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Trailing-Edge - PDP-10 Archives - decuslib20-02 - 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.