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decus/20-0026/lbvp.doc
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SUBROUTINE LBVP
PURPOSE
TO SOLVE A LINEAR BOUNDARY VALUE PROBLEM, WHICH CONSISTS OF
A SYSTEM OF NDIM LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS
DY/DX=A(X)*Y(X)+F(X)
AND NDIM LINEAR BOUNDARY CONDITIONS
B*Y(XL)+C*Y(XU)=R.
USAGE
CALL LBVP (PRMT,B,C,R,Y,DERY,NDIM,IHLF,AFCT,FCT,DFCT,OUTP,
AUX,A)
PARAMETERS AFCT,FCT,DFCT,OUTP REQUIRE AN EXTERNAL STATEMENT.
DESCRIPTION OF PARAMETERS
PRMT - AN INPUT AND OUTPUT VECTOR WITH DIMENSION GREATER
OR EQUAL TO 5, WHICH SPECIFIES THE PARAMETERS OF
THE INTERVAL AND OF ACCURACY AND WHICH SERVES FOR
COMMUNICATION BETWEEN OUTPUT SUBROUTINE (FURNISHED
BY THE USER) AND SUBROUTINE LBVP.
THE COMPONENTS ARE
PRMT(1)- LOWER BOUND XL OF THE INTERVAL (INPUT),
PRMT(1)- UPPER BOUND XU OF THE INTERVAL (INPUT),
PRMT(3)- INITIAL INCREMENT OF THE INDEPENDENT VARIABLE
(INPUT),
PRMT(4)- UPPER ERROR BOUND (INPUT). IF RELATIVE ERROR IS
GREATER THAN PRMT(4), INCREMENT GETS HALVED.
IF INCREMENT IS LESS THAN PRMT(3) AND RELATIVE
ERROR LESS THAN PRMT(4)/50, INCREMENT GETS DOUBLED.
THE USER MAY CHANGE PRMT(4) BY MEANS OF HIS
OUTPUT SUBROUTINE.
PRMT(5)- NO INPUT PARAMETER. SUBROUTINE LBVP INITIALIZES
PRMT(5)=0. IF THE USER WANTS TO TERMINATE
SUBROUTINE LBVP AT ANY OUTPUT POINT, HE HAS TO
CHANGE PRMT(5) TO NON-ZERO BY MEANS OF SUBROUTINE
OUTP. FURTHER COMPONENTS OF VECTOR PRMT ARE
FEASIBLE IF ITS DIMENSION IS DEFINED GREATER
THAN 5. HOWEVER SUBROUTINE LBVP DOES NOT REQUIRE
AND CHANGE THEM. NEVERTHELESS THEY MAY BE USEFUL
FOR HANDING RESULT VALUES TO THE MAIN PROGRAM
(CALLING LBVP) WHICH ARE OBTAINED BY SPECIAL
MANIPULATIONS WITH OUTPUT DATA IN SUBROUTINE OUTP.
B - AN NDIM BY NDIM INPUT MATRIX. (DESTROYED)
IT IS THE COEFFICIENT MATRIX OF Y(XL) IN
THE BOUNDARY CONDITIONS.
C - AN NDIM BY NDIM INPUT MATRIX (POSSIBLY DESTROYED).
IT IS THE COEFFICIENT MATRIX OF Y(XU) IN
THE BOUNDARY CONDITIONS.
R - AN INPUT VECTOR WITH DIMENSION NDIM. (DESTROYED)
IT SPECIFIES THE RIGHT HAND SIDE OF THE
BOUNDARY CONDITIONS.
Y - AN AUXILIARY VECTOR WITH DIMENSION NDIM.
IT IS USED AS STORAGE LOCATION FOR THE RESULTING
VALUES OF DEPENDENT VARIABLES COMPUTED AT
INTERMEDIATE POINTS.
DERY - INPUT VECTOR OF ERROR WEIGHTS. (DESTROYED)
ITS MAXIMAL COMPONENT SHOULD BE EQUAL TO 1.
LATERON DERY IS THE VECTOR OF DERIVATIVES, WHICH
BELONG TO FUNCTION VALUES Y AT INTERMEDIATE POINTS.
NDIM - AN INPUT VALUE, WHICH SPECIFIES THE NUMBER OF
DIFFERENTIAL EQUATIONS IN THE SYSTEM.
IHLF - AN OUTPUT VALUE, WHICH SPECIFIES THE NUMBER OF
BISECTIONS OF THE INITIAL INCREMENT. IF IHLF GETS
GREATER THAN 10, SUBROUTINE LBVP RETURNS WITH
ERROR MESSAGE IHLF=11 INTO MAIN PROGRAM.
ERROR MESSAGE IHLF=12 OR IHLF=13 APPEARS IN CASE
PRMT(3)=0 OR IN CASE SIGN(PRMT(3)).NE.SIGN(PRMT(2)-
PRMT(1)) RESPECTIVELY. FINALLY ERROR MESSAGE
IHLF=14 INDICATES, THAT THERE IS NO SOLUTION OR
THAT THERE ARE MORE THAN ONE SOLUTION OF THE
PROBLEM.
A NEGATIVE VALUE OF IHLF HANDED TO SUBROUTINE OUTP
TOGETHER WITH INITIAL VALUES OF FINALLY GENERATED
INITIAL VALUE PROBLEM INDICATES, THAT THERE WAS
POSSIBLE LOSS OF SIGNIFICANCE IN THE SOLUTION OF
THE SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS FOR
THESE INITIAL VALUES. THE ABSOLUTE VALUE OF IHLF
SHOWS, AFTER WHICH ELIMINATION STEP OF GAUSS
ALGORITHM POSSIBLE LOSS OF SIGNIFICANCE WAS
DETECTED.
AFCT - THE NAME OF AN EXTERNAL SUBROUTINE USED. IT
COMPUTES THE COEFFICIENT MATRIX A OF VECTOR Y ON
THE RIGHT HAND SIDE OF THE SYSTEM OF DIFFERENTIAL
EQUATIONS FOR A GIVEN X-VALUE. ITS PARAMETER LIST
MUST BE X,A. SUBROUTINE AFCT SHOULD NOT DESTROY X.
FCT - THE NAME OF AN EXTERNAL SUBROUTINE USED. IT
COMPUTES VECTOR F (INHOMOGENEOUS PART OF THE
RIGHT HAND SIDE OF THE SYSTEM OF DIFFERENTIAL
EQUATIONS) FOR A GIVEN X-VALUE. ITS PARAMETER LIST
MUST BE X,F. SUBROUTINE FCT SHOULD NOT DESTROY X.
DFCT - THE NAME OF AN EXTERNAL SUBROUTINE USED. IT
COMPUTES VECTOR DF (DERIVATIVE OF THE INHOMOGENEOUS
PART ON THE RIGHT HAND SIDE OF THE SYSTEM OF
DIFFERENTIAL EQUATIONS) FOR A GIVEN X-VALUE. ITS
PARAMETER LIST MUST BE X,DF. SUBROUTINE DFCT
SHOULD NOT DESTROY X.
OUTP - THE NAME OF AN EXTERNAL OUTPUT SUBROUTINE USED.
ITS PARAMETER LIST MUST BE X,Y,DERY,IHLF,NDIM,PRMT.
NONE OF THESE PARAMETERS (EXCEPT, IF NECESSARY,
PRMT(4),PRMT(5),...) SHOULD BE CHANGED BY
SUBROUTINE OUTP. IF PRMT(5) IS CHANGED TO NON-ZERO,
SUBROUTINE LBVP IS TERMINATED.
AUX - AN AUXILIARY STORAGE ARRAY WIRH 20 ROWS AND
NDIM COLUMNS.
A - AN NDIM BY NDIM MATRIX, WHICH IS USED AS AUXILIARY
STORAGE ARRAY.
REMARKS
THE PROCEDURE TERMINATES AND RETURNS TO CALLING PROGRAM, IF
(1) MORE THAN 10 BISECTIONS OF THE INITIAL INCREMENT ARE
NECESSARY TO GET SATISFACTORY ACCURACY (ERROR MESSAGE
IHLF=11),
(2) INITIAL INCREMENT IS EQUAL TO 0 OR IF IT HAS WRONG SIGN
(ERROR MESSAGES IHLF=12 OR IHLF=13),
(3) THERE IS NO OR MORE THAN ONE SOLUTION OF THE PROBLEM
(ERROR MESSAGE IHLF=14),
(4) THE WHOLE INTEGRATION INTERVAL IS WORKED THROUGH,
(5) SUBROUTINE OUTP HAS CHANGED PRMT(5) TO NON-ZERO.
SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
SUBROUTINE GELG SYSTEM OF LINEAR EQUATIONS.
THE EXTERNAL SUBROUTINES AFCT(X,A), FCT(X,F), DFCT(X,DF),
AND OUTP(X,Y,DERY,IHLF,NDIM,PRMT) MUST BE FURNISHED
BY THE USER.
METHOD
EVALUATION IS DONE USING THE METHOD OF ADJOINT EQUATIONS.
HAMMINGS FOURTH ORDER MODIFIED PREDICTOR-CORRECTOR METHOD
IS USED TO SOLVE THE ADJOINT INITIAL VALUE PROBLEMS AND FI-
NALLY TO SOLVE THE GENERATED INITIAL VALUE PROBLEM FOR Y(X).
THE INITIAL INCREMENT PRMT(3) IS AUTOMATICALLY ADJUSTED.
FOR COMPUTATION OF INTEGRAL SUM, A FOURTH ORDER HERMITEAN
INTEGRATION FORMULA IS USED.
FOR REFERENCE, SEE
(1) LANCE, NUMERICAL METHODS FOR HIGH SPEED COMPUTERS,
ILIFFE, LONDON, 1960, PP.64-67.
(2) RALSTON/WILF, MATHEMATICAL METHODS FOR DIGITAL
COMPUTERS, WILEY, NEW YORK/LONDON, 1960, PP.95-109.
(3) RALSTON, RUNGE-KUTTA METHODS WITH MINIMUM ERROR BOUNDS,
MTAC, VOL.16, ISS.80 (1962), PP.431-437.
(4) ZURMUEHL, PRAKTISCHE MATHEMATIK FUER INGENIEURE UND
PHYSIKER, SPRINGER, BERLIN/GOETTINGEN/HEIDELBERG, 1963,
PP.227-232.