*DECK STOD SUBROUTINE STOD (NEQ, Y, YH, NYH, YH1, EWT, SAVF, ACOR, WM, IWM, + F, JAC, RPAR, IPAR) C***BEGIN PROLOGUE STOD C***SUBSIDIARY C***PURPOSE Subsidiary to DEBDF C***LIBRARY SLATEC C***TYPE SINGLE PRECISION (STOD-S, DSTOD-D) C***AUTHOR Watts, H. A., (SNLA) C***DESCRIPTION C C STOD integrates a system of first order odes over one step in the C integrator package DEBDF. C ---------------------------------------------------------------------- C STOD performs one step of the integration of an initial value C problem for a system of ordinary differential equations. C Note.. STOD is independent of the value of the iteration method C indicator MITER, when this is .NE. 0, and hence is independent C of the type of chord method used, or the Jacobian structure. C Communication with STOD is done with the following variables.. C C Y = An array of length .GE. n used as the Y argument in C all calls to F and JAC. C NEQ = Integer array containing problem size in NEQ(1), and C passed as the NEQ argument in all calls to F and JAC. C YH = An NYH by LMAX array containing the dependent variables C and their approximate scaled derivatives, where C LMAX = MAXORD + 1. YH(I,J+1) contains the approximate C J-th derivative of Y(I), scaled by H**J/Factorial(j) C (J = 0,1,...,NQ). On entry for the first step, the first C two columns of YH must be set from the initial values. C NYH = A constant integer .GE. N, the first dimension of YH. C YH1 = A one-dimensional array occupying the same space as YH. C EWT = An array of N elements with which the estimated local C errors in YH are compared. C SAVF = An array of working storage, of length N. C ACOR = A work array of length N, used for the accumulated C corrections. On a successful return, ACOR(I) contains C the estimated one-step local error in Y(I). C WM,IWM = Real and integer work arrays associated with matrix C operations in chord iteration (MITER .NE. 0). C PJAC = Name of routine to evaluate and preprocess Jacobian matrix C if a chord method is being used. C SLVS = Name of routine to solve linear system in chord iteration. C H = The step size to be attempted on the next step. C H is altered by the error control algorithm during the C problem. H can be either positive or negative, but its C sign must remain constant throughout the problem. C HMIN = The minimum absolute value of the step size H to be used. C HMXI = Inverse of the maximum absolute value of H to be used. C HMXI = 0.0 is allowed and corresponds to an infinite HMAX. C HMIN and HMXI may be changed at any time, but will not C take effect until the next change of H is considered. C TN = The independent variable. TN is updated on each step taken. C JSTART = An integer used for input only, with the following C values and meanings.. C 0 Perform the first step. C .GT.0 Take a new step continuing from the last. C -1 Take the next step with a new value of H, MAXORD, C N, METH, MITER, and/or matrix parameters. C -2 Take the next step with a new value of H, C but with other inputs unchanged. C On return, JSTART is set to 1 to facilitate continuation. C KFLAG = a completion code with the following meanings.. C 0 The step was successful. C -1 The requested error could not be achieved. C -2 Corrector convergence could not be achieved. C A return with KFLAG = -1 or -2 means either C ABS(H) = HMIN or 10 consecutive failures occurred. C On a return with KFLAG negative, the values of TN and C the YH array are as of the beginning of the last C step, and H is the last step size attempted. C MAXORD = The maximum order of integration method to be allowed. C METH/MITER = The method flags. See description in driver. C N = The number of first-order differential equations. C ---------------------------------------------------------------------- C C***SEE ALSO DEBDF C***ROUTINES CALLED CFOD, PJAC, SLVS, VNWRMS C***COMMON BLOCKS DEBDF1 C***REVISION HISTORY (YYMMDD) C 800901 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 891214 Prologue converted to Version 4.0 format. (BAB) C 900328 Added TYPE section. (WRB) C 910722 Updated AUTHOR section. (ALS) C 920422 Changed DIMENSION statement. (WRB) C***END PROLOGUE STOD EXTERNAL F, JAC C CLLL. OPTIMIZE INTEGER NEQ, NYH, IWM, I, I1, IALTH, IER, IOWND, IREDO, IRET, 1 IPUP, J, JB, JSTART, KFLAG, L, LMAX, M, MAXORD, MEO, METH, 2 MITER, N, NCF, NEWQ, NFE, NJE, NQ, NQNYH, NQU, NST, NSTEPJ REAL Y, YH, YH1, EWT, SAVF, ACOR, WM, 1 ROWND, CONIT, CRATE, EL, ELCO, HOLD, RC, RMAX, TESCO, 2 EL0, H, HMIN, HMXI, HU, TN, UROUND, 3 DCON, DDN, DEL, DELP, DSM, DUP, EXDN, EXSM, EXUP, 4 R, RH, RHDN, RHSM, RHUP, TOLD, VNWRMS DIMENSION Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*), 1 ACOR(*), WM(*), IWM(*), RPAR(*), IPAR(*) COMMON /DEBDF1/ ROWND, CONIT, CRATE, EL(13), ELCO(13,12), 1 HOLD, RC, RMAX, TESCO(3,12), 2 EL0, H, HMIN, HMXI, HU, TN, UROUND, IOWND(7), KSTEPS, IOD(6), 3 IALTH, IPUP, LMAX, MEO, NQNYH, NSTEPJ, 4 IER, JSTART, KFLAG, L, METH, MITER, MAXORD, N, NQ, NST, NFE, 5 NJE, NQU C C C***FIRST EXECUTABLE STATEMENT STOD KFLAG = 0 TOLD = TN NCF = 0 IF (JSTART .GT. 0) GO TO 200 IF (JSTART .EQ. -1) GO TO 100 IF (JSTART .EQ. -2) GO TO 160 C----------------------------------------------------------------------- C ON THE FIRST CALL, THE ORDER IS SET TO 1, AND OTHER VARIABLES ARE C INITIALIZED. RMAX IS THE MAXIMUM RATIO BY WHICH H CAN BE INCREASED C IN A SINGLE STEP. IT IS INITIALLY 1.E4 TO COMPENSATE FOR THE SMALL C INITIAL H, BUT THEN IS NORMALLY EQUAL TO 10. IF A FAILURE C OCCURS (IN CORRECTOR CONVERGENCE OR ERROR TEST), RMAX IS SET AT 2 C FOR THE NEXT INCREASE. C----------------------------------------------------------------------- LMAX = MAXORD + 1 NQ = 1 L = 2 IALTH = 2 RMAX = 10000.0E0 RC = 0.0E0 EL0 = 1.0E0 CRATE = 0.7E0 DELP = 0.0E0 HOLD = H MEO = METH NSTEPJ = 0 IRET = 3 GO TO 140 C----------------------------------------------------------------------- C THE FOLLOWING BLOCK HANDLES PRELIMINARIES NEEDED WHEN JSTART = -1. C IPUP IS SET TO MITER TO FORCE A MATRIX UPDATE. C IF AN ORDER INCREASE IS ABOUT TO BE CONSIDERED (IALTH = 1), C IALTH IS RESET TO 2 TO POSTPONE CONSIDERATION ONE MORE STEP. C IF THE CALLER HAS CHANGED METH, CFOD IS CALLED TO RESET C THE COEFFICIENTS OF THE METHOD. C IF THE CALLER HAS CHANGED MAXORD TO A VALUE LESS THAN THE CURRENT C ORDER NQ, NQ IS REDUCED TO MAXORD, AND A NEW H CHOSEN ACCORDINGLY. C IF H IS TO BE CHANGED, YH MUST BE RESCALED. C IF H OR METH IS BEING CHANGED, IALTH IS RESET TO L = NQ + 1 C TO PREVENT FURTHER CHANGES IN H FOR THAT MANY STEPS. C----------------------------------------------------------------------- 100 IPUP = MITER LMAX = MAXORD + 1 IF (IALTH .EQ. 1) IALTH = 2 IF (METH .EQ. MEO) GO TO 110 CALL CFOD (METH, ELCO, TESCO) MEO = METH IF (NQ .GT. MAXORD) GO TO 120 IALTH = L IRET = 1 GO TO 150 110 IF (NQ .LE. MAXORD) GO TO 160 120 NQ = MAXORD L = LMAX DO 125 I = 1,L 125 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5E0/(NQ+2) DDN = VNWRMS (N, SAVF, EWT)/TESCO(1,L) EXDN = 1.0E0/L RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0) RH = MIN(RHDN,1.0E0) IREDO = 3 IF (H .EQ. HOLD) GO TO 170 RH = MIN(RH,ABS(H/HOLD)) H = HOLD GO TO 175 C----------------------------------------------------------------------- C CFOD IS CALLED TO GET ALL THE INTEGRATION COEFFICIENTS FOR THE C CURRENT METH. THEN THE EL VECTOR AND RELATED CONSTANTS ARE RESET C WHENEVER THE ORDER NQ IS CHANGED, OR AT THE START OF THE PROBLEM. C----------------------------------------------------------------------- 140 CALL CFOD (METH, ELCO, TESCO) 150 DO 155 I = 1,L 155 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5E0/(NQ+2) GO TO (160, 170, 200), IRET C----------------------------------------------------------------------- C IF H IS BEING CHANGED, THE H RATIO RH IS CHECKED AGAINST C RMAX, HMIN, AND HMXI, AND THE YH ARRAY RESCALED. IALTH IS SET TO C L = NQ + 1 TO PREVENT A CHANGE OF H FOR THAT MANY STEPS, UNLESS C FORCED BY A CONVERGENCE OR ERROR TEST FAILURE. C----------------------------------------------------------------------- 160 IF (H .EQ. HOLD) GO TO 200 RH = H/HOLD H = HOLD IREDO = 3 GO TO 175 170 RH = MAX(RH,HMIN/ABS(H)) 175 RH = MIN(RH,RMAX) RH = RH/MAX(1.0E0,ABS(H)*HMXI*RH) R = 1.0E0 DO 180 J = 2,L R = R*RH DO 180 I = 1,N 180 YH(I,J) = YH(I,J)*R H = H*RH RC = RC*RH IALTH = L IF (IREDO .EQ. 0) GO TO 680 C----------------------------------------------------------------------- C THIS SECTION COMPUTES THE PREDICTED VALUES BY EFFECTIVELY C MULTIPLYING THE YH ARRAY BY THE PASCAL TRIANGLE MATRIX. C RC IS THE RATIO OF NEW TO OLD VALUES OF THE COEFFICIENT H*EL(1). C WHEN RC DIFFERS FROM 1 BY MORE THAN 30 PERCENT, IPUP IS SET TO MITER C TO FORCE PJAC TO BE CALLED, IF A JACOBIAN IS INVOLVED. C IN ANY CASE, PJAC IS CALLED AT LEAST EVERY 20-TH STEP. C----------------------------------------------------------------------- 200 IF (ABS(RC-1.0E0) .GT. 0.3E0) IPUP = MITER IF (NST .GE. NSTEPJ+20) IPUP = MITER TN = TN + H I1 = NQNYH + 1 DO 215 JB = 1,NQ I1 = I1 - NYH DO 210 I = I1,NQNYH 210 YH1(I) = YH1(I) + YH1(I+NYH) 215 CONTINUE KSTEPS = KSTEPS + 1 C----------------------------------------------------------------------- C UP TO 3 CORRECTOR ITERATIONS ARE TAKEN. A CONVERGENCE TEST IS C MADE ON THE R.M.S. NORM OF EACH CORRECTION, WEIGHTED BY THE ERROR C WEIGHT VECTOR EWT. THE SUM OF THE CORRECTIONS IS ACCUMULATED IN THE C VECTOR ACOR(I). THE YH ARRAY IS NOT ALTERED IN THE CORRECTOR LOOP. C----------------------------------------------------------------------- 220 M = 0 DO 230 I = 1,N 230 Y(I) = YH(I,1) CALL F (TN, Y, SAVF, RPAR, IPAR) NFE = NFE + 1 IF (IPUP .LE. 0) GO TO 250 C----------------------------------------------------------------------- C IF INDICATED, THE MATRIX P = I - H*EL(1)*J IS REEVALUATED AND C PREPROCESSED BEFORE STARTING THE CORRECTOR ITERATION. IPUP IS SET C TO 0 AS AN INDICATOR THAT THIS HAS BEEN DONE. C----------------------------------------------------------------------- IPUP = 0 RC = 1.0E0 NSTEPJ = NST CRATE = 0.7E0 CALL PJAC (NEQ, Y, YH, NYH, EWT, ACOR, SAVF, WM, IWM, F, JAC, 1 RPAR, IPAR) IF (IER .NE. 0) GO TO 430 250 DO 260 I = 1,N 260 ACOR(I) = 0.0E0 270 IF (MITER .NE. 0) GO TO 350 C----------------------------------------------------------------------- C IN THE CASE OF FUNCTIONAL ITERATION, UPDATE Y DIRECTLY FROM C THE RESULT OF THE LAST FUNCTION EVALUATION. C----------------------------------------------------------------------- DO 290 I = 1,N SAVF(I) = H*SAVF(I) - YH(I,2) 290 Y(I) = SAVF(I) - ACOR(I) DEL = VNWRMS (N, Y, EWT) DO 300 I = 1,N Y(I) = YH(I,1) + EL(1)*SAVF(I) 300 ACOR(I) = SAVF(I) GO TO 400 C----------------------------------------------------------------------- C IN THE CASE OF THE CHORD METHOD, COMPUTE THE CORRECTOR ERROR, C AND SOLVE THE LINEAR SYSTEM WITH THAT AS RIGHT-HAND SIDE AND C P AS COEFFICIENT MATRIX. C----------------------------------------------------------------------- 350 DO 360 I = 1,N 360 Y(I) = H*SAVF(I) - (YH(I,2) + ACOR(I)) CALL SLVS (WM, IWM, Y, SAVF) IF (IER .NE. 0) GO TO 410 DEL = VNWRMS (N, Y, EWT) DO 380 I = 1,N ACOR(I) = ACOR(I) + Y(I) 380 Y(I) = YH(I,1) + EL(1)*ACOR(I) C----------------------------------------------------------------------- C TEST FOR CONVERGENCE. IF M.GT.0, AN ESTIMATE OF THE CONVERGENCE C RATE CONSTANT IS STORED IN CRATE, AND THIS IS USED IN THE TEST. C----------------------------------------------------------------------- 400 IF (M .NE. 0) CRATE = MAX(0.2E0*CRATE,DEL/DELP) DCON = DEL*MIN(1.0E0,1.5E0*CRATE)/(TESCO(2,NQ)*CONIT) IF (DCON .LE. 1.0E0) GO TO 450 M = M + 1 IF (M .EQ. 3) GO TO 410 IF (M .GE. 2 .AND. DEL .GT. 2.0E0*DELP) GO TO 410 DELP = DEL CALL F (TN, Y, SAVF, RPAR, IPAR) NFE = NFE + 1 GO TO 270 C----------------------------------------------------------------------- C THE CORRECTOR ITERATION FAILED TO CONVERGE IN 3 TRIES. C IF MITER .NE. 0 AND THE JACOBIAN IS OUT OF DATE, PJAC IS CALLED FOR C THE NEXT TRY. OTHERWISE THE YH ARRAY IS RETRACTED TO ITS VALUES C BEFORE PREDICTION, AND H IS REDUCED, IF POSSIBLE. IF H CANNOT BE C REDUCED OR 10 FAILURES HAVE OCCURRED, EXIT WITH KFLAG = -2. C----------------------------------------------------------------------- 410 IF (IPUP .EQ. 0) GO TO 430 IPUP = MITER GO TO 220 430 TN = TOLD NCF = NCF + 1 RMAX = 2.0E0 I1 = NQNYH + 1 DO 445 JB = 1,NQ I1 = I1 - NYH DO 440 I = I1,NQNYH 440 YH1(I) = YH1(I) - YH1(I+NYH) 445 CONTINUE IF (ABS(H) .LE. HMIN*1.00001E0) GO TO 670 IF (NCF .EQ. 10) GO TO 670 RH = 0.25E0 IPUP = MITER IREDO = 1 GO TO 170 C----------------------------------------------------------------------- C THE CORRECTOR HAS CONVERGED. IPUP IS SET TO -1 IF MITER .NE. 0, C TO SIGNAL THAT THE JACOBIAN INVOLVED MAY NEED UPDATING LATER. C THE LOCAL ERROR TEST IS MADE AND CONTROL PASSES TO STATEMENT 500 C IF IT FAILS. C----------------------------------------------------------------------- 450 IF (MITER .NE. 0) IPUP = -1 IF (M .EQ. 0) DSM = DEL/TESCO(2,NQ) IF (M .GT. 0) DSM = VNWRMS (N, ACOR, EWT)/TESCO(2,NQ) IF (DSM .GT. 1.0E0) GO TO 500 C----------------------------------------------------------------------- C AFTER A SUCCESSFUL STEP, UPDATE THE YH ARRAY. C CONSIDER CHANGING H IF IALTH = 1. OTHERWISE DECREASE IALTH BY 1. C IF IALTH IS THEN 1 AND NQ .LT. MAXORD, THEN ACOR IS SAVED FOR C USE IN A POSSIBLE ORDER INCREASE ON THE NEXT STEP. C IF A CHANGE IN H IS CONSIDERED, AN INCREASE OR DECREASE IN ORDER C BY ONE IS CONSIDERED ALSO. A CHANGE IN H IS MADE ONLY IF IT IS BY A C FACTOR OF AT LEAST 1.1. IF NOT, IALTH IS SET TO 3 TO PREVENT C TESTING FOR THAT MANY STEPS. C----------------------------------------------------------------------- KFLAG = 0 IREDO = 0 NST = NST + 1 HU = H NQU = NQ DO 470 J = 1,L DO 470 I = 1,N 470 YH(I,J) = YH(I,J) + EL(J)*ACOR(I) IALTH = IALTH - 1 IF (IALTH .EQ. 0) GO TO 520 IF (IALTH .GT. 1) GO TO 690 IF (L .EQ. LMAX) GO TO 690 DO 490 I = 1,N 490 YH(I,LMAX) = ACOR(I) GO TO 690 C----------------------------------------------------------------------- C THE ERROR TEST FAILED. KFLAG KEEPS TRACK OF MULTIPLE FAILURES. C RESTORE TN AND THE YH ARRAY TO THEIR PREVIOUS VALUES, AND PREPARE C TO TRY THE STEP AGAIN. COMPUTE THE OPTIMUM STEP SIZE FOR THIS OR C ONE LOWER ORDER. AFTER 2 OR MORE FAILURES, H IS FORCED TO DECREASE C BY A FACTOR OF 0.2 OR LESS. C----------------------------------------------------------------------- 500 KFLAG = KFLAG - 1 TN = TOLD I1 = NQNYH + 1 DO 515 JB = 1,NQ I1 = I1 - NYH DO 510 I = I1,NQNYH 510 YH1(I) = YH1(I) - YH1(I+NYH) 515 CONTINUE RMAX = 2.0E0 IF (ABS(H) .LE. HMIN*1.00001E0) GO TO 660 IF (KFLAG .LE. -3) GO TO 640 IREDO = 2 RHUP = 0.0E0 GO TO 540 C----------------------------------------------------------------------- C REGARDLESS OF THE SUCCESS OR FAILURE OF THE STEP, FACTORS C RHDN, RHSM, AND RHUP ARE COMPUTED, BY WHICH H COULD BE MULTIPLIED C AT ORDER NQ - 1, ORDER NQ, OR ORDER NQ + 1, RESPECTIVELY. C IN THE CASE OF FAILURE, RHUP = 0.0 TO AVOID AN ORDER INCREASE. C THE LARGEST OF THESE IS DETERMINED AND THE NEW ORDER CHOSEN C ACCORDINGLY. IF THE ORDER IS TO BE INCREASED, WE COMPUTE ONE C ADDITIONAL SCALED DERIVATIVE. C----------------------------------------------------------------------- 520 RHUP = 0.0E0 IF (L .EQ. LMAX) GO TO 540 DO 530 I = 1,N 530 SAVF(I) = ACOR(I) - YH(I,LMAX) DUP = VNWRMS (N, SAVF, EWT)/TESCO(3,NQ) EXUP = 1.0E0/(L+1) RHUP = 1.0E0/(1.4E0*DUP**EXUP + 0.0000014E0) 540 EXSM = 1.0E0/L RHSM = 1.0E0/(1.2E0*DSM**EXSM + 0.0000012E0) RHDN = 0.0E0 IF (NQ .EQ. 1) GO TO 560 DDN = VNWRMS (N, YH(1,L), EWT)/TESCO(1,NQ) EXDN = 1.0E0/NQ RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0) 560 IF (RHSM .GE. RHUP) GO TO 570 IF (RHUP .GT. RHDN) GO TO 590 GO TO 580 570 IF (RHSM .LT. RHDN) GO TO 580 NEWQ = NQ RH = RHSM GO TO 620 580 NEWQ = NQ - 1 RH = RHDN IF (KFLAG .LT. 0 .AND. RH .GT. 1.0E0) RH = 1.0E0 GO TO 620 590 NEWQ = L RH = RHUP IF (RH .LT. 1.1E0) GO TO 610 R = EL(L)/L DO 600 I = 1,N 600 YH(I,NEWQ+1) = ACOR(I)*R GO TO 630 610 IALTH = 3 GO TO 690 620 IF ((KFLAG .EQ. 0) .AND. (RH .LT. 1.1E0)) GO TO 610 IF (KFLAG .LE. -2) RH = MIN(RH,0.2E0) C----------------------------------------------------------------------- C IF THERE IS A CHANGE OF ORDER, RESET NQ, L, AND THE COEFFICIENTS. C IN ANY CASE H IS RESET ACCORDING TO RH AND THE YH ARRAY IS RESCALED. C THEN EXIT FROM 680 IF THE STEP WAS OK, OR REDO THE STEP OTHERWISE. C----------------------------------------------------------------------- IF (NEWQ .EQ. NQ) GO TO 170 630 NQ = NEWQ L = NQ + 1 IRET = 2 GO TO 150 C----------------------------------------------------------------------- C CONTROL REACHES THIS SECTION IF 3 OR MORE FAILURES HAVE OCCURRED. C IF 10 FAILURES HAVE OCCURRED, EXIT WITH KFLAG = -1. C IT IS ASSUMED THAT THE DERIVATIVES THAT HAVE ACCUMULATED IN THE C YH ARRAY HAVE ERRORS OF THE WRONG ORDER. HENCE THE FIRST C DERIVATIVE IS RECOMPUTED, AND THE ORDER IS SET TO 1. THEN C H IS REDUCED BY A FACTOR OF 10, AND THE STEP IS RETRIED, C UNTIL IT SUCCEEDS OR H REACHES HMIN. C----------------------------------------------------------------------- 640 IF (KFLAG .EQ. -10) GO TO 660 RH = 0.1E0 RH = MAX(HMIN/ABS(H),RH) H = H*RH DO 645 I = 1,N 645 Y(I) = YH(I,1) CALL F (TN, Y, SAVF, RPAR, IPAR) NFE = NFE + 1 DO 650 I = 1,N 650 YH(I,2) = H*SAVF(I) IPUP = MITER IALTH = 5 IF (NQ .EQ. 1) GO TO 200 NQ = 1 L = 2 IRET = 3 GO TO 150 C----------------------------------------------------------------------- C ALL RETURNS ARE MADE THROUGH THIS SECTION. H IS SAVED IN HOLD C TO ALLOW THE CALLER TO CHANGE H ON THE NEXT STEP. C----------------------------------------------------------------------- 660 KFLAG = -1 GO TO 700 670 KFLAG = -2 GO TO 700 680 RMAX = 10.0E0 690 R = 1.0E0/TESCO(2,NQU) DO 695 I = 1,N 695 ACOR(I) = ACOR(I)*R 700 HOLD = H JSTART = 1 RETURN C----------------------- END OF SUBROUTINE STOD ----------------------- END