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decus/20-0026/hpcg.doc
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SUBROUTINE HPCG
PURPOSE
TO SOLVE A SYSTEM OF FIRST ORDER ORDINARY GENERAL
DIFFERENTIAL EQUATIONS WITH GIVEN INITIAL VALUES.
USAGE
CALL HPCG (PRMT,Y,DERY,NDIM,IHLF,FCT,OUTP,AUX)
PARAMETERS FCT AND 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 HPCG. EXCEPT PRMT(5)
THE COMPONENTS ARE NOT DESTROYED BY SUBROUTINE
HPCG AND THEY ARE
PRMT(1)- LOWER BOUND OF THE INTERVAL (INPUT),
PRMT(2)- UPPER BOUND OF THE INTERVAL (INPUT),
PRMT(3)- INITIAL INCREMENT OF THE INDEPENDENT VARIABLE
(INPUT),
PRMT(4)- UPPER ERROR BOUND (INPUT). IF ABSOLUTE ERROR IS
GREATER THAN PRMT(4), INCREMENT GETS HALVED.
IF INCREMENT IS LESS THAN PRMT(3) AND ABSOLUTE
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 HPCG INITIALIZES
PRMT(5)=0. IF THE USER WANTS TO TERMINATE
SUBROUTINE HPCG 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 HPCG DOES NOT REQUIRE
AND CHANGE THEM. NEVERTHELESS THEY MAY BE USEFUL
FOR HANDING RESULT VALUES TO THE MAIN PROGRAM
(CALLING HPCG) WHICH ARE OBTAINED BY SPECIAL
MANIPULATIONS WITH OUTPUT DATA IN SUBROUTINE OUTP.
Y - INPUT VECTOR OF INITIAL VALUES. (DESTROYED)
LATERON Y IS THE RESULTING VECTOR OF DEPENDENT
VARIABLES COMPUTED AT INTERMEDIATE POINTS X.
DERY - INPUT VECTOR OF ERROR WEIGHTS. (DESTROYED)
THE SUM OF ITS COMPONENTS MUST BE EQUAL TO 1.
LATERON DERY IS THE VECTOR OF DERIVATIVES, WHICH
BELONG TO FUNCTION VALUES Y AT A POINT X.
NDIM - AN INPUT VALUE, WHICH SPECIFIES THE NUMBER OF
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 HPCG 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.
FCT - THE NAME OF AN EXTERNAL SUBROUTINE USED. IT
COMPUTES THE RIGHT HAND SIDES DERY OF THE SYSTEM
TO GIVEN VALUES OF X AND Y. ITS PARAMETER LIST
MUST BE X,Y,DERY. THE SUBROUTINE SHOULD NOT
DESTROY X AND Y.
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 HPCG IS TERMINATED.
AUX - AN AUXILIARY STORAGE ARRAY WITH 16 ROWS AND NDIM
COLUMNS.
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 HAS WRONG SIGN
(ERROR MESSAGES IHLF=12 OR IHLF=13),
(3) THE WHOLE INTEGRATION INTERVAL IS WORKED THROUGH,
(4) SUBROUTINE OUTP HAS CHANGED PRMT(5) TO NON-ZERO.
SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
THE EXTERNAL SUBROUTINES FCT(X,Y,DERY) AND
OUTP(X,Y,DERY,IHLF,NDIM,PRMT) MUST BE FURNISHED BY THE USER.
METHOD
EVALUATION IS DONE BY MEANS OF HAMMINGS MODIFIED PREDICTOR-
CORRECTOR METHOD. IT IS A FOURTH ORDER METHOD, USING 4
PRECEEDING POINTS FOR COMPUTATION OF A NEW VECTOR Y OF THE
DEPENDENT VARIABLES.
FOURTH ORDER RUNGE-KUTTA METHOD SUGGESTED BY RALSTON IS
USED FOR ADJUSTMENT OF THE INITIAL INCREMENT AND FOR
COMPUTATION OF STARTING VALUES.
SUBROUTINE HPCG AUTOMATICALLY ADJUSTS THE INCREMENT DURING
THE WHOLE COMPUTATION BY HALVING OR DOUBLING.
TO GET FULL FLEXIBILITY IN OUTPUT, AN OUTPUT SUBROUTINE
MUST BE CODED BY THE USER.
FOR REFERENCE, SEE
(1) RALSTON/WILF, MATHEMATICAL METHODS FOR DIGITAL
COMPUTERS, WILEY, NEW YORK/LONDON, 1960, PP.95-109.
(2) RALSTON, RUNGE-KUTTA METHODS WITH MINIMUM ERROR BOUNDS,
MTAC, VOL.16, ISS.80 (1962), PP.431-437.