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decus/20-0026/drkgs.doc
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SUBROUTINE DRKGS
PURPOSE
TO SOLVE A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL
EQUATIONS WITH GIVEN INITIAL VALUES.
USAGE
CALL DRKGS (PRMT,Y,DERY,NDIM,IHLF,FCT,OUTP,AUX)
PARAMETERS FCT AND 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 DRKGS. EXCEPT PRMT(5) THE COMPONENTS
ARE NOT DESTROYED BY SUBROUTINE DRKGS 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 DRKGS INITIALIZES
PRMT(5)=0. IF THE USER WANTS TO TERMINATE
SUBROUTINE DRKGS 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 DRKGS DOES NOT REQUIRE
AND CHANGE THEM. NEVERTHELESS THEY MAY BE USEFUL
FOR HANDING RESULT VALUES TO THE MAIN PROGRAM
(CALLING DRKGS) WHICH ARE OBTAINED BY SPECIAL
MANIPULATIONS WITH OUTPUT DATA IN SUBROUTINE OUTP.
Y - DOUBLE PRECISION INPUT VECTOR OF INITIAL VALUES
(DESTROYED). LATERON Y IS THE RESULTING VECTOR OF
DEPENDENT VARIABLES COMPUTED AT INTERMEDIATE
POINTS X.
DERY - DOUBLE PRECISION 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
INTERMEDIATE POINTS 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 DRKGS 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. THIS
SUBROUTINE COMPUTES THE RIGHT HAND SIDES DERY OF
THE SYSTEM TO GIVEN VALUES X AND Y. ITS PARAMETER
LIST MUST BE X,Y,DERY. SUBROUTINE FCT 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 DRKGS IS TERMINATED.
AUX - DOUBLE PRECISION AUXILIARY STORAGE ARRAY WITH 8
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 FOURTH ORDER RUNGE-KUTTA
FORMULAE IN THE MODIFICATION DUE TO GILL. ACCURACY IS
TESTED COMPARING THE RESULTS OF THE PROCEDURE WITH SINGLE
AND DOUBLE INCREMENT.
SUBROUTINE DRKGS AUTOMATICALLY ADJUSTS THE INCREMENT DURING
THE WHOLE COMPUTATION BY HALVING OR DOUBLING. IF MORE THAN
10 BISECTIONS OF THE INCREMENT ARE NECESSARY TO GET
SATISFACTORY ACCURACY, THE SUBROUTINE RETURNS WITH
ERROR MESSAGE IHLF=11 INTO MAIN PROGRAM.
TO GET FULL FLEXIBILITY IN OUTPUT, AN OUTPUT SUBROUTINE
MUST BE FURNISHED BY THE USER.
FOR REFERENCE, SEE
RALSTON/WILF, MATHEMATICAL METHODS FOR DIGITAL COMPUTERS,
WILEY, NEW YORK/LONDON, 1960, PP.110-120.