1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694
|
SUBROUTINE DOPRI5(N,FCN,X,Y,XEND,
& RTOL,ATOL,ITOL,
& SOLOUT,IOUT,
& WORK,LWORK,IWORK,LIWORK,RPAR,IPAR,IDID)
C ----------------------------------------------------------
C NUMERICAL SOLUTION OF A SYSTEM OF FIRST 0RDER
C ORDINARY DIFFERENTIAL EQUATIONS Y'=F(X,Y).
C THIS IS AN EXPLICIT RUNGE-KUTTA METHOD OF ORDER (4)5
C DUE TO DORMAND & PRINCE (WITH STEPSIZE CONTROL AND
C DENSE OUTPUT).
C
C AUTHORS: E. HAIRER AND G. WANNER
C UNIVERSITE DE GENEVE, DEPT. DE MATHEMATIQUES
C CH-1211 GENEVE 24, SWITZERLAND
C E-MAIL: Ernst.Hairer@math.unige.ch
C Gerhard.Wanner@math.unige.ch
C
C THIS CODE IS DESCRIBED IN:
C E. HAIRER, S.P. NORSETT AND G. WANNER, SOLVING ORDINARY
C DIFFERENTIAL EQUATIONS I. NONSTIFF PROBLEMS. 2ND EDITION.
C SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
C SPRINGER-VERLAG (1993)
C
C VERSION OF APRIL 25, 1996
C (latest correction of a small bug: August 8, 2005)
C
C INPUT PARAMETERS
C ----------------
C N DIMENSION OF THE SYSTEM
C
C FCN NAME (EXTERNAL) OF SUBROUTINE COMPUTING THE
C VALUE OF F(X,Y):
C SUBROUTINE FCN(N,X,Y,F,RPAR,IPAR)
C DOUBLE PRECISION X,Y(N),F(N)
C F(1)=... ETC.
C
C X INITIAL X-VALUE
C
C Y(N) INITIAL VALUES FOR Y
C
C XEND FINAL X-VALUE (XEND-X MAY BE POSITIVE OR NEGATIVE)
C
C RTOL,ATOL RELATIVE AND ABSOLUTE ERROR TOLERANCES. THEY
C CAN BE BOTH SCALARS OR ELSE BOTH VECTORS OF LENGTH N.
C
C ITOL SWITCH FOR RTOL AND ATOL:
C ITOL=0: BOTH RTOL AND ATOL ARE SCALARS.
C THE CODE KEEPS, ROUGHLY, THE LOCAL ERROR OF
C Y(I) BELOW RTOL*ABS(Y(I))+ATOL
C ITOL=1: BOTH RTOL AND ATOL ARE VECTORS.
C THE CODE KEEPS THE LOCAL ERROR OF Y(I) BELOW
C RTOL(I)*ABS(Y(I))+ATOL(I).
C
C SOLOUT NAME (EXTERNAL) OF SUBROUTINE PROVIDING THE
C NUMERICAL SOLUTION DURING INTEGRATION.
C IF IOUT.GE.1, IT IS CALLED AFTER EVERY SUCCESSFUL STEP.
C SUPPLY A DUMMY SUBROUTINE IF IOUT=0.
C IT MUST HAVE THE FORM
C SUBROUTINE SOLOUT (NR,XOLD,X,Y,N,CON,ICOMP,ND,
C RPAR,IPAR,IRTRN)
C DIMENSION Y(N),CON(5*ND),ICOMP(ND)
C ....
C SOLOUT FURNISHES THE SOLUTION "Y" AT THE NR-TH
C GRID-POINT "X" (THEREBY THE INITIAL VALUE IS
C THE FIRST GRID-POINT).
C "XOLD" IS THE PRECEEDING GRID-POINT.
C "IRTRN" SERVES TO INTERRUPT THE INTEGRATION. IF IRTRN
C IS SET <0, DOPRI5 WILL RETURN TO THE CALLING PROGRAM.
C IF THE NUMERICAL SOLUTION IS ALTERED IN SOLOUT,
C SET IRTRN = 2
C
C ----- CONTINUOUS OUTPUT: -----
C DURING CALLS TO "SOLOUT", A CONTINUOUS SOLUTION
C FOR THE INTERVAL [XOLD,X] IS AVAILABLE THROUGH
C THE FUNCTION
C >>> CONTD5(I,S,CON,ICOMP,ND) <<<
C WHICH PROVIDES AN APPROXIMATION TO THE I-TH
C COMPONENT OF THE SOLUTION AT THE POINT S. THE VALUE
C S SHOULD LIE IN THE INTERVAL [XOLD,X].
C
C IOUT SWITCH FOR CALLING THE SUBROUTINE SOLOUT:
C IOUT=0: SUBROUTINE IS NEVER CALLED
C IOUT=1: SUBROUTINE IS USED FOR OUTPUT.
C IOUT=2: DENSE OUTPUT IS PERFORMED IN SOLOUT
C (IN THIS CASE WORK(5) MUST BE SPECIFIED)
C
C WORK ARRAY OF WORKING SPACE OF LENGTH "LWORK".
C WORK(1),...,WORK(20) SERVE AS PARAMETERS FOR THE CODE.
C FOR STANDARD USE, SET THEM TO ZERO BEFORE CALLING.
C "LWORK" MUST BE AT LEAST 8*N+5*NRDENS+21
C WHERE NRDENS = IWORK(5)
C
C LWORK DECLARED LENGHT OF ARRAY "WORK".
C
C IWORK INTEGER WORKING SPACE OF LENGHT "LIWORK".
C IWORK(1),...,IWORK(20) SERVE AS PARAMETERS FOR THE CODE.
C FOR STANDARD USE, SET THEM TO ZERO BEFORE CALLING.
C "LIWORK" MUST BE AT LEAST NRDENS+21 .
C
C LIWORK DECLARED LENGHT OF ARRAY "IWORK".
C
C RPAR, IPAR REAL AND INTEGER PARAMETERS (OR PARAMETER ARRAYS) WHICH
C CAN BE USED FOR COMMUNICATION BETWEEN YOUR CALLING
C PROGRAM AND THE FCN, JAC, MAS, SOLOUT SUBROUTINES.
C
C-----------------------------------------------------------------------
C
C SOPHISTICATED SETTING OF PARAMETERS
C -----------------------------------
C SEVERAL PARAMETERS (WORK(1),...,IWORK(1),...) ALLOW
C TO ADAPT THE CODE TO THE PROBLEM AND TO THE NEEDS OF
C THE USER. FOR ZERO INPUT, THE CODE CHOOSES DEFAULT VALUES.
C
C WORK(1) UROUND, THE ROUNDING UNIT, DEFAULT 2.3D-16.
C
C WORK(2) THE SAFETY FACTOR IN STEP SIZE PREDICTION,
C DEFAULT 0.9D0.
C
C WORK(3), WORK(4) PARAMETERS FOR STEP SIZE SELECTION
C THE NEW STEP SIZE IS CHOSEN SUBJECT TO THE RESTRICTION
C WORK(3) <= HNEW/HOLD <= WORK(4)
C DEFAULT VALUES: WORK(3)=0.2D0, WORK(4)=10.D0
C
C WORK(5) IS THE "BETA" FOR STABILIZED STEP SIZE CONTROL
C (SEE SECTION IV.2). LARGER VALUES OF BETA ( <= 0.1 )
C MAKE THE STEP SIZE CONTROL MORE STABLE. DOPRI5 NEEDS
C A LARGER BETA THAN HIGHAM & HALL. NEGATIVE WORK(5)
C PROVOKE BETA=0.
C DEFAULT 0.04D0.
C
C WORK(6) MAXIMAL STEP SIZE, DEFAULT XEND-X.
C
C WORK(7) INITIAL STEP SIZE, FOR WORK(7)=0.D0 AN INITIAL GUESS
C IS COMPUTED WITH HELP OF THE FUNCTION HINIT
C
C IWORK(1) THIS IS THE MAXIMAL NUMBER OF ALLOWED STEPS.
C THE DEFAULT VALUE (FOR IWORK(1)=0) IS 100000.
C
C IWORK(2) SWITCH FOR THE CHOICE OF THE COEFFICIENTS
C IF IWORK(2).EQ.1 METHOD DOPRI5 OF DORMAND AND PRINCE
C (TABLE 5.2 OF SECTION II.5).
C AT THE MOMENT THIS IS THE ONLY POSSIBLE CHOICE.
C THE DEFAULT VALUE (FOR IWORK(2)=0) IS IWORK(2)=1.
C
C IWORK(3) SWITCH FOR PRINTING ERROR MESSAGES
C IF IWORK(3).LT.0 NO MESSAGES ARE BEING PRINTED
C IF IWORK(3).GT.0 MESSAGES ARE PRINTED WITH
C WRITE (IWORK(3),*) ...
C DEFAULT VALUE (FOR IWORK(3)=0) IS IWORK(3)=6
C
C IWORK(4) TEST FOR STIFFNESS IS ACTIVATED AFTER STEP NUMBER
C J*IWORK(4) (J INTEGER), PROVIDED IWORK(4).GT.0.
C FOR NEGATIVE IWORK(4) THE STIFFNESS TEST IS
C NEVER ACTIVATED; DEFAULT VALUE IS IWORK(4)=1000
C
C IWORK(5) = NRDENS = NUMBER OF COMPONENTS, FOR WHICH DENSE OUTPUT
C IS REQUIRED; DEFAULT VALUE IS IWORK(5)=0;
C FOR 0 < NRDENS < N THE COMPONENTS (FOR WHICH DENSE
C OUTPUT IS REQUIRED) HAVE TO BE SPECIFIED IN
C IWORK(21),...,IWORK(NRDENS+20);
C FOR NRDENS=N THIS IS DONE BY THE CODE.
C
C----------------------------------------------------------------------
C
C OUTPUT PARAMETERS
C -----------------
C X X-VALUE FOR WHICH THE SOLUTION HAS BEEN COMPUTED
C (AFTER SUCCESSFUL RETURN X=XEND).
C
C Y(N) NUMERICAL SOLUTION AT X
C
C H PREDICTED STEP SIZE OF THE LAST ACCEPTED STEP
C
C IDID REPORTS ON SUCCESSFULNESS UPON RETURN:
C IDID= 1 COMPUTATION SUCCESSFUL,
C IDID= 2 COMPUT. SUCCESSFUL (INTERRUPTED BY SOLOUT)
C IDID=-1 INPUT IS NOT CONSISTENT,
C IDID=-2 LARGER NMAX IS NEEDED,
C IDID=-3 STEP SIZE BECOMES TOO SMALL.
C IDID=-4 PROBLEM IS PROBABLY STIFF (INTERRUPTED).
C
C IWORK(17) NFCN NUMBER OF FUNCTION EVALUATIONS
C IWORK(18) NSTEP NUMBER OF COMPUTED STEPS
C IWORK(19) NACCPT NUMBER OF ACCEPTED STEPS
C IWORK(20) NREJCT NUMBER OF REJECTED STEPS (DUE TO ERROR TEST),
C (STEP REJECTIONS IN THE FIRST STEP ARE NOT COUNTED)
C-----------------------------------------------------------------------
C *** *** *** *** *** *** *** *** *** *** *** *** ***
C DECLARATIONS
C *** *** *** *** *** *** *** *** *** *** *** *** ***
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION Y(N),ATOL(*),RTOL(*),WORK(LWORK),IWORK(LIWORK)
DIMENSION RPAR(*),IPAR(*)
LOGICAL ARRET
EXTERNAL FCN,SOLOUT
C *** *** *** *** *** *** ***
C SETTING THE PARAMETERS
C *** *** *** *** *** *** ***
NFCN=0
NSTEP=0
NACCPT=0
NREJCT=0
ARRET=.FALSE.
C -------- IPRINT FOR MONITORING THE PRINTING
IF(IWORK(3).EQ.0)THEN
IPRINT=6
ELSE
IPRINT=IWORK(3)
END IF
C -------- NMAX , THE MAXIMAL NUMBER OF STEPS -----
IF(IWORK(1).EQ.0)THEN
NMAX=100000
ELSE
NMAX=IWORK(1)
IF(NMAX.LE.0)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' WRONG INPUT IWORK(1)=',IWORK(1)
ARRET=.TRUE.
END IF
END IF
C -------- METH COEFFICIENTS OF THE METHOD
IF(IWORK(2).EQ.0)THEN
METH=1
ELSE
METH=IWORK(2)
IF(METH.LE.0.OR.METH.GE.4)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' CURIOUS INPUT IWORK(2)=',IWORK(2)
ARRET=.TRUE.
END IF
END IF
C -------- NSTIFF PARAMETER FOR STIFFNESS DETECTION
NSTIFF=IWORK(4)
IF (NSTIFF.EQ.0) NSTIFF=1000
IF (NSTIFF.LT.0) NSTIFF=NMAX+10
C -------- NRDENS NUMBER OF DENSE OUTPUT COMPONENTS
NRDENS=IWORK(5)
IF(NRDENS.LT.0.OR.NRDENS.GT.N)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' CURIOUS INPUT IWORK(5)=',IWORK(5)
ARRET=.TRUE.
ELSE
IF(NRDENS.GT.0.AND.IOUT.LT.2)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' WARNING: PUT IOUT=2 FOR DENSE OUTPUT '
END IF
IF (NRDENS.EQ.N) THEN
DO 16 I=1,NRDENS
16 IWORK(20+I)=I
END IF
END IF
C -------- UROUND SMALLEST NUMBER SATISFYING 1.D0+UROUND>1.D0
IF(WORK(1).EQ.0.D0)THEN
UROUND=2.3D-16
ELSE
UROUND=WORK(1)
IF(UROUND.LE.1.D-35.OR.UROUND.GE.1.D0)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' WHICH MACHINE DO YOU HAVE? YOUR UROUND WAS:',WORK(1)
ARRET=.TRUE.
END IF
END IF
C ------- SAFETY FACTOR -------------
IF(WORK(2).EQ.0.D0)THEN
SAFE=0.9D0
ELSE
SAFE=WORK(2)
IF(SAFE.GE.1.D0.OR.SAFE.LE.1.D-4)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' CURIOUS INPUT FOR SAFETY FACTOR WORK(2)=',WORK(2)
ARRET=.TRUE.
END IF
END IF
C ------- FAC1,FAC2 PARAMETERS FOR STEP SIZE SELECTION
IF(WORK(3).EQ.0.D0)THEN
FAC1=0.2D0
ELSE
FAC1=WORK(3)
END IF
IF(WORK(4).EQ.0.D0)THEN
FAC2=10.D0
ELSE
FAC2=WORK(4)
END IF
C --------- BETA FOR STEP CONTROL STABILIZATION -----------
IF(WORK(5).EQ.0.D0)THEN
BETA=0.04D0
ELSE
IF(WORK(5).LT.0.D0)THEN
BETA=0.D0
ELSE
BETA=WORK(5)
IF(BETA.GT.0.2D0)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' CURIOUS INPUT FOR BETA: WORK(5)=',WORK(5)
ARRET=.TRUE.
END IF
END IF
END IF
C -------- MAXIMAL STEP SIZE
IF(WORK(6).EQ.0.D0)THEN
HMAX=XEND-X
ELSE
HMAX=WORK(6)
END IF
C -------- INITIAL STEP SIZE
H=WORK(7)
C ------- PREPARE THE ENTRY-POINTS FOR THE ARRAYS IN WORK -----
IEY1=21
IEK1=IEY1+N
IEK2=IEK1+N
IEK3=IEK2+N
IEK4=IEK3+N
IEK5=IEK4+N
IEK6=IEK5+N
IEYS=IEK6+N
IECO=IEYS+N
C ------ TOTAL STORAGE REQUIREMENT -----------
ISTORE=IEYS+5*NRDENS-1
IF(ISTORE.GT.LWORK)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' INSUFFICIENT STORAGE FOR WORK, MIN. LWORK=',ISTORE
ARRET=.TRUE.
END IF
ICOMP=21
ISTORE=ICOMP+NRDENS-1
IF(ISTORE.GT.LIWORK)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' INSUFFICIENT STORAGE FOR IWORK, MIN. LIWORK=',ISTORE
ARRET=.TRUE.
END IF
C ------ WHEN A FAIL HAS OCCURED, WE RETURN WITH IDID=-1
IF (ARRET) THEN
IDID=-1
RETURN
END IF
C -------- CALL TO CORE INTEGRATOR ------------
CALL DOPCOR(N,FCN,X,Y,XEND,HMAX,H,RTOL,ATOL,ITOL,IPRINT,
& SOLOUT,IOUT,IDID,NMAX,UROUND,METH,NSTIFF,SAFE,BETA,FAC1,FAC2,
& WORK(IEY1),WORK(IEK1),WORK(IEK2),WORK(IEK3),WORK(IEK4),
& WORK(IEK5),WORK(IEK6),WORK(IEYS),WORK(IECO),IWORK(ICOMP),
& NRDENS,RPAR,IPAR,NFCN,NSTEP,NACCPT,NREJCT)
WORK(7)=H
IWORK(17)=NFCN
IWORK(18)=NSTEP
IWORK(19)=NACCPT
IWORK(20)=NREJCT
C ----------- RETURN -----------
RETURN
END
C
C
C
C ----- ... AND HERE IS THE CORE INTEGRATOR ----------
C
SUBROUTINE DOPCOR(N,FCN,X,Y,XEND,HMAX,H,RTOL,ATOL,ITOL,IPRINT,
& SOLOUT,IOUT,IDID,NMAX,UROUND,METH,NSTIFF,SAFE,BETA,FAC1,FAC2,
& Y1,K1,K2,K3,K4,K5,K6,YSTI,CONT,ICOMP,NRD,RPAR,IPAR,
& NFCN,NSTEP,NACCPT,NREJCT)
C ----------------------------------------------------------
C CORE INTEGRATOR FOR DOPRI5
C PARAMETERS SAME AS IN DOPRI5 WITH WORKSPACE ADDED
C ----------------------------------------------------------
C DECLARATIONS
C ----------------------------------------------------------
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DOUBLE PRECISION K1(N),K2(N),K3(N),K4(N),K5(N),K6(N)
DIMENSION Y(N),Y1(N),YSTI(N),ATOL(*),RTOL(*),RPAR(*),IPAR(*)
DIMENSION CONT(5*NRD),ICOMP(NRD)
LOGICAL REJECT,LAST
EXTERNAL FCN
COMMON /CONDO5/XOLD,HOUT
C *** *** *** *** *** *** ***
C INITIALISATIONS
C *** *** *** *** *** *** ***
IF (METH.EQ.1) CALL CDOPRI(C2,C3,C4,C5,E1,E3,E4,E5,E6,E7,
& A21,A31,A32,A41,A42,A43,A51,A52,A53,A54,
& A61,A62,A63,A64,A65,A71,A73,A74,A75,A76,
& D1,D3,D4,D5,D6,D7)
FACOLD=1.D-4
EXPO1=0.2D0-BETA*0.75D0
FACC1=1.D0/FAC1
FACC2=1.D0/FAC2
POSNEG=SIGN(1.D0,XEND-X)
C --- INITIAL PREPARATIONS
ATOLI=ATOL(1)
RTOLI=RTOL(1)
LAST=.FALSE.
HLAMB=0.D0
IASTI=0
CALL FCN(N,X,Y,K1,RPAR,IPAR)
HMAX=ABS(HMAX)
IORD=5
IF (H.EQ.0.D0) H=HINIT(N,FCN,X,Y,XEND,POSNEG,K1,K2,K3,IORD,
& HMAX,ATOL,RTOL,ITOL,RPAR,IPAR)
NFCN=NFCN+2
REJECT=.FALSE.
XOLD=X
IF (IOUT.NE.0) THEN
IRTRN=1
HOUT=H
CALL SOLOUT(NACCPT+1,XOLD,X,Y,N,CONT,ICOMP,NRD,
& RPAR,IPAR,IRTRN)
IF (IRTRN.LT.0) GOTO 79
ELSE
IRTRN=0
END IF
C --- BASIC INTEGRATION STEP
1 CONTINUE
IF (NSTEP.GT.NMAX) GOTO 78
IF (0.1D0*ABS(H).LE.ABS(X)*UROUND)GOTO 77
IF ((X+1.01D0*H-XEND)*POSNEG.GT.0.D0) THEN
H=XEND-X
LAST=.TRUE.
END IF
NSTEP=NSTEP+1
C --- THE FIRST 6 STAGES
IF (IRTRN.GE.2) THEN
CALL FCN(N,X,Y,K1,RPAR,IPAR)
END IF
DO 22 I=1,N
22 Y1(I)=Y(I)+H*A21*K1(I)
CALL FCN(N,X+C2*H,Y1,K2,RPAR,IPAR)
DO 23 I=1,N
23 Y1(I)=Y(I)+H*(A31*K1(I)+A32*K2(I))
CALL FCN(N,X+C3*H,Y1,K3,RPAR,IPAR)
DO 24 I=1,N
24 Y1(I)=Y(I)+H*(A41*K1(I)+A42*K2(I)+A43*K3(I))
CALL FCN(N,X+C4*H,Y1,K4,RPAR,IPAR)
DO 25 I=1,N
25 Y1(I)=Y(I)+H*(A51*K1(I)+A52*K2(I)+A53*K3(I)+A54*K4(I))
CALL FCN(N,X+C5*H,Y1,K5,RPAR,IPAR)
DO 26 I=1,N
26 YSTI(I)=Y(I)+H*(A61*K1(I)+A62*K2(I)+A63*K3(I)+A64*K4(I)+A65*K5(I))
XPH=X+H
CALL FCN(N,XPH,YSTI,K6,RPAR,IPAR)
DO 27 I=1,N
27 Y1(I)=Y(I)+H*(A71*K1(I)+A73*K3(I)+A74*K4(I)+A75*K5(I)+A76*K6(I))
CALL FCN(N,XPH,Y1,K2,RPAR,IPAR)
IF (IOUT.GE.2) THEN
DO 40 J=1,NRD
I=ICOMP(J)
CONT(4*NRD+J)=H*(D1*K1(I)+D3*K3(I)+D4*K4(I)+D5*K5(I)
& +D6*K6(I)+D7*K2(I))
40 CONTINUE
END IF
DO 28 I=1,N
28 K4(I)=(E1*K1(I)+E3*K3(I)+E4*K4(I)+E5*K5(I)+E6*K6(I)+E7*K2(I))*H
NFCN=NFCN+6
C --- ERROR ESTIMATION
ERR=0.D0
IF (ITOL.EQ.0) THEN
DO 41 I=1,N
SK=ATOLI+RTOLI*MAX(ABS(Y(I)),ABS(Y1(I)))
41 ERR=ERR+(K4(I)/SK)**2
ELSE
DO 42 I=1,N
SK=ATOL(I)+RTOL(I)*MAX(ABS(Y(I)),ABS(Y1(I)))
42 ERR=ERR+(K4(I)/SK)**2
END IF
ERR=SQRT(ERR/N)
C --- COMPUTATION OF HNEW
FAC11=ERR**EXPO1
C --- LUND-STABILIZATION
FAC=FAC11/FACOLD**BETA
C --- WE REQUIRE FAC1 <= HNEW/H <= FAC2
FAC=MAX(FACC2,MIN(FACC1,FAC/SAFE))
HNEW=H/FAC
IF(ERR.LE.1.D0)THEN
C --- STEP IS ACCEPTED
FACOLD=MAX(ERR,1.0D-4)
NACCPT=NACCPT+1
C ------- STIFFNESS DETECTION
IF (MOD(NACCPT,NSTIFF).EQ.0.OR.IASTI.GT.0) THEN
STNUM=0.D0
STDEN=0.D0
DO 64 I=1,N
STNUM=STNUM+(K2(I)-K6(I))**2
STDEN=STDEN+(Y1(I)-YSTI(I))**2
64 CONTINUE
IF (STDEN.GT.0.D0) HLAMB=H*SQRT(STNUM/STDEN)
IF (HLAMB.GT.3.25D0) THEN
NONSTI=0
IASTI=IASTI+1
IF (IASTI.EQ.15) THEN
IF (IPRINT.GT.0) WRITE (IPRINT,*)
& ' THE PROBLEM SEEMS TO BECOME STIFF AT X = ',X
IF (IPRINT.LE.0) GOTO 76
END IF
ELSE
NONSTI=NONSTI+1
IF (NONSTI.EQ.6) IASTI=0
END IF
END IF
IF (IOUT.GE.2) THEN
DO 43 J=1,NRD
I=ICOMP(J)
YD0=Y(I)
YDIFF=Y1(I)-YD0
BSPL=H*K1(I)-YDIFF
CONT(J)=Y(I)
CONT(NRD+J)=YDIFF
CONT(2*NRD+J)=BSPL
CONT(3*NRD+J)=-H*K2(I)+YDIFF-BSPL
43 CONTINUE
END IF
DO 44 I=1,N
K1(I)=K2(I)
44 Y(I)=Y1(I)
XOLD=X
X=XPH
IF (IOUT.NE.0) THEN
HOUT=H
CALL SOLOUT(NACCPT+1,XOLD,X,Y,N,CONT,ICOMP,NRD,
& RPAR,IPAR,IRTRN)
IF (IRTRN.LT.0) GOTO 79
END IF
C ------- NORMAL EXIT
IF (LAST) THEN
H=HNEW
IDID=1
RETURN
END IF
IF(ABS(HNEW).GT.HMAX)HNEW=POSNEG*HMAX
IF(REJECT)HNEW=POSNEG*MIN(ABS(HNEW),ABS(H))
REJECT=.FALSE.
ELSE
C --- STEP IS REJECTED
HNEW=H/MIN(FACC1,FAC11/SAFE)
REJECT=.TRUE.
IF(NACCPT.GE.1)NREJCT=NREJCT+1
LAST=.FALSE.
END IF
H=HNEW
GOTO 1
C --- FAIL EXIT
76 CONTINUE
IDID=-4
RETURN
77 CONTINUE
IF (IPRINT.GT.0) WRITE(IPRINT,979)X
IF (IPRINT.GT.0) WRITE(IPRINT,*)' STEP SIZE T0O SMALL, H=',H
IDID=-3
RETURN
78 CONTINUE
IF (IPRINT.GT.0) WRITE(IPRINT,979)X
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' MORE THAN NMAX =',NMAX,'STEPS ARE NEEDED'
IDID=-2
RETURN
79 CONTINUE
IF (IPRINT.GT.0) WRITE(IPRINT,979)X
979 FORMAT(' EXIT OF DOPRI5 AT X=',E18.4)
IDID=2
RETURN
END
C
FUNCTION HINIT(N,FCN,X,Y,XEND,POSNEG,F0,F1,Y1,IORD,
& HMAX,ATOL,RTOL,ITOL,RPAR,IPAR)
C ----------------------------------------------------------
C ---- COMPUTATION OF AN INITIAL STEP SIZE GUESS
C ----------------------------------------------------------
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION Y(N),Y1(N),F0(N),F1(N),ATOL(*),RTOL(*)
DIMENSION RPAR(*),IPAR(*)
C ---- COMPUTE A FIRST GUESS FOR EXPLICIT EULER AS
C ---- H = 0.01 * NORM (Y0) / NORM (F0)
C ---- THE INCREMENT FOR EXPLICIT EULER IS SMALL
C ---- COMPARED TO THE SOLUTION
DNF=0.0D0
DNY=0.0D0
ATOLI=ATOL(1)
RTOLI=RTOL(1)
IF (ITOL.EQ.0) THEN
DO 10 I=1,N
SK=ATOLI+RTOLI*ABS(Y(I))
DNF=DNF+(F0(I)/SK)**2
10 DNY=DNY+(Y(I)/SK)**2
ELSE
DO 11 I=1,N
SK=ATOL(I)+RTOL(I)*ABS(Y(I))
DNF=DNF+(F0(I)/SK)**2
11 DNY=DNY+(Y(I)/SK)**2
END IF
IF (DNF.LE.1.D-10.OR.DNY.LE.1.D-10) THEN
H=1.0D-6
ELSE
H=SQRT(DNY/DNF)*0.01D0
END IF
H=MIN(H,HMAX)
H=SIGN(H,POSNEG)
C ---- PERFORM AN EXPLICIT EULER STEP
DO 12 I=1,N
12 Y1(I)=Y(I)+H*F0(I)
CALL FCN(N,X+H,Y1,F1,RPAR,IPAR)
C ---- ESTIMATE THE SECOND DERIVATIVE OF THE SOLUTION
DER2=0.0D0
IF (ITOL.EQ.0) THEN
DO 15 I=1,N
SK=ATOLI+RTOLI*ABS(Y(I))
15 DER2=DER2+((F1(I)-F0(I))/SK)**2
ELSE
DO 16 I=1,N
SK=ATOL(I)+RTOL(I)*ABS(Y(I))
16 DER2=DER2+((F1(I)-F0(I))/SK)**2
END IF
DER2=SQRT(DER2)/H
C ---- STEP SIZE IS COMPUTED SUCH THAT
C ---- H**IORD * MAX ( NORM (F0), NORM (DER2)) = 0.01
DER12=MAX(ABS(DER2),SQRT(DNF))
IF (DER12.LE.1.D-15) THEN
H1=MAX(1.0D-6,ABS(H)*1.0D-3)
ELSE
H1=(0.01D0/DER12)**(1.D0/IORD)
END IF
H=MIN(100*ABS(H),H1,HMAX)
HINIT=SIGN(H,POSNEG)
RETURN
END
C
FUNCTION CONTD5(II,X,CON,ICOMP,ND)
C ----------------------------------------------------------
C THIS FUNCTION CAN BE USED FOR CONTINUOUS OUTPUT IN CONNECTION
C WITH THE OUTPUT-SUBROUTINE FOR DOPRI5. IT PROVIDES AN
C APPROXIMATION TO THE II-TH COMPONENT OF THE SOLUTION AT X.
C ----------------------------------------------------------
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION CON(5*ND),ICOMP(ND)
COMMON /CONDO5/XOLD,H
C ----- COMPUTE PLACE OF II-TH COMPONENT
I=0
DO 5 J=1,ND
IF (ICOMP(J).EQ.II) I=J
5 CONTINUE
IF (I.EQ.0) THEN
WRITE (6,*) ' NO DENSE OUTPUT AVAILABLE FOR COMP.',II
CONTD5=-1
RETURN
END IF
THETA=(X-XOLD)/H
THETA1=1.D0-THETA
CONTD5=CON(I)+THETA*(CON(ND+I)+THETA1*(CON(2*ND+I)+THETA*
& (CON(3*ND+I)+THETA1*CON(4*ND+I))))
RETURN
END
C
SUBROUTINE CDOPRI(C2,C3,C4,C5,E1,E3,E4,E5,E6,E7,
& A21,A31,A32,A41,A42,A43,A51,A52,A53,A54,
& A61,A62,A63,A64,A65,A71,A73,A74,A75,A76,
& D1,D3,D4,D5,D6,D7)
C ----------------------------------------------------------
C RUNGE-KUTTA COEFFICIENTS OF DORMAND AND PRINCE (1980)
C ----------------------------------------------------------
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
C2=0.2D0
C3=0.3D0
C4=0.8D0
C5=8.D0/9.D0
A21=0.2D0
A31=3.D0/40.D0
A32=9.D0/40.D0
A41=44.D0/45.D0
A42=-56.D0/15.D0
A43=32.D0/9.D0
A51=19372.D0/6561.D0
A52=-25360.D0/2187.D0
A53=64448.D0/6561.D0
A54=-212.D0/729.D0
A61=9017.D0/3168.D0
A62=-355.D0/33.D0
A63=46732.D0/5247.D0
A64=49.D0/176.D0
A65=-5103.D0/18656.D0
A71=35.D0/384.D0
A73=500.D0/1113.D0
A74=125.D0/192.D0
A75=-2187.D0/6784.D0
A76=11.D0/84.D0
E1=71.D0/57600.D0
E3=-71.D0/16695.D0
E4=71.D0/1920.D0
E5=-17253.D0/339200.D0
E6=22.D0/525.D0
E7=-1.D0/40.D0
C ---- DENSE OUTPUT OF SHAMPINE (1986)
D1=-12715105075.D0/11282082432.D0
D3=87487479700.D0/32700410799.D0
D4=-10690763975.D0/1880347072.D0
D5=701980252875.D0/199316789632.D0
D6=-1453857185.D0/822651844.D0
D7=69997945.D0/29380423.D0
RETURN
END
|