SUBROUTINE DDRIV2 (N,T,Y,F,TOUT,MSTATE,NROOT,EPS,EWT,MINT,WORK, 8 LENW,IWORK,LENIW,G) C***BEGIN PROLOGUE DDRIV2 C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 871105 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***KEYWORDS ODE,STIFF,ORDINARY DIFFERENTIAL EQUATIONS, C INITIAL VALUE PROBLEMS,GEAR'S METHOD, C DOUBLE PRECISION C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***PURPOSE The function of DDRIV2 is to solve N ordinary differential C equations of the form dY(I)/dT = F(Y(I),T), given the C initial conditions Y(I) = YI. The program has options to C allow the solution of both stiff and non-stiff differential C equations. DDRIV2 uses double precision arithmetic. C***DESCRIPTION C From the book "Numerical Methods and Software" C by D. Kahaner, C. Moler, S. Nash C Prentice Hall 1988 C C I. ABSTRACT ....................................................... C C The function of DDRIV2 is to solve N ordinary differential C equations of the form dY(I)/dT = F(Y(I),T), given the initial C conditions Y(I) = YI. The program has options to allow the C solution of both stiff and non-stiff differential equations. C DDRIV2 is to be called once for each output point of T. C C II. PARAMETERS .................................................... C C (REMEMBER--To run DDRIV2 correctly in double precision, ALL C non-integer arguments in the call sequence, including C arrays, MUST be declared double precision.) C C The user should use parameter names in the call sequence of DDRIV2 C for those quantities whose value may be altered by DDRIV2. The C parameters in the call sequence are: C C N = (Input) The number of differential equations. C C T = The independent variable. On input for the first call, T C is the initial point. On output, T is the point at which C the solution is given. C C Y = The vector of dependent variables. Y is used as input on C the first call, to set the initial values. On output, Y C is the computed solution vector. This array Y is passed C in the call sequence of the user-provided routines F and C G. Thus parameters required by F and G can be stored in C this array in components N+1 and above. (Note: Changes C by the user to the first N components of this array will C take effect only after a restart, i.e., after setting C MSTATE to +1(-1).) C C F = A subroutine supplied by the user. The name must be C declared EXTERNAL in the user's calling program. This C subroutine is of the form: C SUBROUTINE F (N, T, Y, YDOT) C DOUBLE PRECISION Y(*), YDOT(*) C . C . C YDOT(1) = ... C . C . C YDOT(N) = ... C END (Sample) C This computes YDOT = F(Y,T), the right hand side of the C differential equations. Here Y is a vector of length at C least N. The actual length of Y is determined by the C user's declaration in the program which calls DDRIV2. C Thus the dimensioning of Y in F, while required by FORTRAN C convention, does not actually allocate any storage. When C this subroutine is called, the first N components of Y are C intermediate approximations to the solution components. C The user should not alter these values. Here YDOT is a C vector of length N. The user should only compute YDOT(I) C for I from 1 to N. Normally a return from F passes C control back to DDRIV2. However, if the user would like C to abort the calculation, i.e., return control to the C program which calls DDRIV2, he should set N to zero. C DDRIV2 will signal this by returning a value of MSTATE C equal to +6(-6). Altering the value of N in F has no C effect on the value of N in the call sequence of DDRIV2. C C TOUT = (Input) The point at which the solution is desired. C C MSTATE = An integer describing the status of integration. The user C must initialize MSTATE to +1 or -1. If MSTATE is C positive, the routine will integrate past TOUT and C interpolate the solution. This is the most efficient C mode. If MSTATE is negative, the routine will adjust its C internal step to reach TOUT exactly (useful if a C singularity exists beyond TOUT.) The meaning of the C magnitude of MSTATE: C 1 (Input) Means the first call to the routine. This C value must be set by the user. On all subsequent C calls the value of MSTATE should be tested by the C user. Unless DDRIV2 is to be reinitialized, only the C sign of MSTATE may be changed by the user. (As a C convenience to the user who may wish to put out the C initial conditions, DDRIV2 can be called with C MSTATE=+1(-1), and TOUT=T. In this case the program C will return with MSTATE unchanged, i.e., C MSTATE=+1(-1).) C 2 (Output) Means a successful integration. If a normal C continuation is desired (i.e., a further integration C in the same direction), simply advance TOUT and call C again. All other parameters are automatically set. C 3 (Output)(Unsuccessful) Means the integrator has taken C 1000 steps without reaching TOUT. The user can C continue the integration by simply calling DDRIV2 C again. Other than an error in problem setup, the C most likely cause for this condition is trying to C integrate a stiff set of equations with the non-stiff C integrator option. (See description of MINT below.) C 4 (Output)(Unsuccessful) Means too much accuracy has C been requested. EPS has been increased to a value C the program estimates is appropriate. The user can C continue the integration by simply calling DDRIV2 C again. C 5 (Output) A root was found at a point less than TOUT. C The user can continue the integration toward TOUT by C simply calling DDRIV2 again. C 6 (Output)(Unsuccessful) N has been set to zero in C SUBROUTINE F. C 7 (Output)(Unsuccessful) N has been set to zero in C FUNCTION G. See description of G below. C C NROOT = (Input) The number of equations whose roots are desired. C If NROOT is zero, the root search is not active. This C option is useful for obtaining output at points which are C not known in advance, but depend upon the solution, e.g., C when some solution component takes on a specified value. C The root search is carried out using the user-written C function G (see description of G below.) DDRIV2 attempts C to find the value of T at which one of the equations C changes sign. DDRIV2 can find at most one root per C equation per internal integration step, and will then C return the solution either at TOUT or at a root, whichever C occurs first in the direction of integration. The index C of the equation whose root is being reported is stored in C the sixth element of IWORK. C NOTE: NROOT is never altered by this program. C C EPS = On input, the requested relative accuracy in all solution C components. EPS = 0 is allowed. On output, the adjusted C relative accuracy if the input value was too small. The C value of EPS should be set as large as is reasonable, C because the amount of work done by DDRIV2 increases as C EPS decreases. C C EWT = (Input) Problem zero, i.e., the smallest physically C meaningful value for the solution. This is used inter- C nally to compute an array YWT(I) = MAX(ABS(Y(I)), EWT). C One step error estimates divided by YWT(I) are kept less C than EPS. Setting EWT to zero provides pure relative C error control. However, setting EWT smaller than C necessary can adversely affect the running time. C C MINT = (Input) The integration method flag. C MINT = 1 Means the Adams methods, and is used for C non-stiff problems. C MINT = 2 Means the stiff methods of Gear (i.e., the C backward differentiation formulas), and is C used for stiff problems. C MINT = 3 Means the program dynamically selects the C Adams methods when the problem is non-stiff C and the Gear methods when the problem is C stiff. C MINT may not be changed without restarting, i.e., setting C the magnitude of MSTATE to 1. C C WORK C LENW = (Input) C WORK is an array of LENW double precision words used C internally for temporary storage. The user must allocate C space for this array in the calling program by a statement C such as C DOUBLE PRECISION WORK(...) C The length of WORK should be at least C 16*N + 2*NROOT + 204 if MINT is 1, or C N*N + 10*N + 2*NROOT + 204 if MINT is 2, or C N*N + 17*N + 2*NROOT + 204 if MINT is 3, C and LENW should be set to the value used. The contents of C WORK should not be disturbed between calls to DDRIV2. C C IWORK C LENIW = (Input) C IWORK is an integer array of length LENIW used internally C for temporary storage. The user must allocate space for C this array in the calling program by a statement such as C INTEGER IWORK(...) C The length of IWORK should be at least C 21 if MINT is 1, or C N+21 if MINT is 2 or 3, C and LENIW should be set to the value used. The contents C of IWORK should not be disturbed between calls to DDRIV2. C C G = A double precision FORTRAN function supplied by the user C if NROOT is not 0. In this case, the name must be C declared EXTERNAL in the user's calling program. G is C repeatedly called with different values of IROOT to C obtain the value of each of the NROOT equations for which C a root is desired. G is of the form: C DOUBLE PRECISION FUNCTION G (N, T, Y, IROOT) C DOUBLE PRECISION Y(*) C GO TO (10, ...), IROOT C 10 G = ... C . C . C END (Sample) C Here, Y is a vector of length at least N, whose first N C components are the solution components at the point T. C The user should not alter these values. The actual length C of Y is determined by the user's declaration in the C program which calls DDRIV2. Thus the dimensioning of Y in C G, while required by FORTRAN convention, does not actually C allocate any storage. Normally a return from G passes C control back to DDRIV2. However, if the user would like C to abort the calculation, i.e., return control to the C program which calls DDRIV2, he should set N to zero. C DDRIV2 will signal this by returning a value of MSTATE C equal to +7(-7). In this case, the index of the equation C being evaluated is stored in the sixth element of IWORK. C Altering the value of N in G has no effect on the value of C N in the call sequence of DDRIV2. C C***LONG DESCRIPTION C C III. OTHER COMMUNICATION TO THE USER .............................. C C A. The solver communicates to the user through the parameters C above. In addition it writes diagnostic messages through the C standard error handling program XERROR. That program will C terminate the user's run if it detects a probable problem setup C error, e.g., insufficient storage allocated by the user for the C WORK array. Messages are written on the standard error message C file. At installations which have this error handling package C the user should determine the standard error handling file from C the local documentation. Otherwise the short but serviceable C routine, XERROR, available with this package, can be used. That C program writes on logical unit 6 to transmit messages. A C complete description of XERROR is given in the Sandia C Laboratories report SAND78-1189 by R. E. Jones. C C B. The first three elements of WORK and the first five elements of C IWORK will contain the following statistical data: C AVGH The average step size used. C HUSED The step size last used (successfully). C AVGORD The average order used. C IMXERR The index of the element of the solution vector that C contributed most to the last error test. C NQUSED The order last used (successfully). C NSTEP The number of steps taken since last initialization. C NFE The number of evaluations of the right hand side. C NJE The number of evaluations of the Jacobian matrix. C C IV. REMARKS ....................................................... C C A. On any return from DDRIV2 all information necessary to continue C the calculation is contained in the call sequence parameters, C including the work arrays. Thus it is possible to suspend one C problem, integrate another, and then return to the first. C C B. If this package is to be used in an overlay situation, the user C must declare in the primary overlay the variables in the call C sequence to DDRIV2. C C C. When the routine G is not required, difficulties associated with C an unsatisfied external can be avoided by using the name of the C routine which calculates the right hand side of the differential C equations in place of G in the call sequence of DDRIV2. C C V. USAGE .......................................................... C C PROGRAM SAMPLE C EXTERNAL F C PARAMETER(MINT = 1, NROOT = 0, N = ..., C 8 LENW = 16*N + 2*NROOT + 204, LENIW = 21) C N is the number of equations C DOUBLE PRECISION EPS, EWT, T, TOUT, WORK(LENW), Y(N) C INTEGER IWORK(LENIW) C OPEN(FILE='TAPE6', UNIT=6, STATUS='NEW') C T = 0. Initial point C DO 10 I = 1,N C 10 Y(I) = ... Set initial conditions C TOUT = T C EWT = ... C MSTATE = 1 C EPS = ... C 20 CALL DDRIV2 (N, T, Y, F, TOUT, MSTATE, NROOT, EPS, EWT, C 8 MINT, WORK, LENW, IWORK, LENIW, F) C Last argument is not the same C as F if rootfinding is used. C IF (MSTATE .GT. 2) STOP C WRITE(6, 100) TOUT, (Y(I), I=1,N) C TOUT = TOUT + 1. C IF (TOUT .LE. 10.) GO TO 20 C 100 FORMAT(...) C END (Sample) C C***REFERENCES GEAR, C. W., "NUMERICAL INITIAL VALUE PROBLEMS IN C ORDINARY DIFFERENTIAL EQUATIONS", PRENTICE-HALL, 1971. C***ROUTINES CALLED DDRIV3,XERROR C***END PROLOGUE DDRIV2 EXTERNAL F, G DOUBLE PRECISION EPS, EWT, EWTCOM(1), G, HMAX, T, TOUT, 8 WORK(*), Y(*) INTEGER IWORK(*) CHARACTER MSG*81 PARAMETER(IMPL = 0, MXSTEP = 1000) C***FIRST EXECUTABLE STATEMENT DDRIV2 IF (MINT .LT. 1 .OR. MINT .GT. 3) THEN WRITE(MSG, '(''DDRIV21FE Illegal input. Improper value for '', 8 ''the integration method flag,'', I8)') MINT CALL XERROR(MSG(1:81), 81, 21, 2) RETURN END IF IF (MSTATE .GE. 0) THEN NSTATE = MSTATE NTASK = 1 ELSE NSTATE = - MSTATE NTASK = 3 END IF EWTCOM(1) = EWT IF (EWT .NE. 0.D0) THEN IERROR = 3 ELSE IERROR = 2 END IF IF (MINT .EQ. 1) THEN MITER = 0 MXORD = 12 ELSE IF (MINT .EQ. 2) THEN MITER = 2 MXORD = 5 ELSE IF (MINT .EQ. 3) THEN MITER = 2 MXORD = 12 END IF HMAX = 2.D0*ABS(TOUT - T) CALL DDRIV3 (N, T, Y, F, NSTATE, TOUT, NTASK, NROOT, EPS, EWTCOM, 8 IERROR, MINT, MITER, IMPL, ML, MU, MXORD, HMAX, WORK, 8 LENW, IWORK, LENIW, F, F, NDE, MXSTEP, G, F) IF (MSTATE .GE. 0) THEN MSTATE = NSTATE ELSE MSTATE = - NSTATE END IF END SUBROUTINE DDRIV1 (N,T,Y,TOUT,MSTATE,EPS,WORK,LENW) C***BEGIN PROLOGUE DDRIV1 C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 880229 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***KEYWORDS ODE,STIFF,ORDINARY DIFFERENTIAL EQUATIONS, C INITIAL VALUE PROBLEMS,GEAR'S METHOD, C DOUBLE PRECISION C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***PURPOSE The function of DDRIV1 is to solve N (200 or fewer) C ordinary differential equations of the form C dY(I)/dT = F(Y(I),T), given the initial conditions C Y(I) = YI. DDRIV1 uses double precision arithmetic. C***DESCRIPTION C From the book "Numerical Methods and Software" C by D. Kahaner, C. Moler, S. Nash C Prentice Hall 1988 C C Version 88.1 C C I. CHOOSING THE CORRECT ROUTINE ................................... C C SDRIV C DDRIV C CDRIV C These are the generic names for three packages for solving C initial value problems for ordinary differential equations. C SDRIV uses single precision arithmetic. DDRIV uses double C precision arithmetic. CDRIV allows complex-valued C differential equations, integrated with respect to a single, C real precision, independent variable. C C As an aid in selecting the proper program, the following is a C discussion of the important options or restrictions associated with C each program: C C A. DDRIV1 should be tried first for those routine problems with C no more than 200 differential equations. Internally this C routine has two important technical defaults: C 1. Numerical approximation of the Jacobian matrix of the C right hand side is used. C 2. The stiff solver option is used. C Most users of DDRIV1 should not have to concern themselves C with these details. C C B. DDRIV2 should be considered for those problems for which C DDRIV1 is inadequate (SDRIV2 has no explicit restriction on C the number of differential equations.) For example, DDRIV1 C may have difficulty with problems having zero initial C conditions and zero derivatives. In this case DDRIV2, with an C appropriate value of the parameter EWT, should perform more C efficiently. DDRIV2 provides three important additional C options: C 1. The nonstiff equation solver (as well as the stiff C solver) is available. C 2. The root-finding option is available. C 3. The program can dynamically select either the non-stiff C or the stiff methods. C Internally this routine also defaults to the numerical C approximation of the Jacobian matrix of the right hand side. C C C. DDRIV3 is the most flexible, and hence the most complex, of C the programs. Its important additional features include: C 1. The ability to exploit band structure in the Jacobian C matrix. C 2. The ability to solve some implicit differential C equations, i.e., those having the form: C A(Y,T)*dY/dT = F(Y,T). C 3. The option of integrating in the one step mode. C 4. The option of allowing the user to provide a routine C which computes the analytic Jacobian matrix of the right C hand side. C 5. The option of allowing the user to provide a routine C which does all the matrix algebra associated with C corrections to the solution components. C C II. ABSTRACT ...................................................... C C The function of DDRIV1 is to solve N (200 or fewer) ordinary C differential equations of the form dY(I)/dT = F(Y(I),T), given the C initial conditions Y(I) = YI. DDRIV1 is to be called once for each C output point. C C III. PARAMETERS ................................................... C C (REMEMBER--To run DDRIV1 correctly in double precision, ALL C non-integer arguments in the call sequence, including C arrays, MUST be declared double precision.) c C The user should use parameter names in the call sequence of DDRIV1 C for those quantities whose value may be altered by DDRIV1. The C parameters in the call sequence are: C C N = (Input) The number of differential equations, N .LE. 200 C C T = The independent variable. On input for the first call, T C is the initial point. On output, T is the point at which C the solution is given. C C Y = The vector of dependent variables. Y is used as input on C the first call, to set the initial values. On output, Y C is the computed solution vector. This array Y is passed C in the call sequence of the user-provided routine F. Thus C parameters required by F can be stored in this array in C components N+1 and above. (Note: Changes by the user to C the first N components of this array will take effect only C after a restart, i.e., after setting MSTATE to +1(-1).) C C TOUT = (Input) The point at which the solution is desired. C C MSTATE = An integer describing the status of integration. The user C must initialize MSTATE to +1 or -1. If MSTATE is C positive, the routine will integrate past TOUT and C interpolate the solution. This is the most efficient C mode. If MSTATE is negative, the routine will adjust its C internal step to reach TOUT exactly (useful if a C singularity exists beyond TOUT.) The meaning of the C magnitude of MSTATE: C 1 (Input) Means the first call to the routine. This C value must be set by the user. On all subsequent C calls the value of MSTATE should be tested by the C user. Unless DDRIV1 is to be reinitialized, only the C sign of MSTATE may be changed by the user. (As a C convenience to the user who may wish to put out the C initial conditions, DDRIV1 can be called with C MSTATE=+1(-1), and TOUT=T. In this case the program C will return with MSTATE unchanged, i.e., C MSTATE=+1(-1).) C 2 (Output) Means a successful integration. If a normal C continuation is desired (i.e., a further integration C in the same direction), simply advance TOUT and call C again. All other parameters are automatically set. C 3 (Output)(Unsuccessful) Means the integrator has taken C 1000 steps without reaching TOUT. The user can C continue the integration by simply calling DDRIV1 C again. C 4 (Output)(Unsuccessful) Means too much accuracy has C been requested. EPS has been increased to a value C the program estimates is appropriate. The user can C continue the integration by simply calling DDRIV1 C again. C 5 (Output)(Unsuccessful) N has been set to zero in C SUBROUTINE F. See description of F in Section IV. C C EPS = On input, the requested relative accuracy in all solution C components. On output, the adjusted relative accuracy if C the input value was too small. The value of EPS should be C set as large as is reasonable, because the amount of work C done by DDRIV1 increases as EPS decreases. C C WORK C LENW = (Input) C WORK is an array of LENW double precision words used C internally for temporary storage. The user must allocate C space for this array in the calling program by a statement C such as C DOUBLE PRECISION WORK(...) C The length of WORK should be at least N*N + 11*N + 225 C and LENW should be set to the value used. The contents of C WORK should not be disturbed between calls to DDRIV1. C C***LONG DESCRIPTION C C IV. USAGE ......................................................... C C PROGRAM SAMPLE C DOUBLE PRECISION ALFA, EPS, T, TOUT CC N is the number of equations C PARAMETER(ALFA = 1.D0, N = 3, C 8 LENW = N*N + 11*N + 225) C DOUBLE PRECISION WORK(LENW), Y(N+1) CC Initial point C T = 0.00001D0 CC Set initial conditions C Y(1) = 10.D0 C Y(2) = 0.D0 C Y(3) = 10.D0 CC Pass parameter C Y(4) = ALFA C TOUT = T C MSTATE = 1 C EPS = .001D0 C 10 CALL DDRIV1 (N, T, Y, TOUT, MSTATE, EPS, WORK, LENW) C IF (MSTATE .GT. 2) STOP C WRITE(*, '(4E12.3)') TOUT, (Y(I), I=1,3) C TOUT = 10.D0*TOUT C IF (TOUT .LT. 50.D0) GO TO 10 C END C C The user must write a subroutine called F to evaluate the right C hand side of the differential equations. It is of the form: C C SUBROUTINE F (N, T, Y, YDOT) C DOUBLE PRECISION ALFA, T, Y(*), YDOT(*) C ALFA = Y(N+1) C YDOT(1) = 1.D0 + ALFA*(Y(2) - Y(1)) - Y(1)*Y(3) C YDOT(2) = ALFA*(Y(1) - Y(2)) - Y(2)*Y(3) C YDOT(3) = 1.D0 - Y(3)*(Y(1) + Y(2)) C END C C This computes YDOT = F(Y,T), the right hand side of the C differential equations. Here Y is a vector of length at least N. C The actual length of Y is determined by the user's declaration in C the program which calls DDRIV1. Thus the dimensioning of Y in F, C while required by FORTRAN convention, does not actually allocate C any storage. When this subroutine is called, the first N C components of Y are intermediate approximations to the solution C components. The user should not alter these values. Here YDOT is C a vector of length N. The user should only compute YDOT(I) for I C from 1 to N. Normally a return from F passes control back to C DDRIV1. However, if the user would like to abort the calculation, C i.e., return control to the program which calls DDRIV1, he should C set N to zero. DDRIV1 will signal this by returning a value of C MSTATE equal to +5(-5). Altering the value of N in F has no effect C on the value of N in the call sequence of DDRIV1. C C V. OTHER COMMUNICATION TO THE USER ................................ C C A. The solver communicates to the user through the parameters C above. In addition it writes diagnostic messages through the C standard error handling program XERROR. That program will C terminate the user's run if it detects a probable problem setup C error, e.g., insufficient storage allocated by the user for the C WORK array. For further information see section III-A of the C writeup for DDRIV3. C C B. The number of evaluations of the right hand side can be found C in the WORK array in the location determined by: C LENW - (N + 21) + 4 C C VI. REMARKS ....................................................... C C For other information, see section IV of the writeup for DDRIV3. C C***REFERENCES GEAR, C. W., "NUMERICAL INITIAL VALUE PROBLEMS IN C ORDINARY DIFFERENTIAL EQUATIONS", PRENTICE-HALL, 1971. C***ROUTINES CALLED DDRIV3,XERROR C***END PROLOGUE DDRIV1 EXTERNAL F DOUBLE PRECISION EPS, EWT, HMAX, T, TOUT, WORK(*), Y(*) PARAMETER(MXN = 200, IDLIW = 21) INTEGER IWORK(IDLIW+MXN) CHARACTER MSG*103 PARAMETER(NROOT = 0, EWT = 1.D0, IERROR = 2, MINT = 2, MITER = 2, 8 IMPL = 0, MXORD = 5, MXSTEP = 1000) C***FIRST EXECUTABLE STATEMENT DDRIV1 IF (N .GT. MXN) THEN WRITE(MSG, '(''DDRIV115FE Illegal input. The number of '', 8 ''equations,'', I8, '', is greater than the maximum allowed.'') 8 ') N CALL XERROR(MSG(1:97), 97, 15, 2) RETURN END IF IF (MSTATE .GT. 0) THEN NSTATE = MSTATE NTASK = 1 ELSE NSTATE = - MSTATE NTASK = 3 END IF HMAX = 2.D0*ABS(TOUT - T) LENIW = N + IDLIW LENWCM = LENW - LENIW IF (LENWCM .LT. (N*N + 10*N + 204)) THEN LNWCHK = N*N + 10*N + 204 + LENIW WRITE(MSG, '(''DDRIV116FE Insufficient storage allocated for '', 8 ''the work array. The required storage is at least'', I8)') 8 LNWCHK CALL XERROR(MSG(1:103), 103, 16, 2) RETURN END IF IF (NSTATE .NE. 1) THEN DO 20 I = 1,LENIW II = I + LENWCM 20 IWORK(I) = INT(WORK(II)) END IF CALL DDRIV3 (N, T, Y, F, NSTATE, TOUT, NTASK, NROOT, EPS, EWT, 8 IERROR, MINT, MITER, IMPL, ML, MU, MXORD, HMAX, WORK, 8 LENWCM, IWORK, LENIW, F, F, NDE, MXSTEP, F, F) DO 40 I = 1,LENIW II = LENWCM + I 40 WORK(II) = DBLE(IWORK(I)) IF (NSTATE .LE. 4) THEN MSTATE = SIGN(NSTATE, MSTATE) ELSE IF (NSTATE .EQ. 6) THEN MSTATE = SIGN(5, MSTATE) END IF END SUBROUTINE DDRIV3 (N,T,Y,F,NSTATE,TOUT,NTASK,NROOT,EPS,EWT,IERROR, 8 MINT,MITER,IMPL,ML,MU,MXORD,HMAX,WORK,LENW,IWORK,LENIW,JACOBN, 8 FA,NDE,MXSTEP,G,USERS) C***BEGIN PROLOGUE DDRIV3 C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 880308 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***KEYWORDS ODE,STIFF,ORDINARY DIFFERENTIAL EQUATIONS, C INITIAL VALUE PROBLEMS,GEAR'S METHOD, C DOUBLE PRECISION C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***PURPOSE The function of DDRIV3 is to solve N ordinary differential C equations of the form dY(I)/dT = F(Y(I),T), given the C initial conditions Y(I) = YI. The program has options to C allow the solution of both stiff and non-stiff differential C equations. Other important options are available. DDRIV3 C uses double precision arithmetic. C***DESCRIPTION C From the book "Numerical Methods and Software" C by D. Kahaner, C. Moler, S. Nash C Prentice Hall 1988 C C I. ABSTRACT ....................................................... C C The primary function of DDRIV3 is to solve N ordinary differential C equations of the form dY(I)/dT = F(Y(I),T), given the initial C conditions Y(I) = YI. The program has options to allow the C solution of both stiff and non-stiff differential equations. In C addition, DDRIV3 may be used to solve: C 1. The initial value problem, A*dY(I)/dT = F(Y(I),T), where A is C a non-singular matrix depending on Y and T. C 2. The hybrid differential/algebraic initial value problem, C A*dY(I)/dT = F(Y(I),T), where A is a vector (whose values may C depend upon Y and T) some of whose components will be zero C corresponding to those equations which are algebraic rather C than differential. C DDRIV3 is to be called once for each output point of T. C C II. PARAMETERS .................................................... C C (REMEMBER--To run DDRIV3 correctly in double precision, ALL C non-integer arguments in the call sequence, including C arrays, MUST be declared double precision.) C C The user should use parameter names in the call sequence of DDRIV3 C for those quantities whose value may be altered by DDRIV3. The C parameters in the call sequence are: C C N = (Input) The number of dependent functions whose solution C is desired. N must not be altered during a problem. C C T = The independent variable. On input for the first call, T C is the initial point. On output, T is the point at which C the solution is given. C C Y = The vector of dependent variables. Y is used as input on C the first call, to set the initial values. On output, Y C is the computed solution vector. This array Y is passed C in the call sequence of the user-provided routines F, C JACOBN, FA, USERS, and G. Thus parameters required by C those routines can be stored in this array in components C N+1 and above. (Note: Changes by the user to the first C N components of this array will take effect only after a C restart, i.e., after setting NSTATE to 1 .) C C F = A subroutine supplied by the user. The name must be C declared EXTERNAL in the user's calling program. This C subroutine is of the form: C SUBROUTINE F (N, T, Y, YDOT) C DOUBLE PRECISION Y(*), YDOT(*) C . C . C YDOT(1) = ... C . C . C YDOT(N) = ... C END (Sample) C This computes YDOT = F(Y,T), the right hand side of the C differential equations. Here Y is a vector of length at C least N. The actual length of Y is determined by the C user's declaration in the program which calls DDRIV3. C Thus the dimensioning of Y in F, while required by FORTRAN C convention, does not actually allocate any storage. When C this subroutine is called, the first N components of Y are C intermediate approximations to the solution components. C The user should not alter these values. Here YDOT is a C vector of length N. The user should only compute YDOT(I) C for I from 1 to N. Normally a return from F passes C control back to DDRIV3. However, if the user would like C to abort the calculation, i.e., return control to the C program which calls DDRIV3, he should set N to zero. C DDRIV3 will signal this by returning a value of NSTATE C equal to 6 . Altering the value of N in F has no effect C on the value of N in the call sequence of DDRIV3. C C NSTATE = An integer describing the status of integration. The C meaning of NSTATE is as follows: C 1 (Input) Means the first call to the routine. This C value must be set by the user. On all subsequent C calls the value of NSTATE should be tested by the C user, but must not be altered. (As a convenience to C the user who may wish to put out the initial C conditions, DDRIV3 can be called with NSTATE=1, and C TOUT=T. In this case the program will return with C NSTATE unchanged, i.e., NSTATE=1.) C 2 (Output) Means a successful integration. If a normal C continuation is desired (i.e., a further integration C in the same direction), simply advance TOUT and call C again. All other parameters are automatically set. C 3 (Output)(Unsuccessful) Means the integrator has taken C MXSTEP steps without reaching TOUT. The user can C continue the integration by simply calling DDRIV3 C again. C 4 (Output)(Unsuccessful) Means too much accuracy has C been requested. EPS has been increased to a value C the program estimates is appropriate. The user can C continue the integration by simply calling DDRIV3 C again. C 5 (Output) A root was found at a point less than TOUT. C The user can continue the integration toward TOUT by C simply calling DDRIV3 again. C 6 (Output)(Unsuccessful) N has been set to zero in C SUBROUTINE F. C 7 (Output)(Unsuccessful) N has been set to zero in C FUNCTION G. See description of G below. C 8 (Output)(Unsuccessful) N has been set to zero in C SUBROUTINE JACOBN. See description of JACOBN below. C 9 (Output)(Unsuccessful) N has been set to zero in C SUBROUTINE FA. See description of FA below. C 10 (Output)(Unsuccessful) N has been set to zero in C SUBROUTINE USERS. See description of USERS below. C C TOUT = (Input) The point at which the solution is desired. The C position of TOUT relative to T on the first call C determines the direction of integration. C C NTASK = (Input) An index specifying the manner of returning the C solution, according to the following: C NTASK = 1 Means DDRIV3 will integrate past TOUT and C interpolate the solution. This is the most C efficient mode. C NTASK = 2 Means DDRIV3 will return the solution after C each internal integration step, or at TOUT, C whichever comes first. In the latter case, C the program integrates exactly to TOUT. C NTASK = 3 Means DDRIV3 will adjust its internal step to C reach TOUT exactly (useful if a singularity C exists beyond TOUT.) C C NROOT = (Input) The number of equations whose roots are desired. C If NROOT is zero, the root search is not active. This C option is useful for obtaining output at points which are C not known in advance, but depend upon the solution, e.g., C when some solution component takes on a specified value. C The root search is carried out using the user-written C function G (see description of G below.) DDRIV3 attempts C to find the value of T at which one of the equations C changes sign. DDRIV3 can find at most one root per C equation per internal integration step, and will then C return the solution either at TOUT or at a root, whichever C occurs first in the direction of integration. The index C of the equation whose root is being reported is stored in C the sixth element of IWORK. C NOTE: NROOT is never altered by this program. C C EPS = On input, the requested relative accuracy in all solution C components. EPS = 0 is allowed. On output, the adjusted C relative accuracy if the input value was too small. The C value of EPS should be set as large as is reasonable, C because the amount of work done by DDRIV3 increases as EPS C decreases. C C EWT = (Input) Problem zero, i.e., the smallest, nonzero, C physically meaningful value for the solution. (Array, C possibly of length one. See following description of C IERROR.) Setting EWT smaller than necessary can adversely C affect the running time. C C IERROR = (Input) Error control indicator. A value of 3 is C suggested for most problems. Other choices and detailed C explanations of EWT and IERROR are given below for those C who may need extra flexibility. C C These last three input quantities EPS, EWT and IERROR C control the accuracy of the computed solution. EWT and C IERROR are used internally to compute an array YWT. One C step error estimates divided by YWT(I) are kept less than C EPS in root mean square norm. C IERROR (Set by the user) = C 1 Means YWT(I) = 1. (Absolute error control) C EWT is ignored. C 2 Means YWT(I) = ABS(Y(I)), (Relative error control) C EWT is ignored. C 3 Means YWT(I) = MAX(ABS(Y(I)), EWT(1)). C 4 Means YWT(I) = MAX(ABS(Y(I)), EWT(I)). C This choice is useful when the solution components C have differing scales. C 5 Means YWT(I) = EWT(I). C If IERROR is 3, EWT need only be dimensioned one. C If IERROR is 4 or 5, the user must dimension EWT at least C N, and set its values. C C MINT = (Input) The integration method indicator. C MINT = 1 Means the Adams methods, and is used for C non-stiff problems. C MINT = 2 Means the stiff methods of Gear (i.e., the C backward differentiation formulas), and is C used for stiff problems. C MINT = 3 Means the program dynamically selects the C Adams methods when the problem is non-stiff C and the Gear methods when the problem is C stiff. When using the Adams methods, the C program uses a value of MITER=0; when using C the Gear methods, the program uses the value C of MITER provided by the user. Only a value C of IMPL = 0 and a value of MITER = 1, 2, 4, or C 5 is allowed for this option. The user may C not alter the value of MINT or MITER without C restarting, i.e., setting NSTATE to 1. C C MITER = (Input) The iteration method indicator. C MITER = 0 Means functional iteration. This value is C suggested for non-stiff problems. C MITER = 1 Means chord method with analytic Jacobian. C In this case, the user supplies subroutine C JACOBN (see description below). C MITER = 2 Means chord method with Jacobian calculated C internally by finite differences. C MITER = 3 Means chord method with corrections computed C by the user-written routine USERS (see C description of USERS below.) This option C allows all matrix algebra and storage C decisions to be made by the user. When using C a value of MITER = 3, the subroutine FA is C not required, even if IMPL is not 0. For C further information on using this option, see C section IV-E below. C MITER = 4 Means the same as MITER = 1 but the A and C Jacobian matrices are assumed to be banded. C MITER = 5 Means the same as MITER = 2 but the A and C Jacobian matrices are assumed to be banded. C C IMPL = (Input) The implicit method indicator. C IMPL = 0 Means solving dY(I)/dT = F(Y(I),T). C IMPL = 1 Means solving A*dY(I)/dT = F(Y(I),T), C non-singular A (see description of FA below.) C Only MINT = 1 or 2, and MITER = 1, 2, 3, 4, or C 5 are allowed for this option. C IMPL = 2 Means solving certain systems of hybrid C differential/algebraic equations (see C description of FA below.) Only MINT = 2 and C MITER = 1, 2, 3, 4, or 5, are allowed for this C option. C The value of IMPL must not be changed during a problem. C C ML = (Input) The lower half-bandwidth in the case of a banded C A or Jacobian matrix. (I.e., maximum(R-C) for nonzero C A(R,C).) C C MU = (Input) The upper half-bandwidth in the case of a banded C A or Jacobian matrix. (I.e., maximum(C-R).) C C MXORD = (Input) The maximum order desired. This is .LE. 12 for C the Adams methods and .LE. 5 for the Gear methods. Normal C value is 12 and 5, respectively. If MINT is 3, the C maximum order used will be MIN(MXORD, 12) when using the C Adams methods, and MIN(MXORD, 5) when using the Gear C methods. MXORD must not be altered during a problem. C C HMAX = (Input) The maximum magnitude of the step size that will C be used for the problem. This is useful for ensuring that C important details are not missed. If this is not the C case, a large value, such as the interval length, is C suggested. C C WORK C LENW = (Input) C WORK is an array of LENW double precision words used C internally for temporary storage. The user must allocate C space for this array in the calling program by a statement C such as C DOUBLE PRECISION WORK(...) C The following table gives the required minimum value for C the length of WORK, depending on the value of IMPL and C MITER. LENW should be set to the value used. The C contents of WORK should not be disturbed between calls to C DDRIV3. C C IMPL = 0 1 2 C --------------------------------------------------------- C MITER = 0 (MXORD+4)*N + Not allowed Not allowed C 2*NROOT + 204 C C 1,2 N*N+(MXORD+5)*N 2*N*N+(MXORD+5)*N N*N+(MXORD+6)*N C + 2*NROOT + 204 + 2*NROOT + 204 + 2*NROOT + 204 C C 3 (MXORD+4)*N + (MXORD+4)*N + (MXORD+4)*N + C 2*NROOT + 204 2*NROOT + 204 2*NROOT + 204 C C 4,5 (2*ML+MU)*N + (4*ML+2*MU)*N + (2*ML+MU)*N + C (MXORD+6)*N + (MXORD+7)*N + (MXORD+7)*N + C 2*NROOT + 204 2*NROOT + 204 2*NROOT + 204 C --------------------------------------------------------- C C IWORK C LENIW = (Input) C IWORK is an integer array of length LENIW used internally C for temporary storage. The user must allocate space for C this array in the calling program by a statement such as C INTEGER IWORK(...) C The length of IWORK should be at least C 21 if MITER is 0 or 3, or C N+21 if MITER is 1, 2, 4, or 5, or MINT is 3, C and LENIW should be set to the value used. The contents C of IWORK should not be disturbed between calls to DDRIV3. C C JACOBN = A subroutine supplied by the user, if MITER is 1 or 4. C If this is the case, the name must be declared EXTERNAL in C the user's calling program. Given a system of N C differential equations, it is meaningful to speak about C the partial derivative of the I-th right hand side with C respect to the J-th dependent variable. In general there C are N*N such quantities. Often however the equations can C be ordered so that the I-th differential equation only C involves dependent variables with index near I, e.g., I+1, C I-2. Such a system is called banded. If, for all I, the C I-th equation depends on at most the variables C Y(I-ML), Y(I-ML+1), ... , Y(I), Y(I+1), ... , Y(I+MU) C then we call ML+MU+1 the bandwith of the system. In a C banded system many of the partial derivatives above are C automatically zero. For the cases MITER = 1, 2, 4, and 5, C some of these partials are needed. For the cases C MITER = 2 and 5 the necessary derivatives are C approximated numerically by DDRIV3, and we only ask the C user to tell DDRIV3 the value of ML and MU if the system C is banded. For the cases MITER = 1 and 4 the user must C derive these partials algebraically and encode them in C subroutine JACOBN. By computing these derivatives the C user can often save 20-30 per cent of the computing time. C Usually, however, the accuracy is not much affected and C most users will probably forego this option. The optional C user-written subroutine JACOBN has the form: C SUBROUTINE JACOBN (N, T, Y, DFDY, MATDIM, ML, MU) C DOUBLE PRECISION Y(*), DFDY(MATDIM,*) C . C . C Calculate values of DFDY C . C . C END (Sample) C Here Y is a vector of length at least N. The actual C length of Y is determined by the user's declaration in the C program which calls DDRIV3. Thus the dimensioning of Y in C JACOBN, while required by FORTRAN convention, does not C actually allocate any storage. When this subroutine is C called, the first N components of Y are intermediate C approximations to the solution components. The user C should not alter these values. If the system is not C banded (MITER=1), the partials of the I-th equation with C respect to the J-th dependent function are to be stored in C DFDY(I,J). Thus partials of the I-th equation are stored C in the I-th row of DFDY. If the system is banded C (MITER=4), then the partials of the I-th equation with C respect to Y(J) are to be stored in DFDY(K,J), where C K=I-J+MU+1 . Normally a return from JACOBN passes control C back to DDRIV3. However, if the user would like to abort C the calculation, i.e., return control to the program which C calls DDRIV3, he should set N to zero. DDRIV3 will signal C this by returning a value of NSTATE equal to +8(-8). C Altering the value of N in JACOBN has no effect on the C value of N in the call sequence of DDRIV3. C C FA = A subroutine supplied by the user if IMPL is 1 or 2, and C MITER is not 3. If so, the name must be declared EXTERNAL C in the user's calling program. This subroutine computes C the array A, where A*dY(I)/dT = F(Y(I),T). C There are two cases: C C IMPL=1. C Subroutine FA is of the form: C SUBROUTINE FA (N, T, Y, A, MATDIM, ML, MU, NDE) C DOUBLE PRECISION Y(*), A(MATDIM,*) C . C . C Calculate ALL values of A C . C . C END (Sample) C In this case A is assumed to be a nonsingular matrix, C with the same structure as DFDY (see JACOBN description C above). Programming considerations prevent complete C generality. If MITER is 1 or 2, A is assumed to be full C and the user must compute and store all values of C A(I,J), I,J=1, ... ,N. If MITER is 4 or 5, A is assumed C to be banded with lower and upper half bandwidth ML and C MU. The left hand side of the I-th equation is a linear C combination of dY(I-ML)/dT, dY(I-ML+1)/dT, ... , C dY(I)/dT, ... , dY(I+MU-1)/dT, dY(I+MU)/dT. Thus in the C I-th equation, the coefficient of dY(J)/dT is to be C stored in A(K,J), where K=I-J+MU+1. C NOTE: The array A will be altered between calls to FA. C C IMPL=2. C Subroutine FA is of the form: C SUBROUTINE FA (N, T, Y, A, MATDIM, ML, MU, NDE) C DOUBLE PRECISION Y(*), A(*) C . C . C Calculate non-zero values of A(1),...,A(NDE) C . C . C END (Sample) C In this case it is assumed that the system is ordered by C the user so that the differential equations appear C first, and the algebraic equations appear last. The C algebraic equations must be written in the form: C 0 = F(Y(I),T). When using this option it is up to the C user to provide initial values for the Y(I) that satisfy C the algebraic equations as well as possible. It is C further assumed that A is a vector of length NDE. All C of the components of A, which may depend on T, Y(I), C etc., must be set by the user to non-zero values. C Here Y is a vector of length at least N. The actual C length of Y is determined by the user's declaration in the C program which calls DDRIV3. Thus the dimensioning of Y in C FA, while required by FORTRAN convention, does not C actually allocate any storage. When this subroutine is C called, the first N components of Y are intermediate C approximations to the solution components. The user C should not alter these values. FA is always called C immediately after calling F, with the same values of T C and Y. Normally a return from FA passes control back to C DDRIV3. However, if the user would like to abort the C calculation, i.e., return control to the program which C calls DDRIV3, he should set N to zero. DDRIV3 will signal C this by returning a value of NSTATE equal to +9(-9). C Altering the value of N in FA has no effect on the value C of N in the call sequence of DDRIV3. C C NDE = (Input) The number of differential equations. This is C required only for IMPL = 2, with NDE .LT. N. C C MXSTEP = (Input) The maximum number of internal steps allowed on C one call to DDRIV3. C C G = A double precision FORTRAN function supplied by the user C if NROOT is not 0. In this case, the name must be C declared EXTERNAL in the user's calling program. G is C repeatedly called with different values of IROOT to obtain C the value of each of the NROOT equations for which a root C is desired. G is of the form: C DOUBLE PRECISION FUNCTION G (N, T, Y, IROOT) C DOUBLE PRECISION Y(*) C GO TO (10, ...), IROOT C 10 G = ... C . C . C END (Sample) C Here, Y is a vector of length at least N, whose first N C components are the solution components at the point T. C The user should not alter these values. The actual length C of Y is determined by the user's declaration in the C program which calls DDRIV3. Thus the dimensioning of Y in C G, while required by FORTRAN convention, does not actually C allocate any storage. Normally a return from G passes C control back to DDRIV3. However, if the user would like C to abort the calculation, i.e., return control to the C program which calls DDRIV3, he should set N to zero. C DDRIV3 will signal this by returning a value of NSTATE C equal to +7(-7). In this case, the index of the equation C being evaluated is stored in the sixth element of IWORK. C Altering the value of N in G has no effect on the value of C N in the call sequence of DDRIV3. C C USERS = A subroutine supplied by the user, if MITER is 3. C If this is the case, the name must be declared EXTERNAL in C the user's calling program. The routine USERS is called C by DDRIV3 when certain linear systems must be solved. The C user may choose any method to form, store and solve these C systems in order to obtain the solution result that is C returned to DDRIV3. In particular, this allows sparse C matrix methods to be used. The call sequence for this C routine is: C C SUBROUTINE USERS (Y, YH, YWT, SAVE1, SAVE2, T, H, EL, C 8 IMPL, N, NDE, IFLAG) C DOUBLE PRECISION Y(*), YH(*), YWT(*), SAVE1(*), C 8 SAVE2(*), T, H, EL C C The input variable IFLAG indicates what action is to be C taken. Subroutine USERS should perform the following C operations, depending on the value of IFLAG and IMPL. C C IFLAG = 0 C IMPL = 0. USERS is not called. C IMPL = 1 or 2. Solve the system A*X = SAVE2, C returning the result in SAVE2. The array SAVE1 can C be used as a work array. C C IFLAG = 1 C IMPL = 0. Compute, decompose and store the matrix C (I - H*EL*J), where I is the identity matrix and J C is the Jacobian matrix of the right hand side. The C array SAVE1 can be used as a work array. C IMPL = 1 or 2. Compute, decompose and store the matrix C (A - H*EL*J). The array SAVE1 can be used as a work C array. C C IFLAG = 2 C IMPL = 0. Solve the system C (I - H*EL*J)*X = H*SAVE2 - YH - SAVE1, C returning the result in SAVE2. C IMPL = 1 or 2. Solve the system C (A - H*EL*J)*X = H*SAVE2 - A*(YH + SAVE1) C returning the result in SAVE2. C The array SAVE1 should not be altered. C Normally a return from USERS passes control back to C DDRIV3. However, if the user would like to abort the C calculation, i.e., return control to the program which C calls DDRIV3, he should set N to zero. DDRIV3 will signal C this by returning a value of NSTATE equal to +10(-10). C Altering the value of N in USERS has no effect on the C value of N in the call sequence of DDRIV3. C C***LONG DESCRIPTION C C III. OTHER COMMUNICATION TO THE USER .............................. C C A. The solver communicates to the user through the parameters C above. In addition it writes diagnostic messages through the C standard error handling program XERROR. That program will C terminate the user's run if it detects a probable problem setup C error, e.g., insufficient storage allocated by the user for the C WORK array. Messages are written on the standard error message C file. At installations which have this error handling package C the user should determine the standard error handling file from C the local documentation. Otherwise the short but serviceable C routine, XERROR, available with this package, can be used. That C program writes on logical unit 6 to transmit messages. A C complete description of XERROR is given in the Sandia C Laboratories report SAND78-1189 by R. E. Jones. Following is a C list of possible errors. Unless otherwise noted, all messages C come from DDRIV3: C C No. Type Message C --- ---- ------- C 1 Fatal From DDRIV2: The integration method flag has C an illegal value. C 2 Warning The output point is inconsistent with the C value of NTASK and T. C 3 Warning Number of steps to reach TOUT exceeds MXSTEP. C 4 Recoverable Requested accuracy is too stringent. C 5 Warning Step size is below the roundoff level. C 6 Fatal EPS is less than zero. C 7 Fatal N is not positive. C 8 Fatal Insufficient work space provided. C 9 Fatal Improper value for NSTATE, MINT, MITER and/or C IMPL. C 10 Fatal The IWORK array is too small. C 11 Fatal The step size has gone to zero. C 12 Fatal Excessive amount of work. C 13 Fatal For IMPL=1 or 2, the matrix A is singular. C 14 Fatal MXORD is not positive. C 15 Fatal From DDRIV1: N is greater than 200. C 16 Fatal From DDRIV1: The WORK array is too small. C C B. The first three elements of WORK and the first five elements of C IWORK will contain the following statistical data: C AVGH The average step size used. C HUSED The step size last used (successfully). C AVGORD The average order used. C IMXERR The index of the element of the solution vector that C contributed most to the last error test. C NQUSED The order last used (successfully). C NSTEP The number of steps taken since last initialization. C NFE The number of evaluations of the right hand side. C NJE The number of evaluations of the Jacobian matrix. C C IV. REMARKS ....................................................... C C A. Other routines used: C DDNTP, DDZRO, DDSTP, DDNTL, DDPST, DDCOR, DDCST, C DDPSC, and DDSCL; C DGEFA, DGESL, DGBFA, DGBSL, and DNRM2 (from LINPACK) C D1MACH (from the Bell Laboratories Machine Constants Package) C XERROR (from the SLATEC Common Math Library) C The last seven routines above, not having been written by the C present authors, are not explicitly part of this package. C C B. On any return from DDRIV3 all information necessary to continue C the calculation is contained in the call sequence parameters, C including the work arrays. Thus it is possible to suspend one C problem, integrate another, and then return to the first. C C C. If this package is to be used in an overlay situation, the user C must declare in the primary overlay the variables in the call C sequence to DDRIV3. C C D. Changing parameters during an integration. C The value of NROOT, EPS, EWT, IERROR, MINT, MITER, or HMAX may C be altered by the user between calls to DDRIV3. For example, if C too much accuracy has been requested (the program returns with C NSTATE = 4 and an increased value of EPS) the user may wish to C increase EPS further. In general, prudence is necessary when C making changes in parameters since such changes are not C implemented until the next integration step, which is not C necessarily the next call to DDRIV3. This can happen if the C program has already integrated to a point which is beyond the C new point TOUT. C C E. As the price for complete control of matrix algebra, the DDRIV3 C USERS option puts all responsibility for Jacobian matrix C evaluation on the user. It is often useful to approximate C numerically all or part of the Jacobian matrix. However this C must be done carefully. The FORTRAN sequence below illustrates C the method we recommend. It can be inserted directly into C subroutine USERS to approximate Jacobian elements in rows I1 C to I2 and columns J1 to J2. C DOUBLE PRECISION DFDY(N,N), EPSJ, H, R, D1MACH, C 8 SAVE1(N), SAVE2(N), T, UROUND, Y(N), YJ, YWT(N) C UROUND = D1MACH(4) C EPSJ = SQRT(UROUND) C DO 30 J = J1,J2 C R = EPSJ*MAX(ABS(YWT(J)), ABS(Y(J))) C IF (R .EQ. 0.D0) R = YWT(J) C YJ = Y(J) C Y(J) = Y(J) + R C CALL F (N, T, Y, SAVE1) C IF (N .EQ. 0) RETURN C Y(J) = YJ C DO 20 I = I1,I2 C 20 DFDY(I,J) = (SAVE1(I) - SAVE2(I))/R C 30 CONTINUE C Many problems give rise to structured sparse Jacobians, e.g., C block banded. It is possible to approximate them with fewer C function evaluations than the above procedure uses; see Curtis, C Powell and Reid, J. Inst. Maths Applics, (1974), Vol. 13, C pp. 117-119. C C F. When any of the routines JACOBN, FA, G, or USERS, is not C required, difficulties associated with unsatisfied externals can C be avoided by using the name of the routine which calculates the C right hand side of the differential equations in place of the C corresponding name in the call sequence of DDRIV3. C C***REFERENCES GEAR, C. W., "NUMERICAL INITIAL VALUE PROBLEMS IN C ORDINARY DIFFERENTIAL EQUATIONS", PRENTICE-HALL, 1971. C***ROUTINES CALLED DDSTP,DDNTP,DDZRO,DGEFA,DGESL,DGBFA,DGBSL,DNRM2, C D1MACH,XERROR C***END PROLOGUE DDRIV3 EXTERNAL F, JACOBN, FA, G, USERS DOUBLE PRECISION AE, BIG, EPS, EWT(*), G, GLAST, H, HMAX, HSIGN, 8 NROUND, RE, D1MACH, SIZE, DNRM2, SUM, T, TLAST, TOUT, TROOT, 8 UROUND, WORK(*), Y(*) INTEGER IWORK(*) LOGICAL CONVRG CHARACTER MSG*205 PARAMETER(NROUND = 20.D0) PARAMETER(IAVGH = 1, IHUSED = 2, IAVGRD = 3, 8 IEL = 4, IH = 160, IHMAX = 161, IHOLD = 162, 8 IHSIGN = 163, IRC = 164, IRMAX = 165, IT = 166, 8 ITOUT = 167, ITQ = 168, ITREND = 204, IYH = 205, 8 INDMXR = 1, INQUSD = 2, INSTEP = 3, INFE = 4, INJE = 5, 8 INROOT = 6, ICNVRG = 7, IJROOT = 8, IJTASK = 9, 8 IMNTLD = 10, IMTRLD = 11, INQ = 12, INRTLD = 13, 8 INDTRT = 14, INWAIT = 15, IMNT = 16, IMTRSV = 17, 8 IMTR = 18, IMXRDS = 19, IMXORD = 20, INDPRT = 21, 8 INDPVT = 22) C***FIRST EXECUTABLE STATEMENT DDRIV3 NPAR = N UROUND = D1MACH (4) IF (NROOT .NE. 0) THEN AE = D1MACH(1) RE = UROUND END IF IF (EPS .LT. 0.D0) THEN WRITE(MSG, '(''DDRIV36FE Illegal input. EPS,'', D16.8, 8 '', is negative.'')') EPS CALL XERROR(MSG(1:60), 60, 6, 2) RETURN END IF IF (N .LE. 0) THEN WRITE(MSG, '(''DDRIV37FE Illegal input. Number of equations,'', 8 I8, '', is not positive.'')') N CALL XERROR(MSG(1:72), 72, 7, 2) RETURN END IF IF (MXORD .LE. 0) THEN WRITE(MSG, '(''DDRIV314FE Illegal input. Maximum order,'', I8, 8 '', is not positive.'')') MXORD CALL XERROR(MSG(1:67), 67, 14, 2) RETURN END IF IF ((MINT .LT. 1 .OR. MINT .GT. 3) .OR. (MINT .EQ. 3 .AND. 8 (MITER .EQ. 0 .OR. MITER .EQ. 3 .OR. IMPL .NE. 0)) 8 .OR. (MITER .LT. 0 .OR. MITER .GT. 5) .OR. 8 (IMPL .NE. 0 .AND. IMPL .NE. 1 .AND. IMPL .NE. 2) .OR. 8 ((IMPL .EQ. 1 .OR. IMPL .EQ. 2) .AND. MITER .EQ. 0) .OR. 8 (IMPL .EQ. 2 .AND. MINT .EQ. 1) .OR. 8 (NSTATE .LT. 1 .OR. NSTATE .GT. 10)) THEN WRITE(MSG, '(''DDRIV39FE Illegal input. Improper value for '', 8 ''NSTATE(MSTATE), MINT, MITER or IMPL.'')') CALL XERROR(MSG(1:81), 81, 9, 2) RETURN END IF IF (MITER .EQ. 0 .OR. MITER .EQ. 3) THEN LIWCHK = INDPVT - 1 ELSE IF (MITER .EQ. 1 .OR. MITER .EQ. 2 .OR. MITER .EQ. 4 .OR. 8 MITER .EQ. 5) THEN LIWCHK = INDPVT + N - 1 END IF IF (LENIW .LT. LIWCHK) THEN WRITE(MSG, '(''DDRIV310FE Illegal input. Insufficient '', 8 ''storage allocated for the IWORK array. Based on the '')') WRITE(MSG(94:), '(''value of the input parameters involved, '', 8 ''the required storage is'', I8)') LIWCHK CALL XERROR(MSG(1:164), 164, 10, 2) RETURN END IF C Allocate the WORK array C IYH is the index of YH in WORK IF (MINT .EQ. 1 .OR. MINT .EQ. 3) THEN MAXORD = MIN(MXORD, 12) ELSE IF (MINT .EQ. 2) THEN MAXORD = MIN(MXORD, 5) END IF IDFDY = IYH + (MAXORD + 1)*N C IDFDY is the index of DFDY C IF (MITER .EQ. 0 .OR. MITER .EQ. 3) THEN IYWT = IDFDY ELSE IF (MITER .EQ. 1 .OR. MITER .EQ. 2) THEN IYWT = IDFDY + N*N ELSE IF (MITER .EQ. 4 .OR. MITER .EQ. 5) THEN IYWT = IDFDY + (2*ML + MU + 1)*N END IF C IYWT is the index of YWT ISAVE1 = IYWT + N C ISAVE1 is the index of SAVE1 ISAVE2 = ISAVE1 + N C ISAVE2 is the index of SAVE2 IGNOW = ISAVE2 + N C IGNOW is the index of GNOW ITROOT = IGNOW + NROOT C ITROOT is the index of TROOT IFAC = ITROOT + NROOT C IFAC is the index of FAC IF (MITER .EQ. 2 .OR. MITER .EQ. 5 .OR. MINT .EQ. 3) THEN IA = IFAC + N ELSE IA = IFAC END IF C IA is the index of A IF (IMPL .EQ. 0 .OR. MITER .EQ. 3) THEN LENCHK = IA - 1 ELSE IF (IMPL .EQ. 1 .AND. (MITER .EQ. 1 .OR. MITER .EQ. 2)) THEN LENCHK = IA - 1 + N*N ELSE IF (IMPL .EQ. 1 .AND. (MITER .EQ. 4 .OR. MITER .EQ. 5)) THEN LENCHK = IA - 1 + (2*ML + MU + 1)*N ELSE IF (IMPL .EQ. 2 .AND. MITER .NE. 3) THEN LENCHK = IA - 1 + N END IF IF (LENW .LT. LENCHK) THEN WRITE(MSG, '(''DDRIV38FE Illegal input. Insufficient '', 8 ''storage allocated for the WORK array. Based on the '')') WRITE(MSG(92:), '(''value of the input parameters involved, '', 8 ''the required storage is'', I8)') LENCHK CALL XERROR(MSG(1:162), 162, 8, 2) RETURN END IF IF (MITER .EQ. 0 .OR. MITER .EQ. 3) THEN MATDIM = 1 ELSE IF (MITER .EQ. 1 .OR. MITER .EQ. 2) THEN MATDIM = N ELSE IF (MITER .EQ. 4 .OR. MITER .EQ. 5) THEN MATDIM = 2*ML + MU + 1 END IF IF (IMPL .EQ. 0 .OR. IMPL .EQ. 1) THEN NDECOM = N ELSE IF (IMPL .EQ. 2) THEN NDECOM = NDE END IF IF (NSTATE .EQ. 1) THEN C Initialize parameters IF (MINT .EQ. 1 .OR. MINT .EQ. 3) THEN IWORK(IMXORD) = MIN(MXORD, 12) ELSE IF (MINT .EQ. 2) THEN IWORK(IMXORD) = MIN(MXORD, 5) END IF IWORK(IMXRDS) = MXORD IF (MINT .EQ. 1 .OR. MINT .EQ. 2) THEN IWORK(IMNT) = MINT IWORK(IMTR) = MITER IWORK(IMNTLD) = MINT IWORK(IMTRLD) = MITER ELSE IF (MINT .EQ. 3) THEN IWORK(IMNT) = 1 IWORK(IMTR) = 0 IWORK(IMNTLD) = IWORK(IMNT) IWORK(IMTRLD) = IWORK(IMTR) IWORK(IMTRSV) = MITER END IF WORK(IHMAX) = HMAX H = (TOUT - T)*(1.D0 - 4.D0*UROUND) H = SIGN(MIN(ABS(H), HMAX), H) WORK(IH) = H HSIGN = SIGN(1.D0, H) WORK(IHSIGN) = HSIGN IWORK(IJTASK) = 0 WORK(IAVGH) = 0.D0 WORK(IHUSED) =0.D0 WORK(IAVGRD) = 0.D0 IWORK(INDMXR) = 0 IWORK(INQUSD) = 0 IWORK(INSTEP) = 0 IWORK(INFE) = 0 IWORK(INJE) = 0 IWORK(INROOT) = 0 WORK(IT) = T IWORK(ICNVRG) = 0 IWORK(INDPRT) = 0 C Set initial conditions DO 30 I = 1,N JYH = I + IYH - 1 30 WORK(JYH) = Y(I) IF (T .EQ. TOUT) RETURN GO TO 180 END IF C On a continuation, check C that output points have C been or will be overtaken. IF (IWORK(ICNVRG) .EQ. 1) THEN CONVRG = .TRUE. ELSE CONVRG = .FALSE. END IF T = WORK(IT) H = WORK(IH) HSIGN = WORK(IHSIGN) IF (IWORK(IJTASK) .EQ. 0) GO TO 180 C C IWORK(IJROOT) flags unreported C roots, and is set to the value of C NTASK when a root was last selected. C It is set to zero when all roots C have been reported. IWORK(INROOT) C contains the index and WORK(ITOUT) C contains the value of the root last C selected to be reported. C IWORK(INRTLD) contains the value of C NROOT and IWORK(INDTRT) contains C the value of ITROOT when the array C of roots was last calculated. IF (NROOT .NE. 0) THEN JROOT = IWORK(IJROOT) IF (JROOT .GT. 0) THEN C TOUT has just been reported. C If TROOT .LE. TOUT, report TROOT. IF (NSTATE .NE. 5) THEN IF (TOUT*HSIGN .GE. WORK(ITOUT)*HSIGN) THEN TROOT = WORK(ITOUT) CALL DDNTP(H, 0, N, IWORK(INQ), T, TROOT, WORK(IYH), Y) T = TROOT NSTATE = 5 GO TO 580 END IF C A root has just been reported. C Select the next root. ELSE TROOT = T IROOT = 0 DO 50 I = 1,IWORK(INRTLD) JTROOT = IWORK(INDTRT) + I - 1 IF (WORK(JTROOT)*HSIGN .LE. TROOT*HSIGN) THEN C C Check for multiple roots. C IF (WORK(JTROOT) .EQ. WORK(ITOUT) .AND. 8 I .GT. IWORK(INROOT)) THEN IROOT = I TROOT = WORK(JTROOT) GO TO 60 END IF IF (WORK(JTROOT)*HSIGN .GT. WORK(ITOUT)*HSIGN) THEN IROOT = I TROOT = WORK(JTROOT) END IF END IF 50 CONTINUE 60 IWORK(INROOT) = IROOT WORK(ITOUT) = TROOT IWORK(IJROOT) = NTASK IF (NTASK .EQ. 1) THEN IF (IROOT .EQ. 0) THEN IWORK(IJROOT) = 0 ELSE IF (TOUT*HSIGN .GE. TROOT*HSIGN) THEN CALL DDNTP(H, 0, N, IWORK(INQ), T, TROOT,WORK(IYH),Y) NSTATE = 5 T = TROOT GO TO 580 END IF END IF ELSE IF (NTASK .EQ. 2 .OR. NTASK .EQ. 3) THEN C C If there are no more roots, or the C user has altered TOUT to be less C than a root, set IJROOT to zero. C IF (IROOT .EQ. 0 .OR. (TOUT*HSIGN .LT. TROOT*HSIGN)) THEN IWORK(IJROOT) = 0 ELSE CALL DDNTP(H, 0, N, IWORK(INQ), T, TROOT, WORK(IYH), Y) NSTATE = 5 T = TROOT GO TO 580 END IF END IF END IF END IF END IF C IF (NTASK .EQ. 1) THEN NSTATE = 2 IF (T*HSIGN .GE. TOUT*HSIGN) THEN CALL DDNTP (H, 0, N, IWORK(INQ), T, TOUT, WORK(IYH), Y) T = TOUT GO TO 580 END IF ELSE IF (NTASK .EQ. 2) THEN C Check if TOUT has C been reset .LT. T IF (T*HSIGN .GT. TOUT*HSIGN) THEN WRITE(MSG, '(''DDRIV32WRN With NTASK='', I1, '' on input, '', 8 ''T,'', D16.8, '', was beyond TOUT,'', D16.8, ''. Solution'', 8 '' obtained by interpolation.'')') NTASK, T, TOUT CALL XERROR(MSG(1:124), 124, 2, 0) CALL DDNTP (H, 0, N, IWORK(INQ), T, TOUT, WORK(IYH), Y) T = TOUT NSTATE = 2 GO TO 580 END IF C Determine if TOUT has been overtaken C IF (ABS(TOUT - T).LE.NROUND*UROUND*MAX(ABS(T), ABS(TOUT))) THEN T = TOUT NSTATE = 2 GO TO 560 END IF C If there are no more roots C to report, report T. IF (NSTATE .EQ. 5) THEN NSTATE = 2 GO TO 560 END IF NSTATE = 2 C See if TOUT will C be overtaken. IF ((T + H)*HSIGN .GT. TOUT*HSIGN) THEN H = TOUT - T IF ((T + H)*HSIGN .GT. TOUT*HSIGN) H = H*(1.D0 - 4.D0*UROUND) WORK(IH) = H IF (H .EQ. 0.D0) GO TO 670 IWORK(IJTASK) = -1 END IF ELSE IF (NTASK .EQ. 3) THEN NSTATE = 2 IF (T*HSIGN .GT. TOUT*HSIGN) THEN WRITE(MSG, '(''DDRIV32WRN With NTASK='', I1, '' on input, '', 8 ''T,'', D16.8, '', was beyond TOUT,'', D16.8, ''. Solution'', 8 '' obtained by interpolation.'')') NTASK, T, TOUT CALL XERROR(MSG(1:124), 124, 2, 0) CALL DDNTP (H, 0, N, IWORK(INQ), T, TOUT, WORK(IYH), Y) T = TOUT GO TO 580 END IF IF (ABS(TOUT - T).LE.NROUND*UROUND*MAX(ABS(T), ABS(TOUT))) THEN T = TOUT GO TO 560 END IF IF ((T + H)*HSIGN .GT. TOUT*HSIGN) THEN H = TOUT - T IF ((T + H)*HSIGN .GT. TOUT*HSIGN) H = H*(1.D0 - 4.D0*UROUND) WORK(IH) = H IF (H .EQ. 0.D0) GO TO 670 IWORK(IJTASK) = -1 END IF END IF C Implement changes in MINT, MITER, and/or HMAX. C IF ((MINT .NE. IWORK(IMNTLD) .OR. MITER .NE. IWORK(IMTRLD)) .AND. 8 MINT .NE. 3 .AND. IWORK(IMNTLD) .NE. 3) IWORK(IJTASK) = -1 IF (HMAX .NE. WORK(IHMAX)) THEN H = SIGN(MIN(ABS(H), HMAX), H) IF (H .NE. WORK(IH)) THEN IWORK(IJTASK) = -1 WORK(IH) = H END IF WORK(IHMAX) = HMAX END IF C 180 NSTEPL = IWORK(INSTEP) DO 190 I = 1,N JYH = IYH + I - 1 190 Y(I) = WORK(JYH) IF (NROOT .NE. 0) THEN DO 200 I = 1,NROOT JGNOW = IGNOW + I - 1 WORK(JGNOW) = G (NPAR, T, Y, I) IF (NPAR .EQ. 0) THEN IWORK(INROOT) = I NSTATE = 7 RETURN END IF 200 CONTINUE END IF IF (IERROR .EQ. 1) THEN DO 230 I = 1,N JYWT = I + IYWT - 1 230 WORK(JYWT) = 1.D0 GO TO 410 ELSE IF (IERROR .EQ. 5) THEN DO 250 I = 1,N JYWT = I + IYWT - 1 250 WORK(JYWT) = EWT(I) GO TO 410 END IF C Reset YWT array. Looping point. 260 IF (IERROR .EQ. 2) THEN DO 280 I = 1,N IF (Y(I) .EQ. 0.D0) GO TO 290 JYWT = I + IYWT - 1 280 WORK(JYWT) = ABS(Y(I)) GO TO 410 290 IF (IWORK(IJTASK) .EQ. 0) THEN CALL F (NPAR, T, Y, WORK(ISAVE2)) IF (NPAR .EQ. 0) THEN NSTATE = 6 RETURN END IF IWORK(INFE) = IWORK(INFE) + 1 IF (MITER .EQ. 3 .AND. IMPL .NE. 0) THEN IFLAG = 0 CALL USERS(Y, WORK(IYH), WORK(IYWT), WORK(ISAVE1), 8 WORK(ISAVE2), T, H, WORK(IEL), IMPL, NPAR, 8 NDECOM, IFLAG) IF (NPAR .EQ. 0) THEN NSTATE = 10 RETURN END IF ELSE IF (IMPL .EQ. 1) THEN IF (MITER .EQ. 1 .OR. MITER .EQ. 2) THEN CALL FA (NPAR, T, Y, WORK(IA), MATDIM, ML, MU, NDECOM) IF (NPAR .EQ. 0) THEN NSTATE = 9 RETURN END IF CALL DGEFA (WORK(IA), MATDIM, N, IWORK(INDPVT), INFO) IF (INFO .NE. 0) GO TO 690 CALL DGESL(WORK(IA),MATDIM,N,IWORK(INDPVT),WORK(ISAVE2),0) ELSE IF (MITER .EQ. 4 .OR. MITER .EQ. 5) THEN JAML = IA + ML CALL FA (NPAR, T, Y, WORK(JAML), MATDIM, ML, MU, NDECOM) IF (NPAR .EQ. 0) THEN NSTATE = 9 RETURN END IF CALL DGBFA (WORK(IA),MATDIM,N,ML,MU,IWORK(INDPVT),INFO) IF (INFO .NE. 0) GO TO 690 CALL DGBSL (WORK(IA), MATDIM, N, ML, MU, IWORK(INDPVT), 8 WORK(ISAVE2), 0) END IF ELSE IF (IMPL .EQ. 2) THEN CALL FA (NPAR, T, Y, WORK(IA), MATDIM, ML, MU, NDECOM) IF (NPAR .EQ. 0) THEN NSTATE = 9 RETURN END IF DO 340 I = 1,NDECOM JA = I + IA - 1 JSAVE2 = I + ISAVE2 - 1 IF (WORK(JA) .EQ. 0.D0) GO TO 690 340 WORK(JSAVE2) = WORK(JSAVE2)/WORK(JA) END IF END IF DO 360 J = I,N JYWT = J + IYWT - 1 IF (Y(J) .NE. 0.D0) THEN WORK(JYWT) = ABS(Y(J)) ELSE IF (IWORK(IJTASK) .EQ. 0) THEN JSAVE2 = J + ISAVE2 - 1 WORK(JYWT) = ABS(H*WORK(JSAVE2)) ELSE JHYP = J + IYH + N - 1 WORK(JYWT) = ABS(WORK(JHYP)) END IF END IF IF (WORK(JYWT) .EQ. 0.D0) WORK(JYWT) = UROUND 360 CONTINUE ELSE IF (IERROR .EQ. 3) THEN DO 380 I = 1,N JYWT = I + IYWT - 1 380 WORK(JYWT) = MAX(EWT(1), ABS(Y(I))) ELSE IF (IERROR .EQ. 4) THEN DO 400 I = 1,N JYWT = I + IYWT - 1 400 WORK(JYWT) = MAX(EWT(I), ABS(Y(I))) END IF C 410 DO 420 I = 1,N JYWT = I + IYWT - 1 JSAVE2 = I + ISAVE2 - 1 420 WORK(JSAVE2) = Y(I)/WORK(JYWT) SUM = DNRM2(N, WORK(ISAVE2), 1)/SQRT(DBLE(N)) IF (EPS .LT. SUM*UROUND) THEN EPS = SUM*UROUND*(1.D0 + 10.D0*UROUND) WRITE(MSG, '(''DDRIV34REC At T,'', D16.8, '', the requested '', 8 ''accuracy, EPS, was not obtainable with the machine '', 8 ''precision. EPS has been increased to'')') T WRITE(MSG(137:), '(D16.8)') EPS CALL XERROR(MSG(1:152), 152, 4, 1) NSTATE = 4 GO TO 560 END IF IF (ABS(H) .GE. UROUND*ABS(T)) THEN IWORK(INDPRT) = 0 ELSE IF (IWORK(INDPRT) .EQ. 0) THEN WRITE(MSG, '(''DDRIV35WRN At T,'', D16.8, '', the step size,'', 8 D16.8, '', is smaller than the roundoff level of T. '')') T, H WRITE(MSG(109:), '(''This may occur if there is an abrupt '', 8 ''change in the right hand side of the differential '', 8 ''equations.'')') CALL XERROR(MSG(1:205), 205, 5, 0) IWORK(INDPRT) = 1 END IF IF (NTASK.NE.2) THEN IF ((IWORK(INSTEP)-NSTEPL) .GT. MXSTEP) THEN WRITE(MSG, '(''DDRIV33WRN At T,'', D16.8, '', '', I8, 8 '' steps have been taken without reaching TOUT,'', D16.8)') 8 T, MXSTEP, TOUT CALL XERROR(MSG(1:103), 103, 3, 0) NSTATE = 3 GO TO 560 END IF END IF C C CALL DDSTP (EPS, F, FA, HMAX, IMPL, JACOBN, MATDIM, MAXORD, C 8 MINT, MITER, ML, MU, N, NDE, YWT, UROUND, USERS, C 8 AVGH, AVGORD, H, HUSED, JTASK, MNTOLD, MTROLD, C 8 NFE, NJE, NQUSED, NSTEP, T, Y, YH, A, CONVRG, C 8 DFDY, EL, FAC, HOLD, IPVT, JSTATE, NQ, NWAIT, RC, C 8 RMAX, SAVE1, SAVE2, TQ, TREND, ISWFLG, MTRSV, MXRDSV) C CALL DDSTP (EPS, F, FA, WORK(IHMAX), IMPL, JACOBN, MATDIM, 8 IWORK(IMXORD), IWORK(IMNT), IWORK(IMTR), ML, MU, NPAR, 8 NDECOM, WORK(IYWT), UROUND, USERS, WORK(IAVGH), 8 WORK(IAVGRD), WORK(IH), WORK(IHUSED), IWORK(IJTASK), 8 IWORK(IMNTLD), IWORK(IMTRLD), IWORK(INFE), IWORK(INJE), 8 IWORK(INQUSD), IWORK(INSTEP), WORK(IT), Y, WORK(IYH), 8 WORK(IA), CONVRG, WORK(IDFDY), WORK(IEL), WORK(IFAC), 8 WORK(IHOLD), IWORK(INDPVT), JSTATE, IWORK(INQ), 8 IWORK(INWAIT), WORK(IRC), WORK(IRMAX), WORK(ISAVE1), 8 WORK(ISAVE2), WORK(ITQ), WORK(ITREND), MINT, 8 IWORK(IMTRSV), IWORK(IMXRDS)) T = WORK(IT) H = WORK(IH) GO TO (470, 670, 680, 690, 690, 660, 660, 660, 660, 660), JSTATE 470 IWORK(IJTASK) = 1 C Determine if a root has been overtaken IF (NROOT .NE. 0) THEN IROOT = 0 DO 500 I = 1,NROOT JTROOT = ITROOT + I - 1 JGNOW = IGNOW + I - 1 GLAST = WORK(JGNOW) WORK(JGNOW) = G (NPAR, T, Y, I) IF (NPAR .EQ. 0) THEN IWORK(INROOT) = I NSTATE = 7 RETURN END IF IF (GLAST*WORK(JGNOW) .GT. 0.D0) THEN WORK(JTROOT) = T + H ELSE IF (WORK(JGNOW) .EQ. 0.D0) THEN WORK(JTROOT) = T IROOT = I ELSE IF (GLAST .EQ. 0.D0) THEN WORK(JTROOT) = T + H ELSE IF (ABS(WORK(IHUSED)) .GE. UROUND*ABS(T)) THEN TLAST = T - WORK(IHUSED) IROOT = I TROOT = T CALL DDZRO (AE, G, H, NPAR, IWORK(INQ), IROOT, RE, T, 8 WORK(IYH), UROUND, TROOT, TLAST, 8 WORK(JGNOW), GLAST, Y) DO 480 J = 1,N 480 Y(J) = WORK(IYH + J -1) IF (NPAR .EQ. 0) THEN IWORK(INROOT) = I NSTATE = 7 RETURN END IF WORK(JTROOT) = TROOT ELSE WORK(JTROOT) = T IROOT = I END IF END IF END IF END IF 500 CONTINUE IF (IROOT .EQ. 0) THEN IWORK(IJROOT) = 0 C Select the first root ELSE IWORK(IJROOT) = NTASK IWORK(INRTLD) = NROOT IWORK(INDTRT) = ITROOT TROOT = T + H DO 510 I = 1,NROOT JTROOT = ITROOT + I - 1 IF (WORK(JTROOT)*HSIGN .LT. TROOT*HSIGN) THEN TROOT = WORK(JTROOT) IROOT = I END IF 510 CONTINUE IWORK(INROOT) = IROOT WORK(ITOUT) = TROOT IF (TROOT*HSIGN .LE. TOUT*HSIGN) THEN CALL DDNTP (H, 0, N, IWORK(INQ), T, TROOT, WORK(IYH), Y) NSTATE = 5 T = TROOT GO TO 580 END IF END IF END IF C Test for NTASK condition to be satisfied NSTATE = 2 IF (NTASK .EQ. 1) THEN IF (T*HSIGN .LT. TOUT*HSIGN) GO TO 260 CALL DDNTP (H, 0, N, IWORK(INQ), T, TOUT, WORK(IYH), Y) T = TOUT GO TO 580 C TOUT is assumed to have been attained C exactly if T is within twenty roundoff C units of TOUT, relative to max(TOUT, T). ELSE IF (NTASK .EQ. 2) THEN IF (ABS(TOUT - T).LE.NROUND*UROUND*MAX(ABS(T), ABS(TOUT))) THEN T = TOUT ELSE IF ((T + H)*HSIGN .GT. TOUT*HSIGN) THEN H = TOUT - T IF ((T + H)*HSIGN.GT.TOUT*HSIGN) H = H*(1.D0 - 4.D0*UROUND) WORK(IH) = H IF (H .EQ. 0.D0) GO TO 670 IWORK(IJTASK) = -1 END IF END IF ELSE IF (NTASK .EQ. 3) THEN IF (ABS(TOUT - T).LE.NROUND*UROUND*MAX(ABS(T), ABS(TOUT))) THEN T = TOUT ELSE IF ((T + H)*HSIGN .GT. TOUT*HSIGN) THEN H = TOUT - T IF ((T + H)*HSIGN.GT.TOUT*HSIGN) H = H*(1.D0 - 4.D0*UROUND) WORK(IH) = H IF (H .EQ. 0.D0) GO TO 670 IWORK(IJTASK) = -1 END IF GO TO 260 END IF END IF C All returns are made through this C section. IMXERR is determined. 560 DO 570 I = 1,N JYH = I + IYH - 1 570 Y(I) = WORK(JYH) 580 IF (CONVRG) THEN IWORK(ICNVRG) = 1 ELSE IWORK(ICNVRG) = 0 END IF IF (IWORK(IJTASK) .EQ. 0) RETURN BIG = 0.D0 IMXERR = 1 IWORK(INDMXR) = IMXERR DO 590 I = 1,N C SIZE = ABS(ERROR(I)/YWT(I)) JYWT = I + IYWT - 1 JERROR = I + ISAVE1 - 1 SIZE = ABS(WORK(JERROR)/WORK(JYWT)) IF (BIG .LT. SIZE) THEN BIG = SIZE IMXERR = I IWORK(INDMXR) = IMXERR END IF 590 CONTINUE RETURN C 660 NSTATE = JSTATE RETURN C Fatal errors are processed here C 670 WRITE(MSG, '(''DDRIV311FE At T,'', D16.8, '', the attempted '', 8 ''step size has gone to zero. Often this occurs if the '', 8 ''problem setup is incorrect.'')') T CALL XERROR(MSG(1:129), 129, 11, 2) RETURN C 680 WRITE(MSG, '(''DDRIV312FE At T,'', D16.8, '', the step size has'', 8 '' been reduced about 50 times without advancing the '')') T WRITE(MSG(103:), '(''solution. Often this occurs if the '', 8 ''problem setup is incorrect.'')') CALL XERROR(MSG(1:165), 165, 12, 2) RETURN C 690 WRITE(MSG, '(''DDRIV313FE At T,'', D16.8, '', while solving'', 8 '' A*YDOT = F, A is singular.'')') T CALL XERROR(MSG(1:74), 74, 13, 2) RETURN END SUBROUTINE DDNTP (H,K,N,NQ,T,TOUT,YH,Y) C***BEGIN PROLOGUE DDNTP C From the book "Numerical Methods and Software" C by D. Kahaner, C. Moler, S. Nash C Prentice Hall 1988 C***REFER TO DDRIV3 C Subroutine DDNTP interpolates the K-th derivative of Y at TOUT, C using the data in the YH array. If K has a value greater than NQ, C the NQ-th derivative is calculated. C***ROUTINES CALLED (NONE) C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 870216 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDNTP DOUBLE PRECISION FACTOR, H, R, T, TOUT, Y(*), YH(N,*) C***FIRST EXECUTABLE STATEMENT DDNTP IF (K .EQ. 0) THEN DO 10 I = 1,N 10 Y(I) = YH(I,NQ+1) R = ((TOUT - T)/H) DO 20 JJ = 1,NQ J = NQ + 1 - JJ DO 20 I = 1,N 20 Y(I) = YH(I,J) + R*Y(I) ELSE KUSED = MIN(K, NQ) FACTOR = 1.D0 DO 40 KK = 1,KUSED 40 FACTOR = FACTOR*DBLE(NQ+1-KK) DO 50 I = 1,N 50 Y(I) = FACTOR*YH(I,NQ+1) DO 80 JJ = KUSED+1,NQ J = K + 1 + NQ - JJ FACTOR = 1.D0 DO 60 KK = 1,KUSED 60 FACTOR = FACTOR*DBLE(J-KK) DO 70 I = 1,N 70 Y(I) = FACTOR*YH(I,J) + R*Y(I) 80 CONTINUE DO 100 I = 1,N 100 Y(I) = Y(I)*H**(-KUSED) END IF END SUBROUTINE DDZRO (AE,F,H,N,NQ,IROOT,RE,T,YH,UROUND,B,C,FB,FC,Y) C***BEGIN PROLOGUE DDZRO C***REFER TO DDRIV3 C This is a special purpose version of ZEROIN, modified for use with C the DDRIV1 package. C C Sandia Mathematical Program Library C Mathematical Computing Services Division 5422 C Sandia Laboratories C P. O. Box 5800 C Albuquerque, New Mexico 87115 C Control Data 6600 Version 4.5, 1 November 1971 C C ABSTRACT C ZEROIN searches for a zero of a function F(N, T, Y, IROOT) C between the given values B and C until the width of the C interval (B, C) has collapsed to within a tolerance specified C by the stopping criterion, ABS(B - C) .LE. 2.*(RW*ABS(B) + AE). C C Description of parameters C F - Name of the external function, which returns a C double precision result. This name must be in an C EXTERNAL statement in the calling program. C B - One end of the interval (B, C). The value returned for C B usually is the better approximation to a zero of F. C C - The other end of the interval (B, C). C RE - Relative error used for RW in the stopping criterion. C If the requested RE is less than machine precision, C then RW is set to approximately machine precision. C AE - Absolute error used in the stopping criterion. If the C given interval (B, C) contains the origin, then a C nonzero value should be chosen for AE. C C REFERENCES C 1. L F Shampine and H A Watts, ZEROIN, A Root-Solving Routine, C SC-TM-70-631, Sept 1970. C 2. T J Dekker, Finding a Zero by Means of Successive Linear C Interpolation, "Constructive Aspects of the Fundamental C Theorem of Algebra", edited by B Dejon and P Henrici, 1969. C***ROUTINES CALLED DDNTP C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 870511 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDZRO DOUBLE PRECISION A, ACBS, ACMB, AE, B, C, CMB, ER, F, FA, FB, FC, 8 H, P, Q, RE, RW, T, TOL, UROUND, Y(*), YH(N,*) C***FIRST EXECUTABLE STATEMENT DDZRO ER = 4.D0*UROUND RW = MAX(RE, ER) IC = 0 ACBS = ABS(B - C) A = C FA = FC KOUNT = 0 C Perform interchange 10 IF (ABS(FC) .LT. ABS(FB)) THEN A = B FA = FB B = C FB = FC C = A FC = FA END IF CMB = 0.5D0*(C - B) ACMB = ABS(CMB) TOL = RW*ABS(B) + AE C Test stopping criterion IF (ACMB .LE. TOL) RETURN IF (KOUNT .GT. 50) RETURN C Calculate new iterate implicitly as C B + P/Q, where we arrange P .GE. 0. C The implicit form is used to prevent overflow. P = (B - A)*FB Q = FA - FB IF (P .LT. 0.D0) THEN P = -P Q = -Q END IF C Update A and check for satisfactory reduction C in the size of our bounding interval. A = B FA = FB IC = IC + 1 IF (IC .GE. 4) THEN IF (8.D0*ACMB .GE. ACBS) THEN C Bisect B = 0.5D0*(C + B) GO TO 20 END IF IC = 0 END IF ACBS = ACMB C Test for too small a change IF (P .LE. ABS(Q)*TOL) THEN C Increment by tolerance B = B + SIGN(TOL, CMB) C Root ought to be between C B and (C + B)/2. ELSE IF (P .LT. CMB*Q) THEN C Interpolate B = B + P/Q ELSE C Bisect B = 0.5D0*(C + B) END IF C Have completed computation C for new iterate B. 20 CALL DDNTP (H, 0, N, NQ, T, B, YH, Y) FB = F(N, B, Y, IROOT) IF (N .EQ. 0) RETURN IF (FB .EQ. 0.D0) RETURN KOUNT = KOUNT + 1 C C Decide whether next step is interpolation or extrapolation C IF (SIGN(1.0D0, FB) .EQ. SIGN(1.0D0, FC)) THEN C = A FC = FA END IF GO TO 10 END SUBROUTINE DDSTP (EPS,F,FA,HMAX,IMPL,JACOBN,MATDIM,MAXORD,MINT, 8 MITER,ML,MU,N,NDE,YWT,UROUND,USERS,AVGH,AVGORD,H,HUSED,JTASK, 8 MNTOLD,MTROLD,NFE,NJE,NQUSED,NSTEP,T,Y,YH,A,CONVRG,DFDY,EL,FAC, 8 HOLD,IPVT,JSTATE,NQ,NWAIT,RC,RMAX,SAVE1,SAVE2,TQ,TREND,ISWFLG, 8 MTRSV,MXRDSV) C***BEGIN PROLOGUE DDSTP C***REFER TO DDRIV3 C DDSTP performs one step of the integration of an initial value C problem for a system of ordinary differential equations. C Communication with DDSTP is done with the following variables: C C YH An N by MAXORD+1 array containing the dependent variables C and their scaled derivatives. MAXORD, the maximum order C used, is currently 12 for the Adams methods and 5 for the C Gear methods. YH(I,J+1) contains the J-th derivative of C Y(I), scaled by H**J/factorial(J). Only Y(I), C 1 .LE. I .LE. N, need be set by the calling program on C the first entry. The YH array should not be altered by C the calling program. When referencing YH as a C 2-dimensional array, use a column length of N, as this is C the value used in DDSTP. C DFDY A block of locations used for partial derivatives if MITER C is not 0. If MITER is 1 or 2 its length must be at least C N*N. If MITER is 4 or 5 its length must be at least C (2*ML+MU+1)*N. C YWT An array of N locations used in convergence and error tests C SAVE1 C SAVE2 Arrays of length N used for temporary storage. C IPVT An integer array of length N used by the linear system C solvers for the storage of row interchange information. C A A block of locations used to store the matrix A, when using C the implicit method. If IMPL is 1, A is a MATDIM by N C array. If MITER is 1 or 2 MATDIM is N, and if MITER is 4 C or 5 MATDIM is 2*ML+MU+1. If IMPL is 2 its length is N. C JTASK An integer used on input. C It has the following values and meanings: C .EQ. 0 Perform the first step. This value enables C the subroutine to initialize itself. C .GT. 0 Take a new step continuing from the last. C Assumes the last step was successful and C user has not changed any parameters. C .LT. 0 Take a new step with a new value of H and/or C MINT and/or MITER. C JSTATE A completion code with the following meanings: C 1 The step was successful. C 2 A solution could not be obtained with H .NE. 0. C 3 A solution was not obtained in MXTRY attempts. C 4 For IMPL .NE. 0, the matrix A is singular. C On a return with JSTATE .GT. 1, the values of T and C the YH array are as of the beginning of the last C step, and H is the last step size attempted. C***ROUTINES CALLED DDNTL,DDPST,DDCOR,DDPSC,DDSCL,DNRM2 C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 870810 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDSTP EXTERNAL F, JACOBN, FA, USERS DOUBLE PRECISION A(MATDIM,*), AVGH, AVGORD, BIAS1, BIAS2, BIAS3, 8 BND, CTEST, D, DENOM, DFDY(MATDIM,*), D1, EL(13,12), EPS, 8 ERDN, ERUP, ETEST, FAC(*), H, HMAX, HN, HOLD, HS, HUSED, 8 NUMER, RC, RCTEST, RH, RH1, RH2, RH3, RMAX, RMFAIL, RMNORM, 8 SAVE1(*), SAVE2(*), DNRM2, T, TOLD, TQ(3,12), TREND, TRSHLD, 8 UROUND, Y(*), YH(N,*), YWT(*), Y0NRM INTEGER IPVT(*) LOGICAL CONVRG, EVALFA, EVALJC, IER, SWITCH PARAMETER(BIAS1 = 1.3D0, BIAS2 = 1.2D0, BIAS3 = 1.4D0, MXFAIL = 3, 8 MXITER = 3, MXTRY = 50, RCTEST = .3D0, RMFAIL = 2.D0, 8 RMNORM = 10.D0, TRSHLD = 1.D0) DATA IER /.FALSE./ C***FIRST EXECUTABLE STATEMENT DDSTP NSV = N BND = 0.D0 SWITCH = .FALSE. NTRY = 0 TOLD = T NFAIL = 0 IF (JTASK .LE. 0) THEN CALL DDNTL (EPS, F, FA, HMAX, HOLD, IMPL, JTASK, MATDIM, 8 MAXORD, MINT, MITER, ML, MU, N, NDE, SAVE1, T, 8 UROUND, USERS, Y, YWT, H, MNTOLD, MTROLD, NFE, RC, 8 YH, A, CONVRG, EL, FAC, IER, IPVT, NQ, NWAIT, RH, 8 RMAX, SAVE2, TQ, TREND, ISWFLG, JSTATE) IF (N .EQ. 0) GO TO 440 IF (H .EQ. 0.D0) GO TO 400 IF (IER) GO TO 420 END IF 100 NTRY = NTRY + 1 IF (NTRY .GT. MXTRY) GO TO 410 T = T + H CALL DDPSC (1, N, NQ, YH) EVALJC = ((ABS(RC - 1.D0) .GT. RCTEST) .AND. (MITER .NE. 0)) EVALFA = .NOT. EVALJC C 110 ITER = 0 DO 115 I = 1,N 115 Y(I) = YH(I,1) CALL F (N, T, Y, SAVE2) IF (N .EQ. 0) THEN JSTATE = 6 GO TO 430 END IF NFE = NFE + 1 IF (EVALJC .OR. IER) THEN CALL DDPST (EL, F, FA, H, IMPL, JACOBN, MATDIM, MITER, ML, 8 MU, N, NDE, NQ, SAVE2, T, USERS, Y, YH, YWT, UROUND, 8 NFE, NJE, A, DFDY, FAC, IER, IPVT, SAVE1, ISWFLG, 8 BND, JSTATE) IF (N .EQ. 0) GO TO 430 IF (IER) GO TO 160 CONVRG = .FALSE. RC = 1.D0 END IF DO 125 I = 1,N 125 SAVE1(I) = 0.D0 C Up to MXITER corrector iterations are taken. C Convergence is tested by requiring the r.m.s. C norm of changes to be less than EPS. The sum of C the corrections is accumulated in the vector C SAVE1(I). It is approximately equal to the L-th C derivative of Y multiplied by C H**L/(factorial(L-1)*EL(L,NQ)), and is thus C proportional to the actual errors to the lowest C power of H present (H**L). The YH array is not C altered in the correction loop. The norm of the C iterate difference is stored in D. If C ITER .GT. 0, an estimate of the convergence rate C constant is stored in TREND, and this is used in C the convergence test. C 130 CALL DDCOR (DFDY, EL, FA, H, IMPL, IPVT, MATDIM, MITER, ML, 8 MU, N, NDE, NQ, T, USERS, Y, YH, YWT, EVALFA, SAVE1, 8 SAVE2, A, D, JSTATE) IF (N .EQ. 0) GO TO 430 IF (ISWFLG .EQ. 3 .AND. MINT .EQ. 1) THEN IF (ITER .EQ. 0) THEN NUMER = DNRM2(N, SAVE1, 1) DO 132 I = 1,N 132 DFDY(1,I) = SAVE1(I) Y0NRM = DNRM2(N, YH, 1) ELSE DENOM = NUMER DO 134 I = 1,N 134 DFDY(1,I) = SAVE1(I) - DFDY(1,I) NUMER = DNRM2(N, DFDY, MATDIM) IF (EL(1,NQ)*NUMER .LE. 100.D0*UROUND*Y0NRM) THEN IF (RMAX .EQ. RMFAIL) THEN SWITCH = .TRUE. GO TO 170 END IF END IF DO 136 I = 1,N 136 DFDY(1,I) = SAVE1(I) IF (DENOM .NE. 0.D0) 8 BND = MAX(BND, NUMER/(DENOM*ABS(H)*EL(1,NQ))) END IF END IF IF (ITER .GT. 0) TREND = MAX(.9D0*TREND, D/D1) D1 = D CTEST = MIN(2.D0*TREND, 1.D0)*D IF (CTEST .LE. EPS) GO TO 170 ITER = ITER + 1 IF (ITER .LT. MXITER) THEN DO 140 I = 1,N 140 Y(I) = YH(I,1) + EL(1,NQ)*SAVE1(I) CALL F (N, T, Y, SAVE2) IF (N .EQ. 0) THEN JSTATE = 6 GO TO 430 END IF NFE = NFE + 1 GO TO 130 END IF C The corrector iteration failed to converge in C MXITER tries. If partials are involved but are C not up to date, they are reevaluated for the next C try. Otherwise the YH array is retracted to its C values before prediction, and H is reduced, if C possible. If not, a no-convergence exit is taken. IF (CONVRG) THEN EVALJC = .TRUE. EVALFA = .FALSE. GO TO 110 END IF 160 T = TOLD CALL DDPSC (-1, N, NQ, YH) NWAIT = NQ + 2 IF (JTASK .NE. 0 .AND. JTASK .NE. 2) RMAX = RMFAIL IF (ITER .EQ. 0) THEN RH = .3D0 ELSE RH = .9D0*(EPS/CTEST)**(.2D0) END IF IF (RH*H .EQ. 0.D0) GO TO 400 CALL DDSCL (HMAX, N, NQ, RMAX, H, RC, RH, YH) GO TO 100 C The corrector has converged. CONVRG is set C to .TRUE. if partial derivatives were used, C to indicate that they may need updating on C subsequent steps. The error test is made. 170 CONVRG = (MITER .NE. 0) DO 180 I = 1,NDE 180 SAVE2(I) = SAVE1(I)/YWT(I) ETEST = DNRM2(NDE, SAVE2, 1)/(TQ(2,NQ)*SQRT(DBLE(NDE))) C C The error test failed. NFAIL keeps track of C multiple failures. Restore T and the YH C array to their previous values, and prepare C to try the step again. Compute the optimum C step size for this or one lower order. IF (ETEST .GT. EPS) THEN T = TOLD CALL DDPSC (-1, N, NQ, YH) NFAIL = NFAIL + 1 IF (NFAIL .LT. MXFAIL) THEN IF (JTASK .NE. 0 .AND. JTASK .NE. 2) RMAX = RMFAIL RH2 = 1.D0/(BIAS2*(ETEST/EPS)**(1.D0/DBLE(NQ+1))) IF (NQ .GT. 1) THEN DO 190 I = 1,NDE 190 SAVE2(I) = YH(I,NQ+1)/YWT(I) ERDN = DNRM2(NDE, SAVE2, 1)/(TQ(1,NQ)*SQRT(DBLE(NDE))) RH1 = 1.D0/MAX(1.D0, BIAS1*(ERDN/EPS)**(1.D0/DBLE(NQ))) IF (RH2 .LT. RH1) THEN NQ = NQ - 1 RC = RC*EL(1,NQ)/EL(1,NQ+1) RH = RH1 ELSE RH = RH2 END IF ELSE RH = RH2 END IF NWAIT = NQ + 2 IF (RH*H .EQ. 0.D0) GO TO 400 CALL DDSCL (HMAX, N, NQ, RMAX, H, RC, RH, YH) GO TO 100 END IF C Control reaches this section if the error test has C failed MXFAIL or more times. It is assumed that the C derivatives that have accumulated in the YH array have C errors of the wrong order. Hence the first derivative C is recomputed, the order is set to 1, and the step is C retried. NFAIL = 0 JTASK = 2 DO 215 I = 1,N 215 Y(I) = YH(I,1) CALL DDNTL (EPS, F, FA, HMAX, HOLD, IMPL, JTASK, MATDIM, 8 MAXORD, MINT, MITER, ML, MU, N, NDE, SAVE1, T, 8 UROUND, USERS, Y, YWT, H, MNTOLD, MTROLD, NFE, RC, 8 YH, A, CONVRG, EL, FAC, IER, IPVT, NQ, NWAIT, RH, 8 RMAX, SAVE2, TQ, TREND, ISWFLG, JSTATE) RMAX = RMNORM IF (N .EQ. 0) GO TO 440 IF (H .EQ. 0.D0) GO TO 400 IF (IER) GO TO 420 GO TO 100 END IF C After a successful step, update the YH array. NSTEP = NSTEP + 1 HUSED = H NQUSED = NQ AVGH = (DBLE(NSTEP-1)*AVGH + H)/DBLE(NSTEP) AVGORD = (DBLE(NSTEP-1)*AVGORD + DBLE(NQ))/DBLE(NSTEP) DO 230 J = 1,NQ+1 DO 230 I = 1,N 230 YH(I,J) = YH(I,J) + EL(J,NQ)*SAVE1(I) DO 235 I = 1,N 235 Y(I) = YH(I,1) C If ISWFLG is 3, consider C changing integration methods. C IF (ISWFLG .EQ. 3) THEN IF (BND .NE. 0.D0) THEN IF (MINT .EQ. 1 .AND. NQ .LE. 5) THEN HN = ABS(H)/MAX(UROUND, (ETEST/EPS)**(1.D0/DBLE(NQ+1))) HN = MIN(HN, 1.D0/(2.D0*EL(1,NQ)*BND)) HS = ABS(H)/MAX(UROUND, 8 (ETEST/(EPS*EL(NQ+1,1)))**(1.D0/DBLE(NQ+1))) IF (HS .GT. 1.2D0*HN) THEN MINT = 2 MNTOLD = MINT MITER = MTRSV MTROLD = MITER MAXORD = MIN(MXRDSV, 5) RC = 0.D0 RMAX = RMNORM TREND = 1.D0 CALL DDCST (MAXORD, MINT, ISWFLG, EL, TQ) NWAIT = NQ + 2 END IF ELSE IF (MINT .EQ. 2) THEN HS = ABS(H)/MAX(UROUND, (ETEST/EPS)**(1.D0/DBLE(NQ+1))) HN = ABS(H)/MAX(UROUND, 8 (ETEST*EL(NQ+1,1)/EPS)**(1.D0/DBLE(NQ+1))) HN = MIN(HN, 1.D0/(2.D0*EL(1,NQ)*BND)) IF (HN .GE. HS) THEN MINT = 1 MNTOLD = MINT MITER = 0 MTROLD = MITER MAXORD = MIN(MXRDSV, 12) RMAX = RMNORM TREND = 1.D0 CONVRG = .FALSE. CALL DDCST (MAXORD, MINT, ISWFLG, EL, TQ) NWAIT = NQ + 2 END IF END IF END IF END IF IF (SWITCH) THEN MINT = 2 MNTOLD = MINT MITER = MTRSV MTROLD = MITER MAXORD = MIN(MXRDSV, 5) NQ = MIN(NQ, MAXORD) RC = 0.D0 RMAX = RMNORM TREND = 1.D0 CALL DDCST (MAXORD, MINT, ISWFLG, EL, TQ) NWAIT = NQ + 2 END IF C Consider changing H if NWAIT = 1. Otherwise C decrease NWAIT by 1. If NWAIT is then 1 and C NQ.LT.MAXORD, then SAVE1 is saved for use in C a possible order increase on the next step. C IF (JTASK .EQ. 0 .OR. JTASK .EQ. 2) THEN RH = 1.D0/MAX(UROUND, BIAS2*(ETEST/EPS)**(1.D0/DBLE(NQ+1))) IF (RH.GT.TRSHLD) CALL DDSCL (HMAX, N, NQ, RMAX, H, RC, RH, YH) ELSE IF (NWAIT .GT. 1) THEN NWAIT = NWAIT - 1 IF (NWAIT .EQ. 1 .AND. NQ .LT. MAXORD) THEN DO 250 I = 1,NDE 250 YH(I,MAXORD+1) = SAVE1(I) END IF C If a change in H is considered, an increase or decrease in C order by one is considered also. A change in H is made C only if it is by a factor of at least TRSHLD. Factors C RH1, RH2, and RH3 are computed, by which H could be C multiplied at order NQ - 1, order NQ, or order NQ + 1, C respectively. The largest of these is determined and the C new order chosen accordingly. If the order is to be C increased, we compute one additional scaled derivative. C If there is a change of order, reset NQ and the C coefficients. In any case H is reset according to RH and C the YH array is rescaled. ELSE IF (NQ .EQ. 1) THEN RH1 = 0.D0 ELSE DO 270 I = 1,NDE 270 SAVE2(I) = YH(I,NQ+1)/YWT(I) ERDN = DNRM2(NDE, SAVE2, 1)/(TQ(1,NQ)*SQRT(DBLE(NDE))) RH1 = 1.D0/MAX(UROUND, BIAS1*(ERDN/EPS)**(1.D0/DBLE(NQ))) END IF RH2 = 1.D0/MAX(UROUND, BIAS2*(ETEST/EPS)**(1.D0/DBLE(NQ+1))) IF (NQ .EQ. MAXORD) THEN RH3 = 0.D0 ELSE DO 290 I = 1,NDE 290 SAVE2(I) = (SAVE1(I) - YH(I,MAXORD+1))/YWT(I) ERUP = DNRM2(NDE, SAVE2, 1)/(TQ(3,NQ)*SQRT(DBLE(NDE))) RH3 = 1.D0/MAX(UROUND, BIAS3*(ERUP/EPS)**(1.D0/DBLE(NQ+2))) END IF IF (RH1 .GT. RH2 .AND. RH1 .GE. RH3) THEN RH = RH1 IF (RH .LE. TRSHLD) GO TO 380 NQ = NQ - 1 RC = RC*EL(1,NQ)/EL(1,NQ+1) ELSE IF (RH2 .GE. RH1 .AND. RH2 .GE. RH3) THEN RH = RH2 IF (RH .LE. TRSHLD) GO TO 380 ELSE RH = RH3 IF (RH .LE. TRSHLD) GO TO 380 DO 360 I = 1,N 360 YH(I,NQ+2) = SAVE1(I)*EL(NQ+1,NQ)/DBLE(NQ+1) NQ = NQ + 1 RC = RC*EL(1,NQ)/EL(1,NQ-1) END IF IF (ISWFLG .EQ. 3 .AND. MINT .EQ. 1) THEN IF (BND.NE.0.D0) RH = MIN(RH, 1.D0/(2.D0*EL(1,NQ)*BND*ABS(H))) END IF CALL DDSCL (HMAX, N, NQ, RMAX, H, RC, RH, YH) RMAX = RMNORM 380 NWAIT = NQ + 2 END IF C All returns are made through this section. H is saved C in HOLD to allow the caller to change H on the next step JSTATE = 1 HOLD = H RETURN C 400 JSTATE = 2 HOLD = H DO 405 I = 1,N 405 Y(I) = YH(I,1) RETURN C 410 JSTATE = 3 HOLD = H RETURN C 420 JSTATE = 4 HOLD = H RETURN C 430 T = TOLD CALL DDPSC (-1, NSV, NQ, YH) DO 435 I = 1,NSV 435 Y(I) = YH(I,1) 440 HOLD = H RETURN END SUBROUTINE DDNTL (EPS,F,FA,HMAX,HOLD,IMPL,JTASK,MATDIM,MAXORD, 8 MINT,MITER,ML,MU,N,NDE,SAVE1,T,UROUND,USERS,Y,YWT,H,MNTOLD, 8 MTROLD,NFE,RC,YH,A,CONVRG,EL,FAC,IER,IPVT,NQ,NWAIT,RH,RMAX, 8 SAVE2,TQ,TREND,ISWFLG,JSTATE) C***BEGIN PROLOGUE DDNTL C***REFER TO DDRIV3 C Subroutine DDNTL is called to set parameters on the first call C to DDSTP, on an internal restart, or when the user has altered C MINT, MITER, and/or H. C On the first call, the order is set to 1 and the initial derivatives C are calculated. RMAX is the maximum ratio by which H can be C increased in one step. It is initially RMINIT to compensate C for the small initial H, but then is normally equal to RMNORM. C If a failure occurs (in corrector convergence or error test), RMAX C is set at RMFAIL for the next increase. C If the caller has changed MINT, or if JTASK = 0, DDCST is called C to set the coefficients of the method. If the caller has changed H, C YH must be rescaled. If H or MINT has been changed, NWAIT is C reset to NQ + 2 to prevent further increases in H for that many C steps. Also, RC is reset. RC is the ratio of new to old values of C the coefficient L(0)*H. If the caller has changed MITER, RC is C set to 0 to force the partials to be updated, if partials are used. C***ROUTINES CALLED DDCST,DDSCL,DGEFA,DGESL,DGBFA,DGBSL,DNRM2 C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 870810 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDNTL DOUBLE PRECISION A(MATDIM,*), EL(13,12), EPS, FAC(*), H, HMAX, 8 HOLD, OLDL0, RC, RH, RMAX, RMINIT, SAVE1(*), SAVE2(*), SMAX, 8 SMIN, DNRM2, SUM, SUM0, T, TQ(3,12), TREND, UROUND, Y(*), 8 YH(N,*), YWT(*) INTEGER IPVT(*) LOGICAL CONVRG, IER PARAMETER(RMINIT = 10000.D0) C***FIRST EXECUTABLE STATEMENT DDNTL IER = .FALSE. IF (JTASK .GE. 0) THEN IF (JTASK .EQ. 0) THEN CALL DDCST (MAXORD, MINT, ISWFLG, EL, TQ) RMAX = RMINIT END IF RC = 0.D0 CONVRG = .FALSE. TREND = 1.D0 NQ = 1 NWAIT = 3 CALL F (N, T, Y, SAVE2) IF (N .EQ. 0) THEN JSTATE = 6 RETURN END IF NFE = NFE + 1 IF (IMPL .NE. 0) THEN IF (MITER .EQ. 3) THEN IFLAG = 0 CALL USERS (Y, YH, YWT, SAVE1, SAVE2, T, H, EL, IMPL, N, 8 NDE, IFLAG) IF (N .EQ. 0) THEN JSTATE = 10 RETURN END IF ELSE IF (IMPL .EQ. 1) THEN IF (MITER .EQ. 1 .OR. MITER .EQ. 2) THEN CALL FA (N, T, Y, A, MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF CALL DGEFA (A, MATDIM, N, IPVT, INFO) IF (INFO .NE. 0) THEN IER = .TRUE. RETURN END IF CALL DGESL (A, MATDIM, N, IPVT, SAVE2, 0) ELSE IF (MITER .EQ. 4 .OR. MITER .EQ. 5) THEN CALL FA (N, T, Y, A(ML+1,1), MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF CALL DGBFA (A, MATDIM, N, ML, MU, IPVT, INFO) IF (INFO .NE. 0) THEN IER = .TRUE. RETURN END IF CALL DGBSL (A, MATDIM, N, ML, MU, IPVT, SAVE2, 0) END IF ELSE IF (IMPL .EQ. 2) THEN CALL FA (N, T, Y, A, MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF DO 150 I = 1,NDE IF (A(I,1) .EQ. 0.D0) THEN IER = .TRUE. RETURN ELSE SAVE2(I) = SAVE2(I)/A(I,1) END IF 150 CONTINUE DO 155 I = NDE+1,N 155 A(I,1) = 0.D0 END IF END IF DO 170 I = 1,NDE 170 SAVE1(I) = SAVE2(I)/YWT(I) SUM = DNRM2(NDE, SAVE1, 1) SUM0 = 1.D0/MAX(1.D0, ABS(T)) SMAX = MAX(SUM0, SUM) SMIN = MIN(SUM0, SUM) SUM = SMAX*SQRT(1.D0 + (SMIN/SMAX)**2)/SQRT(DBLE(NDE)) H = SIGN(MIN(2.D0*EPS/SUM, ABS(H)), H) DO 180 I = 1,N 180 YH(I,2) = H*SAVE2(I) IF (MITER .EQ. 2 .OR. MITER .EQ. 5 .OR. ISWFLG .EQ. 3) THEN DO 20 I = 1,N 20 FAC(I) = SQRT(UROUND) END IF ELSE IF (MITER .NE. MTROLD) THEN MTROLD = MITER RC = 0.D0 CONVRG = .FALSE. END IF IF (MINT .NE. MNTOLD) THEN MNTOLD = MINT OLDL0 = EL(1,NQ) CALL DDCST (MAXORD, MINT, ISWFLG, EL, TQ) RC = RC*EL(1,NQ)/OLDL0 NWAIT = NQ + 2 END IF IF (H .NE. HOLD) THEN NWAIT = NQ + 2 RH = H/HOLD CALL DDSCL (HMAX, N, NQ, RMAX, HOLD, RC, RH, YH) END IF END IF END SUBROUTINE DDPST (EL,F,FA,H,IMPL,JACOBN,MATDIM,MITER,ML,MU,N,NDE, 8 NQ,SAVE2,T,USERS,Y,YH,YWT,UROUND,NFE,NJE,A,DFDY,FAC,IER,IPVT, 8 SAVE1,ISWFLG,BND,JSTATE) C***BEGIN PROLOGUE DDPST C***REFER TO DDRIV3 C Subroutine DDPST is called to reevaluate the partials. C If MITER is 1, 2, 4, or 5, the matrix C P = I - L(0)*H*Jacobian is stored in DFDY and subjected to LU C decomposition, with the results also stored in DFDY. C***ROUTINES CALLED DGEFA,DGBFA,DNRM2 C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 870401 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDPST DOUBLE PRECISION A(MATDIM,*), BL, BND, BP, BR, BU, DFDY(MATDIM,*), 8 DFDYMX, DIFF, DY, EL(13,12), FAC(*), FACMAX, FACMIN, FACTOR, 8 H, SAVE1(*), SAVE2(*), SCALE, DNRM2, T, UROUND, Y(*), 8 YH(N,*), YJ, YS, YWT(*) INTEGER IPVT(*) LOGICAL IER PARAMETER(FACMAX = .5D0) C***FIRST EXECUTABLE STATEMENT DDPST NJE = NJE + 1 IER = .FALSE. IF (MITER .EQ. 1 .OR. MITER .EQ. 2) THEN IF (MITER .EQ. 1) THEN CALL JACOBN (N, T, Y, DFDY, MATDIM, ML, MU) IF (N .EQ. 0) THEN JSTATE = 8 RETURN END IF IF (ISWFLG .EQ. 3) BND = DNRM2(N*N, DFDY, 1) FACTOR = -EL(1,NQ)*H DO 110 J = 1,N DO 110 I = 1,N 110 DFDY(I,J) = FACTOR*DFDY(I,J) ELSE IF (MITER .EQ. 2) THEN BR = UROUND**(.875D0) BL = UROUND**(.75D0) BU = UROUND**(.25D0) BP = UROUND**(-.15D0) FACMIN = UROUND**(.78D0) DO 170 J = 1,N YS = MAX(ABS(YWT(J)), ABS(Y(J))) 120 DY = FAC(J)*YS IF (DY .EQ. 0.D0) THEN IF (FAC(J) .LT. FACMAX) THEN FAC(J) = MIN(100.D0*FAC(J), FACMAX) GO TO 120 ELSE DY = YS END IF END IF IF (NQ .EQ. 1) THEN DY = SIGN(DY, SAVE2(J)) ELSE DY = SIGN(DY, YH(J,3)) END IF DY = (Y(J) + DY) - Y(J) YJ = Y(J) Y(J) = Y(J) + DY CALL F (N, T, Y, SAVE1) IF (N .EQ. 0) THEN JSTATE = 6 RETURN END IF Y(J) = YJ FACTOR = -EL(1,NQ)*H/DY DO 140 I = 1,N 140 DFDY(I,J) = (SAVE1(I) - SAVE2(I))*FACTOR C Step 1 DIFF = ABS(SAVE2(1) - SAVE1(1)) IMAX = 1 DO 150 I = 2,N IF (ABS(SAVE2(I) - SAVE1(I)) .GT. DIFF) THEN IMAX = I DIFF = ABS(SAVE2(I) - SAVE1(I)) END IF 150 CONTINUE C Step 2 IF (MIN(ABS(SAVE2(IMAX)), ABS(SAVE1(IMAX))) .GT. 0.D0) THEN SCALE = MAX(ABS(SAVE2(IMAX)), ABS(SAVE1(IMAX))) C Step 3 IF (DIFF .GT. BU*SCALE) THEN FAC(J) = MAX(FACMIN, FAC(J)*.1D0) ELSE IF (BR*SCALE .LE. DIFF .AND. DIFF .LE. BL*SCALE) THEN FAC(J) = MIN(FAC(J)*10.D0, FACMAX) C Step 4 ELSE IF (DIFF .LT. BR*SCALE) THEN FAC(J) = MIN(BP*FAC(J), FACMAX) END IF END IF 170 CONTINUE IF (ISWFLG .EQ. 3) BND = DNRM2(N*N, DFDY, 1)/(-EL(1,NQ)*H) NFE = NFE + N END IF IF (IMPL .EQ. 0) THEN DO 190 I = 1,N 190 DFDY(I,I) = DFDY(I,I) + 1.D0 ELSE IF (IMPL .EQ. 1) THEN CALL FA (N, T, Y, A, MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF DO 210 J = 1,N DO 210 I = 1,N 210 DFDY(I,J) = DFDY(I,J) + A(I,J) ELSE IF (IMPL .EQ. 2) THEN CALL FA (N, T, Y, A, MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF DO 230 I = 1,NDE 230 DFDY(I,I) = DFDY(I,I) + A(I,1) END IF CALL DGEFA (DFDY, MATDIM, N, IPVT, INFO) IF (INFO .NE. 0) IER = .TRUE. ELSE IF (MITER .EQ. 4 .OR. MITER .EQ. 5) THEN IF (MITER .EQ. 4) THEN CALL JACOBN (N, T, Y, DFDY(ML+1,1), MATDIM, ML, MU) IF (N .EQ. 0) THEN JSTATE = 8 RETURN END IF FACTOR = -EL(1,NQ)*H MW = ML + MU + 1 DO 260 J = 1,N I1 = MAX(ML+1, MW+1-J) I2 = MIN(MW+N-J, MW+ML) DO 260 I = I1,I2 260 DFDY(I,J) = FACTOR*DFDY(I,J) ELSE IF (MITER .EQ. 5) THEN BR = UROUND**(.875D0) BL = UROUND**(.75D0) BU = UROUND**(.25D0) BP = UROUND**(-.15D0) FACMIN = UROUND**(.78D0) MW = ML + MU + 1 J2 = MIN(MW, N) DO 340 J = 1,J2 DO 290 K = J,N,MW YS = MAX(ABS(YWT(K)), ABS(Y(K))) 280 DY = FAC(K)*YS IF (DY .EQ. 0.D0) THEN IF (FAC(K) .LT. FACMAX) THEN FAC(K) = MIN(100.D0*FAC(K), FACMAX) GO TO 280 ELSE DY = YS END IF END IF IF (NQ .EQ. 1) THEN DY = SIGN(DY, SAVE2(K)) ELSE DY = SIGN(DY, YH(K,3)) END IF DY = (Y(K) + DY) - Y(K) DFDY(MW,K) = Y(K) 290 Y(K) = Y(K) + DY CALL F (N, T, Y, SAVE1) IF (N .EQ. 0) THEN JSTATE = 6 RETURN END IF DO 330 K = J,N,MW Y(K) = DFDY(MW,K) YS = MAX(ABS(YWT(K)), ABS(Y(K))) DY = FAC(K)*YS IF (DY .EQ. 0.D0) DY = YS IF (NQ .EQ. 1) THEN DY = SIGN(DY, SAVE2(K)) ELSE DY = SIGN(DY, YH(K,3)) END IF DY = (Y(K) + DY) - Y(K) FACTOR = -EL(1,NQ)*H/DY I1 = MAX(ML+1, MW+1-K) I2 = MIN(MW+N-K, MW+ML) DO 300 I = I1,I2 I3 = K + I - MW 300 DFDY(I,K) = FACTOR*(SAVE1(I3) - SAVE2(I3)) C Step 1 IMAX = MAX(1, K - MU) DIFF = ABS(SAVE2(IMAX) - SAVE1(IMAX)) I1 = IMAX I2 = MIN(K + ML, N) DO 310 I = I1+1,I2 IF (ABS(SAVE2(I) - SAVE1(I)) .GT. DIFF) THEN IMAX = I DIFF = ABS(SAVE2(I) - SAVE1(I)) END IF 310 CONTINUE C Step 2 IF (MIN(ABS(SAVE2(IMAX)), ABS(SAVE1(IMAX))) .GT.0.D0) THEN SCALE = MAX(ABS(SAVE2(IMAX)), ABS(SAVE1(IMAX))) C Step 3 IF (DIFF .GT. BU*SCALE) THEN FAC(K) = MAX(FACMIN, FAC(K)*.1D0) ELSE IF (BR*SCALE .LE.DIFF .AND. DIFF .LE.BL*SCALE) THEN FAC(K) = MIN(FAC(K)*10.D0, FACMAX) C Step 4 ELSE IF (DIFF .LT. BR*SCALE) THEN FAC(K) = MIN(BP*FAC(K), FACMAX) END IF END IF 330 CONTINUE 340 CONTINUE NFE = NFE + J2 END IF IF (ISWFLG .EQ. 3) THEN DFDYMX = 0.D0 DO 345 J = 1,N I1 = MAX(ML+1, MW+1-J) I2 = MIN(MW+N-J, MW+ML) DO 345 I = I1,I2 345 DFDYMX = MAX(DFDYMX, ABS(DFDY(I,J))) BND = 0.D0 IF (DFDYMX .NE. 0.D0) THEN DO 350 J = 1,N I1 = MAX(ML+1, MW+1-J) I2 = MIN(MW+N-J, MW+ML) DO 350 I = I1,I2 350 BND = BND + (DFDY(I,J)/DFDYMX)**2 BND = DFDYMX*SQRT(BND)/(-EL(1,NQ)*H) END IF END IF IF (IMPL .EQ. 0) THEN DO 360 J = 1,N 360 DFDY(MW,J) = DFDY(MW,J) + 1.D0 ELSE IF (IMPL .EQ. 1) THEN CALL FA (N, T, Y, A(ML+1,1), MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF DO 380 J = 1,N I1 = MAX(ML+1, MW+1-J) I2 = MIN(MW+N-J, MW+ML) DO 380 I = I1,I2 380 DFDY(I,J) = DFDY(I,J) + A(I,J) ELSE IF (IMPL .EQ. 2) THEN CALL FA (N, T, Y, A, MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF DO 400 J = 1,NDE 400 DFDY(MW,J) = DFDY(MW,J) + A(J,1) END IF CALL DGBFA (DFDY, MATDIM, N, ML, MU, IPVT, INFO) IF (INFO .NE. 0) IER = .TRUE. ELSE IF (MITER .EQ. 3) THEN IFLAG = 1 CALL USERS (Y, YH(1,2), YWT, SAVE1, SAVE2, T, H, EL(1,NQ), IMPL, 8 N, NDE, IFLAG) IF (N .EQ. 0) THEN JSTATE = 10 RETURN END IF END IF END SUBROUTINE DDCOR (DFDY,EL,FA,H,IMPL,IPVT,MATDIM,MITER,ML,MU,N, 8 NDE,NQ,T,USERS,Y,YH,YWT,EVALFA,SAVE1,SAVE2,A,D,JSTATE) C***BEGIN PROLOGUE DDCOR C***REFER TO DDRIV3 C Subroutine DDCOR is called to compute corrections to the Y array. C In the case of functional iteration, update Y directly from the C result of the last call to F. C In the case of the chord method, compute the corrector error and C solve the linear system with that as right hand side and DFDY as C coefficient matrix, using the LU decomposition if MITER is 1, 2, 4, C or 5. C***ROUTINES CALLED DGESL,DGBSL,DNRM2 C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 870401 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDCOR DOUBLE PRECISION A(MATDIM,*), D, DFDY(MATDIM,*), EL(13,12), H, 8 SAVE1(*), SAVE2(*), DNRM2, T, Y(*), YH(N,*), YWT(*) INTEGER IPVT(*) LOGICAL EVALFA C***FIRST EXECUTABLE STATEMENT DDCOR IF (MITER .EQ. 0) THEN DO 100 I = 1,N 100 SAVE1(I) = (H*SAVE2(I) - YH(I,2) - SAVE1(I))/YWT(I) D = DNRM2(N, SAVE1, 1)/SQRT(DBLE(N)) DO 105 I = 1,N 105 SAVE1(I) = H*SAVE2(I) - YH(I,2) ELSE IF (MITER .EQ. 1 .OR. MITER .EQ. 2) THEN IF (IMPL .EQ. 0) THEN DO 130 I = 1,N 130 SAVE2(I) = H*SAVE2(I) - YH(I,2) - SAVE1(I) ELSE IF (IMPL .EQ. 1) THEN IF (EVALFA) THEN CALL FA (N, T, Y, A, MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF ELSE EVALFA = .TRUE. END IF DO 150 I = 1,N 150 SAVE2(I) = H*SAVE2(I) DO 160 J = 1,N DO 160 I = 1,N 160 SAVE2(I) = SAVE2(I) - A(I,J)*(YH(J,2) + SAVE1(J)) ELSE IF (IMPL .EQ. 2) THEN IF (EVALFA) THEN CALL FA (N, T, Y, A, MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF ELSE EVALFA = .TRUE. END IF DO 180 I = 1,N 180 SAVE2(I) = H*SAVE2(I) - A(I,1)*(YH(I,2) + SAVE1(I)) END IF CALL DGESL (DFDY, MATDIM, N, IPVT, SAVE2, 0) DO 200 I = 1,N SAVE1(I) = SAVE1(I) + SAVE2(I) 200 SAVE2(I) = SAVE2(I)/YWT(I) D = DNRM2(N, SAVE2, 1)/SQRT(DBLE(N)) ELSE IF (MITER .EQ. 4 .OR. MITER .EQ. 5) THEN IF (IMPL .EQ. 0) THEN DO 230 I = 1,N 230 SAVE2(I) = H*SAVE2(I) - YH(I,2) - SAVE1(I) ELSE IF (IMPL .EQ. 1) THEN IF (EVALFA) THEN CALL FA (N, T, Y, A(ML+1,1), MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF ELSE EVALFA = .TRUE. END IF DO 250 I = 1,N 250 SAVE2(I) = H*SAVE2(I) MW = ML + 1 + MU DO 260 J = 1,N I1 = MAX(ML+1, MW+1-J) I2 = MIN(MW+N-J, MW+ML) DO 260 I = I1,I2 I3 = I + J - MW 260 SAVE2(I3) = SAVE2(I3) - A(I,J)*(YH(J,2) + SAVE1(J)) ELSE IF (IMPL .EQ. 2) THEN IF (EVALFA) THEN CALL FA (N, T, Y, A, MATDIM, ML, MU, NDE) IF (N .EQ. 0) THEN JSTATE = 9 RETURN END IF ELSE EVALFA = .TRUE. END IF DO 280 I = 1,N 280 SAVE2(I) = H*SAVE2(I) - A(I,1)*(YH(I,2) + SAVE1(I)) END IF CALL DGBSL (DFDY, MATDIM, N, ML, MU, IPVT, SAVE2, 0) DO 300 I = 1,N SAVE1(I) = SAVE1(I) + SAVE2(I) 300 SAVE2(I) = SAVE2(I)/YWT(I) D = DNRM2(N, SAVE2, 1)/SQRT(DBLE(N)) ELSE IF (MITER .EQ. 3) THEN IFLAG = 2 CALL USERS (Y, YH(1,2), YWT, SAVE1, SAVE2, T, H, EL(1,NQ), IMPL, 8 N, NDE, IFLAG) IF (N .EQ. 0) THEN JSTATE = 10 RETURN END IF DO 320 I = 1,N SAVE1(I) = SAVE1(I) + SAVE2(I) 320 SAVE2(I) = SAVE2(I)/YWT(I) D = DNRM2(N, SAVE2, 1)/SQRT(DBLE(N)) END IF END SUBROUTINE DDCST (MAXORD,MINT,ISWFLG,EL,TQ) C***BEGIN PROLOGUE DDCST C***REFER TO DDRIV3 C DDCST is called by DDNTL and sets coefficients used by the core C integrator DDSTP. The array EL determines the basic method. C The array TQ is involved in adjusting the step size in relation C to truncation error. EL and TQ depend upon MINT, and are calculated C for orders 1 to MAXORD(.LE. 12). For each order NQ, the coefficients C EL are calculated from the generating polynomial: C L(T) = EL(1,NQ) + EL(2,NQ)*T + ... + EL(NQ+1,NQ)*T**NQ. C For the implicit Adams methods, L(T) is given by C dL/dT = (1+T)*(2+T)* ... *(NQ-1+T)/K, L(-1) = 0, C where K = factorial(NQ-1). C For the Gear methods, C L(T) = (1+T)*(2+T)* ... *(NQ+T)/K, C where K = factorial(NQ)*(1 + 1/2 + ... + 1/NQ). C For each order NQ, there are three components of TQ. C***ROUTINES CALLED (NONE) C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 870216 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDCST DOUBLE PRECISION EL(13,12), FACTRL(12), GAMMA(14), SUM, TQ(3,12) C***FIRST EXECUTABLE STATEMENT DDCST FACTRL(1) = 1.D0 DO 10 I = 2,MAXORD 10 FACTRL(I) = DBLE(I)*FACTRL(I-1) C COMPUTE ADAMS COEFFICIENTS IF (MINT .EQ. 1) THEN GAMMA(1) = 1.D0 DO 40 I = 1,MAXORD+1 SUM = 0.D0 DO 30 J = 1,I 30 SUM = SUM - GAMMA(J)/DBLE(I-J+2) 40 GAMMA(I+1) = SUM EL(1,1) = 1.D0 EL(2,1) = 1.D0 EL(2,2) = 1.D0 EL(3,2) = 1.D0 DO 60 J = 3,MAXORD EL(2,J) = FACTRL(J-1) DO 50 I = 3,J 50 EL(I,J) = DBLE(J-1)*EL(I,J-1) + EL(I-1,J-1) 60 EL(J+1,J) = 1.D0 DO 80 J = 2,MAXORD EL(1,J) = EL(1,J-1) + GAMMA(J) EL(2,J) = 1.D0 DO 80 I = 3,J+1 80 EL(I,J) = EL(I,J)/(DBLE(I-1)*FACTRL(J-1)) DO 100 J = 1,MAXORD TQ(1,J) = -1.D0/(FACTRL(J)*GAMMA(J)) TQ(2,J) = -1.D0/GAMMA(J+1) 100 TQ(3,J) = -1.D0/GAMMA(J+2) C COMPUTE GEAR COEFFICIENTS ELSE IF (MINT .EQ. 2) THEN EL(1,1) = 1.D0 EL(2,1) = 1.D0 DO 130 J = 2,MAXORD EL(1,J) = FACTRL(J) DO 120 I = 2,J 120 EL(I,J) = DBLE(J)*EL(I,J-1) + EL(I-1,J-1) 130 EL(J+1,J) = 1.D0 SUM = 1.D0 DO 150 J = 2,MAXORD SUM = SUM + 1.D0/DBLE(J) DO 150 I = 1,J+1 150 EL(I,J) = EL(I,J)/(FACTRL(J)*SUM) DO 170 J = 1,MAXORD IF (J .GT. 1) TQ(1,J) = 1.D0/FACTRL(J-1) TQ(2,J) = DBLE(J+1)/EL(1,J) 170 TQ(3,J) = DBLE(J+2)/EL(1,J) END IF C Compute constants used in the stiffness test. C These are the ratio of TQ(2,NQ) for the Gear C methods to those for the Adams methods. IF (ISWFLG .EQ. 3) THEN MXRD = MIN(MAXORD, 5) IF (MINT .EQ. 2) THEN GAMMA(1) = 1.D0 DO 190 I = 1,MXRD SUM = 0.D0 DO 180 J = 1,I 180 SUM = SUM - GAMMA(J)/DBLE(I-J+2) 190 GAMMA(I+1) = SUM END IF SUM = 1.D0 DO 200 I = 2,MXRD SUM = SUM + 1.D0/DBLE(I) 200 EL(1+I,1) = -DBLE(I+1)*SUM*GAMMA(I+1) END IF END SUBROUTINE DDSCL (HMAX,N,NQ,RMAX,H,RC,RH,YH) C***BEGIN PROLOGUE DDSCL C***REFER TO DDRIV3 C This subroutine rescales the YH array whenever the step size C is changed. C***ROUTINES CALLED (NONE) C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 850319 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDSCL DOUBLE PRECISION H, HMAX, RC, RH, RMAX, R1, YH(N,*) C***FIRST EXECUTABLE STATEMENT DDSCL IF (H .LT. 1.D0) THEN RH = MIN(ABS(H)*RH, ABS(H)*RMAX, HMAX)/ABS(H) ELSE RH = MIN(RH, RMAX, HMAX/ABS(H)) END IF R1 = 1.D0 DO 10 J = 1,NQ R1 = R1*RH DO 10 I = 1,N 10 YH(I,J+1) = YH(I,J+1)*R1 H = H*RH RC = RC*RH END SUBROUTINE DDPSC (KSGN,N,NQ,YH) C***BEGIN PROLOGUE DDPSC C***REFER TO DDRIV3 C This subroutine computes the predicted YH values by effectively C multiplying the YH array by the Pascal triangle matrix when KSGN C is +1, and performs the inverse function when KSGN is -1. C***ROUTINES CALLED (NONE) C***DATE WRITTEN 790601 (YYMMDD) C***REVISION DATE 841119 (YYMMDD) C***CATEGORY NO. I1A2,I1A1B C***AUTHOR KAHANER, D. K., NATIONAL BUREAU OF STANDARDS, C SUTHERLAND, C. D., LOS ALAMOS NATIONAL LABORATORY C***END PROLOGUE DDPSC DOUBLE PRECISION YH(N,*) C***FIRST EXECUTABLE STATEMENT DDPSC IF (KSGN .GT. 0) THEN DO 10 J1 = 1,NQ DO 10 J2 = J1,NQ J = NQ - J2 + J1 DO 10 I = 1,N 10 YH(I,J) = YH(I,J) + YH(I,J+1) ELSE DO 30 J1 = 1,NQ DO 30 J2 = J1,NQ J = NQ - J2 + J1 DO 30 I = 1,N 30 YH(I,J) = YH(I,J) - YH(I,J+1) END IF END SUBROUTINE DGBFA(ABD,LDA,N,ML,MU,IPVT,INFO) C***BEGIN PROLOGUE DGBFA C***DATE WRITTEN 780814 (YYMMDD) C***REVISION DATE 820801 (YYMMDD) C***CATEGORY NO. D2A2 C***KEYWORDS BANDED,DOUBLE PRECISION,FACTOR,LINEAR ALGEBRA,LINPACK, C MATRIX C***AUTHOR MOLER, C. B., (U. OF NEW MEXICO) C***PURPOSE Factors a double precision BAND matrix by elimination. C***DESCRIPTION C From the book "Numerical Methods and Software" C by D. Kahaner, C. Moler, S. Nash C Prentice Hall 1988 C C DGBFA factors a double precision band matrix by elimination. C C DGBFA is usually called by DGBCO, but it can be called C directly with a saving in time if RCOND is not needed. C C On Entry C C ABD DOUBLE PRECISION(LDA, N) C contains the matrix in band storage. The columns C of the matrix are stored in the columns of ABD and C the diagonals of the matrix are stored in rows C ML+1 through 2*ML+MU+1 of ABD . C See the comments below for details. C C LDA INTEGER C the leading dimension of the array ABD . C LDA must be .GE. 2*ML + MU + 1 . C C N INTEGER C the order of the original matrix. C C ML INTEGER C number of diagonals below the main diagonal. C 0 .LE. ML .LT. N . C C MU INTEGER C number of diagonals above the main diagonal. C 0 .LE. MU .LT. N . C More efficient if ML .LE. MU . C On Return C C ABD an upper triangular matrix in band storage and C the multipliers which were used to obtain it. C The factorization can be written A = L*U where C L is a product of permutation and unit lower C triangular matrices and U is upper triangular. C C IPVT INTEGER(N) C an integer vector of pivot indices. C C INFO INTEGER C = 0 normal value. C = K if U(K,K) .EQ. 0.0 . This is not an error C condition for this subroutine, but it does C indicate that DGBSL will divide by zero if C called. Use RCOND in DGBCO for a reliable C indication of singularity. C C Band Storage C C If A is a band matrix, the following program segment C will set up the input. C C ML = (band width below the diagonal) C MU = (band width above the diagonal) C M = ML + MU + 1 C DO 20 J = 1, N C I1 = MAX0(1, J-MU) C I2 = MIN0(N, J+ML) C DO 10 I = I1, I2 C K = I - J + M C ABD(K,J) = A(I,J) C 10 CONTINUE C 20 CONTINUE C C This uses rows ML+1 through 2*ML+MU+1 of ABD . C In addition, the first ML rows in ABD are used for C elements generated during the triangularization. C The total number of rows needed in ABD is 2*ML+MU+1 . C The ML+MU by ML+MU upper left triangle and the C ML by ML lower right triangle are not referenced. C C LINPACK. This version dated 08/14/78 . C Cleve Moler, University of New Mexico, Argonne National Lab. C C Subroutines and Functions C C BLAS DAXPY,DSCAL,IDAMAX C Fortran MAX0,MIN0 C***REFERENCES DONGARRA J.J., BUNCH J.R., MOLER C.B., STEWART G.W., C *LINPACK USERS GUIDE*, SIAM, 1979. C***ROUTINES CALLED DAXPY,DSCAL,IDAMAX C***END PROLOGUE DGBFA INTEGER LDA,N,ML,MU,IPVT(1),INFO DOUBLE PRECISION ABD(LDA,1) C DOUBLE PRECISION T INTEGER I,IDAMAX,I0,J,JU,JZ,J0,J1,K,KP1,L,LM,M,MM,NM1 C C***FIRST EXECUTABLE STATEMENT DGBFA M = ML + MU + 1 INFO = 0 C C ZERO INITIAL FILL-IN COLUMNS C J0 = MU + 2 J1 = MIN0(N,M) - 1 IF (J1 .LT. J0) GO TO 30 DO 20 JZ = J0, J1 I0 = M + 1 - JZ DO 10 I = I0, ML ABD(I,JZ) = 0.0D0 10 CONTINUE 20 CONTINUE 30 CONTINUE JZ = J1 JU = 0 C C GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING C NM1 = N - 1 IF (NM1 .LT. 1) GO TO 130 DO 120 K = 1, NM1 KP1 = K + 1 C C ZERO NEXT FILL-IN COLUMN C JZ = JZ + 1 IF (JZ .GT. N) GO TO 50 IF (ML .LT. 1) GO TO 50 DO 40 I = 1, ML ABD(I,JZ) = 0.0D0 40 CONTINUE 50 CONTINUE C C FIND L = PIVOT INDEX C LM = MIN0(ML,N-K) L = IDAMAX(LM+1,ABD(M,K),1) + M - 1 IPVT(K) = L + K - M C C ZERO PIVOT IMPLIES THIS COLUMN ALREADY TRIANGULARIZED C IF (ABD(L,K) .EQ. 0.0D0) GO TO 100 C C INTERCHANGE IF NECESSARY C IF (L .EQ. M) GO TO 60 T = ABD(L,K) ABD(L,K) = ABD(M,K) ABD(M,K) = T 60 CONTINUE C C COMPUTE MULTIPLIERS C T = -1.0D0/ABD(M,K) CALL DSCAL(LM,T,ABD(M+1,K),1) C C ROW ELIMINATION WITH COLUMN INDEXING C JU = MIN0(MAX0(JU,MU+IPVT(K)),N) MM = M IF (JU .LT. KP1) GO TO 90 DO 80 J = KP1, JU L = L - 1 MM = MM - 1 T = ABD(L,J) IF (L .EQ. MM) GO TO 70 ABD(L,J) = ABD(MM,J) ABD(MM,J) = T 70 CONTINUE CALL DAXPY(LM,T,ABD(M+1,K),1,ABD(MM+1,J),1) 80 CONTINUE 90 CONTINUE GO TO 110 100 CONTINUE INFO = K 110 CONTINUE 120 CONTINUE 130 CONTINUE IPVT(N) = N IF (ABD(M,N) .EQ. 0.0D0) INFO = N RETURN END SUBROUTINE DGBSL(ABD,LDA,N,ML,MU,IPVT,B,JOB) C***BEGIN PROLOGUE DGBSL C***DATE WRITTEN 780814 (YYMMDD) C***REVISION DATE 820801 (YYMMDD) C***CATEGORY NO. D2A2 C***KEYWORDS BANDED,DOUBLE PRECISION,LINEAR ALGEBRA,LINPACK,MATRIX, C SOLVE C***AUTHOR MOLER, C. B., (U. OF NEW MEXICO) C***PURPOSE Solves the double precision BAND system A*X=B or C TRANS(A)*X=B using the factors computed by DGBCO or DGBFA. C***DESCRIPTION C From the book "Numerical Methods and Software" C by D. Kahaner, C. Moler, S. Nash C Prentice Hall 1988 C C DGBSL solves the double precision band system C A * X = B or TRANS(A) * X = B C using the factors computed by DGBCO or DGBFA. C C On Entry C C ABD DOUBLE PRECISION(LDA, N) C the output from DGBCO or DGBFA. C C LDA INTEGER C the leading dimension of the array ABD . C C N INTEGER C the order of the original matrix. C C ML INTEGER C number of diagonals below the main diagonal. C C MU INTEGER C number of diagonals above the main diagonal. C C IPVT INTEGER(N) C the pivot vector from DGBCO or DGBFA. C C B DOUBLE PRECISION(N) C the right hand side vector. C C JOB INTEGER C = 0 to solve A*X = B , C = nonzero to solve TRANS(A)*X = B , where C TRANS(A) is the transpose. C C On Return C C B the solution vector X . C C Error Condition C C A division by zero will occur if the input factor contains a C zero on the diagonal. Technically this indicates singularity C but it is often caused by improper arguments or improper C setting of LDA . It will not occur if the subroutines are C called correctly and if DGBCO has set RCOND .GT. 0.0 C or DGBFA has set INFO .EQ. 0 . C C To compute INVERSE(A) * C where C is a matrix C with P columns C CALL DGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z) C IF (RCOND is too small) GO TO ... C DO 10 J = 1, P C CALL DGBSL(ABD,LDA,N,ML,MU,IPVT,C(1,J),0) C 10 CONTINUE C C LINPACK. This version dated 08/14/78 . C Cleve Moler, University of New Mexico, Argonne National Lab. C C Subroutines and Functions C C BLAS DAXPY,DDOT C Fortran MIN0 C***REFERENCES DONGARRA J.J., BUNCH J.R., MOLER C.B., STEWART G.W., C *LINPACK USERS GUIDE*, SIAM, 1979. C***ROUTINES CALLED DAXPY,DDOT C***END PROLOGUE DGBSL INTEGER LDA,N,ML,MU,IPVT(1),JOB DOUBLE PRECISION ABD(LDA,1),B(1) C DOUBLE PRECISION DDOT,T INTEGER K,KB,L,LA,LB,LM,M,NM1 C***FIRST EXECUTABLE STATEMENT DGBSL M = MU + ML + 1 NM1 = N - 1 IF (JOB .NE. 0) GO TO 50 C C JOB = 0 , SOLVE A * X = B C FIRST SOLVE L*Y = B C IF (ML .EQ. 0) GO TO 30 IF (NM1 .LT. 1) GO TO 30 DO 20 K = 1, NM1 LM = MIN0(ML,N-K) L = IPVT(K) T = B(L) IF (L .EQ. K) GO TO 10 B(L) = B(K) B(K) = T 10 CONTINUE CALL DAXPY(LM,T,ABD(M+1,K),1,B(K+1),1) 20 CONTINUE 30 CONTINUE C C NOW SOLVE U*X = Y C DO 40 KB = 1, N K = N + 1 - KB B(K) = B(K)/ABD(M,K) LM = MIN0(K,M) - 1 LA = M - LM LB = K - LM T = -B(K) CALL DAXPY(LM,T,ABD(LA,K),1,B(LB),1) 40 CONTINUE GO TO 100 50 CONTINUE C C JOB = NONZERO, SOLVE TRANS(A) * X = B C FIRST SOLVE TRANS(U)*Y = B C DO 60 K = 1, N LM = MIN0(K,M) - 1 LA = M - LM LB = K - LM T = DDOT(LM,ABD(LA,K),1,B(LB),1) B(K) = (B(K) - T)/ABD(M,K) 60 CONTINUE C C NOW SOLVE TRANS(L)*X = Y C IF (ML .EQ. 0) GO TO 90 IF (NM1 .LT. 1) GO TO 90 DO 80 KB = 1, NM1 K = N - KB LM = MIN0(ML,N-K) B(K) = B(K) + DDOT(LM,ABD(M+1,K),1,B(K+1),1) L = IPVT(K) IF (L .EQ. K) GO TO 70 T = B(L) B(L) = B(K) B(K) = T 70 CONTINUE 80 CONTINUE 90 CONTINUE 100 CONTINUE RETURN END