SUBROUTINE DASKR (RES, NEQ, T, Y, YPRIME, TOUT, INFO, RTOL, ATOL, * IDID, RWORK, LRW, IWORK, LIW, RPAR, IPAR, JAC, PSOL, * RT, NRT, JROOT) C C***BEGIN PROLOGUE DDASKR C***REVISION HISTORY (YYMMDD) C 020815 DATE WRITTEN C 021105 Changed yprime argument in DRCHEK calls to YPRIME. C 021217 Modified error return for zeros found too close together. C 021217 Added root direction output in JROOT. C 040518 Changed adjustment to X2 in Subr. DROOTS. C 050511 Revised stopping tests in statements 530 - 580; reordered C to test for tn at tstop before testing for tn past tout. C 060712 In DMATD, changed minimum D.Q. increment to 1/EWT(j). C 071003 In DRCHEK, fixed bug in TEMP2 (HMINR) below 110. C 110608 In DRCHEK, fixed bug in setting of T1 at 300. C***CATEGORY NO. I1A2 C***KEYWORDS DIFFERENTIAL/ALGEBRAIC, BACKWARD DIFFERENTIATION FORMULAS, C IMPLICIT DIFFERENTIAL SYSTEMS, KRYLOV ITERATION C***AUTHORS Linda R. Petzold, Peter N. Brown, Alan C. Hindmarsh, and C Clement W. Ulrich C Center for Computational Sciences & Engineering, L-316 C Lawrence Livermore National Laboratory C P.O. Box 808, C Livermore, CA 94551 C***PURPOSE This code solves a system of differential/algebraic C equations of the form C G(t,y,y') = 0 , C using a combination of Backward Differentiation Formula C (BDF) methods and a choice of two linear system solution C methods: direct (dense or band) or Krylov (iterative). C This version is in double precision. C----------------------------------------------------------------------- C***DESCRIPTION C C *Usage: C C IMPLICIT DOUBLE PRECISION(A-H,O-Z) C INTEGER NEQ, INFO(N), IDID, LRW, LIW, IWORK(LIW), IPAR(*) C DOUBLE PRECISION T, Y(*), YPRIME(*), TOUT, RTOL(*), ATOL(*), C RWORK(LRW), RPAR(*) C EXTERNAL RES, JAC, PSOL, RT C C CALL DDASKR (RES, NEQ, T, Y, YPRIME, TOUT, INFO, RTOL, ATOL, C * IDID, RWORK, LRW, IWORK, LIW, RPAR, IPAR, JAC, PSOL, C * RT, NRT, JROOT) C C Quantities which may be altered by the code are: C T, Y(*), YPRIME(*), INFO(1), RTOL, ATOL, IDID, RWORK(*), IWORK(*) C C C *Arguments: C C RES:EXT This is the name of a subroutine which you C provide to define the residual function G(t,y,y') C of the differential/algebraic system. C C NEQ:IN This is the number of equations in the system. C C T:INOUT This is the current value of the independent C variable. C C Y(*):INOUT This array contains the solution components at T. C C YPRIME(*):INOUT This array contains the derivatives of the solution C components at T. C C TOUT:IN This is a point at which a solution is desired. C C INFO(N):IN This is an integer array used to communicate details C of how the solution is to be carried out, such as C tolerance type, matrix structure, step size and C order limits, and choice of nonlinear system method. C N must be at least 20. C C RTOL,ATOL:INOUT These quantities represent absolute and relative C error tolerances (on local error) which you provide C to indicate how accurately you wish the solution to C be computed. You may choose them to be both scalars C or else both arrays of length NEQ. C C IDID:OUT This integer scalar is an indicator reporting what C the code did. You must monitor this variable to C decide what action to take next. C C RWORK:WORK A real work array of length LRW which provides the C code with needed storage space. C C LRW:IN The length of RWORK. C C IWORK:WORK An integer work array of length LIW which provides C the code with needed storage space. C C LIW:IN The length of IWORK. C C RPAR,IPAR:IN These are real and integer parameter arrays which C you can use for communication between your calling C program and the RES, JAC, and PSOL subroutines. C C JAC:EXT This is the name of a subroutine which you may C provide (optionally) for calculating Jacobian C (partial derivative) data involved in solving linear C systems within DDASKR. C C PSOL:EXT This is the name of a subroutine which you must C provide for solving linear systems if you selected C a Krylov method. The purpose of PSOL is to solve C linear systems involving a left preconditioner P. C C RT:EXT This is the name of the subroutine for defining C constraint functions Ri(T,Y,Y')) whose roots are C desired during the integration. This name must be C declared external in the calling program. C C NRT:IN This is the number of constraint functions C Ri(T,Y,Y'). If there are no constraints, set C NRT = 0, and pass a dummy name for RT. C C JROOT:OUT This is an integer array of length NRT for output C of root information. C C *Overview C C The DDASKR solver uses the backward differentiation formulas of C orders one through five to solve a system of the form G(t,y,y') = 0 C for y = Y and y' = YPRIME. Values for Y and YPRIME at the initial C time must be given as input. These values should be consistent, C that is, if T, Y, YPRIME are the given initial values, they should C satisfy G(T,Y,YPRIME) = 0. However, if consistent values are not C known, in many cases you can have DDASKR solve for them -- see C INFO(11). (This and other options are described in detail below.) C C Normally, DDASKR solves the system from T to TOUT. It is easy to C continue the solution to get results at additional TOUT. This is C the interval mode of operation. Intermediate results can also be C obtained easily by specifying INFO(3). C C On each step taken by DDASKR, a sequence of nonlinear algebraic C systems arises. These are solved by one of two types of C methods: C * a Newton iteration with a direct method for the linear C systems involved (INFO(12) = 0), or C * a Newton iteration with a preconditioned Krylov iterative C method for the linear systems involved (INFO(12) = 1). C C The direct method choices are dense and band matrix solvers, C with either a user-supplied or an internal difference quotient C Jacobian matrix, as specified by INFO(5) and INFO(6). C In the band case, INFO(6) = 1, you must supply half-bandwidths C in IWORK(1) and IWORK(2). C C The Krylov method is the Generalized Minimum Residual (GMRES) C method, in either complete or incomplete form, and with C scaling and preconditioning. The method is implemented C in an algorithm called SPIGMR. Certain options in the Krylov C method case are specified by INFO(13) and INFO(15). C C If the Krylov method is chosen, you may supply a pair of routines, C JAC and PSOL, to apply preconditioning to the linear system. C If the system is A*x = b, the matrix is A = dG/dY + CJ*dG/dYPRIME C (of order NEQ). This system can then be preconditioned in the form C (P-inverse)*A*x = (P-inverse)*b, with left preconditioner P. C (DDASKR does not allow right preconditioning.) C Then the Krylov method is applied to this altered, but equivalent, C linear system, hopefully with much better performance than without C preconditioning. (In addition, a diagonal scaling matrix based on C the tolerances is also introduced into the altered system.) C C The JAC routine evaluates any data needed for solving systems C with coefficient matrix P, and PSOL carries out that solution. C In any case, in order to improve convergence, you should try to C make P approximate the matrix A as much as possible, while keeping C the system P*x = b reasonably easy and inexpensive to solve for x, C given a vector b. C C While integrating the given DAE system, DDASKR also searches for C roots of the given constraint functions Ri(T,Y,Y') given by RT. C If DDASKR detects a sign change in any Ri(T,Y,Y'), it will return C the intermediate value of T and Y for which Ri(T,Y,Y') = 0. C Caution: If some Ri has a root at or very near the initial time, C DDASKR may fail to find it, or may find extraneous roots there, C because it does not yet have a sufficient history of the solution. C C *Description C C------INPUT - WHAT TO DO ON THE FIRST CALL TO DDASKR------------------- C C C The first call of the code is defined to be the start of each new C problem. Read through the descriptions of all the following items, C provide sufficient storage space for designated arrays, set C appropriate variables for the initialization of the problem, and C give information about how you want the problem to be solved. C C C RES -- Provide a subroutine of the form C C SUBROUTINE RES (T, Y, YPRIME, CJ, DELTA, IRES, RPAR, IPAR) C C to define the system of differential/algebraic C equations which is to be solved. For the given values C of T, Y and YPRIME, the subroutine should return C the residual of the differential/algebraic system C DELTA = G(T,Y,YPRIME) C DELTA is a vector of length NEQ which is output from RES. C C Subroutine RES must not alter T, Y, YPRIME, or CJ. C You must declare the name RES in an EXTERNAL C statement in your program that calls DDASKR. C You must dimension Y, YPRIME, and DELTA in RES. C C The input argument CJ can be ignored, or used to rescale C constraint equations in the system (see Ref. 2, p. 145). C Note: In this respect, DDASKR is not downward-compatible C with DDASSL, which does not have the RES argument CJ. C C IRES is an integer flag which is always equal to zero C on input. Subroutine RES should alter IRES only if it C encounters an illegal value of Y or a stop condition. C Set IRES = -1 if an input value is illegal, and DDASKR C will try to solve the problem without getting IRES = -1. C If IRES = -2, DDASKR will return control to the calling C program with IDID = -11. C C RPAR and IPAR are real and integer parameter arrays which C you can use for communication between your calling program C and subroutine RES. They are not altered by DDASKR. If you C do not need RPAR or IPAR, ignore these parameters by treat- C ing them as dummy arguments. If you do choose to use them, C dimension them in your calling program and in RES as arrays C of appropriate length. C C NEQ -- Set it to the number of equations in the system (NEQ .GE. 1). C C T -- Set it to the initial point of the integration. (T must be C a variable.) C C Y(*) -- Set this array to the initial values of the NEQ solution C components at the initial point. You must dimension Y of C length at least NEQ in your calling program. C C YPRIME(*) -- Set this array to the initial values of the NEQ first C derivatives of the solution components at the initial C point. You must dimension YPRIME at least NEQ in your C calling program. C C TOUT - Set it to the first point at which a solution is desired. C You cannot take TOUT = T. Integration either forward in T C (TOUT .GT. T) or backward in T (TOUT .LT. T) is permitted. C C The code advances the solution from T to TOUT using step C sizes which are automatically selected so as to achieve the C desired accuracy. If you wish, the code will return with the C solution and its derivative at intermediate steps (the C intermediate-output mode) so that you can monitor them, C but you still must provide TOUT in accord with the basic C aim of the code. C C The first step taken by the code is a critical one because C it must reflect how fast the solution changes near the C initial point. The code automatically selects an initial C step size which is practically always suitable for the C problem. By using the fact that the code will not step past C TOUT in the first step, you could, if necessary, restrict the C length of the initial step. C C For some problems it may not be permissible to integrate C past a point TSTOP, because a discontinuity occurs there C or the solution or its derivative is not defined beyond C TSTOP. When you have declared a TSTOP point (see INFO(4) C and RWORK(1)), you have told the code not to integrate past C TSTOP. In this case any tout beyond TSTOP is invalid input. C C INFO(*) - Use the INFO array to give the code more details about C how you want your problem solved. This array should be C dimensioned of length 20, though DDASKR uses only the C first 15 entries. You must respond to all of the following C items, which are arranged as questions. The simplest use C of DDASKR corresponds to setting all entries of INFO to 0. C C INFO(1) - This parameter enables the code to initialize itself. C You must set it to indicate the start of every new C problem. C C **** Is this the first call for this problem ... C yes - set INFO(1) = 0 C no - not applicable here. C See below for continuation calls. **** C C INFO(2) - How much accuracy you want of your solution C is specified by the error tolerances RTOL and ATOL. C The simplest use is to take them both to be scalars. C To obtain more flexibility, they can both be arrays. C The code must be told your choice. C C **** Are both error tolerances RTOL, ATOL scalars ... C yes - set INFO(2) = 0 C and input scalars for both RTOL and ATOL C no - set INFO(2) = 1 C and input arrays for both RTOL and ATOL **** C C INFO(3) - The code integrates from T in the direction of TOUT C by steps. If you wish, it will return the computed C solution and derivative at the next intermediate step C (the intermediate-output mode) or TOUT, whichever comes C first. This is a good way to proceed if you want to C see the behavior of the solution. If you must have C solutions at a great many specific TOUT points, this C code will compute them efficiently. C C **** Do you want the solution only at C TOUT (and not at the next intermediate step) ... C yes - set INFO(3) = 0 (interval-output mode) C no - set INFO(3) = 1 (intermediate-output mode) **** C C INFO(4) - To handle solutions at a great many specific C values TOUT efficiently, this code may integrate past C TOUT and interpolate to obtain the result at TOUT. C Sometimes it is not possible to integrate beyond some C point TSTOP because the equation changes there or it is C not defined past TSTOP. Then you must tell the code C this stop condition. C C **** Can the integration be carried out without any C restrictions on the independent variable T ... C yes - set INFO(4) = 0 C no - set INFO(4) = 1 C and define the stopping point TSTOP by C setting RWORK(1) = TSTOP **** C C INFO(5) - used only when INFO(12) = 0 (direct methods). C To solve differential/algebraic systems you may wish C to use a matrix of partial derivatives of the C system of differential equations. If you do not C provide a subroutine to evaluate it analytically (see C description of the item JAC in the call list), it will C be approximated by numerical differencing in this code. C Although it is less trouble for you to have the code C compute partial derivatives by numerical differencing, C the solution will be more reliable if you provide the C derivatives via JAC. Usually numerical differencing is C more costly than evaluating derivatives in JAC, but C sometimes it is not - this depends on your problem. C C **** Do you want the code to evaluate the partial deriv- C atives automatically by numerical differences ... C yes - set INFO(5) = 0 C no - set INFO(5) = 1 C and provide subroutine JAC for evaluating the C matrix of partial derivatives **** C C INFO(6) - used only when INFO(12) = 0 (direct methods). C DDASKR will perform much better if the matrix of C partial derivatives, dG/dY + CJ*dG/dYPRIME (here CJ is C a scalar determined by DDASKR), is banded and the code C is told this. In this case, the storage needed will be C greatly reduced, numerical differencing will be performed C much cheaper, and a number of important algorithms will C execute much faster. The differential equation is said C to have half-bandwidths ML (lower) and MU (upper) if C equation i involves only unknowns Y(j) with C i-ML .le. j .le. i+MU . C For all i=1,2,...,NEQ. Thus, ML and MU are the widths C of the lower and upper parts of the band, respectively, C with the main diagonal being excluded. If you do not C indicate that the equation has a banded matrix of partial C derivatives the code works with a full matrix of NEQ**2 C elements (stored in the conventional way). Computations C with banded matrices cost less time and storage than with C full matrices if 2*ML+MU .lt. NEQ. If you tell the C code that the matrix of partial derivatives has a banded C structure and you want to provide subroutine JAC to C compute the partial derivatives, then you must be careful C to store the elements of the matrix in the special form C indicated in the description of JAC. C C **** Do you want to solve the problem using a full (dense) C matrix (and not a special banded structure) ... C yes - set INFO(6) = 0 C no - set INFO(6) = 1 C and provide the lower (ML) and upper (MU) C bandwidths by setting C IWORK(1)=ML C IWORK(2)=MU **** C C INFO(7) - You can specify a maximum (absolute value of) C stepsize, so that the code will avoid passing over very C large regions. C C **** Do you want the code to decide on its own the maximum C stepsize ... C yes - set INFO(7) = 0 C no - set INFO(7) = 1 C and define HMAX by setting C RWORK(2) = HMAX **** C C INFO(8) - Differential/algebraic problems may occasionally C suffer from severe scaling difficulties on the first C step. If you know a great deal about the scaling of C your problem, you can help to alleviate this problem C by specifying an initial stepsize H0. C C **** Do you want the code to define its own initial C stepsize ... C yes - set INFO(8) = 0 C no - set INFO(8) = 1 C and define H0 by setting C RWORK(3) = H0 **** C C INFO(9) - If storage is a severe problem, you can save some C storage by restricting the maximum method order MAXORD. C The default value is 5. For each order decrease below 5, C the code requires NEQ fewer locations, but it is likely C to be slower. In any case, you must have C 1 .le. MAXORD .le. 5. C **** Do you want the maximum order to default to 5 ... C yes - set INFO(9) = 0 C no - set INFO(9) = 1 C and define MAXORD by setting C IWORK(3) = MAXORD **** C C INFO(10) - If you know that certain components of the C solutions to your equations are always nonnegative C (or nonpositive), it may help to set this C parameter. There are three options that are C available: C 1. To have constraint checking only in the initial C condition calculation. C 2. To enforce nonnegativity in Y during the integration. C 3. To enforce both options 1 and 2. C C When selecting option 2 or 3, it is probably best to try C the code without using this option first, and only use C this option if that does not work very well. C C **** Do you want the code to solve the problem without C invoking any special inequality constraints ... C yes - set INFO(10) = 0 C no - set INFO(10) = 1 to have option 1 enforced C no - set INFO(10) = 2 to have option 2 enforced C no - set INFO(10) = 3 to have option 3 enforced **** C C If you have specified INFO(10) = 1 or 3, then you C will also need to identify how each component of Y C in the initial condition calculation is constrained. C You must set: C IWORK(40+I) = +1 if Y(I) must be .GE. 0, C IWORK(40+I) = +2 if Y(I) must be .GT. 0, C IWORK(40+I) = -1 if Y(I) must be .LE. 0, while C IWORK(40+I) = -2 if Y(I) must be .LT. 0, while C IWORK(40+I) = 0 if Y(I) is not constrained. C C INFO(11) - DDASKR normally requires the initial T, Y, and C YPRIME to be consistent. That is, you must have C G(T,Y,YPRIME) = 0 at the initial T. If you do not know C the initial conditions precisely, in some cases C DDASKR may be able to compute it. C C Denoting the differential variables in Y by Y_d C and the algebraic variables by Y_a, DDASKR can solve C one of two initialization problems: C 1. Given Y_d, calculate Y_a and Y'_d, or C 2. Given Y', calculate Y. C In either case, initial values for the given C components are input, and initial guesses for C the unknown components must also be provided as input. C C **** Are the initial T, Y, YPRIME consistent ... C C yes - set INFO(11) = 0 C no - set INFO(11) = 1 to calculate option 1 above, C or set INFO(11) = 2 to calculate option 2 **** C C If you have specified INFO(11) = 1, then you C will also need to identify which are the C differential and which are the algebraic C components (algebraic components are components C whose derivatives do not appear explicitly C in the function G(T,Y,YPRIME)). You must set: C IWORK(LID+I) = +1 if Y(I) is a differential variable C IWORK(LID+I) = -1 if Y(I) is an algebraic variable, C where LID = 40 if INFO(10) = 0 or 2 and LID = 40+NEQ C if INFO(10) = 1 or 3. C C INFO(12) - Except for the addition of the RES argument CJ, C DDASKR by default is downward-compatible with DDASSL, C which uses only direct (dense or band) methods to solve C the linear systems involved. You must set INFO(12) to C indicate whether you want the direct methods or the C Krylov iterative method. C **** Do you want DDASKR to use standard direct methods C (dense or band) or the Krylov (iterative) method ... C direct methods - set INFO(12) = 0. C Krylov method - set INFO(12) = 1, C and check the settings of INFO(13) and INFO(15). C C INFO(13) - used when INFO(12) = 1 (Krylov methods). C DDASKR uses scalars MAXL, KMP, NRMAX, and EPLI for the C iterative solution of linear systems. INFO(13) allows C you to override the default values of these parameters. C These parameters and their defaults are as follows: C MAXL = maximum number of iterations in the SPIGMR C algorithm (MAXL .le. NEQ). The default is C MAXL = MIN(5,NEQ). C KMP = number of vectors on which orthogonalization is C done in the SPIGMR algorithm. The default is C KMP = MAXL, which corresponds to complete GMRES C iteration, as opposed to the incomplete form. C NRMAX = maximum number of restarts of the SPIGMR C algorithm per nonlinear iteration. The default is C NRMAX = 5. C EPLI = convergence test constant in SPIGMR algorithm. C The default is EPLI = 0.05. C Note that the length of RWORK depends on both MAXL C and KMP. See the definition of LRW below. C **** Are MAXL, KMP, and EPLI to be given their C default values ... C yes - set INFO(13) = 0 C no - set INFO(13) = 1, C and set all of the following: C IWORK(24) = MAXL (1 .le. MAXL .le. NEQ) C IWORK(25) = KMP (1 .le. KMP .le. MAXL) C IWORK(26) = NRMAX (NRMAX .ge. 0) C RWORK(10) = EPLI (0 .lt. EPLI .lt. 1.0) **** C C INFO(14) - used with INFO(11) > 0 (initial condition C calculation is requested). In this case, you may C request control to be returned to the calling program C immediately after the initial condition calculation, C before proceeding to the integration of the system C (e.g. to examine the computed Y and YPRIME). C If this is done, and if the initialization succeeded C (IDID = 4), you should reset INFO(11) to 0 for the C next call, to prevent the solver from repeating the C initialization (and to avoid an infinite loop). C **** Do you want to proceed to the integration after C the initial condition calculation is done ... C yes - set INFO(14) = 0 C no - set INFO(14) = 1 **** C C INFO(15) - used when INFO(12) = 1 (Krylov methods). C When using preconditioning in the Krylov method, C you must supply a subroutine, PSOL, which solves the C associated linear systems using P. C The usage of DDASKR is simpler if PSOL can carry out C the solution without any prior calculation of data. C However, if some partial derivative data is to be C calculated in advance and used repeatedly in PSOL, C then you must supply a JAC routine to do this, C and set INFO(15) to indicate that JAC is to be called C for this purpose. For example, P might be an C approximation to a part of the matrix A which can be C calculated and LU-factored for repeated solutions of C the preconditioner system. The arrays WP and IWP C (described under JAC and PSOL) can be used to C communicate data between JAC and PSOL. C **** Does PSOL operate with no prior preparation ... C yes - set INFO(15) = 0 (no JAC routine) C no - set INFO(15) = 1 C and supply a JAC routine to evaluate and C preprocess any required Jacobian data. **** C C INFO(16) - option to exclude algebraic variables from C the error test. C **** Do you wish to control errors locally on C all the variables... C yes - set INFO(16) = 0 C no - set INFO(16) = 1 C If you have specified INFO(16) = 1, then you C will also need to identify which are the C differential and which are the algebraic C components (algebraic components are components C whose derivatives do not appear explicitly C in the function G(T,Y,YPRIME)). You must set: C IWORK(LID+I) = +1 if Y(I) is a differential C variable, and C IWORK(LID+I) = -1 if Y(I) is an algebraic C variable, C where LID = 40 if INFO(10) = 0 or 2 and C LID = 40 + NEQ if INFO(10) = 1 or 3. C C INFO(17) - used when INFO(11) > 0 (DDASKR is to do an C initial condition calculation). C DDASKR uses several heuristic control quantities in the C initial condition calculation. They have default values, C but can also be set by the user using INFO(17). C These parameters and their defaults are as follows: C MXNIT = maximum number of Newton iterations C per Jacobian or preconditioner evaluation. C The default is: C MXNIT = 5 in the direct case (INFO(12) = 0), and C MXNIT = 15 in the Krylov case (INFO(12) = 1). C MXNJ = maximum number of Jacobian or preconditioner C evaluations. The default is: C MXNJ = 6 in the direct case (INFO(12) = 0), and C MXNJ = 2 in the Krylov case (INFO(12) = 1). C MXNH = maximum number of values of the artificial C stepsize parameter H to be tried if INFO(11) = 1. C The default is MXNH = 5. C NOTE: the maximum number of Newton iterations C allowed in all is MXNIT*MXNJ*MXNH if INFO(11) = 1, C and MXNIT*MXNJ if INFO(11) = 2. C LSOFF = flag to turn off the linesearch algorithm C (LSOFF = 0 means linesearch is on, LSOFF = 1 means C it is turned off). The default is LSOFF = 0. C STPTOL = minimum scaled step in linesearch algorithm. C The default is STPTOL = (unit roundoff)**(2/3). C EPINIT = swing factor in the Newton iteration convergence C test. The test is applied to the residual vector, C premultiplied by the approximate Jacobian (in the C direct case) or the preconditioner (in the Krylov C case). For convergence, the weighted RMS norm of C this vector (scaled by the error weights) must be C less than EPINIT*EPCON, where EPCON = .33 is the C analogous test constant used in the time steps. C The default is EPINIT = .01. C **** Are the initial condition heuristic controls to be C given their default values... C yes - set INFO(17) = 0 C no - set INFO(17) = 1, C and set all of the following: C IWORK(32) = MXNIT (.GT. 0) C IWORK(33) = MXNJ (.GT. 0) C IWORK(34) = MXNH (.GT. 0) C IWORK(35) = LSOFF ( = 0 or 1) C RWORK(14) = STPTOL (.GT. 0.0) C RWORK(15) = EPINIT (.GT. 0.0) **** C C INFO(18) - option to get extra printing in initial condition C calculation. C **** Do you wish to have extra printing... C no - set INFO(18) = 0 C yes - set INFO(18) = 1 for minimal printing, or C set INFO(18) = 2 for full printing. C If you have specified INFO(18) .ge. 1, data C will be printed with the error handler routines. C To print to a non-default unit number L, include C the line CALL XSETUN(L) in your program. **** C C RTOL, ATOL -- You must assign relative (RTOL) and absolute (ATOL) C error tolerances to tell the code how accurately you C want the solution to be computed. They must be defined C as variables because the code may change them. C you have two choices -- C Both RTOL and ATOL are scalars (INFO(2) = 0), or C both RTOL and ATOL are vectors (INFO(2) = 1). C In either case all components must be non-negative. C C The tolerances are used by the code in a local error C test at each step which requires roughly that C abs(local error in Y(i)) .le. EWT(i) , C where EWT(i) = RTOL*abs(Y(i)) + ATOL is an error weight C quantity, for each vector component. C (More specifically, a root-mean-square norm is used to C measure the size of vectors, and the error test uses the C magnitude of the solution at the beginning of the step.) C C The true (global) error is the difference between the C true solution of the initial value problem and the C computed approximation. Practically all present day C codes, including this one, control the local error at C each step and do not even attempt to control the global C error directly. C C Usually, but not always, the true accuracy of C the computed Y is comparable to the error tolerances. C This code will usually, but not always, deliver a more C accurate solution if you reduce the tolerances and C integrate again. By comparing two such solutions you C can get a fairly reliable idea of the true error in the C solution at the larger tolerances. C C Setting ATOL = 0. results in a pure relative error test C on that component. Setting RTOL = 0. results in a pure C absolute error test on that component. A mixed test C with non-zero RTOL and ATOL corresponds roughly to a C relative error test when the solution component is C much bigger than ATOL and to an absolute error test C when the solution component is smaller than the C threshold ATOL. C C The code will not attempt to compute a solution at an C accuracy unreasonable for the machine being used. It C will advise you if you ask for too much accuracy and C inform you as to the maximum accuracy it believes C possible. C C RWORK(*) -- a real work array, which should be dimensioned in your C calling program with a length equal to the value of C LRW (or greater). C C LRW -- Set it to the declared length of the RWORK array. The C minimum length depends on the options you have selected, C given by a base value plus additional storage as C described below. C C If INFO(12) = 0 (standard direct method), the base value C is BASE = 60 + max(MAXORD+4,7)*NEQ + 3*NRT. C The default value is MAXORD = 5 (see INFO(9)). With the C default MAXORD, BASE = 60 + 9*NEQ + 3*NRT. C Additional storage must be added to the base value for C any or all of the following options: C If INFO(6) = 0 (dense matrix), add NEQ**2. C If INFO(6) = 1 (banded matrix), then: C if INFO(5) = 0, add (2*ML+MU+1)*NEQ C + 2*[NEQ/(ML+MU+1) + 1], and C if INFO(5) = 1, add (2*ML+MU+1)*NEQ. C If INFO(16) = 1, add NEQ. C C If INFO(12) = 1 (Krylov method), the base value is C BASE = 60 + (MAXORD+5)*NEQ + 3*NRT C + [MAXL + 3 + min(1,MAXL-KMP)]*NEQ C + (MAXL+3)*MAXL + 1 + LENWP. C See PSOL for description of LENWP. The default values C are: MAXORD = 5 (see INFO(9)), MAXL = min(5,NEQ) and C KMP = MAXL (see INFO(13)). With these default values, C BASE = 101 + 18*NEQ + 3*NRT + LENWP. C Additional storage must be added to the base value for C the following option: C If INFO(16) = 1, add NEQ. C C C IWORK(*) -- an integer work array, which should be dimensioned in C your calling program with a length equal to the value C of LIW (or greater). C C LIW -- Set it to the declared length of the IWORK array. The C minimum length depends on the options you have selected, C given by a base value plus additions as described below. C C If INFO(12) = 0 (standard direct method), the base value C is BASE = 40 + NEQ. C IF INFO(10) = 1 or 3, add NEQ to the base value. C If INFO(11) = 1 or INFO(16) =1, add NEQ to the base value. C C If INFO(12) = 1 (Krylov method), the base value is C BASE = 40 + LENIWP. See PSOL for description of LENIWP. C If INFO(10) = 1 or 3, add NEQ to the base value. C If INFO(11) = 1 or INFO(16) =1, add NEQ to the base value. C C C RPAR, IPAR -- These are arrays of double precision and integer type, C respectively, which are available for you to use C for communication between your program that calls C DDASKR and the RES subroutine (and the JAC and PSOL C subroutines). They are not altered by DDASKR. C If you do not need RPAR or IPAR, ignore these C parameters by treating them as dummy arguments. C If you do choose to use them, dimension them in C your calling program and in RES (and in JAC and PSOL) C as arrays of appropriate length. C C JAC -- This is the name of a routine that you may supply C (optionally) that relates to the Jacobian matrix of the C nonlinear system that the code must solve at each T step. C The role of JAC (and its call sequence) depends on whether C a direct (INFO(12) = 0) or Krylov (INFO(12) = 1) method C is selected. C C **** INFO(12) = 0 (direct methods): C If you are letting the code generate partial derivatives C numerically (INFO(5) = 0), then JAC can be absent C (or perhaps a dummy routine to satisfy the loader). C Otherwise you must supply a JAC routine to compute C the matrix A = dG/dY + CJ*dG/dYPRIME. It must have C the form C C SUBROUTINE JAC (T, Y, YPRIME, PD, CJ, RPAR, IPAR) C C The JAC routine must dimension Y, YPRIME, and PD (and RPAR C and IPAR if used). CJ is a scalar which is input to JAC. C For the given values of T, Y, and YPRIME, the JAC routine C must evaluate the nonzero elements of the matrix A, and C store these values in the array PD. The elements of PD are C set to zero before each call to JAC, so that only nonzero C elements need to be defined. C The way you store the elements into the PD array depends C on the structure of the matrix indicated by INFO(6). C *** INFO(6) = 0 (full or dense matrix) *** C Give PD a first dimension of NEQ. When you evaluate the C nonzero partial derivatives of equation i (i.e. of G(i)) C with respect to component j (of Y and YPRIME), you must C store the element in PD according to C PD(i,j) = dG(i)/dY(j) + CJ*dG(i)/dYPRIME(j). C *** INFO(6) = 1 (banded matrix with half-bandwidths ML, MU C as described under INFO(6)) *** C Give PD a first dimension of 2*ML+MU+1. When you C evaluate the nonzero partial derivatives of equation i C (i.e. of G(i)) with respect to component j (of Y and C YPRIME), you must store the element in PD according to C IROW = i - j + ML + MU + 1 C PD(IROW,j) = dG(i)/dY(j) + CJ*dG(i)/dYPRIME(j). C C **** INFO(12) = 1 (Krylov method): C If you are not calculating Jacobian data in advance for use C in PSOL (INFO(15) = 0), JAC can be absent (or perhaps a C dummy routine to satisfy the loader). Otherwise, you may C supply a JAC routine to compute and preprocess any parts of C of the Jacobian matrix A = dG/dY + CJ*dG/dYPRIME that are C involved in the preconditioner matrix P. C It is to have the form C C SUBROUTINE JAC (RES, IRES, NEQ, T, Y, YPRIME, REWT, SAVR, C WK, H, CJ, WP, IWP, IER, RPAR, IPAR) C C The JAC routine must dimension Y, YPRIME, REWT, SAVR, WK, C and (if used) WP, IWP, RPAR, and IPAR. C The Y, YPRIME, and SAVR arrays contain the current values C of Y, YPRIME, and the residual G, respectively. C The array WK is work space of length NEQ. C H is the step size. CJ is a scalar, input to JAC, that is C normally proportional to 1/H. REWT is an array of C reciprocal error weights, 1/EWT(i), where EWT(i) is C RTOL*abs(Y(i)) + ATOL (unless you supplied routine DDAWTS C instead), for use in JAC if needed. For example, if JAC C computes difference quotient approximations to partial C derivatives, the REWT array may be useful in setting the C increments used. The JAC routine should do any C factorization operations called for, in preparation for C solving linear systems in PSOL. The matrix P should C be an approximation to the Jacobian, C A = dG/dY + CJ*dG/dYPRIME. C C WP and IWP are real and integer work arrays which you may C use for communication between your JAC routine and your C PSOL routine. These may be used to store elements of the C preconditioner P, or related matrix data (such as factored C forms). They are not altered by DDASKR. C If you do not need WP or IWP, ignore these parameters by C treating them as dummy arguments. If you do use them, C dimension them appropriately in your JAC and PSOL routines. C See the PSOL description for instructions on setting C the lengths of WP and IWP. C C On return, JAC should set the error flag IER as follows.. C IER = 0 if JAC was successful, C IER .ne. 0 if JAC was unsuccessful (e.g. if Y or YPRIME C was illegal, or a singular matrix is found). C (If IER .ne. 0, a smaller stepsize will be tried.) C IER = 0 on entry to JAC, so need be reset only on a failure. C If RES is used within JAC, then a nonzero value of IRES will C override any nonzero value of IER (see the RES description). C C Regardless of the method type, subroutine JAC must not C alter T, Y(*), YPRIME(*), H, CJ, or REWT(*). C You must declare the name JAC in an EXTERNAL statement in C your program that calls DDASKR. C C PSOL -- This is the name of a routine you must supply if you have C selected a Krylov method (INFO(12) = 1) with preconditioning. C In the direct case (INFO(12) = 0), PSOL can be absent C (a dummy routine may have to be supplied to satisfy the C loader). Otherwise, you must provide a PSOL routine to C solve linear systems arising from preconditioning. C When supplied with INFO(12) = 1, the PSOL routine is to C have the form C C SUBROUTINE PSOL (NEQ, T, Y, YPRIME, SAVR, WK, CJ, WGHT, C WP, IWP, B, EPLIN, IER, RPAR, IPAR) C C The PSOL routine must solve linear systems of the form C P*x = b where P is the left preconditioner matrix. C C The right-hand side vector b is in the B array on input, and C PSOL must return the solution vector x in B. C The Y, YPRIME, and SAVR arrays contain the current values C of Y, YPRIME, and the residual G, respectively. C C Work space required by JAC and/or PSOL, and space for data to C be communicated from JAC to PSOL is made available in the form C of arrays WP and IWP, which are parts of the RWORK and IWORK C arrays, respectively. The lengths of these real and integer C work spaces WP and IWP must be supplied in LENWP and LENIWP, C respectively, as follows.. C IWORK(27) = LENWP = length of real work space WP C IWORK(28) = LENIWP = length of integer work space IWP. C C WK is a work array of length NEQ for use by PSOL. C CJ is a scalar, input to PSOL, that is normally proportional C to 1/H (H = stepsize). If the old value of CJ C (at the time of the last JAC call) is needed, it must have C been saved by JAC in WP. C C WGHT is an array of weights, to be used if PSOL uses an C iterative method and performs a convergence test. (In terms C of the argument REWT to JAC, WGHT is REWT/sqrt(NEQ).) C If PSOL uses an iterative method, it should use EPLIN C (a heuristic parameter) as the bound on the weighted norm of C the residual for the computed solution. Specifically, the C residual vector R should satisfy C SQRT (SUM ( (R(i)*WGHT(i))**2 ) ) .le. EPLIN C C PSOL must not alter NEQ, T, Y, YPRIME, SAVR, CJ, WGHT, EPLIN. C C On return, PSOL should set the error flag IER as follows.. C IER = 0 if PSOL was successful, C IER .lt. 0 if an unrecoverable error occurred, meaning C control will be passed to the calling routine, C IER .gt. 0 if a recoverable error occurred, meaning that C the step will be retried with the same step size C but with a call to JAC to update necessary data, C unless the Jacobian data is current, in which case C the step will be retried with a smaller step size. C IER = 0 on entry to PSOL so need be reset only on a failure. C C You must declare the name PSOL in an EXTERNAL statement in C your program that calls DDASKR. C C RT -- This is the name of the subroutine for defining the vector C R(T,Y,Y') of constraint functions Ri(T,Y,Y'), whose roots C are desired during the integration. It is to have the form C SUBROUTINE RT(NEQ, T, Y, YP, NRT, RVAL, RPAR, IPAR) C DIMENSION Y(NEQ), YP(NEQ), RVAL(NRT), C where NEQ, T, Y and NRT are INPUT, and the array RVAL is C output. NEQ, T, Y, and YP have the same meaning as in the C RES routine, and RVAL is an array of length NRT. C For i = 1,...,NRT, this routine is to load into RVAL(i) the C value at (T,Y,Y') of the i-th constraint function Ri(T,Y,Y'). C DDASKR will find roots of the Ri of odd multiplicity C (that is, sign changes) as they occur during the integration. C RT must be declared EXTERNAL in the calling program. C C CAUTION.. Because of numerical errors in the functions Ri C due to roundoff and integration error, DDASKR may return C false roots, or return the same root at two or more nearly C equal values of T. If such false roots are suspected, C the user should consider smaller error tolerances and/or C higher precision in the evaluation of the Ri. C C If a root of some Ri defines the end of the problem, C the input to DDASKR should nevertheless allow C integration to a point slightly past that root, so C that DDASKR can locate the root by interpolation. C C NRT -- The number of constraint functions Ri(T,Y,Y'). If there are C no constraints, set NRT = 0 and pass a dummy name for RT. C C JROOT -- This is an integer array of length NRT, used only for output. C On a return where one or more roots were found (IDID = 5), C JROOT(i) = 1 or -1 if Ri(T,Y,Y') has a root at T, and C JROOT(i) = 0 if not. If nonzero, JROOT(i) shows the direction C of the sign change in Ri in the direction of integration: C JROOT(i) = 1 means Ri changed from negative to positive. C JROOT(i) = -1 means Ri changed from positive to negative. C C C OPTIONALLY REPLACEABLE SUBROUTINE: C C DDASKR uses a weighted root-mean-square norm to measure the C size of various error vectors. The weights used in this norm C are set in the following subroutine: C C SUBROUTINE DDAWTS (NEQ, IWT, RTOL, ATOL, Y, EWT, RPAR, IPAR) C DIMENSION RTOL(*), ATOL(*), Y(*), EWT(*), RPAR(*), IPAR(*) C C A DDAWTS routine has been included with DDASKR which sets the C weights according to C EWT(I) = RTOL*ABS(Y(I)) + ATOL C in the case of scalar tolerances (IWT = 0) or C EWT(I) = RTOL(I)*ABS(Y(I)) + ATOL(I) C in the case of array tolerances (IWT = 1). (IWT is INFO(2).) C In some special cases, it may be appropriate for you to define C your own error weights by writing a subroutine DDAWTS to be C called instead of the version supplied. However, this should C be attempted only after careful thought and consideration. C If you supply this routine, you may use the tolerances and Y C as appropriate, but do not overwrite these variables. You C may also use RPAR and IPAR to communicate data as appropriate. C ***Note: Aside from the values of the weights, the choice of C norm used in DDASKR (weighted root-mean-square) is not subject C to replacement by the user. In this respect, DDASKR is not C downward-compatible with the original DDASSL solver (in which C the norm routine was optionally user-replaceable). C C C------OUTPUT - AFTER ANY RETURN FROM DDASKR---------------------------- C C The principal aim of the code is to return a computed solution at C T = TOUT, although it is also possible to obtain intermediate C results along the way. To find out whether the code achieved its C goal or if the integration process was interrupted before the task C was completed, you must check the IDID parameter. C C C T -- The output value of T is the point to which the solution C was successfully advanced. C C Y(*) -- contains the computed solution approximation at T. C C YPRIME(*) -- contains the computed derivative approximation at T. C C IDID -- reports what the code did, described as follows: C C *** TASK COMPLETED *** C Reported by positive values of IDID C C IDID = 1 -- A step was successfully taken in the C interval-output mode. The code has not C yet reached TOUT. C C IDID = 2 -- The integration to TSTOP was successfully C completed (T = TSTOP) by stepping exactly to TSTOP. C C IDID = 3 -- The integration to TOUT was successfully C completed (T = TOUT) by stepping past TOUT. C Y(*) and YPRIME(*) are obtained by interpolation. C C IDID = 4 -- The initial condition calculation, with C INFO(11) > 0, was successful, and INFO(14) = 1. C No integration steps were taken, and the solution C is not considered to have been started. C C IDID = 5 -- The integration was successfully completed C by finding one or more roots of R(T,Y,Y') at T. C C *** TASK INTERRUPTED *** C Reported by negative values of IDID C C IDID = -1 -- A large amount of work has been expended C (about 500 steps). C C IDID = -2 -- The error tolerances are too stringent. C C IDID = -3 -- The local error test cannot be satisfied C because you specified a zero component in ATOL C and the corresponding computed solution component C is zero. Thus, a pure relative error test is C impossible for this component. C C IDID = -5 -- There were repeated failures in the evaluation C or processing of the preconditioner (in JAC). C C IDID = -6 -- DDASKR had repeated error test failures on the C last attempted step. C C IDID = -7 -- The nonlinear system solver in the time C integration could not converge. C C IDID = -8 -- The matrix of partial derivatives appears C to be singular (direct method). C C IDID = -9 -- The nonlinear system solver in the integration C failed to achieve convergence, and there were C repeated error test failures in this step. C C IDID =-10 -- The nonlinear system solver in the integration C failed to achieve convergence because IRES was C equal to -1. C C IDID =-11 -- IRES = -2 was encountered and control is C being returned to the calling program. C C IDID =-12 -- DDASKR failed to compute the initial Y, YPRIME. C C IDID =-13 -- An unrecoverable error was encountered inside C the user's PSOL routine, and control is being C returned to the calling program. C C IDID =-14 -- The Krylov linear system solver could not C achieve convergence. C C IDID =-15,..,-32 -- Not applicable for this code. C C *** TASK TERMINATED *** C reported by the value of IDID=-33 C C IDID = -33 -- The code has encountered trouble from which C it cannot recover. A message is printed C explaining the trouble and control is returned C to the calling program. For example, this occurs C when invalid input is detected. C C RTOL, ATOL -- these quantities remain unchanged except when C IDID = -2. In this case, the error tolerances have been C increased by the code to values which are estimated to C be appropriate for continuing the integration. However, C the reported solution at T was obtained using the input C values of RTOL and ATOL. C C RWORK, IWORK -- contain information which is usually of no interest C to the user but necessary for subsequent calls. C However, you may be interested in the performance data C listed below. These quantities are accessed in RWORK C and IWORK but have internal mnemonic names, as follows.. C C RWORK(3)--contains H, the step size h to be attempted C on the next step. C C RWORK(4)--contains TN, the current value of the C independent variable, i.e. the farthest point C integration has reached. This will differ C from T if interpolation has been performed C (IDID = 3). C C RWORK(7)--contains HOLD, the stepsize used on the last C successful step. If INFO(11) = INFO(14) = 1, C this contains the value of H used in the C initial condition calculation. C C IWORK(7)--contains K, the order of the method to be C attempted on the next step. C C IWORK(8)--contains KOLD, the order of the method used C on the last step. C C IWORK(11)--contains NST, the number of steps (in T) C taken so far. C C IWORK(12)--contains NRE, the number of calls to RES C so far. C C IWORK(13)--contains NJE, the number of calls to JAC so C far (Jacobian or preconditioner evaluations). C C IWORK(14)--contains NETF, the total number of error test C failures so far. C C IWORK(15)--contains NCFN, the total number of nonlinear C convergence failures so far (includes counts C of singular iteration matrix or singular C preconditioners). C C IWORK(16)--contains NCFL, the number of convergence C failures of the linear iteration so far. C C IWORK(17)--contains LENIW, the length of IWORK actually C required. This is defined on normal returns C and on an illegal input return for C insufficient storage. C C IWORK(18)--contains LENRW, the length of RWORK actually C required. This is defined on normal returns C and on an illegal input return for C insufficient storage. C C IWORK(19)--contains NNI, the total number of nonlinear C iterations so far (each of which calls a C linear solver). C C IWORK(20)--contains NLI, the total number of linear C (Krylov) iterations so far. C C IWORK(21)--contains NPS, the number of PSOL calls so C far, for preconditioning solve operations or C for solutions with the user-supplied method. C C IWORK(36)--contains the total number of calls to the C constraint function routine RT so far. C C Note: The various counters in IWORK do not include C counts during a prior call made with INFO(11) > 0 and C INFO(14) = 1. C C C------INPUT - WHAT TO DO TO CONTINUE THE INTEGRATION ----------------- C (CALLS AFTER THE FIRST) C C This code is organized so that subsequent calls to continue the C integration involve little (if any) additional effort on your C part. You must monitor the IDID parameter in order to determine C what to do next. C C Recalling that the principal task of the code is to integrate C from T to TOUT (the interval mode), usually all you will need C to do is specify a new TOUT upon reaching the current TOUT. C C Do not alter any quantity not specifically permitted below. In C particular do not alter NEQ, T, Y(*), YPRIME(*), RWORK(*), C IWORK(*), or the differential equation in subroutine RES. Any C such alteration constitutes a new problem and must be treated C as such, i.e. you must start afresh. C C You cannot change from array to scalar error control or vice C versa (INFO(2)), but you can change the size of the entries of C RTOL or ATOL. Increasing a tolerance makes the equation easier C to integrate. Decreasing a tolerance will make the equation C harder to integrate and should generally be avoided. C C You can switch from the intermediate-output mode to the C interval mode (INFO(3)) or vice versa at any time. C C If it has been necessary to prevent the integration from going C past a point TSTOP (INFO(4), RWORK(1)), keep in mind that the C code will not integrate to any TOUT beyond the currently C specified TSTOP. Once TSTOP has been reached, you must change C the value of TSTOP or set INFO(4) = 0. You may change INFO(4) C or TSTOP at any time but you must supply the value of TSTOP in C RWORK(1) whenever you set INFO(4) = 1. C C Do not change INFO(5), INFO(6), INFO(12-17) or their associated C IWORK/RWORK locations unless you are going to restart the code. C C *** FOLLOWING A COMPLETED TASK *** C C If.. C IDID = 1, call the code again to continue the integration C another step in the direction of TOUT. C C IDID = 2 or 3, define a new TOUT and call the code again. C TOUT must be different from T. You cannot change C the direction of integration without restarting. C C IDID = 4, reset INFO(11) = 0 and call the code again to begin C the integration. (If you leave INFO(11) > 0 and C INFO(14) = 1, you may generate an infinite loop.) C In this situation, the next call to DDASKR is C considered to be the first call for the problem, C in that all initializations are done. C C IDID = 5, call the code again to continue the integration in the C direction of TOUT. You may change the functions C Ri defined by RT after a return with IDID = 5, but C the number of constraint functions NRT must remain C the same. If you wish to change the functions in C RES or in RT, then you must restart the code. C C *** FOLLOWING AN INTERRUPTED TASK *** C C To show the code that you realize the task was interrupted and C that you want to continue, you must take appropriate action and C set INFO(1) = 1. C C If.. C IDID = -1, the code has taken about 500 steps. If you want to C continue, set INFO(1) = 1 and call the code again. C An additional 500 steps will be allowed. C C C IDID = -2, the error tolerances RTOL, ATOL have been increased C to values the code estimates appropriate for C continuing. You may want to change them yourself. C If you are sure you want to continue with relaxed C error tolerances, set INFO(1) = 1 and call the code C again. C C IDID = -3, a solution component is zero and you set the C corresponding component of ATOL to zero. If you C are sure you want to continue, you must first alter C the error criterion to use positive values of ATOL C for those components corresponding to zero solution C components, then set INFO(1) = 1 and call the code C again. C C IDID = -4 --- cannot occur with this code. C C IDID = -5, your JAC routine failed with the Krylov method. Check C for errors in JAC and restart the integration. C C IDID = -6, repeated error test failures occurred on the last C attempted step in DDASKR. A singularity in the C solution may be present. If you are absolutely C certain you want to continue, you should restart C the integration. (Provide initial values of Y and C YPRIME which are consistent.) C C IDID = -7, repeated convergence test failures occurred on the last C attempted step in DDASKR. An inaccurate or ill- C conditioned Jacobian or preconditioner may be the C problem. If you are absolutely certain you want C to continue, you should restart the integration. C C C IDID = -8, the matrix of partial derivatives is singular, with C the use of direct methods. Some of your equations C may be redundant. DDASKR cannot solve the problem C as stated. It is possible that the redundant C equations could be removed, and then DDASKR could C solve the problem. It is also possible that a C solution to your problem either does not exist C or is not unique. C C IDID = -9, DDASKR had multiple convergence test failures, preceded C by multiple error test failures, on the last C attempted step. It is possible that your problem is C ill-posed and cannot be solved using this code. Or, C there may be a discontinuity or a singularity in the C solution. If you are absolutely certain you want to C continue, you should restart the integration. C C IDID = -10, DDASKR had multiple convergence test failures C because IRES was equal to -1. If you are C absolutely certain you want to continue, you C should restart the integration. C C IDID = -11, there was an unrecoverable error (IRES = -2) from RES C inside the nonlinear system solver. Determine the C cause before trying again. C C IDID = -12, DDASKR failed to compute the initial Y and YPRIME C vectors. This could happen because the initial C approximation to Y or YPRIME was not very good, or C because no consistent values of these vectors exist. C The problem could also be caused by an inaccurate or C singular iteration matrix, or a poor preconditioner. C C IDID = -13, there was an unrecoverable error encountered inside C your PSOL routine. Determine the cause before C trying again. C C IDID = -14, the Krylov linear system solver failed to achieve C convergence. This may be due to ill-conditioning C in the iteration matrix, or a singularity in the C preconditioner (if one is being used). C Another possibility is that there is a better C choice of Krylov parameters (see INFO(13)). C Possibly the failure is caused by redundant equations C in the system, or by inconsistent equations. C In that case, reformulate the system to make it C consistent and non-redundant. C C IDID = -15,..,-32 --- Cannot occur with this code. C C *** FOLLOWING A TERMINATED TASK *** C C If IDID = -33, you cannot continue the solution of this problem. C An attempt to do so will result in your run being C terminated. C C --------------------------------------------------------------------- C C***REFERENCES C 1. L. R. Petzold, A Description of DASSL: A Differential/Algebraic C System Solver, in Scientific Computing, R. S. Stepleman et al. C (Eds.), North-Holland, Amsterdam, 1983, pp. 65-68. C 2. K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical C Solution of Initial-Value Problems in Differential-Algebraic C Equations, Elsevier, New York, 1989. C 3. P. N. Brown and A. C. Hindmarsh, Reduced Storage Matrix Methods C in Stiff ODE Systems, J. Applied Mathematics and Computation, C 31 (1989), pp. 40-91. C 4. P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Using Krylov C Methods in the Solution of Large-Scale Differential-Algebraic C Systems, SIAM J. Sci. Comp., 15 (1994), pp. 1467-1488. C 5. P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Consistent C Initial Condition Calculation for Differential-Algebraic C Systems, SIAM J. Sci. Comp. 19 (1998), pp. 1495-1512. C C***ROUTINES CALLED C C The following are all the subordinate routines used by DDASKR. C C DRCHEK does preliminary checking for roots, and serves as an C interface between Subroutine DDASKR and Subroutine DROOTS. C DROOTS finds the leftmost root of a set of functions. C DDASIC computes consistent initial conditions. C DYYPNW updates Y and YPRIME in linesearch for initial condition C calculation. C DDSTP carries out one step of the integration. C DCNSTR/DCNST0 check the current solution for constraint violations. C DDAWTS sets error weight quantities. C DINVWT tests and inverts the error weights. C DDATRP performs interpolation to get an output solution. C DDWNRM computes the weighted root-mean-square norm of a vector. C D1MACH provides the unit roundoff of the computer. C XERRWD/XSETF/XSETUN/IXSAV is a package to handle error messages. C DDASID nonlinear equation driver to initialize Y and YPRIME using C direct linear system solver methods. Interfaces to Newton C solver (direct case). C DNSID solves the nonlinear system for unknown initial values by C modified Newton iteration and direct linear system methods. C DLINSD carries out linesearch algorithm for initial condition C calculation (direct case). C DFNRMD calculates weighted norm of preconditioned residual in C initial condition calculation (direct case). C DNEDD nonlinear equation driver for direct linear system solver C methods. Interfaces to Newton solver (direct case). C DMATD assembles the iteration matrix (direct case). C DNSD solves the associated nonlinear system by modified C Newton iteration and direct linear system methods. C DSLVD interfaces to linear system solver (direct case). C DDASIK nonlinear equation driver to initialize Y and YPRIME using C Krylov iterative linear system methods. Interfaces to C Newton solver (Krylov case). C DNSIK solves the nonlinear system for unknown initial values by C Newton iteration and Krylov iterative linear system methods. C DLINSK carries out linesearch algorithm for initial condition C calculation (Krylov case). C DFNRMK calculates weighted norm of preconditioned residual in C initial condition calculation (Krylov case). C DNEDK nonlinear equation driver for iterative linear system solver C methods. Interfaces to Newton solver (Krylov case). C DNSK solves the associated nonlinear system by Inexact Newton C iteration and (linear) Krylov iteration. C DSLVK interfaces to linear system solver (Krylov case). C DSPIGM solves a linear system by SPIGMR algorithm. C DATV computes matrix-vector product in Krylov algorithm. C DORTH performs orthogonalization of Krylov basis vectors. C DHEQR performs QR factorization of Hessenberg matrix. C DHELS finds least-squares solution of Hessenberg linear system. C DGEFA, DGESL, DGBFA, DGBSL are LINPACK routines for solving C linear systems (dense or band direct methods). C DAXPY, DCOPY, DDOT, DNRM2, DSCAL are Basic Linear Algebra (BLAS) C routines. C C The routines called directly by DDASKR are: C DCNST0, DDAWTS, DINVWT, D1MACH, DDWNRM, DDASIC, DDATRP, DDSTP, C DRCHEK, XERRWD C C***END PROLOGUE DDASKR C C IMPLICIT DOUBLE PRECISION(A-H,O-Z) LOGICAL DONE, LAVL, LCFN, LCFL, LWARN DIMENSION Y(*),YPRIME(*) DIMENSION INFO(20) DIMENSION RWORK(LRW),IWORK(LIW) DIMENSION RTOL(*),ATOL(*) DIMENSION RPAR(*),IPAR(*) CHARACTER MSG*80 EXTERNAL RES, JAC, PSOL, RT, DDASID, DDASIK, DNEDD, DNEDK C C Set pointers into IWORK. C PARAMETER (LML=1, LMU=2, LMTYPE=4, * LIWM=1, LMXORD=3, LJCALC=5, LPHASE=6, LK=7, LKOLD=8, * LNS=9, LNSTL=10, LNST=11, LNRE=12, LNJE=13, LETF=14, LNCFN=15, * LNCFL=16, LNIW=17, LNRW=18, LNNI=19, LNLI=20, LNPS=21, * LNPD=22, LMITER=23, LMAXL=24, LKMP=25, LNRMAX=26, LLNWP=27, * LLNIWP=28, LLOCWP=29, LLCIWP=30, LKPRIN=31, LMXNIT=32, * LMXNJ=33, LMXNH=34, LLSOFF=35, LNRTE=36, LIRFND=37, LICNS=41) C C Set pointers into RWORK. C PARAMETER (LTSTOP=1, LHMAX=2, LH=3, LTN=4, LCJ=5, LCJOLD=6, * LHOLD=7, LS=8, LROUND=9, LEPLI=10, LSQRN=11, LRSQRN=12, * LEPCON=13, LSTOL=14, LEPIN=15, LALPHA=21, LBETA=27, * LGAMMA=33, LPSI=39, LSIGMA=45, LT0=51, LTLAST=52, LDELTA=61) C SAVE LID, LENID, NONNEG, NCPHI C C C***FIRST EXECUTABLE STATEMENT DDASKR C C IF(INFO(1).NE.0) GO TO 100 C C----------------------------------------------------------------------- C This block is executed for the initial call only. C It contains checking of inputs and initializations. C----------------------------------------------------------------------- C C First check INFO array to make sure all elements of INFO C Are within the proper range. (INFO(1) is checked later, because C it must be tested on every call.) ITEMP holds the location C within INFO which may be out of range. C DO 10 I=2,9 ITEMP = I IF (INFO(I) .NE. 0 .AND. INFO(I) .NE. 1) GO TO 701 10 CONTINUE ITEMP = 10 IF(INFO(10).LT.0 .OR. INFO(10).GT.3) GO TO 701 ITEMP = 11 IF(INFO(11).LT.0 .OR. INFO(11).GT.2) GO TO 701 DO 15 I=12,17 ITEMP = I IF (INFO(I) .NE. 0 .AND. INFO(I) .NE. 1) GO TO 701 15 CONTINUE ITEMP = 18 IF(INFO(18).LT.0 .OR. INFO(18).GT.2) GO TO 701 C C Check NEQ to see if it is positive. C IF (NEQ .LE. 0) GO TO 702 C C Check and compute maximum order. C MXORD=5 IF (INFO(9) .NE. 0) THEN MXORD=IWORK(LMXORD) IF (MXORD .LT. 1 .OR. MXORD .GT. 5) GO TO 703 ENDIF IWORK(LMXORD)=MXORD C C Set and/or check inputs for constraint checking (INFO(10) .NE. 0). C Set values for ICNFLG, NONNEG, and pointer LID. C ICNFLG = 0 NONNEG = 0 LID = LICNS IF (INFO(10) .EQ. 0) GO TO 20 IF (INFO(10) .EQ. 1) THEN ICNFLG = 1 NONNEG = 0 LID = LICNS + NEQ ELSEIF (INFO(10) .EQ. 2) THEN ICNFLG = 0 NONNEG = 1 ELSE ICNFLG = 1 NONNEG = 1 LID = LICNS + NEQ ENDIF C 20 CONTINUE C C Set and/or check inputs for Krylov solver (INFO(12) .NE. 0). C If indicated, set default values for MAXL, KMP, NRMAX, and EPLI. C Otherwise, verify inputs required for iterative solver. C IF (INFO(12) .EQ. 0) GO TO 25 C IWORK(LMITER) = INFO(12) IF (INFO(13) .EQ. 0) THEN IWORK(LMAXL) = MIN(5,NEQ) IWORK(LKMP) = IWORK(LMAXL) IWORK(LNRMAX) = 5 RWORK(LEPLI) = 0.05D0 ELSE IF(IWORK(LMAXL) .LT. 1 .OR. IWORK(LMAXL) .GT. NEQ) GO TO 720 IF(IWORK(LKMP) .LT. 1 .OR. IWORK(LKMP) .GT. IWORK(LMAXL)) 1 GO TO 721 IF(IWORK(LNRMAX) .LT. 0) GO TO 722 IF(RWORK(LEPLI).LE.0.0D0 .OR. RWORK(LEPLI).GE.1.0D0)GO TO 723 ENDIF C 25 CONTINUE C C Set and/or check controls for the initial condition calculation C (INFO(11) .GT. 0). If indicated, set default values. C Otherwise, verify inputs required for iterative solver. C IF (INFO(11) .EQ. 0) GO TO 30 IF (INFO(17) .EQ. 0) THEN IWORK(LMXNIT) = 5 IF (INFO(12) .GT. 0) IWORK(LMXNIT) = 15 IWORK(LMXNJ) = 6 IF (INFO(12) .GT. 0) IWORK(LMXNJ) = 2 IWORK(LMXNH) = 5 IWORK(LLSOFF) = 0 RWORK(LEPIN) = 0.01D0 ELSE IF (IWORK(LMXNIT) .LE. 0) GO TO 725 IF (IWORK(LMXNJ) .LE. 0) GO TO 725 IF (IWORK(LMXNH) .LE. 0) GO TO 725 LSOFF = IWORK(LLSOFF) IF (LSOFF .LT. 0 .OR. LSOFF .GT. 1) GO TO 725 IF (RWORK(LEPIN) .LE. 0.0D0) GO TO 725 ENDIF C 30 CONTINUE C C Below is the computation and checking of the work array lengths C LENIW and LENRW, using direct methods (INFO(12) = 0) or C the Krylov methods (INFO(12) = 1). C LENIC = 0 IF (INFO(10) .EQ. 1 .OR. INFO(10) .EQ. 3) LENIC = NEQ LENID = 0 IF (INFO(11) .EQ. 1 .OR. INFO(16) .EQ. 1) LENID = NEQ IF (INFO(12) .EQ. 0) THEN C C Compute MTYPE, etc. Check ML and MU. C NCPHI = MAX(MXORD + 1, 4) IF(INFO(6).EQ.0) THEN LENPD = NEQ**2 LENRW = 60 + 3*NRT + (NCPHI+3)*NEQ + LENPD IF(INFO(5).EQ.0) THEN IWORK(LMTYPE)=2 ELSE IWORK(LMTYPE)=1 ENDIF ELSE IF(IWORK(LML).LT.0.OR.IWORK(LML).GE.NEQ)GO TO 717 IF(IWORK(LMU).LT.0.OR.IWORK(LMU).GE.NEQ)GO TO 718 LENPD=(2*IWORK(LML)+IWORK(LMU)+1)*NEQ IF(INFO(5).EQ.0) THEN IWORK(LMTYPE)=5 MBAND=IWORK(LML)+IWORK(LMU)+1 MSAVE=(NEQ/MBAND)+1 LENRW = 60 + 3*NRT + (NCPHI+3)*NEQ + LENPD + 2*MSAVE ELSE IWORK(LMTYPE)=4 LENRW = 60 + 3*NRT + (NCPHI+3)*NEQ + LENPD ENDIF ENDIF C C Compute LENIW, LENWP, LENIWP. C LENIW = 40 + LENIC + LENID + NEQ LENWP = 0 LENIWP = 0 C ELSE IF (INFO(12) .EQ. 1) THEN NCPHI = MXORD + 1 MAXL = IWORK(LMAXL) LENWP = IWORK(LLNWP) LENIWP = IWORK(LLNIWP) LENPD = (MAXL+3+MIN0(1,MAXL-IWORK(LKMP)))*NEQ 1 + (MAXL+3)*MAXL + 1 + LENWP LENRW = 60 + 3*NRT + (MXORD+5)*NEQ + LENPD LENIW = 40 + LENIC + LENID + LENIWP C ENDIF IF(INFO(16) .NE. 0) LENRW = LENRW + NEQ C C Check lengths of RWORK and IWORK. C IWORK(LNIW)=LENIW IWORK(LNRW)=LENRW IWORK(LNPD)=LENPD IWORK(LLOCWP) = LENPD-LENWP+1 IF(LRW.LT.LENRW)GO TO 704 IF(LIW.LT.LENIW)GO TO 705 C C Check ICNSTR for legality. C IF (LENIC .GT. 0) THEN DO 40 I = 1,NEQ ICI = IWORK(LICNS-1+I) IF (ICI .LT. -2 .OR. ICI .GT. 2) GO TO 726 40 CONTINUE ENDIF C C Check Y for consistency with constraints. C IF (LENIC .GT. 0) THEN CALL DCNST0(NEQ,Y,IWORK(LICNS),IRET) IF (IRET .NE. 0) GO TO 727 ENDIF C C Check ID for legality and set INDEX = 0 or 1. C INDEX = 1 IF (LENID .GT. 0) THEN INDEX = 0 DO 50 I = 1,NEQ IDI = IWORK(LID-1+I) IF (IDI .NE. 1 .AND. IDI .NE. -1) GO TO 724 IF (IDI .EQ. -1) INDEX = 1 50 CONTINUE ENDIF C C Check to see that TOUT is different from T, and NRT .ge. 0. C IF(TOUT .EQ. T)GO TO 719 IF(NRT .LT. 0) GO TO 730 C C Check HMAX. C IF(INFO(7) .NE. 0) THEN HMAX = RWORK(LHMAX) IF (HMAX .LE. 0.0D0) GO TO 710 ENDIF C C Initialize counters and other flags. C IWORK(LNST)=0 IWORK(LNRE)=0 IWORK(LNJE)=0 IWORK(LETF)=0 IWORK(LNCFN)=0 IWORK(LNNI)=0 IWORK(LNLI)=0 IWORK(LNPS)=0 IWORK(LNCFL)=0 IWORK(LNRTE)=0 IWORK(LKPRIN)=INFO(18) IDID=1 GO TO 200 C C----------------------------------------------------------------------- C This block is for continuation calls only. C Here we check INFO(1), and if the last step was interrupted, C we check whether appropriate action was taken. C----------------------------------------------------------------------- C 100 CONTINUE IF(INFO(1).EQ.1)GO TO 110 ITEMP = 1 IF(INFO(1).NE.-1)GO TO 701 C C If we are here, the last step was interrupted by an error C condition from DDSTP, and appropriate action was not taken. C This is a fatal error. C MSG = 'DASKR-- THE LAST STEP TERMINATED WITH A NEGATIVE' CALL XERRWD(MSG,49,201,0,0,0,0,0,0.0D0,0.0D0) MSG = 'DASKR-- VALUE (=I1) OF IDID AND NO APPROPRIATE' CALL XERRWD(MSG,47,202,0,1,IDID,0,0,0.0D0,0.0D0) MSG = 'DASKR-- ACTION WAS TAKEN. RUN TERMINATED' CALL XERRWD(MSG,41,203,1,0,0,0,0,0.0D0,0.0D0) RETURN 110 CONTINUE C C----------------------------------------------------------------------- C This block is executed on all calls. C C Counters are saved for later checks of performance. C Then the error tolerance parameters are checked, and the C work array pointers are set. C----------------------------------------------------------------------- C 200 CONTINUE C C Save counters for use later. C IWORK(LNSTL)=IWORK(LNST) NLI0 = IWORK(LNLI) NNI0 = IWORK(LNNI) NCFN0 = IWORK(LNCFN) NCFL0 = IWORK(LNCFL) NWARN = 0 C C Check RTOL and ATOL. C NZFLG = 0 RTOLI = RTOL(1) ATOLI = ATOL(1) DO 210 I=1,NEQ IF (INFO(2) .EQ. 1) RTOLI = RTOL(I) IF (INFO(2) .EQ. 1) ATOLI = ATOL(I) IF (RTOLI .GT. 0.0D0 .OR. ATOLI .GT. 0.0D0) NZFLG = 1 IF (RTOLI .LT. 0.0D0) GO TO 706 IF (ATOLI .LT. 0.0D0) GO TO 707 210 CONTINUE IF (NZFLG .EQ. 0) GO TO 708 C C Set pointers to RWORK and IWORK segments. C For direct methods, SAVR is not used. C IWORK(LLCIWP) = LID + LENID LSAVR = LDELTA IF (INFO(12) .NE. 0) LSAVR = LDELTA + NEQ LE = LSAVR + NEQ LWT = LE + NEQ LVT = LWT IF (INFO(16) .NE. 0) LVT = LWT + NEQ LPHI = LVT + NEQ LR0 = LPHI + NCPHI*NEQ LR1 = LR0 + NRT LRX = LR1 + NRT LWM = LRX + NRT IF (INFO(1) .EQ. 1) GO TO 400 C C----------------------------------------------------------------------- C This block is executed on the initial call only. C Set the initial step size, the error weight vector, and PHI. C Compute unknown initial components of Y and YPRIME, if requested. C----------------------------------------------------------------------- C 300 CONTINUE TN=T IDID=1 C C Set error weight array WT and altered weight array VT. C CALL DDAWTS(NEQ,INFO(2),RTOL,ATOL,Y,RWORK(LWT),RPAR,IPAR) CALL DINVWT(NEQ,RWORK(LWT),IER) IF (IER .NE. 0) GO TO 713 IF (INFO(16) .NE. 0) THEN DO 305 I = 1, NEQ 305 RWORK(LVT+I-1) = MAX(IWORK(LID+I-1),0)*RWORK(LWT+I-1) ENDIF C C Compute unit roundoff and HMIN. C UROUND = D1MACH(4) RWORK(LROUND) = UROUND HMIN = 4.0D0*UROUND*MAX(ABS(T),ABS(TOUT)) C C Set/check STPTOL control for initial condition calculation. C IF (INFO(11) .NE. 0) THEN IF( INFO(17) .EQ. 0) THEN RWORK(LSTOL) = UROUND**.6667D0 ELSE IF (RWORK(LSTOL) .LE. 0.0D0) GO TO 725 ENDIF ENDIF C C Compute EPCON and square root of NEQ and its reciprocal, used C inside iterative solver. C RWORK(LEPCON) = 0.33D0 FLOATN = NEQ RWORK(LSQRN) = SQRT(FLOATN) RWORK(LRSQRN) = 1.D0/RWORK(LSQRN) C C Check initial interval to see that it is long enough. C TDIST = ABS(TOUT - T) IF(TDIST .LT. HMIN) GO TO 714 C C Check H0, if this was input. C IF (INFO(8) .EQ. 0) GO TO 310 H0 = RWORK(LH) IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 711 IF (H0 .EQ. 0.0D0) GO TO 712 GO TO 320 310 CONTINUE C C Compute initial stepsize, to be used by either C DDSTP or DDASIC, depending on INFO(11). C H0 = 0.001D0*TDIST YPNORM = DDWNRM(NEQ,YPRIME,RWORK(LVT),RPAR,IPAR) IF (YPNORM .GT. 0.5D0/H0) H0 = 0.5D0/YPNORM H0 = SIGN(H0,TOUT-T) C C Adjust H0 if necessary to meet HMAX bound. C 320 IF (INFO(7) .EQ. 0) GO TO 330 RH = ABS(H0)/RWORK(LHMAX) IF (RH .GT. 1.0D0) H0 = H0/RH C C Check against TSTOP, if applicable. C 330 IF (INFO(4) .EQ. 0) GO TO 340 TSTOP = RWORK(LTSTOP) IF ((TSTOP - T)*H0 .LT. 0.0D0) GO TO 715 IF ((T + H0 - TSTOP)*H0 .GT. 0.0D0) H0 = TSTOP - T IF ((TSTOP - TOUT)*H0 .LT. 0.0D0) GO TO 709 C 340 IF (INFO(11) .EQ. 0) GO TO 370 C C Compute unknown components of initial Y and YPRIME, depending C on INFO(11) and INFO(12). INFO(12) represents the nonlinear C solver type (direct/Krylov). Pass the name of the specific C nonlinear solver, depending on INFO(12). The location of the work C arrays SAVR, YIC, YPIC, PWK also differ in the two cases. C For use in stopping tests, pass TSCALE = TDIST if INDEX = 0. C NWT = 1 EPCONI = RWORK(LEPIN)*RWORK(LEPCON) TSCALE = 0.0D0 IF (INDEX .EQ. 0) TSCALE = TDIST 350 IF (INFO(12) .EQ. 0) THEN LYIC = LPHI + 2*NEQ LYPIC = LYIC + NEQ LPWK = LYPIC CALL DDASIC(TN,Y,YPRIME,NEQ,INFO(11),IWORK(LID), * RES,JAC,PSOL,H0,TSCALE,RWORK(LWT),NWT,IDID,RPAR,IPAR, * RWORK(LPHI),RWORK(LSAVR),RWORK(LDELTA),RWORK(LE), * RWORK(LYIC),RWORK(LYPIC),RWORK(LPWK),RWORK(LWM),IWORK(LIWM), * RWORK(LROUND),RWORK(LEPLI),RWORK(LSQRN),RWORK(LRSQRN), * EPCONI,RWORK(LSTOL),INFO(15),ICNFLG,IWORK(LICNS),DDASID) ELSE IF (INFO(12) .EQ. 1) THEN LYIC = LWM LYPIC = LYIC + NEQ LPWK = LYPIC + NEQ CALL DDASIC(TN,Y,YPRIME,NEQ,INFO(11),IWORK(LID), * RES,JAC,PSOL,H0,TSCALE,RWORK(LWT),NWT,IDID,RPAR,IPAR, * RWORK(LPHI),RWORK(LSAVR),RWORK(LDELTA),RWORK(LE), * RWORK(LYIC),RWORK(LYPIC),RWORK(LPWK),RWORK(LWM),IWORK(LIWM), * RWORK(LROUND),RWORK(LEPLI),RWORK(LSQRN),RWORK(LRSQRN), * EPCONI,RWORK(LSTOL),INFO(15),ICNFLG,IWORK(LICNS),DDASIK) ENDIF C IF (IDID .LT. 0) GO TO 600 C C DDASIC was successful. If this was the first call to DDASIC, C update the WT array (with the current Y) and call it again. C IF (NWT .EQ. 2) GO TO 355 NWT = 2 CALL DDAWTS(NEQ,INFO(2),RTOL,ATOL,Y,RWORK(LWT),RPAR,IPAR) CALL DINVWT(NEQ,RWORK(LWT),IER) IF (IER .NE. 0) GO TO 713 GO TO 350 C C If INFO(14) = 1, return now with IDID = 4. C 355 IF (INFO(14) .EQ. 1) THEN IDID = 4 H = H0 IF (INFO(11) .EQ. 1) RWORK(LHOLD) = H0 GO TO 590 ENDIF C C Update the WT and VT arrays one more time, with the new Y. C CALL DDAWTS(NEQ,INFO(2),RTOL,ATOL,Y,RWORK(LWT),RPAR,IPAR) CALL DINVWT(NEQ,RWORK(LWT),IER) IF (IER .NE. 0) GO TO 713 IF (INFO(16) .NE. 0) THEN DO 357 I = 1, NEQ 357 RWORK(LVT+I-1) = MAX(IWORK(LID+I-1),0)*RWORK(LWT+I-1) ENDIF C C Reset the initial stepsize to be used by DDSTP. C Use H0, if this was input. Otherwise, recompute H0, C and adjust it if necessary to meet HMAX bound. C IF (INFO(8) .NE. 0) THEN H0 = RWORK(LH) GO TO 360 ENDIF C H0 = 0.001D0*TDIST YPNORM = DDWNRM(NEQ,YPRIME,RWORK(LVT),RPAR,IPAR) IF (YPNORM .GT. 0.5D0/H0) H0 = 0.5D0/YPNORM H0 = SIGN(H0,TOUT-T) C 360 IF (INFO(7) .NE. 0) THEN RH = ABS(H0)/RWORK(LHMAX) IF (RH .GT. 1.0D0) H0 = H0/RH ENDIF C C Check against TSTOP, if applicable. C IF (INFO(4) .NE. 0) THEN TSTOP = RWORK(LTSTOP) IF ((T + H0 - TSTOP)*H0 .GT. 0.0D0) H0 = TSTOP - T ENDIF C C Load H and RWORK(LH) with H0. C 370 H = H0 RWORK(LH) = H C C Load Y and H*YPRIME into PHI(*,1) and PHI(*,2). C ITEMP = LPHI + NEQ DO 380 I = 1,NEQ RWORK(LPHI + I - 1) = Y(I) 380 RWORK(ITEMP + I - 1) = H*YPRIME(I) C C Initialize T0 in RWORK; check for a zero of R near initial T. C RWORK(LT0) = T IWORK(LIRFND) = 0 RWORK(LPSI)=H RWORK(LPSI+1)=2.0D0*H IWORK(LKOLD)=1 IF (NRT .EQ. 0) GO TO 390 CALL DRCHEK(1,RT,NRT,NEQ,T,TOUT,Y,YPRIME,RWORK(LPHI), * RWORK(LPSI),IWORK(LKOLD),RWORK(LR0),RWORK(LR1), * RWORK(LRX),JROOT,IRT,RWORK(LROUND),INFO(3), * RWORK,IWORK,RPAR,IPAR) IF (IRT .LT. 0) GO TO 731 C 390 GO TO 500 C C----------------------------------------------------------------------- C This block is for continuation calls only. C Its purpose is to check stop conditions before taking a step. C Adjust H if necessary to meet HMAX bound. C----------------------------------------------------------------------- C 400 CONTINUE UROUND=RWORK(LROUND) DONE = .FALSE. TN=RWORK(LTN) H=RWORK(LH) IF(NRT .EQ. 0) GO TO 405 C C Check for a zero of R near TN. C CALL DRCHEK(2,RT,NRT,NEQ,TN,TOUT,Y,YPRIME,RWORK(LPHI), * RWORK(LPSI),IWORK(LKOLD),RWORK(LR0),RWORK(LR1), * RWORK(LRX),JROOT,IRT,RWORK(LROUND),INFO(3), * RWORK,IWORK,RPAR,IPAR) IF (IRT .LT. 0) GO TO 731 IF (IRT .NE. 1) GO TO 405 IWORK(LIRFND) = 1 IDID = 5 T = RWORK(LT0) DONE = .TRUE. GO TO 490 405 CONTINUE C IF(INFO(7) .EQ. 0) GO TO 410 RH = ABS(H)/RWORK(LHMAX) IF(RH .GT. 1.0D0) H = H/RH 410 CONTINUE IF(T .EQ. TOUT) GO TO 719 IF((T - TOUT)*H .GT. 0.0D0) GO TO 711 IF(INFO(4) .EQ. 1) GO TO 430 IF(INFO(3) .EQ. 1) GO TO 420 IF((TN-TOUT)*H.LT.0.0D0)GO TO 490 CALL DDATRP(TN,TOUT,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T=TOUT IDID = 3 DONE = .TRUE. GO TO 490 420 IF((TN-T)*H .LE. 0.0D0) GO TO 490 IF((TN - TOUT)*H .GE. 0.0D0) GO TO 425 CALL DDATRP(TN,TN,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TN IDID = 1 DONE = .TRUE. GO TO 490 425 CONTINUE CALL DDATRP(TN,TOUT,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TOUT IDID = 3 DONE = .TRUE. GO TO 490 430 IF(INFO(3) .EQ. 1) GO TO 440 TSTOP=RWORK(LTSTOP) IF((TN-TSTOP)*H.GT.0.0D0) GO TO 715 IF((TSTOP-TOUT)*H.LT.0.0D0)GO TO 709 IF((TN-TOUT)*H.LT.0.0D0)GO TO 450 CALL DDATRP(TN,TOUT,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T=TOUT IDID = 3 DONE = .TRUE. GO TO 490 440 TSTOP = RWORK(LTSTOP) IF((TN-TSTOP)*H .GT. 0.0D0) GO TO 715 IF((TSTOP-TOUT)*H .LT. 0.0D0) GO TO 709 IF((TN-T)*H .LE. 0.0D0) GO TO 450 IF((TN - TOUT)*H .GE. 0.0D0) GO TO 445 CALL DDATRP(TN,TN,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TN IDID = 1 DONE = .TRUE. GO TO 490 445 CONTINUE CALL DDATRP(TN,TOUT,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TOUT IDID = 3 DONE = .TRUE. GO TO 490 450 CONTINUE C C Check whether we are within roundoff of TSTOP. C IF(ABS(TN-TSTOP).GT.100.0D0*UROUND* * (ABS(TN)+ABS(H)))GO TO 460 CALL DDATRP(TN,TSTOP,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) IDID=2 T=TSTOP DONE = .TRUE. GO TO 490 460 TNEXT=TN+H IF((TNEXT-TSTOP)*H.LE.0.0D0)GO TO 490 H=TSTOP-TN RWORK(LH)=H C 490 IF (DONE) GO TO 590 C C----------------------------------------------------------------------- C The next block contains the call to the one-step integrator DDSTP. C This is a looping point for the integration steps. C Check for too many steps. C Check for poor Newton/Krylov performance. C Update WT. Check for too much accuracy requested. C Compute minimum stepsize. C----------------------------------------------------------------------- C 500 CONTINUE C C Check for too many steps. C IF((IWORK(LNST)-IWORK(LNSTL)).LT.500) GO TO 505 IDID=-1 GO TO 527 C C Check for poor Newton/Krylov performance. C 505 IF (INFO(12) .EQ. 0) GO TO 510 NSTD = IWORK(LNST) - IWORK(LNSTL) NNID = IWORK(LNNI) - NNI0 IF (NSTD .LT. 10 .OR. NNID .EQ. 0) GO TO 510 AVLIN = REAL(IWORK(LNLI) - NLI0)/REAL(NNID) RCFN = REAL(IWORK(LNCFN) - NCFN0)/REAL(NSTD) RCFL = REAL(IWORK(LNCFL) - NCFL0)/REAL(NNID) FMAXL = IWORK(LMAXL) LAVL = AVLIN .GT. FMAXL LCFN = RCFN .GT. 0.9D0 LCFL = RCFL .GT. 0.9D0 LWARN = LAVL .OR. LCFN .OR. LCFL IF (.NOT.LWARN) GO TO 510 NWARN = NWARN + 1 IF (NWARN .GT. 10) GO TO 510 IF (LAVL) THEN MSG = 'DASKR-- Warning. Poor iterative algorithm performance ' CALL XERRWD (MSG, 56, 501, 0, 0, 0, 0, 0, 0.0D0, 0.0D0) MSG = ' at T = R1. Average no. of linear iterations = R2 ' CALL XERRWD (MSG, 56, 501, 0, 0, 0, 0, 2, TN, AVLIN) ENDIF IF (LCFN) THEN MSG = 'DASKR-- Warning. Poor iterative algorithm performance ' CALL XERRWD (MSG, 56, 502, 0, 0, 0, 0, 0, 0.0D0, 0.0D0) MSG = ' at T = R1. Nonlinear convergence failure rate = R2' CALL XERRWD (MSG, 56, 502, 0, 0, 0, 0, 2, TN, RCFN) ENDIF IF (LCFL) THEN MSG = 'DASKR-- Warning. Poor iterative algorithm performance ' CALL XERRWD (MSG, 56, 503, 0, 0, 0, 0, 0, 0.0D0, 0.0D0) MSG = ' at T = R1. Linear convergence failure rate = R2 ' CALL XERRWD (MSG, 56, 503, 0, 0, 0, 0, 2, TN, RCFL) ENDIF C C Update WT and VT, if this is not the first call. C 510 CALL DDAWTS(NEQ,INFO(2),RTOL,ATOL,RWORK(LPHI),RWORK(LWT), * RPAR,IPAR) CALL DINVWT(NEQ,RWORK(LWT),IER) IF (IER .NE. 0) THEN IDID = -3 GO TO 527 ENDIF IF (INFO(16) .NE. 0) THEN DO 515 I = 1, NEQ 515 RWORK(LVT+I-1) = MAX(IWORK(LID+I-1),0)*RWORK(LWT+I-1) ENDIF C C Test for too much accuracy requested. C R = DDWNRM(NEQ,RWORK(LPHI),RWORK(LWT),RPAR,IPAR)*100.0D0*UROUND IF (R .LE. 1.0D0) GO TO 525 C C Multiply RTOL and ATOL by R and return. C IF(INFO(2).EQ.1)GO TO 523 RTOL(1)=R*RTOL(1) ATOL(1)=R*ATOL(1) IDID=-2 GO TO 527 523 DO 524 I=1,NEQ RTOL(I)=R*RTOL(I) 524 ATOL(I)=R*ATOL(I) IDID=-2 GO TO 527 525 CONTINUE C C Compute minimum stepsize. C HMIN=4.0D0*UROUND*MAX(ABS(TN),ABS(TOUT)) C C Test H vs. HMAX IF (INFO(7) .NE. 0) THEN RH = ABS(H)/RWORK(LHMAX) IF (RH .GT. 1.0D0) H = H/RH ENDIF C C Call the one-step integrator. C Note that INFO(12) represents the nonlinear solver type. C Pass the required nonlinear solver, depending upon INFO(12). C IF (INFO(12) .EQ. 0) THEN CALL DDSTP(TN,Y,YPRIME,NEQ, * RES,JAC,PSOL,H,RWORK(LWT),RWORK(LVT),INFO(1),IDID,RPAR,IPAR, * RWORK(LPHI),RWORK(LSAVR),RWORK(LDELTA),RWORK(LE), * RWORK(LWM),IWORK(LIWM), * RWORK(LALPHA),RWORK(LBETA),RWORK(LGAMMA), * RWORK(LPSI),RWORK(LSIGMA), * RWORK(LCJ),RWORK(LCJOLD),RWORK(LHOLD),RWORK(LS),HMIN, * RWORK(LROUND), RWORK(LEPLI),RWORK(LSQRN),RWORK(LRSQRN), * RWORK(LEPCON), IWORK(LPHASE),IWORK(LJCALC),INFO(15), * IWORK(LK), IWORK(LKOLD),IWORK(LNS),NONNEG,INFO(12), * DNEDD) ELSE IF (INFO(12) .EQ. 1) THEN CALL DDSTP(TN,Y,YPRIME,NEQ, * RES,JAC,PSOL,H,RWORK(LWT),RWORK(LVT),INFO(1),IDID,RPAR,IPAR, * RWORK(LPHI),RWORK(LSAVR),RWORK(LDELTA),RWORK(LE), * RWORK(LWM),IWORK(LIWM), * RWORK(LALPHA),RWORK(LBETA),RWORK(LGAMMA), * RWORK(LPSI),RWORK(LSIGMA), * RWORK(LCJ),RWORK(LCJOLD),RWORK(LHOLD),RWORK(LS),HMIN, * RWORK(LROUND), RWORK(LEPLI),RWORK(LSQRN),RWORK(LRSQRN), * RWORK(LEPCON), IWORK(LPHASE),IWORK(LJCALC),INFO(15), * IWORK(LK), IWORK(LKOLD),IWORK(LNS),NONNEG,INFO(12), * DNEDK) ENDIF C 527 IF(IDID.LT.0)GO TO 600 C C----------------------------------------------------------------------- C This block handles the case of a successful return from DDSTP C (IDID=1). Test for stop conditions. C----------------------------------------------------------------------- C IF(NRT .EQ. 0) GO TO 530 C C Check for a zero of R near TN. C CALL DRCHEK(3,RT,NRT,NEQ,TN,TOUT,Y,YPRIME,RWORK(LPHI), * RWORK(LPSI),IWORK(LKOLD),RWORK(LR0),RWORK(LR1), * RWORK(LRX),JROOT,IRT,RWORK(LROUND),INFO(3), * RWORK,IWORK,RPAR,IPAR) IF(IRT .NE. 1) GO TO 530 IWORK(LIRFND) = 1 IDID = 5 T = RWORK(LT0) GO TO 580 C 530 IF (INFO(4) .EQ. 0) THEN C Stopping tests for the case of no TSTOP. ---------------------- IF ( (TN-TOUT)*H .GE. 0.0D0) THEN CALL DDATRP(TN,TOUT,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TOUT IDID = 3 GO TO 580 ENDIF IF (INFO(3) .EQ. 0) GO TO 500 T = TN IDID = 1 GO TO 580 ENDIF C 540 IF (INFO(3) .NE. 0) GO TO 550 C Stopping tests for the TSTOP case, interval-output mode. --------- IF (ABS(TN-TSTOP) .LE. 100.0D0*UROUND*(ABS(TN)+ABS(H))) THEN CALL DDATRP(TN,TSTOP,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TSTOP IDID = 2 GO TO 580 ENDIF IF ( (TN-TOUT)*H .GE. 0.0D0) THEN CALL DDATRP(TN,TOUT,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TOUT IDID = 3 GO TO 580 ENDIF TNEXT = TN + H IF ((TNEXT-TSTOP)*H .LE. 0.0D0) GO TO 500 H = TSTOP - TN GO TO 500 C 550 CONTINUE C Stopping tests for the TSTOP case, intermediate-output mode. ----- IF (ABS(TN-TSTOP) .LE. 100.0D0*UROUND*(ABS(TN)+ABS(H))) THEN CALL DDATRP(TN,TSTOP,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TSTOP IDID = 2 GO TO 580 ENDIF IF ( (TN-TOUT)*H .GE. 0.0D0) THEN CALL DDATRP(TN,TOUT,Y,YPRIME,NEQ,IWORK(LKOLD), * RWORK(LPHI),RWORK(LPSI)) T = TOUT IDID = 3 GO TO 580 ENDIF T = TN IDID = 1 C 580 CONTINUE C C----------------------------------------------------------------------- C All successful returns from DDASKR are made from this block. C----------------------------------------------------------------------- C 590 CONTINUE RWORK(LTN)=TN RWORK(LTLAST)=T RWORK(LH)=H RETURN C C----------------------------------------------------------------------- C This block handles all unsuccessful returns other than for C illegal input. C----------------------------------------------------------------------- C 600 CONTINUE ITEMP = -IDID GO TO (610,620,630,700,655,640,650,660,670,675, * 680,685,690,695), ITEMP C C The maximum number of steps was taken before C reaching tout. C 610 MSG = 'DASKR-- AT CURRENT T (=R1) 500 STEPS' CALL XERRWD(MSG,38,610,0,0,0,0,1,TN,0.0D0) MSG = 'DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT' CALL XERRWD(MSG,48,611,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C Too much accuracy for machine precision. C 620 MSG = 'DASKR-- AT T (=R1) TOO MUCH ACCURACY REQUESTED' CALL XERRWD(MSG,47,620,0,0,0,0,1,TN,0.0D0) MSG = 'DASKR-- FOR PRECISION OF MACHINE. RTOL AND ATOL' CALL XERRWD(MSG,48,621,0,0,0,0,0,0.0D0,0.0D0) MSG = 'DASKR-- WERE INCREASED BY A FACTOR R (=R1)' CALL XERRWD(MSG,43,622,0,0,0,0,1,R,0.0D0) GO TO 700 C C WT(I) .LE. 0.0D0 for some I (not at start of problem). C 630 MSG = 'DASKR-- AT T (=R1) SOME ELEMENT OF WT' CALL XERRWD(MSG,38,630,0,0,0,0,1,TN,0.0D0) MSG = 'DASKR-- HAS BECOME .LE. 0.0' CALL XERRWD(MSG,28,631,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C Error test failed repeatedly or with H=HMIN. C 640 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2) THE' CALL XERRWD(MSG,44,640,0,0,0,0,2,TN,H) MSG='DASKR-- ERROR TEST FAILED REPEATEDLY OR WITH ABS(H)=HMIN' CALL XERRWD(MSG,57,641,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C Nonlinear solver failed to converge repeatedly or with H=HMIN. C 650 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2) THE' CALL XERRWD(MSG,44,650,0,0,0,0,2,TN,H) MSG = 'DASKR-- NONLINEAR SOLVER FAILED TO CONVERGE' CALL XERRWD(MSG,44,651,0,0,0,0,0,0.0D0,0.0D0) MSG = 'DASKR-- REPEATEDLY OR WITH ABS(H)=HMIN' CALL XERRWD(MSG,40,652,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C The preconditioner had repeated failures. C 655 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2) THE' CALL XERRWD(MSG,44,655,0,0,0,0,2,TN,H) MSG = 'DASKR-- PRECONDITIONER HAD REPEATED FAILURES.' CALL XERRWD(MSG,46,656,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C The iteration matrix is singular. C 660 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2) THE' CALL XERRWD(MSG,44,660,0,0,0,0,2,TN,H) MSG = 'DASKR-- ITERATION MATRIX IS SINGULAR.' CALL XERRWD(MSG,38,661,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C Nonlinear system failure preceded by error test failures. C 670 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2) THE' CALL XERRWD(MSG,44,670,0,0,0,0,2,TN,H) MSG = 'DASKR-- NONLINEAR SOLVER COULD NOT CONVERGE.' CALL XERRWD(MSG,45,671,0,0,0,0,0,0.0D0,0.0D0) MSG = 'DASKR-- ALSO, THE ERROR TEST FAILED REPEATEDLY.' CALL XERRWD(MSG,49,672,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C Nonlinear system failure because IRES = -1. C 675 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2) THE' CALL XERRWD(MSG,44,675,0,0,0,0,2,TN,H) MSG = 'DASKR-- NONLINEAR SYSTEM SOLVER COULD NOT CONVERGE' CALL XERRWD(MSG,51,676,0,0,0,0,0,0.0D0,0.0D0) MSG = 'DASKR-- BECAUSE IRES WAS EQUAL TO MINUS ONE' CALL XERRWD(MSG,44,677,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C Failure because IRES = -2. C 680 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2)' CALL XERRWD(MSG,40,680,0,0,0,0,2,TN,H) MSG = 'DASKR-- IRES WAS EQUAL TO MINUS TWO' CALL XERRWD(MSG,36,681,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C Failed to compute initial YPRIME. C 685 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2) THE' CALL XERRWD(MSG,44,685,0,0,0,0,0,0.0D0,0.0D0) MSG = 'DASKR-- INITIAL (Y,YPRIME) COULD NOT BE COMPUTED' CALL XERRWD(MSG,49,686,0,0,0,0,2,TN,H0) GO TO 700 C C Failure because IER was negative from PSOL. C 690 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2)' CALL XERRWD(MSG,40,690,0,0,0,0,2,TN,H) MSG = 'DASKR-- IER WAS NEGATIVE FROM PSOL' CALL XERRWD(MSG,35,691,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C Failure because the linear system solver could not converge. C 695 MSG = 'DASKR-- AT T (=R1) AND STEPSIZE H (=R2) THE' CALL XERRWD(MSG,44,695,0,0,0,0,2,TN,H) MSG = 'DASKR-- LINEAR SYSTEM SOLVER COULD NOT CONVERGE.' CALL XERRWD(MSG,50,696,0,0,0,0,0,0.0D0,0.0D0) GO TO 700 C C 700 CONTINUE INFO(1)=-1 T=TN RWORK(LTN)=TN RWORK(LH)=H RETURN C C----------------------------------------------------------------------- C This block handles all error returns due to illegal input, C as detected before calling DDSTP. C First the error message routine is called. If this happens C twice in succession, execution is terminated. C----------------------------------------------------------------------- C 701 MSG = 'DASKR-- ELEMENT (=I1) OF INFO VECTOR IS NOT VALID' CALL XERRWD(MSG,50,1,0,1,ITEMP,0,0,0.0D0,0.0D0) GO TO 750 702 MSG = 'DASKR-- NEQ (=I1) .LE. 0' CALL XERRWD(MSG,25,2,0,1,NEQ,0,0,0.0D0,0.0D0) GO TO 750 703 MSG = 'DASKR-- MAXORD (=I1) NOT IN RANGE' CALL XERRWD(MSG,34,3,0,1,MXORD,0,0,0.0D0,0.0D0) GO TO 750 704 MSG='DASKR-- RWORK LENGTH NEEDED, LENRW (=I1), EXCEEDS LRW (=I2)' CALL XERRWD(MSG,60,4,0,2,LENRW,LRW,0,0.0D0,0.0D0) GO TO 750 705 MSG='DASKR-- IWORK LENGTH NEEDED, LENIW (=I1), EXCEEDS LIW (=I2)' CALL XERRWD(MSG,60,5,0,2,LENIW,LIW,0,0.0D0,0.0D0) GO TO 750 706 MSG = 'DASKR-- SOME ELEMENT OF RTOL IS .LT. 0' CALL XERRWD(MSG,39,6,0,0,0,0,0,0.0D0,0.0D0) GO TO 750 707 MSG = 'DASKR-- SOME ELEMENT OF ATOL IS .LT. 0' CALL XERRWD(MSG,39,7,0,0,0,0,0,0.0D0,0.0D0) GO TO 750 708 MSG = 'DASKR-- ALL ELEMENTS OF RTOL AND ATOL ARE ZERO' CALL XERRWD(MSG,47,8,0,0,0,0,0,0.0D0,0.0D0) GO TO 750 709 MSG='DASKR-- INFO(4) = 1 AND TSTOP (=R1) BEHIND TOUT (=R2)' CALL XERRWD(MSG,54,9,0,0,0,0,2,TSTOP,TOUT) GO TO 750 710 MSG = 'DASKR-- HMAX (=R1) .LT. 0.0' CALL XERRWD(MSG,28,10,0,0,0,0,1,HMAX,0.0D0) GO TO 750 711 MSG = 'DASKR-- TOUT (=R1) BEHIND T (=R2)' CALL XERRWD(MSG,34,11,0,0,0,0,2,TOUT,T) GO TO 750 712 MSG = 'DASKR-- INFO(8)=1 AND H0=0.0' CALL XERRWD(MSG,29,12,0,0,0,0,0,0.0D0,0.0D0) GO TO 750 713 MSG = 'DASKR-- SOME ELEMENT OF WT IS .LE. 0.0' CALL XERRWD(MSG,39,13,0,0,0,0,0,0.0D0,0.0D0) GO TO 750 714 MSG='DASKR-- TOUT (=R1) TOO CLOSE TO T (=R2) TO START INTEGRATION' CALL XERRWD(MSG,60,14,0,0,0,0,2,TOUT,T) GO TO 750 715 MSG = 'DASKR-- INFO(4)=1 AND TSTOP (=R1) BEHIND T (=R2)' CALL XERRWD(MSG,49,15,0,0,0,0,2,TSTOP,T) GO TO 750 717 MSG = 'DASKR-- ML (=I1) ILLEGAL. EITHER .LT. 0 OR .GT. NEQ' CALL XERRWD(MSG,52,17,0,1,IWORK(LML),0,0,0.0D0,0.0D0) GO TO 750 718 MSG = 'DASKR-- MU (=I1) ILLEGAL. EITHER .LT. 0 OR .GT. NEQ' CALL XERRWD(MSG,52,18,0,1,IWORK(LMU),0,0,0.0D0,0.0D0) GO TO 750 719 MSG = 'DASKR-- TOUT (=R1) IS EQUAL TO T (=R2)' CALL XERRWD(MSG,39,19,0,0,0,0,2,TOUT,T) GO TO 750 720 MSG = 'DASKR-- MAXL (=I1) ILLEGAL. EITHER .LT. 1 OR .GT. NEQ' CALL XERRWD(MSG,54,20,0,1,IWORK(LMAXL),0,0,0.0D0,0.0D0) GO TO 750 721 MSG = 'DASKR-- KMP (=I1) ILLEGAL. EITHER .LT. 1 OR .GT. MAXL' CALL XERRWD(MSG,54,21,0,1,IWORK(LKMP),0,0,0.0D0,0.0D0) GO TO 750 722 MSG = 'DASKR-- NRMAX (=I1) ILLEGAL. .LT. 0' CALL XERRWD(MSG,36,22,0,1,IWORK(LNRMAX),0,0,0.0D0,0.0D0) GO TO 750 723 MSG = 'DASKR-- EPLI (=R1) ILLEGAL. EITHER .LE. 0.D0 OR .GE. 1.D0' CALL XERRWD(MSG,58,23,0,0,0,0,1,RWORK(LEPLI),0.0D0) GO TO 750 724 MSG = 'DASKR-- ILLEGAL IWORK VALUE FOR INFO(11) .NE. 0' CALL XERRWD(MSG,48,24,0,0,0,0,0,0.0D0,0.0D0) GO TO 750 725 MSG = 'DASKR-- ONE OF THE INPUTS FOR INFO(17) = 1 IS ILLEGAL' CALL XERRWD(MSG,54,25,0,0,0,0,0,0.0D0,0.0D0) GO TO 750 726 MSG = 'DASKR-- ILLEGAL IWORK VALUE FOR INFO(10) .NE. 0' CALL XERRWD(MSG,48,26,0,0,0,0,0,0.0D0,0.0D0) GO TO 750 727 MSG = 'DASKR-- Y(I) AND IWORK(40+I) (I=I1) INCONSISTENT' CALL XERRWD(MSG,49,27,0,1,IRET,0,0,0.0D0,0.0D0) GO TO 750 730 MSG = 'DASKR-- NRT (=I1) .LT. 0' CALL XERRWD(MSG,25,30,1,1,NRT,0,0,0.0D0,0.0D0) GO TO 750 731 MSG = 'DASKR-- R IS ILL-DEFINED. ZERO VALUES WERE FOUND AT TWO' CALL XERRWD(MSG,57,31,1,0,0,0,0,0.0D0,0.0D0) MSG = ' VERY CLOSE T VALUES, AT T = R1' CALL XERRWD(MSG,39,31,1,0,0,0,1,RWORK(LT0),0.0D0) C 750 IF(INFO(1).EQ.-1) GO TO 760 INFO(1)=-1 IDID=-33 RETURN 760 MSG = 'DASKR-- REPEATED OCCURRENCES OF ILLEGAL INPUT' CALL XERRWD(MSG,46,701,0,0,0,0,0,0.0D0,0.0D0) 770 MSG = 'DASKR-- RUN TERMINATED. APPARENT INFINITE LOOP' CALL XERRWD(MSG,47,702,1,0,0,0,0,0.0D0,0.0D0) RETURN C C------END OF SUBROUTINE DDASKR----------------------------------------- END SUBROUTINE DRCHEK (JOB, RT, NRT, NEQ, TN, TOUT, Y, YP, PHI, PSI, * KOLD, R0, R1, RX, JROOT, IRT, UROUND, INFO3, RWORK, IWORK, * RPAR, IPAR) C C***BEGIN PROLOGUE DRCHEK C***REFER TO DDASKR C***ROUTINES CALLED DDATRP, DROOTS, DCOPY, RT C***REVISION HISTORY (YYMMDD) C 020815 DATE WRITTEN C 021217 Added test for roots close when JOB = 2. C 050510 Changed T increment after 110 so that TEMP1/H .ge. 0.1. C 071003 Fixed bug in TEMP2 (HMINR) below 110. C 110608 Fixed bug in setting of T1 at 300. C***END PROLOGUE DRCHEK C IMPLICIT DOUBLE PRECISION(A-H,O-Z) C Pointers into IWORK: PARAMETER (LNRTE=36, LIRFND=37) C Pointers into RWORK: PARAMETER (LT0=51, LTLAST=52) EXTERNAL RT INTEGER JOB, NRT, NEQ, KOLD, JROOT, IRT, INFO3, IWORK, IPAR DOUBLE PRECISION TN, TOUT, Y, YP, PHI, PSI, R0, R1, RX, UROUND, * RWORK, RPAR DIMENSION Y(*), YP(*), PHI(NEQ,*), PSI(*), * R0(*), R1(*), RX(*), JROOT(*), RWORK(*), IWORK(*) INTEGER I, JFLAG DOUBLE PRECISION H DOUBLE PRECISION HMINR, T1, TEMP1, TEMP2, X, ZERO LOGICAL ZROOT DATA ZERO/0.0D0/ C----------------------------------------------------------------------- C This routine checks for the presence of a root of R(T,Y,Y') in the C vicinity of the current T, in a manner depending on the C input flag JOB. It calls subroutine DROOTS to locate the root C as precisely as possible. C C In addition to variables described previously, DRCHEK C uses the following for communication.. C JOB = integer flag indicating type of call.. C JOB = 1 means the problem is being initialized, and DRCHEK C is to look for a root at or very near the initial T. C JOB = 2 means a continuation call to the solver was just C made, and DRCHEK is to check for a root in the C relevant part of the step last taken. C JOB = 3 means a successful step was just taken, and DRCHEK C is to look for a root in the interval of the step. C R0 = array of length NRT, containing the value of R at T = T0. C R0 is input for JOB .ge. 2 and on output in all cases. C R1,RX = arrays of length NRT for work space. C IRT = completion flag.. C IRT = 0 means no root was found. C IRT = -1 means JOB = 1 and a zero was found both at T0 and C and very close to T0. C IRT = -2 means JOB = 2 and some Ri was found to have a zero C both at T0 and very close to T0. C IRT = 1 means a legitimate root was found (JOB = 2 or 3). C On return, T0 is the root location, and Y is the C corresponding solution vector. C T0 = value of T at one endpoint of interval of interest. Only C roots beyond T0 in the direction of integration are sought. C T0 is input if JOB .ge. 2, and output in all cases. C T0 is updated by DRCHEK, whether a root is found or not. C Stored in the global array RWORK. C TLAST = last value of T returned by the solver (input only). C Stored in the global array RWORK. C TOUT = final output time for the solver. C IRFND = input flag showing whether the last step taken had a root. C IRFND = 1 if it did, = 0 if not. C Stored in the global array IWORK. C INFO3 = copy of INFO(3) (input only). C----------------------------------------------------------------------- C H = PSI(1) IRT = 0 DO 10 I = 1,NRT 10 JROOT(I) = 0 HMINR = (ABS(TN) + ABS(H))*UROUND*100.0D0 C GO TO (100, 200, 300), JOB C C Evaluate R at initial T (= RWORK(LT0)); check for zero values.-------- 100 CONTINUE CALL DDATRP(TN,RWORK(LT0),Y,YP,NEQ,KOLD,PHI,PSI) CALL RT (NEQ, RWORK(LT0), Y, YP, NRT, R0, RPAR, IPAR) IWORK(LNRTE) = 1 ZROOT = .FALSE. DO 110 I = 1,NRT 110 IF (ABS(R0(I)) .EQ. ZERO) ZROOT = .TRUE. IF (.NOT. ZROOT) GO TO 190 C R has a zero at T. Look at R at T + (small increment). -------------- TEMP2 = MAX(HMINR/ABS(H), 0.1D0) TEMP1 = TEMP2*H RWORK(LT0) = RWORK(LT0) + TEMP1 DO 120 I = 1,NEQ 120 Y(I) = Y(I) + TEMP2*PHI(I,2) CALL RT (NEQ, RWORK(LT0), Y, YP, NRT, R0, RPAR, IPAR) IWORK(LNRTE) = IWORK(LNRTE) + 1 ZROOT = .FALSE. DO 130 I = 1,NRT 130 IF (ABS(R0(I)) .EQ. ZERO) ZROOT = .TRUE. IF (.NOT. ZROOT) GO TO 190 C R has a zero at T and also close to T. Take error return. ----------- IRT = -1 RETURN C 190 CONTINUE RETURN C 200 CONTINUE IF (IWORK(LIRFND) .EQ. 0) GO TO 260 C If a root was found on the previous step, evaluate R0 = R(T0). ------- CALL DDATRP (TN, RWORK(LT0), Y, YP, NEQ, KOLD, PHI, PSI) CALL RT (NEQ, RWORK(LT0), Y, YP, NRT, R0, RPAR, IPAR) IWORK(LNRTE) = IWORK(LNRTE) + 1 ZROOT = .FALSE. DO 210 I = 1,NRT IF (ABS(R0(I)) .EQ. ZERO) THEN ZROOT = .TRUE. JROOT(I) = 1 ENDIF 210 CONTINUE IF (.NOT. ZROOT) GO TO 260 C R has a zero at T0. Look at R at T0+ = T0 + (small increment). ------ TEMP1 = SIGN(HMINR,H) RWORK(LT0) = RWORK(LT0) + TEMP1 IF ((RWORK(LT0) - TN)*H .LT. ZERO) GO TO 230 TEMP2 = TEMP1/H DO 220 I = 1,NEQ 220 Y(I) = Y(I) + TEMP2*PHI(I,2) GO TO 240 230 CALL DDATRP (TN, RWORK(LT0), Y, YP, NEQ, KOLD, PHI, PSI) 240 CALL RT (NEQ, RWORK(LT0), Y, YP, NRT, R0, RPAR, IPAR) IWORK(LNRTE) = IWORK(LNRTE) + 1 DO 250 I = 1,NRT IF (ABS(R0(I)) .GT. ZERO) GO TO 250 C If Ri has a zero at both T0+ and T0, return an error flag. ----------- IF (JROOT(I) .EQ. 1) THEN IRT = -2 RETURN ELSE C If Ri has a zero at T0+, but not at T0, return valid root. ----------- JROOT(I) = -SIGN(1.0D0,R0(I)) IRT = 1 ENDIF 250 CONTINUE IF (IRT .EQ. 1) RETURN C R0 has no zero components. Proceed to check relevant interval. ------ 260 IF (TN .EQ. RWORK(LTLAST)) RETURN C 300 CONTINUE C Set T1 to TN or TOUT, whichever comes first, and get R at T1. -------- IF ((TOUT - TN)*H .GE. ZERO) THEN T1 = TN GO TO 330 ENDIF T1 = TOUT IF ((T1 - RWORK(LT0))*H .LE. ZERO) GO TO 390 330 CALL DDATRP (TN, T1, Y, YP, NEQ, KOLD, PHI, PSI) CALL RT (NEQ, T1, Y, YP, NRT, R1, RPAR, IPAR) IWORK(LNRTE) = IWORK(LNRTE) + 1 C Call DROOTS to search for root in interval from T0 to T1. ------------ JFLAG = 0 350 CONTINUE CALL DROOTS (NRT, HMINR, JFLAG, RWORK(LT0),T1, R0,R1,RX, X, JROOT) IF (JFLAG .GT. 1) GO TO 360 CALL DDATRP (TN, X, Y, YP, NEQ, KOLD, PHI, PSI) CALL RT (NEQ, X, Y, YP, NRT, RX, RPAR, IPAR) IWORK(LNRTE) = IWORK(LNRTE) + 1 GO TO 350 360 RWORK(LT0) = X CALL DCOPY (NRT, RX, 1, R0, 1) IF (JFLAG .EQ. 4) GO TO 390 C Found a root. Interpolate to X and return. -------------------------- CALL DDATRP (TN, X, Y, YP, NEQ, KOLD, PHI, PSI) IRT = 1 RETURN C 390 CONTINUE RETURN C---------------------- END OF SUBROUTINE DRCHEK ----------------------- END SUBROUTINE DROOTS (NRT, HMIN, JFLAG, X0, X1, R0, R1, RX, X, JROOT) C C***BEGIN PROLOGUE DROOTS C***REFER TO DRCHEK C***ROUTINES CALLED DCOPY C***REVISION HISTORY (YYMMDD) C 020815 DATE WRITTEN C 021217 Added root direction information in JROOT. C 040518 Changed adjustment to X2 at 180 to avoid infinite loop. C***END PROLOGUE DROOTS C INTEGER NRT, JFLAG, JROOT DOUBLE PRECISION HMIN, X0, X1, R0, R1, RX, X DIMENSION R0(NRT), R1(NRT), RX(NRT), JROOT(NRT) C----------------------------------------------------------------------- C This subroutine finds the leftmost root of a set of arbitrary C functions Ri(x) (i = 1,...,NRT) in an interval (X0,X1). Only roots C of odd multiplicity (i.e. changes of sign of the Ri) are found. C Here the sign of X1 - X0 is arbitrary, but is constant for a given C problem, and -leftmost- means nearest to X0. C The values of the vector-valued function R(x) = (Ri, i=1...NRT) C are communicated through the call sequence of DROOTS. C The method used is the Illinois algorithm. C C Reference: C Kathie L. Hiebert and Lawrence F. Shampine, Implicitly Defined C Output Points for Solutions of ODEs, Sandia Report SAND80-0180, C February 1980. C C Description of parameters. C C NRT = number of functions Ri, or the number of components of C the vector valued function R(x). Input only. C C HMIN = resolution parameter in X. Input only. When a root is C found, it is located only to within an error of HMIN in X. C Typically, HMIN should be set to something on the order of C 100 * UROUND * MAX(ABS(X0),ABS(X1)), C where UROUND is the unit roundoff of the machine. C C JFLAG = integer flag for input and output communication. C C On input, set JFLAG = 0 on the first call for the problem, C and leave it unchanged until the problem is completed. C (The problem is completed when JFLAG .ge. 2 on return.) C C On output, JFLAG has the following values and meanings: C JFLAG = 1 means DROOTS needs a value of R(x). Set RX = R(X) C and call DROOTS again. C JFLAG = 2 means a root has been found. The root is C at X, and RX contains R(X). (Actually, X is the C rightmost approximation to the root on an interval C (X0,X1) of size HMIN or less.) C JFLAG = 3 means X = X1 is a root, with one or more of the Ri C being zero at X1 and no sign changes in (X0,X1). C RX contains R(X) on output. C JFLAG = 4 means no roots (of odd multiplicity) were C found in (X0,X1) (no sign changes). C C X0,X1 = endpoints of the interval where roots are sought. C X1 and X0 are input when JFLAG = 0 (first call), and C must be left unchanged between calls until the problem is C completed. X0 and X1 must be distinct, but X1 - X0 may be C of either sign. However, the notion of -left- and -right- C will be used to mean nearer to X0 or X1, respectively. C When JFLAG .ge. 2 on return, X0 and X1 are output, and C are the endpoints of the relevant interval. C C R0,R1 = arrays of length NRT containing the vectors R(X0) and R(X1), C respectively. When JFLAG = 0, R0 and R1 are input and C none of the R0(i) should be zero. C When JFLAG .ge. 2 on return, R0 and R1 are output. C C RX = array of length NRT containing R(X). RX is input C when JFLAG = 1, and output when JFLAG .ge. 2. C C X = independent variable value. Output only. C When JFLAG = 1 on output, X is the point at which R(x) C is to be evaluated and loaded into RX. C When JFLAG = 2 or 3, X is the root. C When JFLAG = 4, X is the right endpoint of the interval, X1. C C JROOT = integer array of length NRT. Output only. C When JFLAG = 2 or 3, JROOT indicates which components C of R(x) have a root at X, and the direction of the sign C change across the root in the direction of integration. C JROOT(i) = 1 if Ri has a root and changes from - to +. C JROOT(i) = -1 if Ri has a root and changes from + to -. C Otherwise JROOT(i) = 0. C----------------------------------------------------------------------- INTEGER I, IMAX, IMXOLD, LAST, NXLAST DOUBLE PRECISION ALPHA, T2, TMAX, X2, FRACINT, FRACSUB, 1 ZERO, TENTH, HALF, FIVE LOGICAL ZROOT, SGNCHG, XROOT SAVE ALPHA, X2, IMAX, LAST DATA ZERO/0.0D0/, TENTH/0.1D0/, HALF/0.5D0/, FIVE/5.0D0/ C IF (JFLAG .EQ. 1) GO TO 200 C JFLAG .ne. 1. Check for change in sign of R or zero at X1. ---------- IMAX = 0 TMAX = ZERO ZROOT = .FALSE. DO 120 I = 1,NRT IF (ABS(R1(I)) .GT. ZERO) GO TO 110 ZROOT = .TRUE. GO TO 120 C At this point, R0(i) has been checked and cannot be zero. ------------ 110 IF (SIGN(1.0D0,R0(I)) .EQ. SIGN(1.0D0,R1(I))) GO TO 120 T2 = ABS(R1(I)/(R1(I)-R0(I))) IF (T2 .LE. TMAX) GO TO 120 TMAX = T2 IMAX = I 120 CONTINUE IF (IMAX .GT. 0) GO TO 130 SGNCHG = .FALSE. GO TO 140 130 SGNCHG = .TRUE. 140 IF (.NOT. SGNCHG) GO TO 400 C There is a sign change. Find the first root in the interval. -------- XROOT = .FALSE. NXLAST = 0 LAST = 1 C C Repeat until the first root in the interval is found. Loop point. --- 150 CONTINUE IF (XROOT) GO TO 300 IF (NXLAST .EQ. LAST) GO TO 160 ALPHA = 1.0D0 GO TO 180 160 IF (LAST .EQ. 0) GO TO 170 ALPHA = 0.5D0*ALPHA GO TO 180 170 ALPHA = 2.0D0*ALPHA 180 X2 = X1 - (X1-X0)*R1(IMAX)/(R1(IMAX) - ALPHA*R0(IMAX)) IF (ABS(X2 - X0) < HALF*HMIN) THEN FRACINT = ABS(X1 - X0)/HMIN IF (FRACINT .GT. FIVE) THEN FRACSUB = TENTH ELSE FRACSUB = HALF/FRACINT ENDIF X2 = X0 + FRACSUB*(X1 - X0) ENDIF IF (ABS(X1 - X2) < HALF*HMIN) THEN FRACINT = ABS(X1 - X0)/HMIN IF (FRACINT .GT. FIVE) THEN FRACSUB = TENTH ELSE FRACSUB = HALF/FRACINT ENDIF X2 = X1 - FRACSUB*(X1 - X0) ENDIF JFLAG = 1 X = X2 C Return to the calling routine to get a value of RX = R(X). ----------- RETURN C Check to see in which interval R changes sign. ----------------------- 200 IMXOLD = IMAX IMAX = 0 TMAX = ZERO ZROOT = .FALSE. DO 220 I = 1,NRT IF (ABS(RX(I)) .GT. ZERO) GO TO 210 ZROOT = .TRUE. GO TO 220 C Neither R0(i) nor RX(i) can be zero at this point. ------------------- 210 IF (SIGN(1.0D0,R0(I)) .EQ. SIGN(1.0D0,RX(I))) GO TO 220 T2 = ABS(RX(I)/(RX(I) - R0(I))) IF (T2 .LE. TMAX) GO TO 220 TMAX = T2 IMAX = I 220 CONTINUE IF (IMAX .GT. 0) GO TO 230 SGNCHG = .FALSE. IMAX = IMXOLD GO TO 240 230 SGNCHG = .TRUE. 240 NXLAST = LAST IF (.NOT. SGNCHG) GO TO 250 C Sign change between X0 and X2, so replace X1 with X2. ---------------- X1 = X2 CALL DCOPY (NRT, RX, 1, R1, 1) LAST = 1 XROOT = .FALSE. GO TO 270 250 IF (.NOT. ZROOT) GO TO 260 C Zero value at X2 and no sign change in (X0,X2), so X2 is a root. ----- X1 = X2 CALL DCOPY (NRT, RX, 1, R1, 1) XROOT = .TRUE. GO TO 270 C No sign change between X0 and X2. Replace X0 with X2. --------------- 260 CONTINUE CALL DCOPY (NRT, RX, 1, R0, 1) X0 = X2 LAST = 0 XROOT = .FALSE. 270 IF (ABS(X1-X0) .LE. HMIN) XROOT = .TRUE. GO TO 150 C C Return with X1 as the root. Set JROOT. Set X = X1 and RX = R1. ----- 300 JFLAG = 2 X = X1 CALL DCOPY (NRT, R1, 1, RX, 1) DO 320 I = 1,NRT JROOT(I) = 0 IF (ABS(R1(I)) .EQ. ZERO) THEN JROOT(I) = -SIGN(1.0D0,R0(I)) GO TO 320 ENDIF IF (SIGN(1.0D0,R0(I)) .NE. SIGN(1.0D0,R1(I))) 1 JROOT(I) = SIGN(1.0D0,R1(I) - R0(I)) 320 CONTINUE RETURN C C No sign change in the interval. Check for zero at right endpoint. --- 400 IF (.NOT. ZROOT) GO TO 420 C C Zero value at X1 and no sign change in (X0,X1). Return JFLAG = 3. --- X = X1 CALL DCOPY (NRT, R1, 1, RX, 1) DO 410 I = 1,NRT JROOT(I) = 0 IF (ABS(R1(I)) .EQ. ZERO) JROOT(I) = -SIGN(1.0D0,R0(I)) 410 CONTINUE JFLAG = 3 RETURN C C No sign changes in this interval. Set X = X1, return JFLAG = 4. ----- 420 CALL DCOPY (NRT, R1, 1, RX, 1) X = X1 JFLAG = 4 RETURN C----------------------- END OF SUBROUTINE DROOTS ---------------------- END SUBROUTINE DDASIC (X, Y, YPRIME, NEQ, ICOPT, ID, RES, JAC, PSOL, * H, TSCALE, WT, NIC, IDID, RPAR, IPAR, PHI, SAVR, DELTA, E, * YIC, YPIC, PWK, WM, IWM, UROUND, EPLI, SQRTN, RSQRTN, * EPCONI, STPTOL, JFLG, ICNFLG, ICNSTR, NLSIC) C C***BEGIN PROLOGUE DDASIC C***REFER TO DDASPK C***DATE WRITTEN 940628 (YYMMDD) C***REVISION DATE 941206 (YYMMDD) C***REVISION DATE 950714 (YYMMDD) C***REVISION DATE 000628 TSCALE argument added. C C----------------------------------------------------------------------- C***DESCRIPTION C C DDASIC is a driver routine to compute consistent initial values C for Y and YPRIME. There are two different options: C Denoting the differential variables in Y by Y_d, and C the algebraic variables by Y_a, the problem solved is either: C 1. Given Y_d, calculate Y_a and Y_d', or C 2. Given Y', calculate Y. C In either case, initial values for the given components C are input, and initial guesses for the unknown components C must also be provided as input. C C The external routine NLSIC solves the resulting nonlinear system. C C The parameters represent C C X -- Independent variable. C Y -- Solution vector at X. C YPRIME -- Derivative of solution vector. C NEQ -- Number of equations to be integrated. C ICOPT -- Flag indicating initial condition option chosen. C ICOPT = 1 for option 1 above. C ICOPT = 2 for option 2. C ID -- Array of dimension NEQ, which must be initialized C if option 1 is chosen. C ID(i) = +1 if Y_i is a differential variable, C ID(i) = -1 if Y_i is an algebraic variable. C RES -- External user-supplied subroutine to evaluate the C residual. See RES description in DDASPK prologue. C JAC -- External user-supplied routine to update Jacobian C or preconditioner information in the nonlinear solver C (optional). See JAC description in DDASPK prologue. C PSOL -- External user-supplied routine to solve C a linear system using preconditioning. C See PSOL in DDASPK prologue. C H -- Scaling factor in iteration matrix. DDASIC may C reduce H to achieve convergence. C TSCALE -- Scale factor in T, used for stopping tests if nonzero. C WT -- Vector of weights for error criterion. C NIC -- Input number of initial condition calculation call C (= 1 or 2). C IDID -- Completion code. See IDID in DDASPK prologue. C RPAR,IPAR -- Real and integer parameter arrays that C are used for communication between the C calling program and external user routines. C They are not altered by DNSK C PHI -- Work space for DDASIC of length at least 2*NEQ. C SAVR -- Work vector for DDASIC of length NEQ. C DELTA -- Work vector for DDASIC of length NEQ. C E -- Work vector for DDASIC of length NEQ. C YIC,YPIC -- Work vectors for DDASIC, each of length NEQ. C PWK -- Work vector for DDASIC of length NEQ. C WM,IWM -- Real and integer arrays storing C information required by the linear solver. C EPCONI -- Test constant for Newton iteration convergence. C ICNFLG -- Flag showing whether constraints on Y are to apply. C ICNSTR -- Integer array of length NEQ with constraint types. C C The other parameters are for use internally by DDASIC. C C----------------------------------------------------------------------- C***ROUTINES CALLED C DCOPY, NLSIC C C***END PROLOGUE DDASIC C C IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION Y(*),YPRIME(*),ID(*),WT(*),PHI(NEQ,*) DIMENSION SAVR(*),DELTA(*),E(*),YIC(*),YPIC(*),PWK(*) DIMENSION WM(*),IWM(*), RPAR(*),IPAR(*), ICNSTR(*) EXTERNAL RES, JAC, PSOL, NLSIC C PARAMETER (LCFN=15) PARAMETER (LMXNH=34) C C The following parameters are data-loaded here: C RHCUT = factor by which H is reduced on retry of Newton solve. C RATEMX = maximum convergence rate for which Newton iteration C is considered converging. C SAVE RHCUT, RATEMX DATA RHCUT/0.1D0/, RATEMX/0.8D0/ C C C----------------------------------------------------------------------- C BLOCK 1. C Initializations. C JSKIP is a flag set to 1 when NIC = 2 and NH = 1, to signal that C the initial call to the JAC routine is to be skipped then. C Save Y and YPRIME in PHI. Initialize IDID, NH, and CJ. C----------------------------------------------------------------------- C MXNH = IWM(LMXNH) IDID = 1 NH = 1 JSKIP = 0 IF (NIC .EQ. 2) JSKIP = 1 CALL DCOPY (NEQ, Y, 1, PHI(1,1), 1) CALL DCOPY (NEQ, YPRIME, 1, PHI(1,2), 1) C IF (ICOPT .EQ. 2) THEN CJ = 0.0D0 ELSE CJ = 1.0D0/H ENDIF C C----------------------------------------------------------------------- C BLOCK 2 C Call the nonlinear system solver to obtain C consistent initial values for Y and YPRIME. C----------------------------------------------------------------------- C 200 CONTINUE CALL NLSIC(X,Y,YPRIME,NEQ,ICOPT,ID,RES,JAC,PSOL,H,TSCALE,WT, * JSKIP,RPAR,IPAR,SAVR,DELTA,E,YIC,YPIC,PWK,WM,IWM,CJ,UROUND, * EPLI,SQRTN,RSQRTN,EPCONI,RATEMX,STPTOL,JFLG,ICNFLG,ICNSTR, * IERNLS) C IF (IERNLS .EQ. 0) RETURN C C----------------------------------------------------------------------- C BLOCK 3 C The nonlinear solver was unsuccessful. Increment NCFN. C Return with IDID = -12 if either C IERNLS = -1: error is considered unrecoverable, C ICOPT = 2: we are doing initialization problem type 2, or C NH = MXNH: the maximum number of H values has been tried. C Otherwise (problem 1 with IERNLS .GE. 1), reduce H and try again. C If IERNLS > 1, restore Y and YPRIME to their original values. C----------------------------------------------------------------------- C IWM(LCFN) = IWM(LCFN) + 1 JSKIP = 0 C IF (IERNLS .EQ. -1) GO TO 350 IF (ICOPT .EQ. 2) GO TO 350 IF (NH .EQ. MXNH) GO TO 350 C NH = NH + 1 H = H*RHCUT CJ = 1.0D0/H C IF (IERNLS .EQ. 1) GO TO 200 C CALL DCOPY (NEQ, PHI(1,1), 1, Y, 1) CALL DCOPY (NEQ, PHI(1,2), 1, YPRIME, 1) GO TO 200 C 350 IDID = -12 RETURN C C------END OF SUBROUTINE DDASIC----------------------------------------- END SUBROUTINE DYYPNW (NEQ, Y, YPRIME, CJ, RL, P, ICOPT, ID, * YNEW, YPNEW) C C***BEGIN PROLOGUE DYYPNW C***REFER TO DLINSK C***DATE WRITTEN 940830 (YYMMDD) C C C----------------------------------------------------------------------- C***DESCRIPTION C C DYYPNW calculates the new (Y,YPRIME) pair needed in the C linesearch algorithm based on the current lambda value. It is C called by DLINSK and DLINSD. Based on the ICOPT and ID values, C the corresponding entry in Y or YPRIME is updated. C C In addition to the parameters described in the calling programs, C the parameters represent C C P -- Array of length NEQ that contains the current C approximate Newton step. C RL -- Scalar containing the current lambda value. C YNEW -- Array of length NEQ containing the updated Y vector. C YPNEW -- Array of length NEQ containing the updated YPRIME C vector. C----------------------------------------------------------------------- C C***ROUTINES CALLED (NONE) C C***END PROLOGUE DYYPNW C C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION Y(*), YPRIME(*), YNEW(*), YPNEW(*), ID(*), P(*) C IF (ICOPT .EQ. 1) THEN DO 10 I=1,NEQ IF(ID(I) .LT. 0) THEN YNEW(I) = Y(I) - RL*P(I) YPNEW(I) = YPRIME(I) ELSE YNEW(I) = Y(I) YPNEW(I) = YPRIME(I) - RL*CJ*P(I) ENDIF 10 CONTINUE ELSE DO 20 I = 1,NEQ YNEW(I) = Y(I) - RL*P(I) YPNEW(I) = YPRIME(I) 20 CONTINUE ENDIF RETURN C----------------------- END OF SUBROUTINE DYYPNW ---------------------- END SUBROUTINE DDSTP(X,Y,YPRIME,NEQ,RES,JAC,PSOL,H,WT,VT, * JSTART,IDID,RPAR,IPAR,PHI,SAVR,DELTA,E,WM,IWM, * ALPHA,BETA,GAMMA,PSI,SIGMA,CJ,CJOLD,HOLD,S,HMIN,UROUND, * EPLI,SQRTN,RSQRTN,EPCON,IPHASE,JCALC,JFLG,K,KOLD,NS,NONNEG, * NTYPE,NLS) C C***BEGIN PROLOGUE DDSTP C***REFER TO DDASPK C***DATE WRITTEN 890101 (YYMMDD) C***REVISION DATE 900926 (YYMMDD) C***REVISION DATE 940909 (YYMMDD) (Reset PSI(1), PHI(*,2) at 690) C C C----------------------------------------------------------------------- C***DESCRIPTION C C DDSTP solves a system of differential/algebraic equations of C the form G(X,Y,YPRIME) = 0, for one step (normally from X to X+H). C C The methods used are modified divided difference, fixed leading C coefficient forms of backward differentiation formulas. C The code adjusts the stepsize and order to control the local error C per step. C C C The parameters represent C X -- Independent variable. C Y -- Solution vector at X. C YPRIME -- Derivative of solution vector C after successful step. C NEQ -- Number of equations to be integrated. C RES -- External user-supplied subroutine C to evaluate the residual. See RES description C in DDASPK prologue. C JAC -- External user-supplied routine to update C Jacobian or preconditioner information in the C nonlinear solver. See JAC description in DDASPK C prologue. C PSOL -- External user-supplied routine to solve C a linear system using preconditioning. C (This is optional). See PSOL in DDASPK prologue. C H -- Appropriate step size for next step. C Normally determined by the code. C WT -- Vector of weights for error criterion used in Newton test. C VT -- Masked vector of weights used in error test. C JSTART -- Integer variable set 0 for C first step, 1 otherwise. C IDID -- Completion code returned from the nonlinear solver. C See IDID description in DDASPK prologue. C RPAR,IPAR -- Real and integer parameter arrays that C are used for communication between the C calling program and external user routines. C They are not altered by DNSK C PHI -- Array of divided differences used by C DDSTP. The length is NEQ*(K+1), where C K is the maximum order. C SAVR -- Work vector for DDSTP of length NEQ. C DELTA,E -- Work vectors for DDSTP of length NEQ. C WM,IWM -- Real and integer arrays storing C information required by the linear solver. C C The other parameters are information C which is needed internally by DDSTP to C continue from step to step. C C----------------------------------------------------------------------- C***ROUTINES CALLED C NLS, DDWNRM, DDATRP C C***END PROLOGUE DDSTP C C IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION Y(*),YPRIME(*),WT(*),VT(*) DIMENSION PHI(NEQ,*),SAVR(*),DELTA(*),E(*) DIMENSION WM(*),IWM(*) DIMENSION PSI(*),ALPHA(*),BETA(*),GAMMA(*),SIGMA(*) DIMENSION RPAR(*),IPAR(*) EXTERNAL RES, JAC, PSOL, NLS C PARAMETER (LMXORD=3) PARAMETER (LNST=11, LETF=14, LCFN=15) C C C----------------------------------------------------------------------- C BLOCK 1. C Initialize. On the first call, set C the order to 1 and initialize C other variables. C----------------------------------------------------------------------- C C Initializations for all calls C XOLD=X NCF=0 NEF=0 IF(JSTART .NE. 0) GO TO 120 C C If this is the first step, perform C other initializations C K=1 KOLD=0 HOLD=0.0D0 PSI(1)=H CJ = 1.D0/H IPHASE = 0 NS=0 120 CONTINUE C C C C C C----------------------------------------------------------------------- C BLOCK 2 C Compute coefficients of formulas for C this step. C----------------------------------------------------------------------- 200 CONTINUE KP1=K+1 KP2=K+2 KM1=K-1 IF(H.NE.HOLD.OR.K .NE. KOLD) NS = 0 NS=MIN0(NS+1,KOLD+2) NSP1=NS+1 IF(KP1 .LT. NS)GO TO 230 C BETA(1)=1.0D0 ALPHA(1)=1.0D0 TEMP1=H GAMMA(1)=0.0D0 SIGMA(1)=1.0D0 DO 210 I=2,KP1 TEMP2=PSI(I-1) PSI(I-1)=TEMP1 BETA(I)=BETA(I-1)*PSI(I-1)/TEMP2 TEMP1=TEMP2+H ALPHA(I)=H/TEMP1 SIGMA(I)=(I-1)*SIGMA(I-1)*ALPHA(I) GAMMA(I)=GAMMA(I-1)+ALPHA(I-1)/H 210 CONTINUE PSI(KP1)=TEMP1 230 CONTINUE C C Compute ALPHAS, ALPHA0 C ALPHAS = 0.0D0 ALPHA0 = 0.0D0 DO 240 I = 1,K ALPHAS = ALPHAS - 1.0D0/I ALPHA0 = ALPHA0 - ALPHA(I) 240 CONTINUE C C Compute leading coefficient CJ C CJLAST = CJ CJ = -ALPHAS/H C C Compute variable stepsize error coefficient CK C CK = ABS(ALPHA(KP1) + ALPHAS - ALPHA0) CK = MAX(CK,ALPHA(KP1)) C C Change PHI to PHI STAR C IF(KP1 .LT. NSP1) GO TO 280 DO 270 J=NSP1,KP1 DO 260 I=1,NEQ 260 PHI(I,J)=BETA(J)*PHI(I,J) 270 CONTINUE 280 CONTINUE C C Update time C X=X+H C C Initialize IDID to 1 C IDID = 1 C C C C C C----------------------------------------------------------------------- C BLOCK 3 C Call the nonlinear system solver to obtain the solution and C derivative. C----------------------------------------------------------------------- C CALL NLS(X,Y,YPRIME,NEQ, * RES,JAC,PSOL,H,WT,JSTART,IDID,RPAR,IPAR,PHI,GAMMA, * SAVR,DELTA,E,WM,IWM,CJ,CJOLD,CJLAST,S, * UROUND,EPLI,SQRTN,RSQRTN,EPCON,JCALC,JFLG,KP1, * NONNEG,NTYPE,IERNLS) C IF(IERNLS .NE. 0)GO TO 600 C C C C C C----------------------------------------------------------------------- C BLOCK 4 C Estimate the errors at orders K,K-1,K-2 C as if constant stepsize was used. Estimate C the local error at order K and test C whether the current step is successful. C----------------------------------------------------------------------- C C Estimate errors at orders K,K-1,K-2 C ENORM = DDWNRM(NEQ,E,VT,RPAR,IPAR) ERK = SIGMA(K+1)*ENORM TERK = (K+1)*ERK EST = ERK KNEW=K IF(K .EQ. 1)GO TO 430 DO 405 I = 1,NEQ 405 DELTA(I) = PHI(I,KP1) + E(I) ERKM1=SIGMA(K)*DDWNRM(NEQ,DELTA,VT,RPAR,IPAR) TERKM1 = K*ERKM1 IF(K .GT. 2)GO TO 410 IF(TERKM1 .LE. 0.5*TERK)GO TO 420 GO TO 430 410 CONTINUE DO 415 I = 1,NEQ 415 DELTA(I) = PHI(I,K) + DELTA(I) ERKM2=SIGMA(K-1)*DDWNRM(NEQ,DELTA,VT,RPAR,IPAR) TERKM2 = (K-1)*ERKM2 IF(MAX(TERKM1,TERKM2).GT.TERK)GO TO 430 C C Lower the order C 420 CONTINUE KNEW=K-1 EST = ERKM1 C C C Calculate the local error for the current step C to see if the step was successful C 430 CONTINUE ERR = CK * ENORM IF(ERR .GT. 1.0D0)GO TO 600 C C C C C C----------------------------------------------------------------------- C BLOCK 5 C The step is successful. Determine C the best order and stepsize for C the next step. Update the differences C for the next step. C----------------------------------------------------------------------- IDID=1 IWM(LNST)=IWM(LNST)+1 KDIFF=K-KOLD KOLD=K HOLD=H C C C Estimate the error at order K+1 unless C already decided to lower order, or C already using maximum order, or C stepsize not constant, or C order raised in previous step C IF(KNEW.EQ.KM1.OR.K.EQ.IWM(LMXORD))IPHASE=1 IF(IPHASE .EQ. 0)GO TO 545 IF(KNEW.EQ.KM1)GO TO 540 IF(K.EQ.IWM(LMXORD)) GO TO 550 IF(KP1.GE.NS.OR.KDIFF.EQ.1)GO TO 550 DO 510 I=1,NEQ 510 DELTA(I)=E(I)-PHI(I,KP2) ERKP1 = (1.0D0/(K+2))*DDWNRM(NEQ,DELTA,VT,RPAR,IPAR) TERKP1 = (K+2)*ERKP1 IF(K.GT.1)GO TO 520 IF(TERKP1.GE.0.5D0*TERK)GO TO 550 GO TO 530 520 IF(TERKM1.LE.MIN(TERK,TERKP1))GO TO 540 IF(TERKP1.GE.TERK.OR.K.EQ.IWM(LMXORD))GO TO 550 C C Raise order C 530 K=KP1 EST = ERKP1 GO TO 550 C C Lower order C 540 K=KM1 EST = ERKM1 GO TO 550 C C If IPHASE = 0, increase order by one and multiply stepsize by C factor two C 545 K = KP1 HNEW = H*2.0D0 H = HNEW GO TO 575 C C C Determine the appropriate stepsize for C the next step. C 550 HNEW=H TEMP2=K+1 R=(2.0D0*EST+0.0001D0)**(-1.0D0/TEMP2) IF(R .LT. 2.0D0) GO TO 555 HNEW = 2.0D0*H GO TO 560 555 IF(R .GT. 1.0D0) GO TO 560 R = MAX(0.5D0,MIN(0.9D0,R)) HNEW = H*R 560 H=HNEW C C C Update differences for next step C 575 CONTINUE IF(KOLD.EQ.IWM(LMXORD))GO TO 585 DO 580 I=1,NEQ 580 PHI(I,KP2)=E(I) 585 CONTINUE DO 590 I=1,NEQ 590 PHI(I,KP1)=PHI(I,KP1)+E(I) DO 595 J1=2,KP1 J=KP1-J1+1 DO 595 I=1,NEQ 595 PHI(I,J)=PHI(I,J)+PHI(I,J+1) JSTART = 1 RETURN C C C C C C----------------------------------------------------------------------- C BLOCK 6 C The step is unsuccessful. Restore X,PSI,PHI C Determine appropriate stepsize for C continuing the integration, or exit with C an error flag if there have been many C failures. C----------------------------------------------------------------------- 600 IPHASE = 1 C C Restore X,PHI,PSI C X=XOLD IF(KP1.LT.NSP1)GO TO 630 DO 620 J=NSP1,KP1 TEMP1=1.0D0/BETA(J) DO 610 I=1,NEQ 610 PHI(I,J)=TEMP1*PHI(I,J) 620 CONTINUE 630 CONTINUE DO 640 I=2,KP1 640 PSI(I-1)=PSI(I)-H C C C Test whether failure is due to nonlinear solver C or error test C IF(IERNLS .EQ. 0)GO TO 660 IWM(LCFN)=IWM(LCFN)+1 C C C The nonlinear solver failed to converge. C Determine the cause of the failure and take appropriate action. C If IERNLS .LT. 0, then return. Otherwise, reduce the stepsize C and try again, unless too many failures have occurred. C IF (IERNLS .LT. 0) GO TO 675 NCF = NCF + 1 R = 0.25D0 H = H*R IF (NCF .LT. 10 .AND. ABS(H) .GE. HMIN) GO TO 690 IF (IDID .EQ. 1) IDID = -7 IF (NEF .GE. 3) IDID = -9 GO TO 675 C C C The nonlinear solver converged, and the cause C of the failure was the error estimate C exceeding the tolerance. C 660 NEF=NEF+1 IWM(LETF)=IWM(LETF)+1 IF (NEF .GT. 1) GO TO 665 C C On first error test failure, keep current order or lower C order by one. Compute new stepsize based on differences C of the solution. C K = KNEW TEMP2 = K + 1 R = 0.90D0*(2.0D0*EST+0.0001D0)**(-1.0D0/TEMP2) R = MAX(0.25D0,MIN(0.9D0,R)) H = H*R IF (ABS(H) .GE. HMIN) GO TO 690 IDID = -6 GO TO 675 C C On second error test failure, use the current order or C decrease order by one. Reduce the stepsize by a factor of C one quarter. C 665 IF (NEF .GT. 2) GO TO 670 K = KNEW R = 0.25D0 H = R*H IF (ABS(H) .GE. HMIN) GO TO 690 IDID = -6 GO TO 675 C C On third and subsequent error test failures, set the order to C one, and reduce the stepsize by a factor of one quarter. C 670 K = 1 R = 0.25D0 H = R*H IF (ABS(H) .GE. HMIN) GO TO 690 IDID = -6 GO TO 675 C C C C C For all crashes, restore Y to its last value, C interpolate to find YPRIME at last X, and return. C C Before returning, verify that the user has not set C IDID to a nonnegative value. If the user has set IDID C to a nonnegative value, then reset IDID to be -7, indicating C a failure in the nonlinear system solver. C 675 CONTINUE CALL DDATRP(X,X,Y,YPRIME,NEQ,K,PHI,PSI) JSTART = 1 IF (IDID .GE. 0) IDID = -7 RETURN C C C Go back and try this step again. C If this is the first step, reset PSI(1) and rescale PHI(*,2). C 690 IF (KOLD .EQ. 0) THEN PSI(1) = H DO 695 I = 1,NEQ 695 PHI(I,2) = R*PHI(I,2) ENDIF GO TO 200 C C------END OF SUBROUTINE DDSTP------------------------------------------ END SUBROUTINE DCNSTR (NEQ, Y, YNEW, ICNSTR, TAU, RLX, IRET, IVAR) C C***BEGIN PROLOGUE DCNSTR C***DATE WRITTEN 950808 (YYMMDD) C***REVISION DATE 950814 (YYMMDD) C C C----------------------------------------------------------------------- C***DESCRIPTION C C This subroutine checks for constraint violations in the proposed C new approximate solution YNEW. C If a constraint violation occurs, then a new step length, TAU, C is calculated, and this value is to be given to the linesearch routine C to calculate a new approximate solution YNEW. C C On entry: C C NEQ -- size of the nonlinear system, and the length of arrays C Y, YNEW and ICNSTR. C C Y -- real array containing the current approximate y. C C YNEW -- real array containing the new approximate y. C C ICNSTR -- INTEGER array of length NEQ containing flags indicating C which entries in YNEW are to be constrained. C if ICNSTR(I) = 2, then YNEW(I) must be .GT. 0, C if ICNSTR(I) = 1, then YNEW(I) must be .GE. 0, C if ICNSTR(I) = -1, then YNEW(I) must be .LE. 0, while C if ICNSTR(I) = -2, then YNEW(I) must be .LT. 0, while C if ICNSTR(I) = 0, then YNEW(I) is not constrained. C C RLX -- real scalar restricting update, if ICNSTR(I) = 2 or -2, C to ABS( (YNEW-Y)/Y ) < FAC2*RLX in component I. C C TAU -- the current size of the step length for the linesearch. C C On return C C TAU -- the adjusted size of the step length if a constraint C violation occurred (otherwise, it is unchanged). it is C the step length to give to the linesearch routine. C C IRET -- output flag. C IRET=0 means that YNEW satisfied all constraints. C IRET=1 means that YNEW failed to satisfy all the C constraints, and a new linesearch step C must be computed. C C IVAR -- index of variable causing constraint to be violated. C C----------------------------------------------------------------------- IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION Y(NEQ), YNEW(NEQ), ICNSTR(NEQ) SAVE FAC, FAC2, ZERO DATA FAC /0.6D0/, FAC2 /0.9D0/, ZERO/0.0D0/ C----------------------------------------------------------------------- C Check constraints for proposed new step YNEW. If a constraint has C been violated, then calculate a new step length, TAU, to be C used in the linesearch routine. C----------------------------------------------------------------------- IRET = 0 RDYMX = ZERO IVAR = 0 DO 100 I = 1,NEQ C IF (ICNSTR(I) .EQ. 2) THEN RDY = ABS( (YNEW(I)-Y(I))/Y(I) ) IF (RDY .GT. RDYMX) THEN RDYMX = RDY IVAR = I ENDIF IF (YNEW(I) .LE. ZERO) THEN TAU = FAC*TAU IVAR = I IRET = 1 RETURN ENDIF C ELSEIF (ICNSTR(I) .EQ. 1) THEN IF (YNEW(I) .LT. ZERO) THEN TAU = FAC*TAU IVAR = I IRET = 1 RETURN ENDIF C ELSEIF (ICNSTR(I) .EQ. -1) THEN IF (YNEW(I) .GT. ZERO) THEN TAU = FAC*TAU IVAR = I IRET = 1 RETURN ENDIF C ELSEIF (ICNSTR(I) .EQ. -2) THEN RDY = ABS( (YNEW(I)-Y(I))/Y(I) ) IF (RDY .GT. RDYMX) THEN RDYMX = RDY IVAR = I ENDIF IF (YNEW(I) .GE. ZERO) THEN TAU = FAC*TAU IVAR = I IRET = 1 RETURN ENDIF C ENDIF 100 CONTINUE IF(RDYMX .GE. RLX) THEN TAU = FAC2*TAU*RLX/RDYMX IRET = 1 ENDIF C RETURN C----------------------- END OF SUBROUTINE DCNSTR ---------------------- END SUBROUTINE DCNST0 (NEQ, Y, ICNSTR, IRET) C C***BEGIN PROLOGUE DCNST0 C***DATE WRITTEN 950808 (YYMMDD) C***REVISION DATE 950808 (YYMMDD) C C C----------------------------------------------------------------------- C***DESCRIPTION C C This subroutine checks for constraint violations in the initial C approximate solution u. C C On entry C C NEQ -- size of the nonlinear system, and the length of arrays C Y and ICNSTR. C C Y -- real array containing the initial approximate root. C C ICNSTR -- INTEGER array of length NEQ containing flags indicating C which entries in Y are to be constrained. C if ICNSTR(I) = 2, then Y(I) must be .GT. 0, C if ICNSTR(I) = 1, then Y(I) must be .GE. 0, C if ICNSTR(I) = -1, then Y(I) must be .LE. 0, while C if ICNSTR(I) = -2, then Y(I) must be .LT. 0, while C if ICNSTR(I) = 0, then Y(I) is not constrained. C C On return C C IRET -- output flag. C IRET=0 means that u satisfied all constraints. C IRET.NE.0 means that Y(IRET) failed to satisfy its C constraint. C C----------------------------------------------------------------------- IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION Y(NEQ), ICNSTR(NEQ) SAVE ZERO DATA ZERO/0.D0/ C----------------------------------------------------------------------- C Check constraints for initial Y. If a constraint has been violated, C set IRET = I to signal an error return to calling routine. C----------------------------------------------------------------------- IRET = 0 DO 100 I = 1,NEQ IF (ICNSTR(I) .EQ. 2) THEN IF (Y(I) .LE. ZERO) THEN IRET = I RETURN ENDIF ELSEIF (ICNSTR(I) .EQ. 1) THEN IF (Y(I) .LT. ZERO) THEN IRET = I RETURN ENDIF ELSEIF (ICNSTR(I) .EQ. -1) THEN IF (Y(I) .GT. ZERO) THEN IRET = I RETURN ENDIF ELSEIF (ICNSTR(I) .EQ. -2) THEN IF (Y(I) .GE. ZERO) THEN IRET = I RETURN ENDIF ENDIF 100 CONTINUE RETURN C----------------------- END OF SUBROUTINE DCNST0 ---------------------- END SUBROUTINE DDAWTS(NEQ,IWT,RTOL,ATOL,Y,WT,RPAR,IPAR) C C***BEGIN PROLOGUE DDAWTS C***REFER TO DDASPK C***ROUTINES CALLED (NONE) C***DATE WRITTEN 890101 (YYMMDD) C***REVISION DATE 900926 (YYMMDD) C***END PROLOGUE DDAWTS C----------------------------------------------------------------------- C This subroutine sets the error weight vector, C WT, according to WT(I)=RTOL(I)*ABS(Y(I))+ATOL(I), C I = 1 to NEQ. C RTOL and ATOL are scalars if IWT = 0, C and vectors if IWT = 1. C----------------------------------------------------------------------- C IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION RTOL(*),ATOL(*),Y(*),WT(*) DIMENSION RPAR(*),IPAR(*) RTOLI=RTOL(1) ATOLI=ATOL(1) DO 20 I=1,NEQ IF (IWT .EQ.0) GO TO 10 RTOLI=RTOL(I) ATOLI=ATOL(I) 10 WT(I)=RTOLI*ABS(Y(I))+ATOLI 20 CONTINUE RETURN C C------END OF SUBROUTINE DDAWTS----------------------------------------- END SUBROUTINE DINVWT(NEQ,WT,IER) C C***BEGIN PROLOGUE DINVWT C***REFER TO DDASPK C***ROUTINES CALLED (NONE) C***DATE WRITTEN 950125 (YYMMDD) C***END PROLOGUE DINVWT C----------------------------------------------------------------------- C This subroutine checks the error weight vector WT, of length NEQ, C for components that are .le. 0, and if none are found, it C inverts the WT(I) in place. This replaces division operations C with multiplications in all norm evaluations. C IER is returned as 0 if all WT(I) were found positive, C and the first I with WT(I) .le. 0.0 otherwise. C----------------------------------------------------------------------- C IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION WT(*) C DO 10 I = 1,NEQ IF (WT(I) .LE. 0.0D0) GO TO 30 10 CONTINUE DO 20 I = 1,NEQ 20 WT(I) = 1.0D0/WT(I) IER = 0 RETURN C 30 IER = I RETURN C C------END OF SUBROUTINE DINVWT----------------------------------------- END SUBROUTINE DDATRP(X,XOUT,YOUT,YPOUT,NEQ,KOLD,PHI,PSI) C C***BEGIN PROLOGUE DDATRP C***REFER TO DDASPK C***ROUTINES CALLED (NONE) C***DATE WRITTEN 890101 (YYMMDD) C***REVISION DATE 900926 (YYMMDD) C***END PROLOGUE DDATRP C C----------------------------------------------------------------------- C The methods in subroutine DDSTP use polynomials C to approximate the solution. DDATRP approximates the C solution and its derivative at time XOUT by evaluating C one of these polynomials, and its derivative, there. C Information defining this polynomial is passed from C DDSTP, so DDATRP cannot be used alone. C C The parameters are C C X The current time in the integration. C XOUT The time at which the solution is desired. C YOUT The interpolated approximation to Y at XOUT. C (This is output.) C YPOUT The interpolated approximation to YPRIME at XOUT. C (This is output.) C NEQ Number of equations. C KOLD Order used on last successful step. C PHI Array of scaled divided differences of Y. C PSI Array of past stepsize history. C----------------------------------------------------------------------- C IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION YOUT(*),YPOUT(*) DIMENSION PHI(NEQ,*),PSI(*) KOLDP1=KOLD+1 TEMP1=XOUT-X DO 10 I=1,NEQ YOUT(I)=PHI(I,1) 10 YPOUT(I)=0.0D0 C=1.0D0 D=0.0D0 GAMMA=TEMP1/PSI(1) DO 30 J=2,KOLDP1 D=D*GAMMA+C/PSI(J-1) C=C*GAMMA GAMMA=(TEMP1+PSI(J-1))/PSI(J) DO 20 I=1,NEQ YOUT(I)=YOUT(I)+C*PHI(I,J) 20 YPOUT(I)=YPOUT(I)+D*PHI(I,J) 30 CONTINUE RETURN C C------END OF SUBROUTINE DDATRP----------------------------------------- END