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SUBROUTINE DLBVP

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 DLBVP (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   - DOUBLE PRECISION INPUT AND OUTPUT VECTOR WITH
	    DIMENSION GREATER THAN 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 DLBVP. EXCEPT PRMT(5) THE COMPONENTS
	    ARE NOT DESTROYED BY SUBROUTINE DLBVP AND THEY 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 DLBVP INITIALIZES
	    PRMT(5)=0. IF THE USER WANTS TO TERMINATE
	    SUBROUTINE DLBVP 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 DLBVP DOES NOT REQUIRE
	    AND CHANGE THEM. NEVERTHELESS THEY MAY BE USEFUL
	    FOR HANDING RESULT VALUES TO THE MAIN PROGRAM
	    (CALLING DLBVP) WHICH ARE OBTAINED BY SPECIAL
	    MANIPULATIONS WITH OUTPUT DATA IN SUBROUTINE OUTP.
   B	  - DOUBLE PRECISION NDIM BY NDIM INPUT MATRIX
	    (DESTROYED). IT IS THE COEFFICIENT MATRIX OF Y(XL)
	    IN THE BOUNDARY CONDITIONS.
   C	  - DOUBLE PRECISION NDIM BY NDIM INPUT MATRIX
	    (POSSIBLY DESTROYED). IT IS THE COEFFICIENT MATRIX
	    OF Y(XU) IN THE BOUNDARY CONDITIONS.
   R	  - DOUBLE PRECISION INPUT VECTOR WITH DIMENSION NDIM
	    (DESTROYED). IT SPECIFIES THE RIGHT HAND SIDE OF
	    THE BOUNDARY CONDITIONS.
   Y	  - DOUBLE PRECISION AUXILIARY VECTOR WITH
	    DIMENSION NDIM. IT IS USED AS STORAGE LOCATION
	    FOR THE RESULTING VALUES OF DEPENDENT VARIABLES
	    COMPUTED AT INTERMEDIATE POINTS X.
   DERY   - DOUBLE PRECISION 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 X.
   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 DLBVP 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 DLBVP IS TERMINATED.
   AUX	  - DOUBLE PRECISION AUXILIARY STORAGE ARRAY WITH 20
	    ROWS AND NDIM COLUMNS.
   A	  - DOUBLE PRECISION 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 DGELG	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.