0121448
.0241525
.0434809
.0722568
.1123770
.1652073
.2312835
.3100540
.3997027
.4970937
.5973740
.6967636
.7880508
.3662899
.9271702
.9684790
.9910023
.9990655
1.0000000
6.5 AND 8
BETA (X)
.0000000
.0000036
.0000459
.0002699
.0010393
.0030568
.0074529
.0158148
.0301336
.0526591
.0856642
.1311349
.1904142
.2638405
.3504332
.4476303
.5514935
.6563866
.7559300
.8435064
.9133635
.9613912
.9889462
.9S88334
1.0000000
3.5
INVERSE
.0400000
.0300000
. 1200000
. 1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400001
,4800000
.5200000
.5600000
.6000000
.6400000
.6800001
.7200000
.7600000
. 3000000
.8399999
.8800000
.9200000
.9600000
1. 0000000
3.5
INVERSE
NOT DEFINED
.0800000
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
. 7600000
. 3000000
.3400000
. 3800000
.9200000
.9600000
1. 0000000
XC ERROR TERM
-.0000000
-.0000000
, ooooooo
-.0000000
-.ooooooo
.0000000
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.0000001
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.0000 001
.ooooooo
.ooooooo
.ooooooo
.0000001
.ooooooo
.ooooooo
.ooooooo
0,0000000
XC ERROR TERM
NOT DEFINED
-.OOOOOOO
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
154
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0*4
.08
.12
.16
.20
,2k
.29
.32
.36
.**0
• *+**
.*t8
.52
.56
.60
.6*4
.68
.72
.76
.80
.8**
.83
.92
.96
1.00
A =
X
.0*4
.08
.12
.16
.20
.2*4
.23
.32
.36
.^0
.*4*4
.**3
.52
.56
.60
• 6<*
.63
.72
.76
.30
.8*4
.83
.92
.96
1.00
7.0 AND B
BETA(X)
.0000000
.0000012
.000018**
• Q00i2**7
.0005361
.0017250
.00*45360
.0102727
.0207221
.0380908
.Q6**83<+9
.103382*4
.155762**
.223171+6
.305552(4
.*401188**
.5065081*
.6160861
.7229391
.8195326
.898*4757
.95*46199
.9866251
.9985659
1.0000000
7.5 AND B
8ETA(X)
.0000000
.000000**
.0000073
.0000571
.00027****
.0009662
.002740*1
,00662*49
.01**1509
.0273677
.0*487533
.0810007
.1266730
.1877*»f+7
.26507*46
.3578791
.*4633101
.5762616
.6895<»32
.79*»557**
.3325820
.9**667**3
,98**0316
.9932606
1.0000000
3.5
INVERSE
NOT DEFINED
.0800000
.1200000
.1600000
.2000000
.2*»00000
.2800000
.3200001
.3600000
.*400000Q
.*4**OOQOQ
.**800000
.5200000
.5600000
.6000000
.6**OOQOO
.6800000
. 7200000
.7599999
.8000000
.8**00000
. 8800000
.9200000
.9600000
1.0000000
3.5
INVERSE
NOT DEFINED
. 0800001
.1200000
.1600000
.2000000
.2**00000
.2799999
.3200000
.3600000
,**000000
.*»**00000
.**800000
.5200000
.5600000
.6000000
.6**00000
.6800000
.7200000
.7600000
.8000000
. 8**00000
. 8799999
.9200000
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
-.0000000
-.0000000
-.0000000
-.0000000
-.0000000
-.0000001
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000 000
.0000000
.0000001
.0000000
.0000000
.0000000
.0000000
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
-.0000001
-.0000000
-.0000000
-.0000000
-.0000000
.0000001
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
.0000000
. ooooooo
.0000000
.ooooooo
.0000001
.ooooooo
.ooooooo
o.ooooooo
155
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
»40
.44
.43
.52
.56
.60
.64
.68
.72
• 76
.30
.34
.83
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.21*
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.75
.30
.84
.33
.92
.96
1.00
8.0 AND 8
BETA (X)
.0000000
.0000001
.0000029
.0000260
.0001395
.0005375
.0016447
.0042448
.0096028
.0195436
.0364455
.0631079
.1024660
.1571463
.2233371
.3178877
.4221910
.5372527
.6558825
.7687675
.3657633
.9380682
.9811592
.9979152
1.0000000
8.5 AND 8
BETA(X)
.0000000
.0000000
.0000011
.0000118
.0000705
.0002972
.0009811
.0027039
.0064792
.0138789
.0270939
.0439148
.0824791
.1309266
.1967878
.2812486
.3333571
.4993415
.6222712
.7423451
.8431068
.9288198
.9730030
.9975274
1.0000000
3.5
INVERSE
NOT DEFINED
. 0300000
. 1200001
.1600000
.2000000
.2400000
.2800001
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
. 3400000
.8800000
.9200000
.9600000
1.0000000
3.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
.2400000
.2800000
. 3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
. 7600000
.3000000
. 8400000
. 3800000
.9200000
.9600000
1. 0000000
XC ERROR TERM
NOT DEFINED
-.0000000
-.0000001
.0000000
-.0000000
-.0000000
-.0000001
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-, ooooooo
.0000000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.OOOOOOO
-.OOOOOOO
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.0000000
.ooooooo
-.0000000
-.ooooooo
.0000 000
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
156
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
,40
.44
.48
.52
.56
,60
,64
.68
.72
.76
.80
.31*
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.2**
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.34
.83
.92
.96
1.00
9.0 AND G
BETA (X)
.0000000
.0000000
.0000004
.0000053
.0000354
.0001634
.0005821
.0017131
.0043489
.0098063
.0200506
.0377350
.0660912
.1036154
.1685133
.2479152
.3469377
.4627551
.5889439
.7154642
.8296990
.9189509
.9745591
.9970952
1.0000000
4.0 AND B
BETA(X)
.0000813
.0011763
.0053693
.0152503
.0333440
.0616955
.1015962
.1534344
.2166517
.2897920
.3706237
.4563199
.5436801
.6293763
.7102080
,7833£»33
.8465656
.8984038
.9333045
.9666560
,9847«»97
.9946307
.9938237
.9999187
1.0000000
- 3.5
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
. 1600000
.?000000
.2400000
.2300000
. 3200000
.3500000
.4000000
.4400000
.*»8QQQOQ
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
.8800000
.9200000
.9599999
1.0000000
= 4.0
INVERSE
.0400000
.0800000
.1200000
.1600000
,2000000
.2400000
.2300000
.3200000
.3600001
,4000000
.^399999
.4800000
.5200000
.5600000
.5000000
.6400000
.6800000
,7200000
.7600000
.8000000
, 8400000
.8300000
.9200000
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.0000000
.0000000
-•ooooooo
-0000000
.0000000
. ooooooo
.0000000
-.0000000
.ooooooo
-.0000000
-.0000000
.ooooooo
-.ooooooo
-.ooooooo
.0000000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
•ooooooo
.0000001
0.0000000
XC ERROR TERM
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
-.0000001
-.ooooooo
.0000001
-.ooooooo
.0000000
.ooooooo
.ooooooo
. 0000 000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
157
-------
INCOMPLETE BETA FUNCTION AND INVERSE
ft =
X
.04
.08
.12
.16
.20
,2k
.23
.32
.36
.40
.44
.1*8
.52
.56
.60
.64
.63
.72
.76
.30
.34
.83
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
if. 5 AND B =
BETA (X)
.0000207
.0004228
.0023572
.0077030
.0187815
.0379325
.0672052
.1080324
.1610168
.2258059
,3010672
.3845578
.4732815
.5637198
.6521187
.7343104
.8085455
.8708067
. 92007*+ 4
.9560096
.9795188
.992662**
.9983649
,9998851
1.0000000
5.0 AND B =
BETA (X)
.0000052
.0001*493
.0010169
.0038303
.QiQ40&<*
.0229548
.0437826
.0749644
.1130242
.1736704
.2416115
.3204741
.4078342
.5003641
.5940 664
.6847209
.7680957
.8405901
.8995638
.9437184
.9733297
.9902784
.9977967
.9998426
i.noooooo
4.0
INVERSE
. 0400000
. 0800000
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
. 7200000
.7600000
.8000000
.8400000
.8799999
.9200000
.9600000
1.0000000
4*0
INVERSE
.0400000
.0800001
.1199999
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4800001
.5200000
.5600000
.6000000
.6400000
.6800000
.7199999
.7600000
.8000000
.8400000
.8800000
.9200000
.9600000
1. 0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-. 0000000
.0000000
-.0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
. 0000 000
.0000000
. ooooooo
. ooooooo
.ooooooo
.ooooooo
.0000001
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
-.ooooooo
-.0000001
.0000001
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.0000000
.ooooooo
-.ooooooo
-.0000000
-.0000001
.ooooooo
.ooooooo
.ooooooo
.ooooooo
. 0000 000
.0000001
.ooooooo
.ooooooo
.000000,0
.ooooooo
.ooooooo
.ooooooo
0.0000000
158
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.34
.88
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.2k
.23
.32
.36
.40
.44
,43
.52
.56
.60
.64
.68
.72
.76
.30
.34
.83
.92
.95
1.00
5.5 AND B -
BETA (X)
.0000013
.0000519
.0004322
.0018757
.0056843
.0137004
.0231456
.0513568
.0354611
.1320365
.1913036
.2643908
.3482078
.4404540
.5372751
.6340695
.7259272
.8031721
.8769531
.9298150
.9661552
.9874494
.9971072
.9997899
1.0000000
6.0 AND 8 =
BETA (X)
.0000003
.0000178
.0001813
.0009068
.0030664
.0030784
.0178821
.0347877
.0612147
.0993526
.1507869
.2161445
.2948105
.3847691
.t,326097
.5837263
.6827203
.7739888
.8524552
.9143533
.9579813
.9841503
.9962849
.9997257
1.0000000
159
4.0
INVERSE
. 0400000
. 0800000
. 1200000
. 1600000
.2000000
.2400000
.2800000
. 3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7199999
.7599999
.8000000
. 3400000
.8300000
.9200000
.9600000
1.0000000
4.0
INVERSE
.0400000
. 0300000
.1200000
Cl&OOQOO
.2000000
. 2 4 0 0 0 0 0
.2300000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
. 8000000
. 3400000
.3300000
.9200000
.9599999
1. 0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
. ooooooo
-.0000000
.ooooooo
-.0000000
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
. ooooooo
.0000001
.0000001
.0000000
.0000000
.ooooooo
.0000000
.ooooooo
0.0000000
XC ERROR TERM
-.ooooooo
-.ooooooo
,0000000
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000001
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.Off
.03
.12
.16
.20
• 2tf
.23
.32
.36
.tfO
.£f<+
.%8
.52
.56
.60
.6%
.68
.72
.76
.30
.8tf
.83
.92
.95
1.00
A =
X
.04
.03
.12
.16
.20
.2*f
.23
.32
.36
.*tO
• <*<+
.^8
.52
.56
.60
.6
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A -
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
»40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.83
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
»75
.80
.34
.83
.92
.96
1.00
7.5 AND 8
BETA (X)
.0000000
.000000?
.0000125
.0000962
.000452?
.0015593
.0043266
.,0102193
.0213070
.0401318
.0697243
.1127123
.1713036
.2464586
.33739'31
.44122^0
.5523130
.6651360
.7700469
.8595623
.9275034
.9712£*15
.9929144
.9994505
1.0000000
3.0 AND B
BETA (X)
.0000000
.0000002
.0000051
.0000448
.0002352
.0008867
.0026526
.0056870
.0147609
.0292815
.0531634
.0395181
.1411628
.2099895
.2962343
.3981P72
,51iO£«27
.6230749
.7
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.21+
.28
.32
.36
.40
.44
.43
.5?
.56
,60
.64
.68
.72
.76
.30
.34
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.30
.84
.88
.92
.96
1.00
8.5 AND E
BETA (X)
.0000000
.0000001
.0000020
.0000207
.0001213
.0005005
.0016152
.0043465
.0101597
.0212045
.0402918
.0706377
.1156913
.1730042
.2539560
.3577136
.4706649
.5912085
.7101882
.8171516
.9024614
.9600077
.9898173
.9991341
1.0000000
9.0 AND 8
•BETA (X)
.0000000
.0000000
.0000008
.0000095
.0000622
.0002807
.0009774
.0028080
.0069513
.0152673
.0303675
.0555222
.0943380
.1501761
.2253373
.3201202
.4319466
.5548303
.6795133
.7945639
.888&315
.9535873
.9879882
.9990218
1.0000000
*».o
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1500000
.2000000
.2^00001
.2800000
.3200000
.3600000
. ff 0 0 0 0 0 0
.^OQOQO
.if799999
.5200000
.5600000
.6000000
.6^00000
.6300000
.7200000
.7600000
.8000000
. 8399999
. 8800000
.9200000
.9600000
1.0000000
= %.o
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
.1600000
.2000000
,2*400000
.2800000
.3200000
.3600000
.4000000
,4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.3400000
.8800000
.9200000
.9600000
1. 0000000
XC ERROR TERM
NOT DEFINED
NOT DEFINED
. 0000000
.0000000
-.0000000
-.0000001
.0000000
.0000000
-.0000000
-.0000000
. ooooooo
.0000001
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
. ooooooo
.ooooooo
.ooooooo
.0000001
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.OOOOOOO
.OOOOOOO
-.ooooooo
.OOOOOOO
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
162
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0*f
.03
,12
.16
.20
.2ff
.28
.32
.36
,*fO
.***»
.52
.56
.60
.6*+
.63
.72
.76
.30
.8%
.83
.92
.96
1.00
A =
X
.0*»
.03
.12
.16
.20
.2k
.23
.32
.35
.1*0
.*+*+
.i»3
.52
.56
.60
.6*4
.63
.72
.76
.30
. 3<+
.33
.92
.96
1.00
**.5 AND 3
BETA (X)
.0000302
.0006058
.0033192
.0106607
.0255017
.0505*405
.0878253
.133*4093
.2021599
.2777228
.3626291
^lll^l
.6373709
.7222772
.7978*401
.8615907
.91217*47
.9**9*4595
.97***f933
.9893393
.9966803
.99939^2
.9999698
1.0000000
5.0 AND 6
BETA (X)
.0000073
.0002201
.001*4731
.005*4*472
.Cl£»52l6
. 0 31*+l*4l
.0537293
.0985090
.1518561
.2136323
.2976060
.3860105
.1^5025*4*4
,575Qil*f
.6637036
.7539831
.828207*4
.8888528
.93*t306*4
.966*4 3UO
.9357306
.995*4772
.9991600
.999957*4
1.0000000
*f.5
INVERSE
. ooooooo
. 0800000
.1200000
.1600000
.2000000
.2*400000
.2300000
.3200000
.3600000
,*4000000
.*»*400000
.5200000
.5600000
.6000000
,6*tOOOQO
.6300000
.7200000
,7600000
.3000000
.8^00000
. 3800000
.9200000
.9600000
1. OOOOOOO
*4.5
INVERSE
,0*+00000
. 0800001
.1199999
.1600000
.2000000
.2*400000
.2799999
.3200000
.3600000
.
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
,24
.23
.32
.36
.40
• kit
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.33
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.50
.64
.63
.72
.76
.30
.34
.33
.92
.96
1.00
5.5 AND 8
BETA (X)
.0000020
.0 000786
.0006430
.0027390
.0081t,12
.0192342
.0337038
.0691489
.1125840
.1700836
.2414569
.3251063
.41BO&12
.5161989
.6146382
.7032644
.7923305
.8630584
.9131533
.9571380
.9814177
.9940047
.9988670
.9999416
1.0000000
6.0 AND 8
BETA(X)
a0000005
.0000277
.0002767
.0013579
.0045021
.0116214
.0251885
.0479466
.0824959
.1308270
,1938778
.2711929
.3607484
.4539901
.5611032
.6615029
.7545007
.8350678
.8995850
.9464452
.9763610
.9922374
.9985074
.9999217
1.0000000
= 4.5
INVERSE
.0400000
.0800000
.1200000
.1,600001
.2000000
.2399999
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
. 3400000
. 8300000
.9200000
.9600000
1. 0000000
= 4.5
INVERSE
. 0400000
. 0300000
. 1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.360000.0
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.5800000
.7200000
.7600000
, 3000000
. 8400000
.8800000
.9200000
. 9600000
1 .0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
-.0000001
-.0000000
.0000001
-.0000000
-. ooooooo
.0000000
-.0000000
-.ooooooo
.0000000
. ooooooo
. ooooooo
.ooooooo
.0000 000
.0000 000
.ooooooo
. ooooooo
.ooooooo
.0000000
.ooooooo
.ooooooo
.ooooooo
o.ooooooo
XC ERROR TERM
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.0000000
.ooooooo
.0000000
-.ooooooo
.ooooooo
.0000 000
.0000 000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000000
o.ooooooo
164
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
• 4 3
.52
.56
.60
• 64
.68
.72
.76
.80
.8**
.88
.92
.96
1.00
A =
X
• 04
.03
.12
.16
.20
.24
.28
.32
.36
.40
• 44
• 48
.52
.56
.60
.64
.68
.72
.76
.30
.88
.92
.96
1.00
6.5 AND B
BETA (X)
.0000001
.0000096
.0001175
.0 00&&49
.0021*597
.0069393
•0162045
.0328817
.0598158
.0996289
•1542130
•2242411
.3087906
•4051612
.5089562
•&144541
.7152626
.8051903
.8792197
.9344117
.9705290
.9901511
.9980733
.9998973
1.0000000
7.0 AND 8
BETA(X)
.0000000
.0000033
.0000 494
.0003220
.0013293
.0041001
.0103187
.0223283
.0 429607
.0751853
.1216138
.1839314
.2623551
.3552313
.<*588829
.5677897
.6751432
.7737530
.8572020
.9210627
.9638993
.9877233
.9975568
.9998675
1.0000000
4.5
INVERSE
.01*00000
.0800000
. 1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
,4000000
. ***» 00 001
.^300000
.5200000
.5600000
.6000000
.6400000
.6800000
.7199999
.7600000
.8000000
. 8400 000
.8800000
.9200000
.9600000
1. 0000000
4.5
INVERSE
NOT DEFINED
.0800000
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.360000.0
.4000000
. 4400 000
.4800000
.5199999
.5600000
.6000000
.6400000
.6300000
.7200000
.7599999
. 3000000
. 3400 000
. 3799999
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000001
.0000000
.0000000
-.0000000
-.0000 000
.0000 000
.0000000
.0000001
.0000000
.00000(30
.0000000
.0000000
.0000000
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
,0000000
-.0000000
.0000000
.0000000
.0000001
-.0000000
-.0000000
-.0000000
.0000 000
.0000000
.0000001
.0000000
.0000000
.0000001
.0000000
.0000000
0.0000000
165
-------
INCOMPLETE 3ETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
• 2k
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.30
.84
.88
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.30
.34
.83
.92
.96
1.00
7.5 AND 8
BETA (X)
,0000000
.0000011
.0000205
.0001544
.0007115
.0023997
.0065104
.0150271
.0305903
.0562722
.0951556
. 1497561
.2213741
.3094930
.4114036
.5220871
.6346378
.7410902
.8336973
.9064403
.9564575
.9849339
.9969500
.9998318
1.0000000
3.0 AND 8
8ETA(X)
.0000000
.0000004
.0000085
.0000734
.0003775
«0013925
.0040734
.0100315
.0216116
.0413003
.0739195
.1211032
.1856083
.2680739
.3668846
.4778250
.5941^85
.7075329
.8033854
.8906036
.9431976
.9817647
.9962450
.9997894
1.0000000
4.5
INVERSE
NOT DEFINED
.0300001
.1200000
. 1600000
.2000000
.2400000
.2300000
. 3200001
.3600000
.4000000
.4400000
.4800000
.5200000
.5600001
.6000 000
.6400000
.6799999
.7200000
.7600000
.8000000
.8400000
,8799999
.9200000
.9600000
1. 0000000
4.5
INVERSE
NOT DEFINED
.0300000
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
,4400000
.4800000
.5200000
.5600000
.6000000
.6399999
.6800000
.7200000
.7600000
.8000000
. 8400000
.8300000
.9200000
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
-.0000001
-.0000000
-.0000000
-.0000000
-.0000000
-.0000000
-.0000001
.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000001
-.0000000
.0000 000
.0000 001
.0000000
.0000000
.0000000
.0000000
.0000001
.0000000
.0000000
0.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
.0000001
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
0.0000000
166
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
,04
.08
.12
.16
.20
.24
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.63
.72
.76
.SO
.84
.83
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.83
.92
.96
1.00
8.5 AND B
BETA (X)
.0000000
.0000001
.0000035
.0000346
.0001987
.0008018
.0025293
.0066473
.0151592
.0308362
.0570439
.0973187
.1547045
.2309266
.3255573
.4353353
.5542280
.6734006
.7829551
.873&217
.9391214
.9732002
.9954342
,9997396
l.OOOQOOO
9.0 AND B
BETA (X)
.0000000
.0000000
.0000014
.0000162
.0001038
.0004534
.0015597
.0043750
.0105635
.0226039
.0437529
.0777517
.1232401
.1979122
. ?375361
.3950574
.5150750
.6339951
.7560994
.8555753
.9292379
.9742270
.9945101
.9996317
1.0000000
= 4.5
INVERSE
NOT DEFINED
. 0800000
.1200000
,1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4300000
.5200000
.5600000
.6000000
.6400000
.6300001
.7200000
.7600000
,8000000
. 8400000
.3300000
,9200000
.9600000
1.0000000
= 4.5
INVERSE
NOT DEFINED
NOT DEFINED
. 1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4799999
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
,7600000
.8000000
. 3400000
.8300000
.9200000
.9600000
1,0000000
167
XC ERROR TERM
NOT DEFINED
-.0000000
,0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
. OOOOOQO
-.0000000
-.0000000
-.0000001
.0000000
.0000000
.0000000
.0000000
.0000000
,0000000
.0000000
o.ooooooo
XC ERROR TERM
NOT DEFINED
NOT DEFINED
.0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000001
-.0000000
.0000000
-.0000000
-.OOQOOOO
-.0000 000
.GOOOQQO
.OOQ010C
.0000000
.0000000
.0000000
.QQQOOOQ
.0000000
O.OOOOOOQ
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.23
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.34
.88
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.50
.64
.68
.72
.76
.30
.84
.88
.92
.96
1.00
5.0 AND 6
BETA (X)
.0000113
,0003136
.0020615
.0074S47
.0195814
.0415503
.0761583
.1251852
.1890360
.2655677
.3551*423
.4508361
.5491139
.6448577
.7334323
.81096^0
.87481^3
.9238^17
.958^it97
.980^186
.9925153
.9979385
.999686tt
.9999887
1.0000000
5.5 AND B
BETA(X)
.0000029
.00011<*9
.0009230
.0038587
.0112506
.0260588
.0513861
.0898962
.1^32612
.2117f»04
.2939650
.3869611
.<+86<*i<»3
.5871«tO^
.6837007
.7710771
.3<*53106
.90it0031
.9t»659^1
.9743tf58
.9900092
.9971976
.9995661
• 99998U
1.0000000
= 5.0
INVERSE
. OfiOOOOO
. 0800001
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.ftOOOOOO
,^00000
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
. 3400000
.8800000
.9199999
.9600000
1.0000000
= 5.0
INVERSE
.0400000
.0800000
.1200000
,1600000
.2000000
.2399999
.2800000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
,6000000
.6400000
.6300000
.7200000
.7600000
.3000000
.8400000
.8800000
.9199999
.9600000
1. 0000000
XC ERROR TERM
-.0000000
-.0000001
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000 000
.0000000
.0000000
.0000000
.0000000
,0000000
.0000000
.0000001
.0000000
o.ooooooo
XC ERROR TERM
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000001
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000001
.0000000
0,0000000
168
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
,03
.12
,16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
A -
X
.Off
.08
.12
.16
.20
,2k
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.88
.92
.96
1.00
6.0 AND B
BETA (X)
.0000007
.0000^15
.0004069
.0019593
.0063694
.0161116
.0341994
.0637149
.1072304
.1662386
.2407033
.3288205
.4270483
.5304187
,6331033
.7291585
.8133446
.8818829
,9330110
.9672065
.9869899
.9952839
.9994143
.9999782
1.0000000
6.5 AND B
BETA (X)
,0000002
.0000147
.0001769
.0009815
.0035589
.0098356
.0224834
.0446299
.0793658
.1291332
.1951471
.2768707
.3718236
.4756587
.5825339
.6858833
.7793242
.8576558
.9177247
.9589624
.9834168
.9951771
.9992263
.9999707
1.0000000
5.0
INVERSE
. 0400000
.0300000
.1200000
. 1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6799899
.7200000
.7600000
. 3000000
.8400000
.8300000
.9199999
.9600000
1.0000000
- 5.0
INVERSE
.0400000
.0800000
. 1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
,4400000
,4800000
,5200000
.5600000
.6000000
.6399999
,6800000
.7200000
.7600000
.3000000
.8400000
.8800000
.9200000
.9600000
1.0000000
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
. ooooooo
.0000001
.ooooooo
.0000000
.ooooooo
.ooooooo
.0000000
.0000001
.ooooooo
o.ooooooo
XC ERROR TERM
-.ooooooo
-.ooooooo
. ooooooo
.ooooooo
-.ooooooo
.ooooooo
.0000000
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
. ooooooo
-.ooooooo
.ooooooo
.0000001
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000000
0.0000000
-------
INCOMPLETE BFTA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.23
.32
.36
.40
•44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.83
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.34
.88
.92
.96
1.00
7.0 ANC B
BETA (X)
.0000000
.0000052
.0000760
.0004853
.0019654
.0059361
.0146187
.0309308
. 0581470
.0993526
.1567813
.2311679
.3212530
.423&090
.5327742
.6418S92
.7436783
.8315324
.9007887
.9495904
.9792542
.9933572
.9989970
.9999614
1.0000000
7.5 AND B
BETA (X)
.0000000
.0000018
.0000323
.0002379
.0010739
.0035460
.0094105
.0212307
.0422081
.0757644
.1249104
.1915033
.2755766
.3748107
.4844800
.5978059
.7068406
.8037502
.8822820
.9390824
.9744715
.9923048
.9987213
.9999499
1.0000000
- 5.0
INVERSE
NOT DEFINED
.0800000
. 1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400001
.4800000
.5200000
.5600000
.6000000
.6400000
.6800000
.7200000
.7600000
.8000000
.8400000
.8800000
.9200000
.9600000
1.0000000
5.0
INVERSE
NOT DEFINED
. 0800001
.1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
.5199999
.5600000
.6000000
.6400000
.6800000
.7199999
.7600000
.8000000
.3400000
.8800000
.9200000
.9600000
1.0000000
170
XC ERROR TERM
NOT DEFINED
-.0000000
-.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000001
.0000000
. ooooooo
-.0000000
.0060000
. ooooooo
.0000 000
.ooooooo
. ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
NOT DEFINED
-.0000001
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.ooooooo
.ooooooo
.0000001
-.ooooooo
-.ooooooo
.ooooooo
.0000 000
.0000001
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
o.oooonoo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.2k
,2%
.32
.36
.40
.44
.43
.52
.56
.60
.64
.68
.72
.76
.80
.34
.88
.92
.96
1.00
A =
X
• 04
.03
.12
.16
.20
.24
.28
.32
.35
.40
.44
.43
.52
.56
.60
,6k
.63
.72
.76
.30
.31*
.83
.92
.96
1.00
8.0 AND B
BETA (X)
.0000000
.0000006
.0000 136
.0001153
.0005812
.0020935
.0060031
.0144450
.0303799
.0573099
.0987553
.1575100
.2348123
.329616**
.4381782
.5541413
.66923*49
.7745643
.8623060
.9274445
.9690432
.9905014
.9983933
.9999360
1.0000000
8.5 AND 8
BETA(X)
.0000000
.0000002
.0000057
.0000554
.0003119
.001231J,
.0037979
.0097495
.0216971
.0430234
.0775240
.1286816
.1983315
.2332153
.39^2704
.5113727
.6312644
.7442385
.8409795
.9146^62
.9629^99
.9834292
.9930090
.9999193
1.0000000
5.0
INVERSE
NOT DEFINED
. 0800000
. 1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4300000
,5200000
.5600000
.6000000
.6400000
.6799999
.7200000
.7600000
.8000000
. 3400000
.8800000
.9200000
.9600000
1.0000000
5.0
INVERSE
NOT DEFINED
. 0800000
,1200000
.1600000
.2000000
.2400000
.2800000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
, f-, 000000
.6400000
.6799999
.7200000
.7599999
. 3000000
.8400000
.3800000
.9200000
.9600000
1.0000000
171
XC ERROR TERM
NOT DEFINED
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
-.0000000
-.0000000
-.0000000
-. ooooooo
.0000001
.0000000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
0.0000000
XC ERROR TERH
NOT DEFINED
-.OOOOOOO
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
.0000 000
.0000 001
. DOOOOOO
. ooooooi
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000000
0.0000000
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.03
.12
.16
.20
• 2k
.28
.32
.36
.40
.44
.43
.52
.56
• 60
.64
.63
.72
.76
.60
.84
.88
.92
.96
1.00
A =
X
.04
.08
.12
.16
.20
• 2k
.23
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.30
• 6k
.33
.92
.96
1.00
9.0 AND 8
n E T A ( x )
.0000000
.0000001
.0000023
.000026*4
.0001660
.0007170
.0023846
.0065319
.0153856
.0320843
.0601+581
.1044764
,1673846
.2506627
.3530418
.4698937
.5933026
.7130388
.8184357
.9008694
.9561774
.9860717
.9975614
.9998994
1.0000000
5.5 AND B
BETA(X)
.0000042
.0001630
.0012853
.0052740
.0150 855
.0342613
.0662141
.1134756
.1770787
.2561949
.3430884
.4483860
.5516140
.6519116
.7438051
.8229213
.8865244
.9337859
.9657337
.9849145
.9947260
.9937147
.9998370
.9999958
1.0000000
5.0
INVERSE
NOT DEFINED
NOT DEFINED
.1200000
. 1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
,4800000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
.7600000
.3000000
.8400000
,8800000
.9200000
.9600000
1. 0000000
= 5.5
INVERSE
.0400000
. 0800000
.1200000
.1600000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5599999
.6000000
.6400000
.6300000
.7200000
.7600000
.8000000
.8400000
.8300000
.9200000
.9600000
1.0000000
172
XC ERROR TERM
NOT DEFINED
NOT DEFINED
. ooooooo
.0000000
-.ooooooo
.0000000
-.OOOOOOO
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.0000 000
. 0000000
.ooooooo
. ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
Q.OOOOOOO
XC ERROR TERM
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.0000001
,0000000
.ooooooo
.0000 000
.0000000
.ooooooo
•ooooooo
.ooooooo
.ooooooo
.ooooooo
,0000000
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.Q<*
.03
.12
.16
.20
.21*
.29
.32
.36
.1*0
.<+«f
.1*3
.52
.55
.50
.61*
.63
.72
.76
.80
.81*
.33
.92
.96
1.00
A =
X
.Qi*
.08
.12
.16
.20
.21*
.28
.32
.36
.1*0
.1*1*
.1*3
.52
.55
.60
.61*
.68
.72
.76
.80
.31*
.83
.92
.96
1.00
6.0 AND B
BETA (X)
.0000011
.0000602
.0005798
.0027392
.0037326
.021&t*97
.01*5011*8
.082101*3
.1352096
.2050159
.2902298
.38751*11*
.%9i939i*
.5973352
.6973970
.7861*539
,8603108
.9168210
.9560953
.9802886
.9929765
.9932562
.99977«*8
,999991*1
1.0000000
6.5 AND 6
BETA(X)
.0000003
.0000219
.0002577
.0011*021*
.OQi*98f*8
.013^966
.0302072
.0586717
.102030^
.1622553
.2395182
.3313306
,^350588
.5^331415
.6i+9365&
.7^78732
.8317207
.8977792
.9<»£»97Jt2
.97^81^0
,99035«*6
.9976871
.9996959
.9999913
1.0000000
- 5.5
INVERSE
.0^00000
.0800000
.1200000
.1600000
.2000000
.2^00000
.2300000
.3200000
.3600000
.1*000000
.f*«fOOOOO
.(+800000
.5200000
.5600000
.6000000
.&<*00000
.6800000
.7200000
.7600000
.8000000
.8399999
. 8799999
.9200000
.9599999
1.0000000
- 5.5
INVERSE
.01*00000
. 0800000
.1200000
.1600000
.2000000
.21*00000
.2300000
.3200000
.3600000
»tf 000000
.<+ 1*00000
,1*300000
,5200000
,5600000
.6000000
.5^00000
,6800000
.7200000
,7600000
, 3000000
, 3399999
,3800000
.9200000
.9599999
1.0000000
173
XC ERROR TERM
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
-.0000000
-.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000000
.0000001
.0000001
.QOOOOOO
.0000001
o.ooooooo
XC ERROR TERM
-.0000000
-.0000000
•0000000
-.0000000
-.0000 000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
.0000000
.0000000
.0000000
,0000000
.0000000
.0000001
.0000000
.0000000
.0000001
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
,01*
.03
.12
-15
.20
.2*4
.28
.32
.36
.tfO
.*4*4
.*43
.52
.56
.60
.6*4
.68
.72
.76
.30
.3*4
.83
.92
.96
1.00
A =
X
.Of*
.03
.12
.16
.20
.2*4
.23
.32
.36
.**0
.*4*4
.**3
.52
.56
.60
.6*4
.63
.72
.76
.30
.8**
.38
.92
.96
1.00
7.0 AND 8
BETA (X)
.0000001
.0000079
.0001130
.0007087
.0028100
.0033120
.0200336
.0*41*4566
.0761703
.1271175
.19580*48
.23166*42
.3817390
.*4908095
.601991**
.70773*t8
.8010572
.3767620
.9323613
. 968*4*43**
.98332*41
.9969923
.999597*4
.9999890
1.0000000
7.5 AND 8
BETA (X)
.0000000
.0000028
.00001,90
.00035**!
.0015661
.0050630
.0131i»55
.0239'?3'+
.0563087
.098665**
.1536751
.2371519
.332ft902
.
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.04
.08
.12
.16
.20
.24
.28
.32
.36
.40
.44
.48
.52
.56
• 60
.64
.63
.72
.76
.30
• 84
.83
.92
.96
1.00
A =
X
.04
.03
.12
.16
.20
.24
.23
.32
.36
.40
• 44
.43
.52
.56
.60
.64
.63
.72
.76
.30
.34
.83
.92
.95
1.00
3.0 AND 6
SETA (X)
.0000000
.0000010
.0000210
.0001750
.0008640
.0030532
.0035423
.0200 876
.041252?
.0759272
.1275491
.1931725
.2875963
.3928058
.5079487
.6249673
.7343624
.8293778
.9027270
.9528754
.9819039
.9951650
.9993292
.9999810
1.0000000
3.5 AND B
BETA (X)
.0000000
.0000003
.0000039
.0000857
.000^722
.0018245
.0055013
.0137980
.0299730
.0579679
.1017616
.1644336
.2471531
.7432599
.4629114
.5333564
.7000378
.8033647
.3857^27
.9436358
.9779543
.9940019
.9991529
.999975-6
1.0000000
5.5
INVERSE
NOT DEFINED
. 0300000
.1200000
. L600000
.2000000
.2400000
.2300000
,3200001
.3600-000
.4000000
.4400000
.4799999
.SlSqcgg
.5600000
.6000000
.6400000
.6800000
.7200000
,7600000
.8000000
.3400000
. 8800000
.9200000
.9600000
1.0000000
5.5
INVERSE
NOT DEFINED
.0300000
. 1200000
.1500000
.2000000
.2400000
.2300000
.3200000
.3600000
.4000000
.2+400000
.4800000
.5200000
.5600000
.6000000
.6400000
.6300000
.7200000
. 7600000
.8000000
.8400000
. 3800000
.9200000
.9600000
1. 0000000
175
XC ERROR TERM
NOT DEFINED
-.0000000
-.0000000
-.0000000
-.0000000
- .0000000
.0000000
-.0000001
.0000000
-.0000000
.0000000
.0000001
.0000001
-. ooooooo
-.0000000
. 0000 000
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.ooooooo
.0000000
.ooooooo
.ooooooo
0.0000000
XC ERROR TERM
NOT DEFINED
-.OOOOOOO
. 0 0 0.0 0 0 0
-.ooooooo
-.ooooooo
-.ooooooo
,0000000
.ooooooo
-.ooooooo
-.ooooooo
.ooooooo
.ooooooo
-.ooooooo
-.ooooooo
-.ooooooo
.0000 000
.0000 000
.ooooooo
.ooooooo
.ooooooo
.QOOOOOO
.ooooooo
. ooooooo
.ooooooo
o.ooooooo
-------
INCOMPLETE BETA FUNCTION AND INVERSE
A =
X
.0*4
.08
.12
.16
.20
.2**
.25
.32
.36
.40
.44
.48
.52
.56
.60
.64
.68
.72
.76
.80
.84
.83
.92
.96
1.00
9.0 AND 8
3ETA (X)
.0000000
.0000001
.0000038
.0000416
.0002559
.0010811
.0035143
.0094030
.0216116
.0439332
.0306232
.1355537
.2111053
.307052&
.4197835
.5421997
.6645304
.7760707
.8675364
.9334155
.973«+773
.9926524
.9989437
.9999690
1.0000000
5.5
INVERSE
NOT DEFINED
. 0800000
.1200000
.1600000
.2000000
.2400001
.2800000
. 3200000
.3600000
.4000000
.4400000
.4800000
.5200000
.5600000
,6000000
.6400000
.6800000
.7199999
.7600000
.8000000
.8400000
.3300000
.9199999
.9600000
1.0000000
XC ERROR TERM
NOT DEFINED
-.0000000
.0000000
.0000000
-.0000000
-.0000001
.0000000
.0000000
-.0000000
-.0000000
.0000000
.0000000
-.0000000
.0000000
-.0000000
-.0000 000
-.0000000
.0000001
.0000000
.0000000
.0000000
.0000000
.0000001
.0000000
o.ooooooo
176
-------
PEARSON'S INCOMPLETE GAMMA FUNCTION
In the field of hydrology the incomplete gamma function is an
extensively used tool for the analyses of flood frequencies and
also in the study of time series.
An excellent discussion of the use of the incomplete gamma function
as related to flood frequency analyses is found in the American
Society of Civil Engineers Transactions. Volume 87, Paper No. 1532,
by H. Alden Foster, dated 1924.
The use of the incomplete gamma function can be found in various
papers concerning stochastic techniques in time series analyses.
The discussion that follows in the subsequent narrative will be
limited to an electronic computer application for the solution of
the incomplete gamma function.
The expression generally given for Pearson's Incomplete Gamma Function
is as follows:
u
tV' dt '
where: x = u /p+1
a = p+1 or p = a-1
177
-------
The expression of the probability function for the solution of a
random variable having a value £ x from a gamma function is:
i a /
p(* 1 x) = FTTT J t3'1 £~bt dt, x ^ o
t3'1 dt
r(a) J
which is often written as:
P(x <_ x) = |- —~ r(a, b-x)
where "a" and "b" are distribution parameters and r is the
incomplete gamma function.
T(a) is the complete gamma function.
The purpose of this electronic computer application is to design a
portable program for the calculation of the incomplete gamma function
under five conditions (five algorithms).
oo
r(A,x) = T(A,x) = / JTU i/""1 du
Condition 1 - if A = 0
then:
r(O.x) = T(O.x) = - Y + £n (x) +
178
-------
Condition 2 - if A - -N for some positive integer N
then:
r(-N,x) = r(-N,x) = zE (x) _ rx f1 .(-I)3 j !
XJ+1
Condition 3 - if x = 0
00
r(A,0) = r(A,0) = t^ £~U uA-l du - T(A)
J0
Condition 4 - if A^O and x < MA I + 1
CO
r(A,x) = r(A,x) = F(A) - XA V
L—i
(-x)n
(A+n)n!
n=o
Condition 5 - if A^O X > Ul + 1
-x A 1 1 - A 2 - A 2_
r(A,x) = £ XA XT 1+ 1+ X+
The electronic computer application is a stand alone set of subroutines
or functions. The discussion of these various subroutines follows in the
subsequent narrative.
PROGRAM GAMINT
This program is simply a little driver that was used to generate values
of Pearson's Incomplete Gamma Function in terms of "u" and "p" for
comparison with the Tables.
179
-------
The variable name A is A in T(A,x) and the variable name
X is X in F(A,x). The variable name U is the transform X = U v'T+l
and the variable name P is the transform P = A-l.
The variable name COX is T(A) and CONGAM(A) is the function name used
for the solution of the complete gamma.
FUNCTION GAMMA (A.X)
In the function labeled GAMMA the arguments A and X are identical to
those described in the previous narrative concerning the five
conditions.
The label EULERS is used to store Euler's constant, which is y is the
previous narrative.
It is in the function (FORTRAN terminology) that the various tests
are made to determine which condition of algorithm will be used in
the solution of the gamma distribution.
Several comments have been included to aid in describing the afore-
mentioned tests.
FUNCTION SMLGAM (A.X)
The function labeled SMLGAM is used to calculate the sum of the
following series:
180
-------
S(a,x) =
n=k
(-x)n
(a+n)n!
n=o
where k = 30 provided enough terms in the series such that:
S (a,x) - S__, (a,x)
n ' ' n+1
< 1 x 10
,-15
This series expansion was used for the solution of Condition 1,
i.e. T(0,X). The series expansion was also used in the solutions
of Conditions 2 and 4.
The arguments labeled A and X are self-explanatory.
The label EPSLON represents the absolute difference of the sums
n and n+1 in the criteria for the series truncation.
All other labels included in this function labeled SMLGAM are by design
self-explanatory.
The series is shown in Reference 1, Chapter 6, Series Developments -
6.5.29, page 262.
FUNCTION CONTER (A.X)
The function labeled CONTER is used for the solution of the following
continued fraction:
181
-------
l_ 1-A 2-A 2_
X+ 1+ 1+ X+
where: X > 0 and A < <»
This series is used as part of the solution for Condition 5. All
labels have been designed to be self-explanatory.
The continued fraction is shown in Reference 1, Chapter 6, 6.5.31,
page 263.
The arguments A and X are self-explanatory.
FUNCTION CONGAM (ALF)
The function labeled CONGAM is used for the solution of the complete
gamma which is often called Euler's Integral. The expression for
the complete gamma is as follows:
00
r(z) = / t2"1 A"' dt
-------
r(z)r(i-z) = - zr(-z)r(z) = n esc nz
where the fundamental identity for CSC IIZ = I/sin HZ.
The following polynomial approximation along with the above recurrence
and reflection formulas are shown in Reference 1, Chapter 6, pages 256
and 257.
T(x+l) = X! = 1 + b,x + b X + ... + bDX8 + e(x)
2 o
where: 0 <_ X <_ | and j e (x) £ 3 x 10~7
The coefficients for the above polynomial are:
b = -0.577191652 b = -0.756704078
1 5
b = 0.988205891 b = 0.482199394
2 6
b3 = -0.897056937 b? = -0.193527818
b, = 0.918206857 b. = 0.035868343
It o
All labels are self-explanatory as well as the argument ALF.
FUNCTION GAMNEG (MINUSN,X)
The function GAMNEG is used for the solution of Condition 2. It
might be well to reiterate the expression for Condition 2.
183
-------
N-l
r K oo - «-* y (-»j .1' i
! L i £—> „}+! J
r(-N,x) = NI
where: E (X) = / £ Xt t~n dt = Xn ^ r(l-n,x)
n J
or:
r°°
E (x) = / £~xt t"1 dt = x° r(o,x) = r(x)
-*- vV
1
which is easily recognized as the complete gamma function.
The argument labeled MINUSN is A = -N for some positive integer N.
The expressions as shown above can be found in Reference 1, Chapter 6.
All labels used in the function labeled GAMNEG are self-explanatory.
Error Messages
The error messages are program-generated and are written on the file
OUTPUT. The error messages were designed to minimize the system-
generated error messages.
184
-------
REFERENCES
1. Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, U.S. Department of Commerce, National Bureau
of Standards, Applied Mathematics Series 55, Issued June 1964,
Fourth Printing, December 1965.
185
-------
PROGR«f GAMINT 7<»/7<» 0"! = ! FTN 4.2+P390 75/03/17. 09.57.05
PROGRAM GflMINTlINPUT, OUTPUT, INOAT,OUT A,TAPE5=INOftT,
I TAPE6=OUTA)
COMMON / ARLE / COT, CIT, COX
INTEGER COT , CIT
5 CIT = 5
COT = 6
P = 0. 0
10 U = 0.0
M WRITE (COT , 20)
oo 10 30 FORMAT (1H1, 8X )
^ DO 100 K = 1, 30
U = U + l.E-1
X = U * SORT (P *• 1.0 )
A = P t 1.0
15 COX = 0.0
COX = CONGAM (A)
REF = 0.0
REF - COX - GAMMA (A,X)
REF = REF / COX
20 WRITE ! COT , 50 ) U , P , REF
50 FORMAT t 8X, t,HU = , 3X , F10.6 » 8Xf «»HP = , 3X , F10.6 t
1 . 8X, 9HI(U,P> = , E19.12 )
100 CONTINUE
P = P * 5-E-l
35 IF 1 P . GE. 10.5 > CALL EXIT
GO TO 10
END
-------
PROGRAM GAMINT
OPT=1
FTN
75/03/17. 09.57.05.
SYMBOLIC REFERENCE MAP (R = l)
ENTRY POINTS
1021i» GAMINT
PAGE
VARIABLES SN
10315 A
0 COT
10313 K
10316 REF
FILE NAMES
1*102 INDAT
MQ2 TAPE5
EXTERNALS
CONGAM
GAMMA
STATEMENT LABELS
10220 10
0 100
TYPE
REAL
INTEGER
INTEGER
REAL
REAL
MODE
TYPE
REAL
REAL
RELOCATION
ABLE
0 INPUT
611»3 TAPE6
ARGS
1
2
102
1 CIT
2 COX
10311 P
10312 U
INTEGER
REAL
REAL
REAL
ABLE
ABLE
61«»3 OUTA
OUTPUT
EXIT
SQRT
REAL
FMT
1 LIBRARY
10276 50
FMT
COMMON BLOCKS
ABLE
LENGTH
STATISTICS
PROGRAM LENGTH
BUFFER LENGTH
CM LABELED COMMON LENGTH
1138
1020^8
38
75
i«228
3
-------
FUNCTION GAMMA 7u/7k OPT = I FTN <..2+P390 75/03/17. 09.57.05.
FUNCTION GAMMA JA,X)
COMMON / ABLE / COT, CIT ,cox P&GE i
INTEGER COT,CIT
GAMMA = 0.0
5 EULEPS = 5. 7721566<4<301533E-1
IF (A.EO.0.0.AND.X.EQ.0.0) GO TO 500
UPDATE = 0.0
IF ( A.LT. 0.0) GO TO
-------
FUNCTION GAHMA
OPT =
FTN <».2+P380
75/03/17. 09.57.05.
SYMBOLIC REFERENCE MAP
PAGE
ENTRY POINTS
l» GAMMA
VARIABLES
0 A
0 COT
13<» CRP
131 GAMMA
136 NS
0 X
EXTERNALS
EXP
SHLGAM
SN TYPE
REAL
INTEGER
REAL
REAL
INTEGER.
REAL
TYPE
REAL
REAL
REAL
INLINE FUNCTIONS TYPE
ABS REAL
STATEMENT LABELS
21 1
53 30
100 60
112 SCO
COMMON BLOCKS
ABLE
LENGTH
3
RELOCATION
F.P.
ABLE
F.P.
ARGS
1 LIBRARY
1 LIBRARY
Z
ARGS
1 INTRIN
1
2
132
135
133
CIT
COX
EULERS
NP
UPDATE
CONTFR
GAMNEG
SQ.RT
INTEGER
REAL
REAL
INTEGER
REAL
REAL
REAL
REAL
ABLE
ABLE
2
2
1 LIBRARY
35
65
101
10
<*0
70
51
70
107
20
50
80
STATISTICS
PROGRAM LENGTH 1373
CM LABELED COMMON LENGTH 38
95
3
-------
FUNCTION SMLGAM 74/7<» 0°T = 1 FTN f».2«-P3SO 75/03/17. 09.57.06.
FUNCTION SMLGAM (A ,X >
COMMON / OSLE / COT, CIT, cox PAGE i
INTEGER COT , CIT
C SET INITIBL VALUES FOR SERIES EXPANSION I.E. N = 0 (6.5.39)
5 SMLGAM = 0.0
IF (A.NE.0.0) SMLGAM = 1.0 / A
ANUM = X
PNEXT = 0.0
FACTN = 1.0
10 SIGN = 1.0
EPSLON = 0.0
DENOH =0.0
DO 10 K = 1, 30
PN = K
_ 15 SIGN = - SIGN
g OFNOM = At PN
FACTN = FACTN » PN
TERM = 0.0
TFRM = ( SIGN * ANUM ) / (OENOM * FACTN >
30 EPSLON = ABS (TERM)
IF (EPSLON. LE.l.E-15) GO TO 100
SMLGAM = SMLGAM + TERM
ANUM = ANUM * X
10 CONTINUE
25 WRITE (COT , 20) EPSLON , A , X
20 FORMATUH1, BX, 35HSERIES FOR SMLGAM DID NOT CONVERGE /
1 3X, 9HEFSLON = , £19.12 ,l»X,«fHA = ,E 19.13 ,
-------
FUNCTION SMLGAM
OPT =
FTN
75/03/17- 09.57.06.
SYMBOLIC REFERENCE MftP (R 1)
PAGE
ENTRY POINTS
<* SKLGAM
VARIABLES SN
o A
1 CIT
2 COX
106 EPSLON
110 K
103 PNEXT
101 SMLGAM
0 X
EXTERNALS
EXIT
INLINE FUNCTIONS
A OS
TYPE RELOCATION
REAL F.P.
INTEGER A3LE
REAL ABLE
REAL
INTEGER
REAL
REAL
REAL F.P.
TYPE ARCS
0
TYPE ARCS
REAL 1 INTRIN
STATEMENT LABELS
0 10
COMMON BLOCKS LENGTH
ABLE 3
STATISTICS
PROGRAM LENGTH
CM LABELED COMMON LENGTH
102 ANUM
0 COT
107 OENCM
FACTN
PN
SIGN
TERM
111
105
1-12
REAL
INTEGER
REAL
REAL
REAL
REAL
REAL
ABLE
6<» 20
FMT
5«« 100
1178
38
79
3
-------
FUNCTION CONTFR 7<»/7<* OPT = I FTN
-------
FUNCTION CONTFR
7
-------
FUNCTION COKGAM
71./7I,
FTN , K = 1,8 ) = -0.577191652 , 0.988205691 ,
1 -0.397056937 , 0.918206857 , - 0.75670<»078 , 0. <<8219939
-------
FUNCTION CONG&M
OPT =
FTN 4.3+P380
60
65
70
FOFX = 1.0
GO TO 60
t»0 J3 = 3
FOFX = 0.0
50 FOFX = FOFX *
FOFX = FOFX »
JB = J3 - 1
IF (JO.GT.O)
FOFX = FOFX *
75/03/17. 09.57.07.
PflGE Z
GCCJB)
X
GO TO 50
1.0
60 CONGftM = FOFX * FB
IF (Mft»K .EQ. 2 ) CONGflM = CONGAM / X
IF ( KSU .EQ. 2 > GO TO 19
100 RETURN
END
-------
FUNCTION CONGAM 7unt> O»T = I FTN it.2+P38Q 75/02/17. 09.57.07.
CARD NR. SEVERITY DETAILS DIAGNOSIS OF PROBLEM
17 I BASIC EXTERNAL OR INTRINSIC FUNCTION CALLED WITH WRONG TYPE ARGUMENT.
PAGE 3
-------
FUNCTION CONGAM
FTN
75/03/17. 09.57.07.
SYMBOLIC REFERENCE MRP
ENTRY POINTS
-------
FUNCTION GAMNEG 7W7«i QPT = 1 FTN («.24-P380 75/03/17. 09.57.08.
FUNCTION GAMNEG
-------
FUNCTION GflMNEG 7<4/Ti OPT = 1 FTN «t.2+P380 75/03/17. 09.57.08.
CftRO NR. SEVERITY DETAILS DIAGNOSIS OF PROBLEM PAGE 2
3 I CONSTANT TOO LONG. HIGH ORDER DIGITS RETAINED, BUT SOME PRECISION LOST.
-------
FUNCTION GAMNEG
OPT=1
FTN
75/03/17. 09.57.08.
SYMBOLIC REFERENCE MAP (R=i)
ENTRY POINTS
i* GAMNEG
PAGE
VARIABLES SN
O
o
64
67
63
0
66
72
70
EULERS
FACTJ
GAMNEO
MINUSN
NLESS1
SUM
XTERM
EXTERNALS
INLINE
ALOG
SMLGAK
FUNCTIONS
FLOAT
TYPE
REAL
REAL
REAL
INTEGER
INTEGER
REAL
REAL
TYPE
REAL
REAL
TYPE
REAL
ARGS
1 L
2
ARGS
1
RELQi
IBRARY
INTRIN
P.P.
STATEMENT LABELS
17 1
STATISTICS
PROGRAM LENGTH
76B
62
75
74
71
65
73
0
E1X
FACTN
J
N
SIGN
X
REAL
REAL
INTEGER
INTEGER
REAL
REAL
EXP
REAL
INACTIVE
F.P.
1 LIBRARY
32 3
-------
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
IT
u
u
u
u
u
u
r
~
r
=
=
=
=
=
=
=
=
=
=
=
=
=
=
—
=
r:
=
—
3
=
=
=
_
=
_
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
•
*
.
*
.
.
*
.
1.
1.
1.
1.
J- *
1.
1.
1.
1.
1.
d. »
2.
2.
2.
a.
2.
2.
2.
2.
2.
3.
.100000000
100000
.100000000
200000
. 100000000
300000
.100000000
<«Q 0000
.100000000
500000
.100000000
600000
.100000000
700000
. 100000000
800000
.100000003
900000
. 100000000
000000
.100000000
1QOOUO
. 1000COOCO
?coooo
.100000000
300000
.100000000
'*OOCOO
. ionoooooo
500003
.100000000
600000
.100000000
700000
.100000000
800000
. 100000000
900000
.100000000
000000
.100000000
130000
.100000000
2COOOO
.100000000
300000
.1000 00000
^00000
. 100000000
500000
.100000000
600000
.100000000
700000
.100000000
800000
.100000000
900000
.100000000
000000
OOOE+01
p =
OODE«-Q1
p =
OOOE+01
p =
QOOE+Ol
p =
OOOE+01
p =
QOOE+Ol
p =
OOOE4-01
p =
OOOF+01
p =
OOOE4-Q1
p =
QOOE+01
p =
OOOE+01
p =
OOOEf 01
p =
OOOE+01
p =
DOOE+01
p =
OOQE+01
P =
OOOE+01
P =
QOOEtOl
p =
OOOE+01
P =
OOOE^Ol
P ~-
OOOE+01
P =
OOOE+01
P -
OOOE+Q1
p =
OOOE+01
P =
OOOE+-01
P =
ODOE+Q1
P =
OOOEtOl
P =
OOOF+Oi
P =
OOOEf 01
P =
OOOE+01
P =
OOOE+01
P =
0
0
0
0
0
0
0
0
0
0
0
G
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
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.240647152284E+00
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.322334688795E+00
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.300000 P =
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.700000000 QOOE+01
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1.200000 P =
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1.500000 P =
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1.600000 P =
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1.700000 P =
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1.800000 P =
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1.900000 P =
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2. (,00000 P =
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2.600000 P =
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2.700000 P =
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2.800000 P =
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2.900000 P =
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3.000000 P =
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1.000000 P =
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2.500000 P=
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2.600000 P =
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KOODOO
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P =
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P =
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P =
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P =
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P =
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P =
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P =
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P =
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P =
OOCE+02
P =
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P =
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P =
9
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P =
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P =
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10 . 1050COOOOOOOE + Q2
U = .100000 P =
10 .105000000000E+02
U = .200000 P =
10 . 105000000000E + 02
U = .300000 P =
10 .105000000000E+02
U = .KOODOO P =
10 .1050000000QOE+02
U = .500000 P =
10 .105000000000E+02
U = .600000 P =
10 .105000000OOOE+Q2
U = .700000 P =
10 .105000000OOOE+02
U = .800000 P =
10 -1050000QOOOOE+02
U = .900000 P =
10 . 105000000QOQE+02
U = 1.000000 P =
10 .io5aoooaooooE+o2
u = i.iooooo P =
10 . 105000000OOQE + 02
U = 1.200000 P =
10 .10500000QOOOE+02
U = 1.300000 P =
10 . 105000000 OOOE-i-02
U = 1.1*00000 P =
10 . 105000000OOOE + 02
U = 1.500000 P =
10 .105000000OOOE+02
U = 1.600000 P =
10 .105000000000E+02
U = 1.700000 P =
10 .105000000QOOE+02
U = 1.800000 P =
10 .105000000000E+02
U = 1.900000 P =
10 .1050000000OOE+02
U = 2.000000 P =
10 .10500COOOOOQE+02
U = 2.100000 P =
10 .10500QQOOQOOE+02
U = 2.200000 P =
10 .105000000QOOE+02
U = 2.300000 P =
10 .1050000000000*02
U = 2.^00000 P =
10 .105000000OOOE+02
U = 2.500000 P =
10 .105000000OOOE+02
U = 2.600000 P =
10 .105000000OOOE+02-
U = 2.700000 P =
10 .105000000000E+02
U = 2.800000 P =
10 .105000000OOOE + 02
U = 2.900000 P =
10 .105000000000E+02
U = 3. 000000 P =
9.500000
9.500000
9.500000
9.500000
9.500000
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9.500000
9.500000
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9.500000
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9.500000
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I(U,P)
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220
-------
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p =
OOOE+02
p =
OOQE+02
p =
OOOE*02
p =
QOOE*02
p =
OOOE+Q2
p =
OOOE+02
P =
OOOE+02
P =
OOOE+02
P =
QQOEt02
p =
OOOE*02
P =
0 0 0 E -i- 0 2
P =
OOOE+02
P =
OQOE*02
P =
OQOE+02
P -
OOOE+02
P =
OOQEtOZ
P =
OOOE+02
p =
OOOE*02
P =
10
10
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10
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10
10
10
10
10
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10
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.000000
.000000
I
I
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-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1, REPORT NO.
EPA-600/2-77-179d
3. RECIPIENT'S ACCESSiON>NO.
4. TITLE AND SUBTITLE
PREDICTION OF MINERAL QUALITY OF IRRIGATION RETURN
FLOW, VOLUME IV, Data Analysis Utility Programs
5. REPORT DATE
August 1977 issuing date
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Bureau of Reclamation
Engineering and Research Center
Denver, Colorado 80225
10. PROGRAM ELEMENT NO.
1HB617
11. CONTRACT/GRANT NO.
EPA-IAG-D4-0371
12. SPONSORING AGENCY NAME AND ADDRESS
Robert S. Kerr Environmental Research Lab. - Ada, OK
Office of Research and Development
U.S. Environmental Protection Agency
Ada. Oklahoma 74820
13. TYPE OF REPORT AND PERIOD COVERED
Final
14. SPONSORING AGENCY CODE
EPA/600/15
15. SUPPLEMENTARY NOTES
VOLUMES I, II, III, V (EPA-600/2-77-179a thru 179c, 179e)
16. ABSTRACT
The development and evaluation of modeling capability to simulate and predict the
effects of irrigation on the quality of return flows are documented in the five
volumes of this report. The report contains two different modeling packages which
represent different levels of detail and sophistication. Volumes I, II and IV
pertain to the model package given in Volume III. Volume V contains the more
sophisticated model. User's manuals are included in Volumes III and V.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
C. COSATI Field/Group
Mathematical Model, digital simulation,
scheduling, Irrigated land, Evapotrans-
piration, Agriculture, Agronomy, water
pollution, water loss
Irrigation Return Flow
02 C/D
3. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
Unclassified
21. NO. OF PAGES
230
20. SECURITY CLASS (This page)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
222
* U.S. GOVERNMENT PRINTING OFFICE: 1977— 757-056/6548
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