da/d6b/ddassl_8f_source.html

      SUBROUTINE ddassl (RES, NEQ, T, Y, YPRIME, TOUT, INFO, RTOL, ATOL,

     +   IDID, RWORK, LRW, IWORK, LIW, RPAR, IPAR, JAC)

C***BEGIN PROLOGUE  DDASSL

C***PURPOSE  This code solves a system of differential/algebraic

C            equations of the form G(T,Y,YPRIME) = 0.

C***LIBRARY   SLATEC (DASSL)

C***CATEGORY  I1A2

C***TYPE      DOUBLE PRECISION (SDASSL-S, DDASSL-D)

C***KEYWORDS  DIFFERENTIAL/ALGEBRAIC, BACKWARD DIFFERENTIATION FORMULAS,

C             IMPLICIT DIFFERENTIAL SYSTEMS

C***AUTHOR  PETZOLD, LINDA R., (LLNL)

C             COMPUTING AND MATHEMATICS RESEARCH DIVISION

C             LAWRENCE LIVERMORE NATIONAL LABORATORY

C             L - 316, P.O. BOX 808,

C             LIVERMORE, CA.    94550

C***DESCRIPTION

C

C *Usage:

C

C      EXTERNAL RES, JAC

C      INTEGER NEQ, INFO(N), IDID, LRW, LIW, IWORK(LIW), IPAR

C      DOUBLE PRECISION T, Y(NEQ), YPRIME(NEQ), TOUT, RTOL, ATOL,

C     *   RWORK(LRW), RPAR

C

C      CALL DDASSL (RES, NEQ, T, Y, YPRIME, TOUT, INFO, RTOL, ATOL,

C     *   IDID, RWORK, LRW, IWORK, LIW, RPAR, IPAR, JAC)

C

C

C *Arguments:

C  (In the following, all real arrays should be type DOUBLE PRECISION.)

C

C  RES:EXT     This is a subroutine which you provide to define the

C              differential/algebraic system.

C

C  NEQ:IN      This is the number of equations to be solved.

C

C  T:INOUT     This is the current value of the independent 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  The basic task of the code is to solve the system from T

C              to TOUT and return an answer at TOUT.  INFO is an integer

C              array which is used to communicate exactly how you want

C              this task to be carried out.  (See below for details.)

C              N must be greater than or equal to 15.

C

C  RTOL,ATOL:INOUT  These quantities represent relative and absolute

C              error tolerances which you provide to indicate how

C              accurately you wish the solution to be computed.  You

C              may choose them to be both scalars or else both vectors.

C              Caution:  In Fortran 77, a scalar is not the same as an

C                        array of length 1.  Some compilers may object

C                        to using scalars for RTOL,ATOL.

C

C  IDID:OUT    This scalar quantity is an indicator reporting what the

C              code did.  You must monitor this integer 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.  (See below for required length.)

C

C  IWORK:WORK  An integer work array of length LIW which probides the

C              code with needed storage space.

C

C  LIW:IN      The length of IWORK.  (See below for required length.)

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 subroutine (and the JAC subroutine)

C

C  JAC:EXT     This is the name of a subroutine which you may choose

C              to provide for defining a matrix of partial derivatives

C              described below.

C

C  Quantities which may be altered by DDASSL are:

C     T, Y(*), YPRIME(*), INFO(1), RTOL, ATOL,

C     IDID, RWORK(*) AND IWORK(*)

C

C *Description

C

C  Subroutine DDASSL uses the backward differentiation formulas of

C  orders one through five to solve a system of the above form for Y and

C  YPRIME.  Values for Y and YPRIME at the initial time must be given as

C  input.  These values must be consistent, (that is, if T,Y,YPRIME are

C  the given initial values, they must satisfy G(T,Y,YPRIME) = 0.).  The

C  subroutine solves the system from T to TOUT.  It is easy to continue

C  the solution to get results at additional TOUT.  This is the interval

C  mode of operation.  Intermediate results can also be obtained easily

C  by using the intermediate-output capability.

C

C  The following detailed description is divided into subsections:

C    1. Input required for the first call to DDASSL.

C    2. Output after any return from DDASSL.

C    3. What to do to continue the integration.

C    4. Error messages.

C

C

C  -------- INPUT -- WHAT TO DO ON THE FIRST CALL TO DDASSL ------------

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             SUBROUTINE RES(T,Y,YPRIME,DELTA,IRES,RPAR,IPAR)

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

C         return the residual of the defferential/algebraic

C         system

C             DELTA = G(T,Y,YPRIME)

C         (DELTA(*) is a vector of length NEQ which is

C         output for RES.)

C

C         Subroutine RES must not alter T,Y or YPRIME.

C         You must declare the name RES in an external

C         statement in your program that calls DDASSL.

C         You must dimension Y,YPRIME and DELTA in RES.

C

C         IRES is an integer flag which is always equal to

C         zero on input. Subroutine RES should alter IRES

C         only if it encounters an illegal value of Y or

C         a stop condition. Set IRES = -1 if an input value

C         is illegal, and DDASSL will try to solve the problem

C         without getting IRES = -1. If IRES = -2, DDASSL

C         will return control to the calling program

C         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 DDASSL. 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 differential equations.

C         (NEQ .GE. 1)

C

C  T -- Set it to the initial point of the integration.

C         T must be defined as a variable.

C

C  Y(*) -- Set this vector 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 vector to the initial values of the NEQ

C         first derivatives of the solution components at the initial

C         point.  You must dimension YPRIME at least NEQ in your

C         calling program. If you do not know initial values of some

C         of the solution components, see the explanation of INFO(11).

C

C  TOUT -- Set it to the first point at which a solution

C         is desired. You can not take TOUT = T.

C         integration either forward in T (TOUT .GT. T) or

C         backward in T (TOUT .LT. T) is permitted.

C

C         The code advances the solution from T to TOUT using

C         step sizes which are automatically selected so as to

C         achieve the desired accuracy. If you wish, the code will

C         return with the solution and its derivative at

C         intermediate steps (intermediate-output mode) so that

C         you can monitor them, but you still must provide TOUT in

C         accord with the basic aim of the code.

C

C         The first step taken by the code is a critical one

C         because it must reflect how fast the solution changes near

C         the initial point. The code automatically selects an

C         initial step size which is practically always suitable for

C         the problem. By using the fact that the code will not step

C         past TOUT in the first step, you could, if necessary,

C         restrict the length of the initial step size.

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

C         past TSTOP. In this case any TOUT beyond TSTOP is invalid

C         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 15, though DDASSL uses only the first

C         eleven entries.  You must respond to all of the following

C         items, which are arranged as questions.  The simplest use

C         of the code corresponds to answering all questions as yes,

C         i.e. setting all entries of INFO to 0.

C

C       INFO(1) - This parameter enables the code to initialize

C              itself. You must set it to indicate the start of every

C              new 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 vectors.

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

C              of TOUT by steps. If you wish, it will return the

C              computed solution and derivative at the next

C              intermediate step (the intermediate-output mode) or

C              TOUT, whichever comes first. This is a good way to

C              proceed if you want to see the behavior of the solution.

C              If you must have solutions at a great many specific

C              TOUT points, this 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

C                  No - Set INFO(3) = 1 ****

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              not to go past.

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) - To solve differential/algebraic problems it is

C              necessary 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. Sometimes numerical differencing

C              is cheaper than evaluating derivatives in JAC and

C              sometimes it is not - this depends on your problem.

C

C           **** Do you want the code to evaluate the partial

C                derivatives 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) - DDASSL will perform much better if the matrix of

C              partial derivatives, DG/DY + CJ*DG/DYPRIME,

C              (here CJ is a scalar determined by DDASSL)

C              is banded and the code is told this. In this

C              case, the storage needed will be greatly reduced,

C              numerical differencing will be performed much cheaper,

C              and a number of important algorithms will execute much

C              faster. The differential equation is said to have

C              half-bandwidths ML (lower) and MU (upper) if equation i

C              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

C               (dense) matrix (and not a special banded

C               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

C        INFO(7) -- You can specify a maximum (absolute value of)

C              stepsize, so that the code

C              will avoid passing over very

C              large regions.

C

C          ****  Do you want the code to decide

C                on its own maximum 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

C              may occaisionally suffer from

C              severe scaling difficulties on the

C              first step. If you know a great deal

C              about the scaling of your problem, you can

C              help to alleviate this problem by

C              specifying an initial stepsize HO.

C

C          ****  Do you want the code to define

C                its own initial stepsize?

C                Yes - Set INFO(8)=0

C                 No - Set INFO(8)=1

C                      and define HO by setting

C                      RWORK(3)=HO ****

C

C        INFO(9) -- If storage is a severe problem,

C              you can save some locations by

C              restricting the maximum order MAXORD.

C              the default value is 5. for each

C              order decrease below 5, the code

C              requires NEQ fewer locations, however

C              it is likely to be slower. In any

C              case, you must have 1 .LE. MAXORD .LE. 5

C          ****  Do you want the maximum order to

C                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 the solutions to your equations

C               will always be nonnegative, it may help to set this

C               parameter. However, it is probably best to

C               try the code without using this option first,

C               and only to use this option if that doesn't

C               work very well.

C           ****  Do you want the code to solve the problem without

C                 invoking any special nonnegativity constraints?

C                  Yes - Set INFO(10)=0

C                   No - Set INFO(10)=1

C

C        INFO(11) --DDASSL normally requires the initial T,

C               Y, and YPRIME to be consistent. That is,

C               you must have G(T,Y,YPRIME) = 0 at the initial

C               time. If you do not know the initial

C               derivative precisely, you can let DDASSL try

C               to compute it.

C          ****   Are the initialHE INITIAL T, Y, YPRIME consistent?

C                 Yes - Set INFO(11) = 0

C                  No - Set INFO(11) = 1,

C                       and set YPRIME to an initial approximation

C                       to YPRIME.  (If you have no idea what

C                       YPRIME should be, set it to zero. Note

C                       that the initial Y should be such

C                       that there must exist a YPRIME so that

C                       G(T,Y,YPRIME) = 0.)

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.  You

C         have two choices --

C               Both RTOL and ATOL are scalars. (INFO(2)=0)

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) .LE. RTOL*ABS(Y)+ATOL

C         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         Usually, but not always, the true accuracy of the

C         computed Y is comparable to the error tolerances. This

C         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 bigger tolerances.

C

C         Setting ATOL=0. results in a pure relative error test on

C         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 much

C         bigger than ATOL and to an absolute error test when the

C         solution component is smaller than the threshhold ATOL.

C

C         The code will not attempt to compute a solution at an

C         accuracy unreasonable for the machine being used.  It will

C         advise you if you ask for too much accuracy and inform

C         you as to the maximum accuracy it believes possible.

C

C  RWORK(*) --  Dimension this real work array of length LRW in your

C         calling program.

C

C  LRW -- Set it to the declared length of the RWORK array.

C               You must have

C                    LRW .GE. 40+(MAXORD+4)*NEQ+NEQ**2

C               for the full (dense) JACOBIAN case (when INFO(6)=0), or

C                    LRW .GE. 40+(MAXORD+4)*NEQ+(2*ML+MU+1)*NEQ

C               for the banded user-defined JACOBIAN case

C               (when INFO(5)=1 and INFO(6)=1), or

C                     LRW .GE. 40+(MAXORD+4)*NEQ+(2*ML+MU+1)*NEQ

C                           +2*(NEQ/(ML+MU+1)+1)

C               for the banded finite-difference-generated JACOBIAN case

C               (when INFO(5)=0 and INFO(6)=1)

C

C  IWORK(*) --  Dimension this integer work array of length LIW in

C         your calling program.

C

C  LIW -- Set it to the declared length of the IWORK array.

C               You must have LIW .GE. 21+NEQ

C

C  RPAR, IPAR -- These are parameter arrays, of real and integer

C         type, respectively.  You can use them for communication

C         between your program that calls DDASSL and the

C         RES subroutine (and the JAC subroutine).  They are not

C         altered by DDASSL.  If you do not need RPAR or IPAR,

C         ignore these parameters by treating them as dummy

C         arguments.  If you do choose to use them, dimension

C         them in your calling program and in RES (and in JAC)

C         as arrays of appropriate length.

C

C  JAC -- If you have set INFO(5)=0, you can ignore this parameter

C         by treating it as a dummy argument.  Otherwise, you must

C         provide a subroutine of the form

C             SUBROUTINE JAC(T,Y,YPRIME,PD,CJ,RPAR,IPAR)

C         to define the matrix of partial derivatives

C             PD=DG/DY+CJ*DG/DYPRIME

C         CJ is a scalar which is input to JAC.

C         For the given values of T,Y,YPRIME, the

C         subroutine must evaluate the non-zero partial

C         derivatives for each equation and each solution

C         component, and store these values in the

C         matrix PD.  The elements of PD are set to zero

C         before each call to JAC so only non-zero elements

C         need to be defined.

C

C         Subroutine JAC must not alter T,Y,(*),YPRIME(*), or CJ.

C         You must declare the name JAC in an EXTERNAL statement in

C         your program that calls DDASSL.  You must dimension Y,

C         YPRIME and PD in JAC.

C

C         The way you must store the elements into the PD matrix

C         depends on the structure of the matrix which you

C         indicated by INFO(6).

C               *** INFO(6)=0 -- Full (dense) matrix ***

C                   Give PD a first dimension of NEQ.

C                   When you evaluate the (non-zero) partial derivative

C                   of equation I with respect to variable J, you must

C                   store it in PD according to

C                   PD(I,J) = "DG(I)/DY(J)+CJ*DG(I)/DYPRIME(J)"

C               *** INFO(6)=1 -- Banded JACOBIAN with ML lower and MU

C                   upper diagonal bands (refer to INFO(6) description

C                   of ML and MU) ***

C                   Give PD a first dimension of 2*ML+MU+1.

C                   when you evaluate the (non-zero) partial derivative

C                   of equation I with respect to variable J, you must

C                   store it 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         RPAR and IPAR are real and integer parameter arrays

C         which you can use for communication between your calling

C         program and your JACOBIAN subroutine JAC. They are not

C         altered by DDASSL. If you do not need RPAR or IPAR,

C         ignore these parameters by treating them as dummy

C         arguments. If you do choose to use them, dimension

C         them in your calling program and in JAC as arrays of

C         appropriate length.

C

C

C  OPTIONALLY REPLACEABLE NORM ROUTINE:

C

C     DDASSL uses a weighted norm DDANRM to measure the size

C     of vectors such as the estimated error in each step.

C     A FUNCTION subprogram

C       DOUBLE PRECISION FUNCTION DDANRM(NEQ,V,WT,RPAR,IPAR)

C       DIMENSION V(NEQ),WT(NEQ)

C     is used to define this norm. Here, V is the vector

C     whose norm is to be computed, and WT is a vector of

C     weights.  A DDANRM routine has been included with DDASSL

C     which computes the weighted root-mean-square norm

C     given by

C       DDANRM=SQRT((1/NEQ)*SUM(V(I)/WT(I))**2)

C     this norm is suitable for most problems. In some

C     special cases, it may be more convenient and/or

C     efficient to define your own norm by writing a function

C     subprogram to be called instead of DDANRM. This should,

C     however, be attempted only after careful thought and

C     consideration.

C

C

C  -------- OUTPUT -- AFTER ANY RETURN FROM DDASSL ---------------------

C

C  The principal aim of the code is to return a computed solution at

C  TOUT, although it is also possible to obtain intermediate results

C  along the way. To find out whether the code achieved its goal

C  or if the integration process was interrupted before the task was

C  completed, you must check the IDID parameter.

C

C

C  T -- The solution was successfully advanced to the

C               output value of T.

C

C  Y(*) -- Contains the computed solution approximation at T.

C

C  YPRIME(*) -- Contains the computed derivative

C               approximation at T.

C

C  IDID -- Reports what the code did.

C

C                     *** Task completed ***

C                Reported by positive values of IDID

C

C           IDID = 1 -- A step was successfully taken in the

C                   intermediate-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(*) is obtained by interpolation.

C                   YPRIME(*) is obtained by interpolation.

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

C                   component is zero. Thus, a pure relative error

C                   test is impossible for this component.

C

C           IDID = -6 -- DDASSL had repeated error test

C                   failures on the last attempted step.

C

C           IDID = -7 -- The corrector could not converge.

C

C           IDID = -8 -- The matrix of partial derivatives

C                   is singular.

C

C           IDID = -9 -- The corrector could not converge.

C                   there were repeated error test failures

C                   in this step.

C

C           IDID =-10 -- The corrector could not converge

C                   because IRES was equal to minus one.

C

C           IDID =-11 -- IRES equal to -2 was encountered

C                   and control is being returned to the

C                   calling program.

C

C           IDID =-12 -- DDASSL failed to compute the initial

C                   YPRIME.

C

C

C

C           IDID = -13,..,-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

C               interest to the user but necessary for subsequent calls.

C               However, you may find use for

C

C               RWORK(3)--Which contains the step size H to be

C                       attempted on the next step.

C

C               RWORK(4)--Which contains the current value of the

C                       independent variable, i.e., the farthest point

C                       integration has reached. This will be different

C                       from T only when interpolation has been

C                       performed (IDID=3).

C

C               RWORK(7)--Which contains the stepsize used

C                       on the last successful step.

C

C               IWORK(7)--Which contains the order of the method to

C                       be attempted on the next step.

C

C               IWORK(8)--Which contains the order of the method used

C                       on the last step.

C

C               IWORK(11)--Which contains the number of steps taken so

C                        far.

C

C               IWORK(12)--Which contains the number of calls to RES

C                        so far.

C

C               IWORK(13)--Which contains the number of evaluations of

C                        the matrix of partial derivatives needed so

C                        far.

C

C               IWORK(14)--Which contains the total number

C                        of error test failures so far.

C

C               IWORK(15)--Which contains the total number

C                        of convergence test failures so far.

C                        (includes singular iteration matrix

C                        failures.)

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,

C  in particular do not alter NEQ,T,Y(*),YPRIME(*),RWORK(*),IWORK(*)

C  or the differential equation in subroutine RES. Any such

C  alteration constitutes a new problem and must be treated as such,

C  i.e., you must start afresh.

C

C  You cannot change from vector to scalar error control or vice

C  versa (INFO(2)), but you can change the size of the entries of

C  RTOL, 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), IWORK(1), or IWORK(2)

C  unless you are going to restart the code.

C

C                 *** Following a completed task ***

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                 *** Following an interrupted task ***

C               To show the code that you realize the task was

C               interrupted and that you want to continue, you

C               must take appropriate action and set INFO(1) = 1

C  If

C    IDID = -1, The code has taken about 500 steps.

C                  If you want to continue, set INFO(1) = 1 and

C                  call the code again. An additional 500 steps

C                  will be allowed.

C

C    IDID = -2, The error tolerances RTOL, ATOL have been

C                  increased to values the code estimates appropriate

C                  for continuing. You may want to change them

C                  yourself. If you are sure you want to continue

C                  with relaxed error tolerances, set INFO(1)=1 and

C                  call the code 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

C                  alter the error criterion to use positive values

C                  for those components of ATOL corresponding to zero

C                  solution components, then set INFO(1)=1 and call

C                  the code again.

C

C    IDID = -4,-5  --- Cannot occur with this code.

C

C    IDID = -6, Repeated error test failures occurred on the

C                  last attempted step in DDASSL. 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

C                  on the last attempted step in DDASSL. An inaccurate

C                  or ill-conditioned JACOBIAN may be the problem. If

C                  you are absolutely certain you want to continue, you

C                  should restart the integration.

C

C    IDID = -8, The matrix of partial derivatives is singular.

C                  Some of your equations may be redundant.

C                  DDASSL cannot solve the problem as stated.

C                  It is possible that the redundant equations

C                  could be removed, and then DDASSL could

C                  solve the problem. It is also possible

C                  that a solution to your problem either

C                  does not exist or is not unique.

C

C    IDID = -9, DDASSL had multiple convergence test

C                  failures, preceeded by multiple error

C                  test failures, on the last attempted step.

C                  It is possible that your problem

C                  is ill-posed, and cannot be solved

C                  using this code. Or, there may be a

C                  discontinuity or a singularity in the

C                  solution. If you are absolutely certain

C                  you want to continue, you should restart

C                  the integration.

C

C    IDID =-10, DDASSL had multiple convergence test failures

C                  because IRES was equal to minus one.

C                  If you are absolutely certain you want

C                  to continue, you should restart the

C                  integration.

C

C    IDID =-11, IRES=-2 was encountered, and control is being

C                  returned to the calling program.

C

C    IDID =-12, DDASSL failed to compute the initial YPRIME.

C                  This could happen because the initial

C                  approximation to YPRIME was not very good, or

C                  if a YPRIME consistent with the initial Y

C                  does not exist. The problem could also be caused

C                  by an inaccurate or singular iteration matrix.

C

C    IDID = -13,..,-32  --- Cannot occur with this code.

C

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

C                  run being terminated.

C

C

C  -------- ERROR MESSAGES ---------------------------------------------

C

C      The SLATEC error print routine XERMSG is called in the event of

C   unsuccessful completion of a task.  Most of these are treated as

C   "recoverable errors", which means that (unless the user has directed

C   otherwise) control will be returned to the calling program for

C   possible action after the message has been printed.

C

C   In the event of a negative value of IDID other than -33, an appro-

C   priate message is printed and the "error number" printed by XERMSG

C   is the value of IDID.  There are quite a number of illegal input

C   errors that can lead to a returned value IDID=-33.  The conditions

C   and their printed "error numbers" are as follows:

C

C   Error number       Condition

C

C        1       Some element of INFO vector is not zero or one.

C        2       NEQ .le. 0

C        3       MAXORD not in range.

C        4       LRW is less than the required length for RWORK.

C        5       LIW is less than the required length for IWORK.

C        6       Some element of RTOL is .lt. 0

C        7       Some element of ATOL is .lt. 0

C        8       All elements of RTOL and ATOL are zero.

C        9       INFO(4)=1 and TSTOP is behind TOUT.

C       10       HMAX .lt. 0.0

C       11       TOUT is behind T.

C       12       INFO(8)=1 and H0=0.0

C       13       Some element of WT is .le. 0.0

C       14       TOUT is too close to T to start integration.

C       15       INFO(4)=1 and TSTOP is behind T.

C       16       --( Not used in this version )--

C       17       ML illegal.  Either .lt. 0 or .gt. NEQ

C       18       MU illegal.  Either .lt. 0 or .gt. NEQ

C       19       TOUT = T.

C

C   If DDASSL is called again without any action taken to remove the

C   cause of an unsuccessful return, XERMSG will be called with a fatal

C   error flag, which will cause unconditional termination of the

C   program.  There are two such fatal errors:

C

C   Error number -998:  The last step was terminated with a negative

C       value of IDID other than -33, and no appropriate action was

C       taken.

C

C   Error number -999:  The previous call was terminated because of

C       illegal input (IDID=-33) and there is illegal input in the

C       present call, as well.  (Suspect infinite loop.)

C

C  ---------------------------------------------------------------------

C

C***REFERENCES  A DESCRIPTION OF DASSL: A DIFFERENTIAL/ALGEBRAIC

C                 SYSTEM SOLVER, L. R. PETZOLD, SAND82-8637,

C                 SANDIA NATIONAL LABORATORIES, SEPTEMBER 1982.

C***ROUTINES CALLED  D1MACH, DDAINI, DDANRM, DDASTP, DDATRP, DDAWTS,

C                    XERMSG

C***REVISION HISTORY  (YYMMDD)

C   830315  DATE WRITTEN

C   880387  Code changes made.  All common statements have been

C           replaced by a DATA statement, which defines pointers into

C           RWORK, and PARAMETER statements which define pointers

C           into IWORK.  As well the documentation has gone through

C           grammatical changes.

C   881005  The prologue has been changed to mixed case.

C           The subordinate routines had revision dates changed to

C           this date, although the documentation for these routines

C           is all upper case.  No code changes.

C   890511  Code changes made.  The DATA statement in the declaration

C           section of DDASSL was replaced with a PARAMETER

C           statement.  Also the statement S = 100.D0 was removed

C           from the top of the Newton iteration in DDASTP.

C           The subordinate routines had revision dates changed to

C           this date.

C   890517  The revision date syntax was replaced with the revision

C           history syntax.  Also the "DECK" comment was added to

C           the top of all subroutines.  These changes are consistent

C           with new SLATEC guidelines.

C           The subordinate routines had revision dates changed to

C           this date.  No code changes.

C   891013  Code changes made.

C           Removed all occurrances of FLOAT or DBLE.  All operations

C           are now performed with "mixed-mode" arithmetic.

C           Also, specific function names were replaced with generic

C           function names to be consistent with new SLATEC guidelines.

C           In particular:

C              Replaced DSQRT with SQRT everywhere.

C              Replaced DABS with ABS everywhere.

C              Replaced DMIN1 with MIN everywhere.

C              Replaced MIN0 with MIN everywhere.

C              Replaced DMAX1 with MAX everywhere.

C              Replaced MAX0 with MAX everywhere.

C              Replaced DSIGN with SIGN everywhere.

C           Also replaced REVISION DATE with REVISION HISTORY in all

C           subordinate routines.

C  901004  Miscellaneous changes to prologue to complete conversion

C          to SLATEC 4.0 format.  No code changes.  (F.N.Fritsch)

C  901009  Corrected GAMS classification code and converted subsidiary

C          routines to 4.0 format.  No code changes.  (F.N.Fritsch)

C  901010  Converted XERRWV calls to XERMSG calls.  (R.Clemens,AFWL)

C  901019  Code changes made.

C          Merged SLATEC 4.0 changes with previous changes made

C          by C. Ulrich.  Below is a history of the changes made by

C          C. Ulrich. (Changes in subsidiary routines are implied

C          by this history)

C          891228  Bug was found and repaired inside the DDASSL

C                  and DDAINI routines.  DDAINI was incorrectly

C                  returning the initial T with Y and YPRIME

C                  computed at T+H.  The routine now returns T+H

C                  rather than the initial T.

C                  Cosmetic changes made to DDASTP.

C          900904  Three modifications were made to fix a bug (inside

C                  DDASSL) re interpolation for continuation calls and

C                  cases where TN is very close to TSTOP:

C

C                  1) In testing for whether H is too large, just

C                     compare H to (TSTOP - TN), rather than

C                     (TSTOP - TN) * (1-4*UROUND), and set H to

C                     TSTOP - TN.  This will force DDASTP to step

C                     exactly to TSTOP under certain situations

C                     (i.e. when H returned from DDASTP would otherwise

C                     take TN beyond TSTOP).

C

C                  2) Inside the DDASTP loop, interpolate exactly to

C                     TSTOP if TN is very close to TSTOP (rather than

C                     interpolating to within roundoff of TSTOP).

C

C                  3) Modified IDID description for IDID = 2 to say that

C                     the solution is returned by stepping exactly to

C                     TSTOP, rather than TOUT.  (In some cases the

C                     solution is actually obtained by extrapolating

C                     over a distance near unit roundoff to TSTOP,

C                     but this small distance is deemed acceptable in

C                     these circumstances.)

C   901026  Added explicit declarations for all variables and minor

C           cosmetic changes to prologue, removed unreferenced labels,

C           and improved XERMSG calls.  (FNF)

C   901030  Added ERROR MESSAGES section and reworked other sections to

C           be of more uniform format.  (FNF)

C   910624  Fixed minor bug related to HMAX (five lines ending in

C           statement 526 in DDASSL).   (LRP)

C

C***END PROLOGUE  DDASSL

C

C**End

C

C     Declare arguments.

C

      INTEGER  NEQ, INFO(15), IDID, LRW, IWORK(*), LIW, IPAR(*)

      double precision

     *   t, y(*), yprime(*), tout, rtol(*), atol(*), rwork(*),

     *   rpar(*)

      EXTERNAL  res, jac

C

C     Declare externals.

C

      EXTERNAL  d1mach, ddaini, ddanrm, ddastp, ddatrp, ddawts, xermsg

      DOUBLE PRECISION  D1MACH, DDANRM

C

C     Declare local variables.

C

      INTEGER  I, ITEMP, LALPHA, LBETA, LCJ, LCJOLD, LCTF, LDELTA,

     *   leniw, lenpd, lenrw, le, letf, lgamma, lh, lhmax, lhold,

     *   lmxstp, lipvt,

     *   ljcalc, lk, lkold, liwm, lml, lmtype, lmu, lmxord, lnje, lnpd,

     *   lnre, lns, lnst, lnstl, lpd, lphase, lphi, lpsi, lround, ls,

     *   lsigma, ltn, ltstop, lwm, lwt, mband, msave, mxord, npd, ntemp,

     *   nzflg

      double precision

     *   atoli, h, hmax, hmin, ho, r, rh, rtoli, tdist, tn, tnext,

     *   tstop, uround, ypnorm

      LOGICAL  DONE

C       Auxiliary variables for conversion of values to be included in

C       error messages.

      CHARACTER*8  XERN1, XERN2

      CHARACTER*16 XERN3, XERN4

C

C     SET POINTERS INTO IWORK

      parameter(lml=1, lmu=2, lmxord=3, lmtype=4, lnst=11,

     *  lnre=12, lnje=13, letf=14, lctf=15, lnpd=16, lmxstp=21,

     *  lipvt=22, ljcalc=5, lphase=6, lk=7, lkold=8,

     *  lns=9, lnstl=10, liwm=1)

C

C     SET RELATIVE OFFSET INTO RWORK

      parameter(npd=1)

C

C     SET POINTERS INTO RWORK

      parameter(ltstop=1, lhmax=2, lh=3, ltn=4,

     *  lcj=5, lcjold=6, lhold=7, ls=8, lround=9,

     *  lalpha=11, lbeta=17, lgamma=23,

     *  lpsi=29, lsigma=35, ldelta=41)

C

C***FIRST EXECUTABLE STATEMENT  DDASSL

      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 EITHER ZERO OR ONE.

      DO 10 i=2,11

         IF(info(i).NE.0.AND.info(i).NE.1)GO TO 701

10       CONTINUE

C

      IF(neq.LE.0)GO TO 702

C

C     CHECK AND COMPUTE MAXIMUM ORDER

      mxord=5

      IF(info(9).EQ.0)GO TO 20

         mxord=iwork(lmxord)

         IF(mxord.LT.1.OR.mxord.GT.5)GO TO 703

20       iwork(lmxord)=mxord

C

C     COMPUTE MTYPE,LENPD,LENRW.CHECK ML AND MU.

      IF(info(6).NE.0)GO TO 40

         lenpd=neq**2

         lenrw=40+(iwork(lmxord)+4)*neq+lenpd

         IF(info(5).NE.0)GO TO 30

            iwork(lmtype)=2

            GO TO 60

30          iwork(lmtype)=1

            GO TO 60

40    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).NE.0)GO TO 50

         iwork(lmtype)=5

         mband=iwork(lml)+iwork(lmu)+1

         msave=(neq/mband)+1

         lenrw=40+(iwork(lmxord)+4)*neq+lenpd+2*msave

         GO TO 60

50       iwork(lmtype)=4

         lenrw=40+(iwork(lmxord)+4)*neq+lenpd

C

C     CHECK LENGTHS OF RWORK AND IWORK

60    leniw=21+neq

      iwork(lnpd)=lenpd

      IF(lrw.LT.lenrw)GO TO 704

      IF(liw.LT.leniw)GO TO 705

C

C     CHECK TO SEE THAT TOUT IS DIFFERENT FROM T

      IF(tout .EQ. t)GO TO 719

C

C     CHECK HMAX

      IF(info(7).EQ.0)GO TO 70

         hmax=rwork(lhmax)

         IF(hmax.LE.0.0d0)GO TO 710

70    CONTINUE

C

C     CHECK AND COMPUTE MAXIMUM STEPS

      mxstp=500

      IF(info(12).EQ.0)GO TO 80

        mxstp=iwork(lmxstp)

        IF(mxstp.LT.0)GO TO 716

80      iwork(lmxstp)=mxstp

C

C     INITIALIZE COUNTERS

      iwork(lnst)=0

      iwork(lnre)=0

      iwork(lnje)=0

C

      iwork(lnstl)=0

      idid=1

      GO TO 200

C

C-----------------------------------------------------------------------

C     THIS BLOCK IS FOR CONTINUATION CALLS

C     ONLY. HERE WE CHECK INFO(1),AND IF THE

C     LAST STEP WAS INTERRUPTED WE CHECK WHETHER

C     APPROPRIATE ACTION WAS TAKEN.

C-----------------------------------------------------------------------

C

100   CONTINUE

      IF(info(1).EQ.1)GO TO 110

      IF(info(1).NE.-1)GO TO 701

C

C     IF WE ARE HERE, THE LAST STEP WAS INTERRUPTED

C     BY AN ERROR CONDITION FROM DDASTP,AND

C     APPROPRIATE ACTION WAS NOT TAKEN. THIS

C     IS A FATAL ERROR.

      WRITE (xern1, '(I8)') idid

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'THE LAST STEP TERMINATED WITH A NEGATIVE VALUE OF IDID = ' //

     *   xern1 // ' AND NO APPROPRIATE ACTION WAS TAKEN.  ' //

     *   'RUN TERMINATED', -998, 2)

      RETURN

110   CONTINUE

      iwork(lnstl)=iwork(lnst)

C

C-----------------------------------------------------------------------

C     THIS BLOCK IS EXECUTED ON ALL CALLS.

C     THE ERROR TOLERANCE PARAMETERS ARE

C     CHECKED, AND THE WORK ARRAY POINTERS

C     ARE SET.

C-----------------------------------------------------------------------

C

200   CONTINUE

C     CHECK RTOL,ATOL

      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 UP RWORK STORAGE.IWORK STORAGE IS FIXED

C     IN DATA STATEMENT.

      le=ldelta+neq

      lwt=le+neq

      lphi=lwt+neq

      lpd=lphi+(iwork(lmxord)+1)*neq

      lwm=lpd

      ntemp=npd+iwork(lnpd)

      IF(info(1).EQ.1)GO TO 400

C

C-----------------------------------------------------------------------

C     THIS BLOCK IS EXECUTED ON THE INITIAL CALL

C     ONLY. SET THE INITIAL STEP SIZE, AND

C     THE ERROR WEIGHT VECTOR, AND PHI.

C     COMPUTE INITIAL YPRIME, IF NECESSARY.

C-----------------------------------------------------------------------

C

      tn=t

      idid=1

C

C     SET ERROR WEIGHT VECTOR WT

      CALL ddawts(neq,info(2),rtol,atol,y,rwork(lwt),rpar,ipar)

      DO 305 i = 1,neq

         IF(rwork(lwt+i-1).LE.0.0d0) GO TO 713

305      CONTINUE

C

C     COMPUTE UNIT ROUNDOFF AND HMIN

      uround = d1mach(4)

      rwork(lround) = uround

      hmin = 4.0d0*uround*max(abs(t),abs(tout))

C

C     CHECK INITIAL INTERVAL TO SEE THAT IT IS LONG ENOUGH

      tdist = abs(tout - t)

      IF(tdist .LT. hmin) GO TO 714

C

C     CHECK HO, IF THIS WAS INPUT

      IF (info(8) .EQ. 0) GO TO 310

         ho = rwork(lh)

         IF ((tout - t)*ho .LT. 0.0d0) GO TO 711

         IF (ho .EQ. 0.0d0) GO TO 712

         GO TO 320

310    CONTINUE

C

C     COMPUTE INITIAL STEPSIZE, TO BE USED BY EITHER

C     DDASTP OR DDAINI, DEPENDING ON INFO(11)

      ho = 0.001d0*tdist

      ypnorm = ddanrm(neq,yprime,rwork(lwt),rpar,ipar)

      IF (ypnorm .GT. 0.5d0/ho) ho = 0.5d0/ypnorm

      ho = sign(ho,tout-t)

C     ADJUST HO IF NECESSARY TO MEET HMAX BOUND

320   IF (info(7) .EQ. 0) GO TO 330

         rh = abs(ho)/rwork(lhmax)

         IF (rh .GT. 1.0d0) ho = ho/rh

C     COMPUTE TSTOP, IF APPLICABLE

330   IF (info(4) .EQ. 0) GO TO 340

         tstop = rwork(ltstop)

         IF ((tstop - t)*ho .LT. 0.0d0) GO TO 715

         IF ((t + ho - tstop)*ho .GT. 0.0d0) ho = tstop - t

         IF ((tstop - tout)*ho .LT. 0.0d0) GO TO 709

C

C     COMPUTE INITIAL DERIVATIVE, UPDATING TN AND Y, IF APPLICABLE

340   IF (info(11) .EQ. 0) GO TO 350

      CALL ddaini(tn,y,yprime,neq,

     *  res,jac,ho,rwork(lwt),idid,rpar,ipar,

     *  rwork(lphi),rwork(ldelta),rwork(le),

     *  rwork(lwm),iwork(liwm),hmin,rwork(lround),

     *  info(10),ntemp)

      IF (idid .LT. 0) GO TO 390

C

C     LOAD H WITH HO.  STORE H IN RWORK(LH)

350   h = ho

      rwork(lh) = h

C

C     LOAD Y AND H*YPRIME INTO PHI(*,1) AND PHI(*,2)

      itemp = lphi + neq

      DO 370 i = 1,neq

         rwork(lphi + i - 1) = y(i)

370      rwork(itemp + i - 1) = h*yprime(i)

C

390   GO TO 500

C

C-------------------------------------------------------

C     THIS BLOCK IS FOR CONTINUATION CALLS ONLY. ITS

C     PURPOSE IS TO CHECK STOP CONDITIONS BEFORE

C     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(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 .GT. 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 .GT. 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     CHECK WHETHER WE ARE WITHIN ROUNDOFF OF TSTOP

      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 580

C

C-------------------------------------------------------

C     THE NEXT BLOCK CONTAINS THE CALL TO THE

C     ONE-STEP INTEGRATOR DDASTP.

C     THIS IS A LOOPING POINT FOR THE INTEGRATION STEPS.

C     CHECK FOR TOO MANY STEPS.

C     UPDATE WT.

C     CHECK FOR TOO MUCH ACCURACY REQUESTED.

C     COMPUTE MINIMUM STEPSIZE.

C-------------------------------------------------------

C

500   CONTINUE

C     CHECK FOR FAILURE TO COMPUTE INITIAL YPRIME

      IF (idid .EQ. -12) GO TO 527

C

C     CHECK FOR TOO MANY STEPS

      IF((iwork(lnst)-iwork(lnstl)).LT.iwork(lmxstp))

     *   GO TO 510

           idid=-1

           GO TO 527

C

C     UPDATE WT

510   CALL ddawts(neq,info(2),rtol,atol,rwork(lphi),

     *  rwork(lwt),rpar,ipar)

      DO 520 i=1,neq

         IF(rwork(i+lwt-1).GT.0.0d0)GO TO 520

           idid=-3

           GO TO 527

520   CONTINUE

C

C     TEST FOR TOO MUCH ACCURACY REQUESTED.

      r=ddanrm(neq,rwork(lphi),rwork(lwt),rpar,ipar)*

     *   100.0d0*uround

      IF(r.LE.1.0d0)GO TO 525

C     MULTIPLY RTOL AND ATOL BY R AND RETURN

      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

      hmin=4.0d0*uround*max(abs(tn),abs(tout))

C

C     TEST H VS. HMAX

      IF (info(7) .EQ. 0) GO TO 526

         rh = abs(h)/rwork(lhmax)

         IF (rh .GT. 1.0d0) h = h/rh

526   CONTINUE

C

      CALL ddastp(tn,y,yprime,neq,

     *   res,jac,h,rwork(lwt),info(1),idid,rpar,ipar,

     *   rwork(lphi),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),

     *   iwork(lphase),iwork(ljcalc),iwork(lk),

     *   iwork(lkold),iwork(lns),info(10),ntemp)

527   IF(idid.LT.0)GO TO 600

C

C--------------------------------------------------------

C     THIS BLOCK HANDLES THE CASE OF A SUCCESSFUL RETURN

C     FROM DDASTP (IDID=1).  TEST FOR STOP CONDITIONS.

C--------------------------------------------------------

C

      IF(info(4).NE.0)GO TO 540

           IF(info(3).NE.0)GO TO 530

             IF((tn-tout)*h.LT.0.0d0)GO TO 500

             CALL ddatrp(tn,tout,y,yprime,neq,

     *         iwork(lkold),rwork(lphi),rwork(lpsi))

             idid=3

             t=tout

             GO TO 580

530          IF((tn-tout)*h.GE.0.0d0)GO TO 535

             t=tn

             idid=1

             GO TO 580

535          CALL ddatrp(tn,tout,y,yprime,neq,

     *         iwork(lkold),rwork(lphi),rwork(lpsi))

             idid=3

             t=tout

             GO TO 580

540   IF(info(3).NE.0)GO TO 550

      IF((tn-tout)*h.LT.0.0d0)GO TO 542

         CALL ddatrp(tn,tout,y,yprime,neq,

     *     iwork(lkold),rwork(lphi),rwork(lpsi))

         t=tout

         idid=3

         GO TO 580

542   IF(abs(tn-tstop).LE.100.0d0*uround*

     *   (abs(tn)+abs(h)))GO TO 545

      tnext=tn+h

      IF((tnext-tstop)*h.LE.0.0d0)GO TO 500

      h=tstop-tn

      GO TO 500

545   CALL ddatrp(tn,tstop,y,yprime,neq,

     *  iwork(lkold),rwork(lphi),rwork(lpsi))

      idid=2

      t=tstop

      GO TO 580

550   IF((tn-tout)*h.GE.0.0d0)GO TO 555

      IF(abs(tn-tstop).LE.100.0d0*uround*(abs(tn)+abs(h)))GO TO 552

      t=tn

      idid=1

      GO TO 580

552   CALL ddatrp(tn,tstop,y,yprime,neq,

     *  iwork(lkold),rwork(lphi),rwork(lpsi))

      idid=2

      t=tstop

      GO TO 580

555   CALL ddatrp(tn,tout,y,yprime,neq,

     *   iwork(lkold),rwork(lphi),rwork(lpsi))

      t=tout

      idid=3

      GO TO 580

C

C--------------------------------------------------------

C     ALL SUCCESSFUL RETURNS FROM DDASSL ARE MADE FROM

C     THIS BLOCK.

C--------------------------------------------------------

C

580   CONTINUE

      rwork(ltn)=tn

      rwork(lh)=h

      RETURN

C

C-----------------------------------------------------------------------

C     THIS BLOCK HANDLES ALL UNSUCCESSFUL

C     RETURNS OTHER THAN FOR ILLEGAL INPUT.

C-----------------------------------------------------------------------

C

600   CONTINUE

      itemp=-idid

      GO TO (610,620,630,690,690,640,650,660,670,675,

     *  680,685), itemp

C

C     THE MAXIMUM NUMBER OF STEPS WAS TAKEN BEFORE

C     REACHING TOUT

610   WRITE (xern3, '(1P,D15.6)') tn

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT CURRENT T = ' // xern3 // ' 500 STEPS TAKEN ON THIS ' //

     *   'CALL BEFORE REACHING TOUT', idid, 1)

      GO TO 690

C

C     TOO MUCH ACCURACY FOR MACHINE PRECISION

620   WRITE (xern3, '(1P,D15.6)') tn

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // ' TOO MUCH ACCURACY REQUESTED FOR ' //

     *   'PRECISION OF MACHINE. RTOL AND ATOL WERE INCREASED TO ' //

     *   'APPROPRIATE VALUES', idid, 1)

      GO TO 690

C

C     WT(I) .LE. 0.0 FOR SOME I (NOT AT START OF PROBLEM)

630   WRITE (xern3, '(1P,D15.6)') tn

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // .LE.' SOME ELEMENT OF WT HAS BECOME  ' //

     *   '0.0', idid, 1)

      GO TO 690

C

C     ERROR TEST FAILED REPEATEDLY OR WITH H=HMIN

640   WRITE (xern3, '(1P,D15.6)') tn

      WRITE (xern4, '(1P,D15.6)') h

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // ' AND STEPSIZE H = ' // xern4 //

     *   ' THE ERROR TEST FAILED REPEATEDLY OR WITH ABS(H)=HMIN',

     *   idid, 1)

      GO TO 690

C

C     CORRECTOR CONVERGENCE FAILED REPEATEDLY OR WITH H=HMIN

650   WRITE (xern3, '(1P,D15.6)') tn

      WRITE (xern4, '(1P,D15.6)') h

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // ' AND STEPSIZE H = ' // xern4 //

     *   ' THE CORRECTOR FAILED TO CONVERGE REPEATEDLY OR WITH ' //

     *   'ABS(H)=HMIN', idid, 1)

      GO TO 690

C

C     THE ITERATION MATRIX IS SINGULAR

660   WRITE (xern3, '(1P,D15.6)') tn

      WRITE (xern4, '(1P,D15.6)') h

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // ' AND STEPSIZE H = ' // xern4 //

     *   ' THE ITERATION MATRIX IS SINGULAR', idid, 1)

      GO TO 690

C

C     CORRECTOR FAILURE PRECEEDED BY ERROR TEST FAILURES.

670   WRITE (xern3, '(1P,D15.6)') tn

      WRITE (xern4, '(1P,D15.6)') h

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // ' AND STEPSIZE H = ' // xern4 //

     *   ' THE CORRECTOR COULD NOT CONVERGE.  ALSO, THE ERROR TEST ' //

     *   'FAILED REPEATEDLY.', idid, 1)

      GO TO 690

C

C     CORRECTOR FAILURE BECAUSE IRES = -1

675   WRITE (xern3, '(1P,D15.6)') tn

      WRITE (xern4, '(1P,D15.6)') h

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // ' AND STEPSIZE H = ' // xern4 //

     *   ' THE CORRECTOR COULD NOT CONVERGE BECAUSE IRES WAS EQUAL ' //

     *   'TO MINUS ONE', idid, 1)

      GO TO 690

C

C     FAILURE BECAUSE IRES = -2

680   WRITE (xern3, '(1P,D15.6)') tn

      WRITE (xern4, '(1P,D15.6)') h

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // ' AND STEPSIZE H = ' // xern4 //

     *   ' IRES WAS EQUAL TO MINUS TWO', idid, 1)

      GO TO 690

C

C     FAILED TO COMPUTE INITIAL YPRIME

685   WRITE (xern3, '(1P,D15.6)') tn

      WRITE (xern4, '(1P,D15.6)') ho

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'AT T = ' // xern3 // ' AND STEPSIZE H = ' // xern4 //

     *   ' THE INITIAL YPRIME COULD NOT BE COMPUTED', idid, 1)

      GO TO 690

C

690   CONTINUE

      info(1)=-1

      t=tn

      rwork(ltn)=tn

      rwork(lh)=h

      RETURN

C

C-----------------------------------------------------------------------

C     THIS BLOCK HANDLES ALL ERROR RETURNS DUE

C     TO ILLEGAL INPUT, AS DETECTED BEFORE CALLING

C     DDASTP. FIRST THE ERROR MESSAGE ROUTINE IS

C     CALLED. IF THIS HAPPENS TWICE IN

C     SUCCESSION, EXECUTION IS TERMINATED

C

C-----------------------------------------------------------------------

701   CALL xermsg ('SLATEC', 'DDASSL',

     *   'SOME ELEMENT OF INFO VECTOR IS NOT ZERO OR ONE', 1, 1)

      GO TO 750

C

702   WRITE (xern1, '(I8)') neq

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'NEQ = ' // xern1 // .LE.'  0', 2, 1)

      GO TO 750

C

703   WRITE (xern1, '(I8)') mxord

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'MAXORD = ' // xern1 // ' NOT IN RANGE', 3, 1)

      GO TO 750

C

704   WRITE (xern1, '(I8)') lenrw

      WRITE (xern2, '(I8)') lrw

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'RWORK LENGTH NEEDED, LENRW = ' // xern1 //

     *   ', EXCEEDS LRW = ' // xern2, 4, 1)

      GO TO 750

C

705   WRITE (xern1, '(I8)') leniw

      WRITE (xern2, '(I8)') liw

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'IWORK LENGTH NEEDED, LENIW = ' // xern1 //

     *   ', EXCEEDS LIW = ' // xern2, 5, 1)

      GO TO 750

C

706   CALL xermsg ('SLATEC', 'DDASSL',

     *   .LT.'SOME ELEMENT OF RTOL IS  0', 6, 1)

      GO TO 750

C

707   CALL xermsg ('SLATEC', 'DDASSL',

     *   .LT.'SOME ELEMENT OF ATOL IS  0', 7, 1)

      GO TO 750

C

708   CALL xermsg ('SLATEC', 'DDASSL',

     *   'ALL ELEMENTS OF RTOL AND ATOL ARE ZERO', 8, 1)

      GO TO 750

C

709   WRITE (xern3, '(1P,D15.6)') tstop

      WRITE (xern4, '(1P,D15.6)') tout

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'INFO(4) = 1 AND TSTOP = ' // xern3 // ' BEHIND TOUT = ' //

     *   xern4, 9, 1)

      GO TO 750

C

710   WRITE (xern3, '(1P,D15.6)') hmax

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'HMAX = ' // xern3 // .LT.'  0.0', 10, 1)

      GO TO 750

C

711   WRITE (xern3, '(1P,D15.6)') tout

      WRITE (xern4, '(1P,D15.6)') t

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'TOUT = ' // xern3 // ' BEHIND T = ' // xern4, 11, 1)

      GO TO 750

C

712   CALL xermsg ('SLATEC', 'DDASSL',

     *   'INFO(8)=1 AND H0=0.0', 12, 1)

      GO TO 750

C

713   CALL xermsg ('SLATEC', 'DDASSL',

     *   .LE.'SOME ELEMENT OF WT IS  0.0', 13, 1)

      GO TO 750

C

714   WRITE (xern3, '(1P,D15.6)') tout

      WRITE (xern4, '(1P,D15.6)') t

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'TOUT = ' // xern3 // ' TOO CLOSE TO T = ' // xern4 //

     *   ' TO START INTEGRATION', 14, 1)

      GO TO 750

C

715   WRITE (xern3, '(1P,D15.6)') tstop

      WRITE (xern4, '(1P,D15.6)') t

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'INFO(4)=1 AND TSTOP = ' // xern3 // ' BEHIND T = ' // xern4,

     *   15, 1)

      GO TO 750

C

716   WRITE (xern1, '(I8)') mxstp

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'INFO(12)=1 AND MXSTP = ' // xern1 // ' ILLEGAL.', 3, 1)

      GO TO 750

C

717   WRITE (xern1, '(I8)') iwork(lml)

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'ML = ' // xern1 // .LT..GT.' ILLEGAL.  EITHER  0 OR  NEQ',

     *   17, 1)

      GO TO 750

C

718   WRITE (xern1, '(I8)') iwork(lmu)

      CALL xermsg ('SLATEC', 'DDASSL',

     *   'MU = ' // xern1 // .LT..GT.' ILLEGAL.  EITHER  0 OR  NEQ',

     *   18, 1)

      GO TO 750

C

719   WRITE (xern3, '(1P,D15.6)') tout

      CALL xermsg ('SLATEC', 'DDASSL',

     *  'TOUT = T = ' // xern3, 19, 1)

      GO TO 750

C

750   idid=-33

      IF(info(1).EQ.-1) THEN

         CALL xermsg ('SLATEC', 'DDASSL',

     *      'REPEATED OCCURRENCES OF ILLEGAL INPUT$$' //

     *      'RUN TERMINATED. APPARENT INFINITE LOOP', -999, 2)

      ENDIF

C

      info(1)=-1

      RETURN

C-----------END OF SUBROUTINE DDASSL------------------------------------


      END

max
charNDArray max(char d, const charNDArray &m)
Definition chNDArray.cc:230

d1mach
double precision function d1mach(i)
Definition d1mach.f:23

ddaini
subroutine ddaini(x, y, yprime, neq, res, jac, h, wt, idid, rpar, ipar, phi, delta, e, wm, iwm, hmin, uround, nonneg, ntemp)
Definition ddaini.f:3

ddanrm
double precision function ddanrm(neq, v, wt, rpar, ipar)
Definition ddanrm.f:2

ddassl
subroutine ddassl(res, neq, t, y, yprime, tout, info, rtol, atol, idid, rwork, lrw, iwork, liw, rpar, ipar, jac)
Definition ddassl.f:3

ddastp
subroutine ddastp(x, y, yprime, neq, res, jac, h, wt, jstart, idid, rpar, ipar, phi, delta, e, wm, iwm, alpha, beta, gamma, psi, sigma, cj, cjold, hold, s, hmin, uround, iphase, jcalc, k, kold, ns, nonneg, ntemp)
Definition ddastp.f:5

ddatrp
subroutine ddatrp(x, xout, yout, ypout, neq, kold, phi, psi)
Definition ddatrp.f:2

ddawts
subroutine ddawts(neq, iwt, rtol, atol, y, wt, rpar, ipar)
Definition ddawts.f:2

lgamma
double lgamma(double x)
Definition lo-specfun.h:336

xermsg
subroutine xermsg(librar, subrou, messg, nerr, level)
Definition xermsg.f:3