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dop853.f
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SUBROUTINE DOP853(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 8(5,3)
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 SMALL CORRECTIONS ON MARCH 4, 2002
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 ATOL SHOULD BE STRICTLY POSITIVE (POSSIBLY VERY SMALL)
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(8*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, DOP853 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 >>> CONTD8(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 11*N+8*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)=0.333D0, WORK(4)=6.D0
C
C WORK(5) IS THE "BETA" FOR STABILIZED STEP SIZE CONTROL
C (SEE SECTION IV.2). POSITIVE VALUES OF BETA ( <= 0.04 )
C MAKE THE STEP SIZE CONTROL MORE STABLE.
C NEGATIVE WORK(5) PROVOKE BETA=0.
C DEFAULT 0.0D0.
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 DOP853 OF DORMAND AND PRINCE
C (SECTION II.6).
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 I=1,NRDENS
IWORK(I+20)=I
END DO
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=0.333D0
ELSE
FAC1=WORK(3)
END IF
IF(WORK(4).EQ.0.D0)THEN
FAC2=6.D0
ELSE
FAC2=WORK(4)
END IF
C --------- BETA FOR STEP CONTROL STABILIZATION -----------
IF(WORK(5).EQ.0.D0)THEN
BETA=0.0D0
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 -----
IEK1=21
IEK2=IEK1+N
IEK3=IEK2+N
IEK4=IEK3+N
IEK5=IEK4+N
IEK6=IEK5+N
IEK7=IEK6+N
IEK8=IEK7+N
IEK9=IEK8+N
IEK10=IEK9+N
IEY1=IEK10+N
IECO=IEY1+N
C ------ TOTAL STORAGE REQUIREMENT -----------
ISTORE=IECO+8*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 DP86CO(N,FCN,X,Y,XEND,HMAX,H,RTOL,ATOL,ITOL,IPRINT,
& SOLOUT,IOUT,IDID,NMAX,UROUND,METH,NSTIFF,SAFE,BETA,FAC1,FAC2,
& WORK(IEK1),WORK(IEK2),WORK(IEK3),WORK(IEK4),WORK(IEK5),
& WORK(IEK6),WORK(IEK7),WORK(IEK8),WORK(IEK9),WORK(IEK10),
& WORK(IEY1),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 DP86CO(N,FCN,X,Y,XEND,HMAX,H,RTOL,ATOL,ITOL,IPRINT,
& SOLOUT,IOUT,IDID,NMAX,UROUND,METH,NSTIFF,SAFE,BETA,FAC1,FAC2,
& K1,K2,K3,K4,K5,K6,K7,K8,K9,K10,Y1,CONT,ICOMP,NRD,RPAR,IPAR,
& NFCN,NSTEP,NACCPT,NREJCT)
C ----------------------------------------------------------
C CORE INTEGRATOR FOR DOP853
C PARAMETERS SAME AS IN DOP853 WITH WORKSPACE ADDED
C ----------------------------------------------------------
C DECLARATIONS
C ----------------------------------------------------------
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
parameter (
& c2 = 0.526001519587677318785587544488D-01,
& c3 = 0.789002279381515978178381316732D-01,
& c4 = 0.118350341907227396726757197510D+00,
& c5 = 0.281649658092772603273242802490D+00,
& c6 = 0.333333333333333333333333333333D+00,
& c7 = 0.25D+00,
& c8 = 0.307692307692307692307692307692D+00,
& c9 = 0.651282051282051282051282051282D+00,
& c10 = 0.6D+00,
& c11 = 0.857142857142857142857142857142D+00,
& c14 = 0.1D+00,
& c15 = 0.2D+00,
& c16 = 0.777777777777777777777777777778D+00)
parameter (
& b1 = 5.42937341165687622380535766363D-2,
& b6 = 4.45031289275240888144113950566D0,
& b7 = 1.89151789931450038304281599044D0,
& b8 = -5.8012039600105847814672114227D0,
& b9 = 3.1116436695781989440891606237D-1,
& b10 = -1.52160949662516078556178806805D-1,
& b11 = 2.01365400804030348374776537501D-1,
& b12 = 4.47106157277725905176885569043D-2)
parameter (
& bhh1 = 0.244094488188976377952755905512D+00,
& bhh2 = 0.733846688281611857341361741547D+00,
& bhh3 = 0.220588235294117647058823529412D-01)
parameter (
& er 1 = 0.1312004499419488073250102996D-01,
& er 6 = -0.1225156446376204440720569753D+01,
& er 7 = -0.4957589496572501915214079952D+00,
& er 8 = 0.1664377182454986536961530415D+01,
& er 9 = -0.3503288487499736816886487290D+00,
& er10 = 0.3341791187130174790297318841D+00,
& er11 = 0.8192320648511571246570742613D-01,
& er12 = -0.2235530786388629525884427845D-01)
parameter (
& a21 = 5.26001519587677318785587544488D-2,
& a31 = 1.97250569845378994544595329183D-2,
& a32 = 5.91751709536136983633785987549D-2,
& a41 = 2.95875854768068491816892993775D-2,
& a43 = 8.87627564304205475450678981324D-2,
& a51 = 2.41365134159266685502369798665D-1,
& a53 = -8.84549479328286085344864962717D-1,
& a54 = 9.24834003261792003115737966543D-1,
& a61 = 3.7037037037037037037037037037D-2,
& a64 = 1.70828608729473871279604482173D-1,
& a65 = 1.25467687566822425016691814123D-1,
& a71 = 3.7109375D-2,
& a74 = 1.70252211019544039314978060272D-1,
& a75 = 6.02165389804559606850219397283D-2,
& a76 = -1.7578125D-2)
parameter (
& a81 = 3.70920001185047927108779319836D-2,
& a84 = 1.70383925712239993810214054705D-1,
& a85 = 1.07262030446373284651809199168D-1,
& a86 = -1.53194377486244017527936158236D-2,
& a87 = 8.27378916381402288758473766002D-3,
& a91 = 6.24110958716075717114429577812D-1,
& a94 = -3.36089262944694129406857109825D0,
& a95 = -8.68219346841726006818189891453D-1,
& a96 = 2.75920996994467083049415600797D1,
& a97 = 2.01540675504778934086186788979D1,
& a98 = -4.34898841810699588477366255144D1,
& a101 = 4.77662536438264365890433908527D-1,
& a104 = -2.48811461997166764192642586468D0,
& a105 = -5.90290826836842996371446475743D-1,
& a106 = 2.12300514481811942347288949897D1,
& a107 = 1.52792336328824235832596922938D1,
& a108 = -3.32882109689848629194453265587D1,
& a109 = -2.03312017085086261358222928593D-2)
parameter (
& a111 = -9.3714243008598732571704021658D-1,
& a114 = 5.18637242884406370830023853209D0,
& a115 = 1.09143734899672957818500254654D0,
& a116 = -8.14978701074692612513997267357D0,
& a117 = -1.85200656599969598641566180701D1,
& a118 = 2.27394870993505042818970056734D1,
& a119 = 2.49360555267965238987089396762D0,
& a1110 = -3.0467644718982195003823669022D0,
& a121 = 2.27331014751653820792359768449D0,
& a124 = -1.05344954667372501984066689879D1,
& a125 = -2.00087205822486249909675718444D0,
& a126 = -1.79589318631187989172765950534D1,
& a127 = 2.79488845294199600508499808837D1,
& a128 = -2.85899827713502369474065508674D0,
& a129 = -8.87285693353062954433549289258D0,
& a1210 = 1.23605671757943030647266201528D1,
& a1211 = 6.43392746015763530355970484046D-1)
parameter (
& a141 = 5.61675022830479523392909219681D-2,
& a147 = 2.53500210216624811088794765333D-1,
& a148 = -2.46239037470802489917441475441D-1,
& a149 = -1.24191423263816360469010140626D-1,
& a1410 = 1.5329179827876569731206322685D-1,
& a1411 = 8.20105229563468988491666602057D-3,
& a1412 = 7.56789766054569976138603589584D-3,
& a1413 = -8.298D-3)
parameter (
& a151 = 3.18346481635021405060768473261D-2,
& a156 = 2.83009096723667755288322961402D-2,
& a157 = 5.35419883074385676223797384372D-2,
& a158 = -5.49237485713909884646569340306D-2,
& a1511 = -1.08347328697249322858509316994D-4,
& a1512 = 3.82571090835658412954920192323D-4,
& a1513 = -3.40465008687404560802977114492D-4,
& a1514 = 1.41312443674632500278074618366D-1,
& a161 = -4.28896301583791923408573538692D-1,
& a166 = -4.69762141536116384314449447206D0,
& a167 = 7.68342119606259904184240953878D0,
& a168 = 4.06898981839711007970213554331D0,
& a169 = 3.56727187455281109270669543021D-1,
& a1613 = -1.39902416515901462129418009734D-3,
& a1614 = 2.9475147891527723389556272149D0,
& a1615 = -9.15095847217987001081870187138D0)
parameter (
& d41 = -0.84289382761090128651353491142D+01,
& d46 = 0.56671495351937776962531783590D+00,
& d47 = -0.30689499459498916912797304727D+01,
& d48 = 0.23846676565120698287728149680D+01,
& d49 = 0.21170345824450282767155149946D+01,
& d410 = -0.87139158377797299206789907490D+00,
& d411 = 0.22404374302607882758541771650D+01,
& d412 = 0.63157877876946881815570249290D+00,
& d413 = -0.88990336451333310820698117400D-01,
& d414 = 0.18148505520854727256656404962D+02,
& d415 = -0.91946323924783554000451984436D+01,
& d416 = -0.44360363875948939664310572000D+01)
parameter (
& d51 = 0.10427508642579134603413151009D+02,
& d56 = 0.24228349177525818288430175319D+03,
& d57 = 0.16520045171727028198505394887D+03,
& d58 = -0.37454675472269020279518312152D+03,
& d59 = -0.22113666853125306036270938578D+02,
& d510 = 0.77334326684722638389603898808D+01,
& d511 = -0.30674084731089398182061213626D+02,
& d512 = -0.93321305264302278729567221706D+01,
& d513 = 0.15697238121770843886131091075D+02,
& d514 = -0.31139403219565177677282850411D+02,
& d515 = -0.93529243588444783865713862664D+01,
& d516 = 0.35816841486394083752465898540D+02)
parameter (
& d61 = 0.19985053242002433820987653617D+02,
& d66 = -0.38703730874935176555105901742D+03,
& d67 = -0.18917813819516756882830838328D+03,
& d68 = 0.52780815920542364900561016686D+03,
& d69 = -0.11573902539959630126141871134D+02,
& d610 = 0.68812326946963000169666922661D+01,
& d611 = -0.10006050966910838403183860980D+01,
& d612 = 0.77771377980534432092869265740D+00,
& d613 = -0.27782057523535084065932004339D+01,
& d614 = -0.60196695231264120758267380846D+02,
& d615 = 0.84320405506677161018159903784D+02,
& d616 = 0.11992291136182789328035130030D+02)
parameter (
& d71 = -0.25693933462703749003312586129D+02,
& d76 = -0.15418974869023643374053993627D+03,
& d77 = -0.23152937917604549567536039109D+03,
& d78 = 0.35763911791061412378285349910D+03,
& d79 = 0.93405324183624310003907691704D+02,
& d710 = -0.37458323136451633156875139351D+02,
& d711 = 0.10409964950896230045147246184D+03,
& d712 = 0.29840293426660503123344363579D+02,
& d713 = -0.43533456590011143754432175058D+02,
& d714 = 0.96324553959188282948394950600D+02,
& d715 = -0.39177261675615439165231486172D+02,
& d716 = -0.14972683625798562581422125276D+03)
DOUBLE PRECISION Y(N),Y1(N),K1(N),K2(N),K3(N),K4(N),K5(N),K6(N)
DOUBLE PRECISION K7(N),K8(N),K9(N),K10(N),ATOL(*),RTOL(*)
DIMENSION CONT(8*NRD),ICOMP(NRD),RPAR(*),IPAR(*)
LOGICAL REJECT,LAST
EXTERNAL FCN
COMMON /CONDO8/XOLD,HOUT
C *** *** *** *** *** *** ***
C INITIALISATIONS
C *** *** *** *** *** *** ***
FACOLD=1.D-4
EXPO1=1.d0/8.d0-BETA*0.2D0
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=8
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.GE.1) THEN
IRTRN=1
HOUT=1.D0
CALL SOLOUT(NACCPT+1,XOLD,X,Y,N,CONT,ICOMP,NRD,
& RPAR,IPAR,IRTRN)
IF (IRTRN.LT.0) GOTO 79
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 TWELVE 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)+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)+A53*K3(I)+A54*K4(I))
CALL FCN(N,X+C5*H,Y1,K5,RPAR,IPAR)
DO 26 I=1,N
26 Y1(I)=Y(I)+H*(A61*K1(I)+A64*K4(I)+A65*K5(I))
CALL FCN(N,X+C6*H,Y1,K6,RPAR,IPAR)
DO 27 I=1,N
27 Y1(I)=Y(I)+H*(A71*K1(I)+A74*K4(I)+A75*K5(I)+A76*K6(I))
CALL FCN(N,X+C7*H,Y1,K7,RPAR,IPAR)
DO 28 I=1,N
28 Y1(I)=Y(I)+H*(A81*K1(I)+A84*K4(I)+A85*K5(I)+A86*K6(I)+A87*K7(I))
CALL FCN(N,X+C8*H,Y1,K8,RPAR,IPAR)
DO 29 I=1,N
29 Y1(I)=Y(I)+H*(A91*K1(I)+A94*K4(I)+A95*K5(I)+A96*K6(I)+A97*K7(I)
& +A98*K8(I))
CALL FCN(N,X+C9*H,Y1,K9,RPAR,IPAR)
DO 30 I=1,N
30 Y1(I)=Y(I)+H*(A101*K1(I)+A104*K4(I)+A105*K5(I)+A106*K6(I)
& +A107*K7(I)+A108*K8(I)+A109*K9(I))
CALL FCN(N,X+C10*H,Y1,K10,RPAR,IPAR)
DO 31 I=1,N
31 Y1(I)=Y(I)+H*(A111*K1(I)+A114*K4(I)+A115*K5(I)+A116*K6(I)
& +A117*K7(I)+A118*K8(I)+A119*K9(I)+A1110*K10(I))
CALL FCN(N,X+C11*H,Y1,K2,RPAR,IPAR)
XPH=X+H
DO 32 I=1,N
32 Y1(I)=Y(I)+H*(A121*K1(I)+A124*K4(I)+A125*K5(I)+A126*K6(I)
& +A127*K7(I)+A128*K8(I)+A129*K9(I)+A1210*K10(I)+A1211*K2(I))
CALL FCN(N,XPH,Y1,K3,RPAR,IPAR)
NFCN=NFCN+11
DO 35 I=1,N
K4(I)=B1*K1(I)+B6*K6(I)+B7*K7(I)+B8*K8(I)+B9*K9(I)
& +B10*K10(I)+B11*K2(I)+B12*K3(I)
35 K5(I)=Y(I)+H*K4(I)
C --- ERROR ESTIMATION
ERR=0.D0
ERR2=0.D0
IF (ITOL.EQ.0) THEN
DO 41 I=1,N
SK=ATOLI+RTOLI*MAX(ABS(Y(I)),ABS(K5(I)))
ERRI=K4(I)-BHH1*K1(I)-BHH2*K9(I)-BHH3*K3(I)
ERR2=ERR2+(ERRI/SK)**2
ERRI=ER1*K1(I)+ER6*K6(I)+ER7*K7(I)+ER8*K8(I)+ER9*K9(I)
& +ER10*K10(I)+ER11*K2(I)+ER12*K3(I)
41 ERR=ERR+(ERRI/SK)**2
ELSE
DO 42 I=1,N
SK=ATOL(I)+RTOL(I)*MAX(ABS(Y(I)),ABS(K5(I)))
ERRI=K4(I)-BHH1*K1(I)-BHH2*K9(I)-BHH3*K3(I)
ERR2=ERR2+(ERRI/SK)**2
ERRI=ER1*K1(I)+ER6*K6(I)+ER7*K7(I)+ER8*K8(I)+ER9*K9(I)
& +ER10*K10(I)+ER11*K2(I)+ER12*K3(I)
42 ERR=ERR+(ERRI/SK)**2
END IF
DENO=ERR+0.01D0*ERR2
IF (DENO.LE.0.D0) DENO=1.D0
ERR=ABS(H)*ERR*SQRT(1.D0/(N*DENO))
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
CALL FCN(N,XPH,K5,K4,RPAR,IPAR)
NFCN=NFCN+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+(K4(I)-K3(I))**2
STDEN=STDEN+(K5(I)-Y1(I))**2
64 CONTINUE
IF (STDEN.GT.0.D0) HLAMB=ABS(H)*SQRT(STNUM/STDEN)
IF (HLAMB.GT.6.1D0) 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
C ------- FINAL PREPARATION FOR DENSE OUTPUT
IF (IOUT.GE.2) THEN
C ---- SAVE THE FIRST FUNCTION EVALUATIONS
DO 62 J=1,NRD
I=ICOMP(J)
CONT(J)=Y(I)
YDIFF=K5(I)-Y(I)
CONT(J+NRD)=YDIFF
BSPL=H*K1(I)-YDIFF
CONT(J+NRD*2)=BSPL
CONT(J+NRD*3)=YDIFF-H*K4(I)-BSPL
CONT(J+NRD*4)=D41*K1(I)+D46*K6(I)+D47*K7(I)+D48*K8(I)
& +D49*K9(I)+D410*K10(I)+D411*K2(I)+D412*K3(I)
CONT(J+NRD*5)=D51*K1(I)+D56*K6(I)+D57*K7(I)+D58*K8(I)
& +D59*K9(I)+D510*K10(I)+D511*K2(I)+D512*K3(I)
CONT(J+NRD*6)=D61*K1(I)+D66*K6(I)+D67*K7(I)+D68*K8(I)
& +D69*K9(I)+D610*K10(I)+D611*K2(I)+D612*K3(I)
CONT(J+NRD*7)=D71*K1(I)+D76*K6(I)+D77*K7(I)+D78*K8(I)
& +D79*K9(I)+D710*K10(I)+D711*K2(I)+D712*K3(I)
62 CONTINUE
C --- THE NEXT THREE FUNCTION EVALUATIONS
DO 51 I=1,N
51 Y1(I)=Y(I)+H*(A141*K1(I)+A147*K7(I)+A148*K8(I)
& +A149*K9(I)+A1410*K10(I)+A1411*K2(I)+A1412*K3(I)
& +A1413*K4(I))
CALL FCN(N,X+C14*H,Y1,K10,RPAR,IPAR)
DO 52 I=1,N
52 Y1(I)=Y(I)+H*(A151*K1(I)+A156*K6(I)+A157*K7(I)
& +A158*K8(I)+A1511*K2(I)+A1512*K3(I)+A1513*K4(I)
& +A1514*K10(I))
CALL FCN(N,X+C15*H,Y1,K2,RPAR,IPAR)
DO 53 I=1,N
53 Y1(I)=Y(I)+H*(A161*K1(I)+A166*K6(I)+A167*K7(I)
& +A168*K8(I)+A169*K9(I)+A1613*K4(I)+A1614*K10(I)
& +A1615*K2(I))
CALL FCN(N,X+C16*H,Y1,K3,RPAR,IPAR)
NFCN=NFCN+3
C --- FINAL PREPARATION
DO 63 J=1,NRD
I=ICOMP(J)
CONT(J+NRD*4)=H*(CONT(J+NRD*4)+D413*K4(I)+D414*K10(I)
& +D415*K2(I)+D416*K3(I))
CONT(J+NRD*5)=H*(CONT(J+NRD*5)+D513*K4(I)+D514*K10(I)
& +D515*K2(I)+D516*K3(I))
CONT(J+NRD*6)=H*(CONT(J+NRD*6)+D613*K4(I)+D614*K10(I)
& +D615*K2(I)+D616*K3(I))
CONT(J+NRD*7)=H*(CONT(J+NRD*7)+D713*K4(I)+D714*K10(I)
& +D715*K2(I)+D716*K3(I))
63 CONTINUE
HOUT=H
END IF
DO 67 I=1,N
K1(I)=K4(I)
67 Y(I)=K5(I)
XOLD=X
X=XPH
IF (IOUT.GE.1) THEN
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 TOO 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 DOP853 AT X=',E18.4)
IDID=2
RETURN
END
C
C
FUNCTION CONTD8(II,X,CON,ICOMP,ND)
C ----------------------------------------------------------
C THIS FUNCTION CAN BE USED FOR CONINUOUS OUTPUT IN CONNECTION
C WITH THE OUTPUT-SUBROUTINE FOR DOP853. 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(8*ND),ICOMP(ND)
COMMON /CONDO8/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
RETURN
END IF
S=(X-XOLD)/H
S1=1.D0-S
CONPAR=CON(I+ND*4)+S*(CON(I+ND*5)+S1*(CON(I+ND*6)+S*CON(I+ND*7)))
CONTD8=CON(I)+S*(CON(I+ND)+S1*(CON(I+ND*2)+S*(CON(I+ND*3)
& +S1*CONPAR)))
RETURN
END