SLATEC Routines --- SDRIV3 ---
*DECK SDRIV3
SUBROUTINE SDRIV3 (N, T, Y, F, NSTATE, TOUT, NTASK, NROOT, EPS,
8 EWT, IERROR, MINT, MITER, IMPL, ML, MU, MXORD, HMAX, WORK,
8 LENW, IWORK, LENIW, JACOBN, FA, NDE, MXSTEP, G, USERS, IERFLG)
C***BEGIN PROLOGUE SDRIV3
C***PURPOSE The function of SDRIV3 is to solve N ordinary differential
C equations of the form dY(I)/dT = F(Y(I),T), given the
C initial conditions Y(I) = YI. The program has options to
C allow the solution of both stiff and non-stiff differential
C equations. Other important options are available. SDRIV3
C uses single precision arithmetic.
C***LIBRARY SLATEC (SDRIVE)
C***CATEGORY I1A2, I1A1B
C***TYPE SINGLE PRECISION (SDRIV3-S, DDRIV3-D, CDRIV3-C)
C***KEYWORDS GEAR'S METHOD, INITIAL VALUE PROBLEMS, ODE,
C ORDINARY DIFFERENTIAL EQUATIONS, SDRIVE, SINGLE PRECISION,
C STIFF
C***AUTHOR Kahaner, D. K., (NIST)
C National Institute of Standards and Technology
C Gaithersburg, MD 20899
C Sutherland, C. D., (LANL)
C Mail Stop D466
C Los Alamos National Laboratory
C Los Alamos, NM 87545
C***DESCRIPTION
C
C I. ABSTRACT .......................................................
C
C The primary function of SDRIV3 is to solve N ordinary differential
C equations of the form dY(I)/dT = F(Y(I),T), given the initial
C conditions Y(I) = YI. The program has options to allow the
C solution of both stiff and non-stiff differential equations. In
C addition, SDRIV3 may be used to solve:
C 1. The initial value problem, A*dY(I)/dT = F(Y(I),T), where A is
C a non-singular matrix depending on Y and T.
C 2. The hybrid differential/algebraic initial value problem,
C A*dY(I)/dT = F(Y(I),T), where A is a vector (whose values may
C depend upon Y and T) some of whose components will be zero
C corresponding to those equations which are algebraic rather
C than differential.
C SDRIV3 is to be called once for each output point of T.
C
C II. PARAMETERS ....................................................
C
C The user should use parameter names in the call sequence of SDRIV3
C for those quantities whose value may be altered by SDRIV3. The
C parameters in the call sequence are:
C
C N = (Input) The number of dependent functions whose solution
C is desired. N must not be altered during a problem.
C
C T = The independent variable. On input for the first call, T
C is the initial point. On output, T is the point at which
C the solution is given.
C
C Y = The vector of dependent variables. Y is used as input on
C the first call, to set the initial values. On output, Y
C is the computed solution vector. This array Y is passed
C in the call sequence of the user-provided routines F,
C JACOBN, FA, USERS, and G. Thus parameters required by
C those routines can be stored in this array in components
C N+1 and above. (Note: Changes by the user to the first
C N components of this array will take effect only after a
C restart, i.e., after setting NSTATE to 1 .)
C
C F = A subroutine supplied by the user. The name must be
C declared EXTERNAL in the user's calling program. This
C subroutine is of the form:
C SUBROUTINE F (N, T, Y, YDOT)
C REAL Y(*), YDOT(*)
C .
C .
C YDOT(1) = ...
C .
C .
C YDOT(N) = ...
C END (Sample)
C This computes YDOT = F(Y,T), the right hand side of the
C differential equations. Here Y is a vector of length at
C least N. The actual length of Y is determined by the
C user's declaration in the program which calls SDRIV3.
C Thus the dimensioning of Y in F, while required by FORTRAN
C convention, does not actually allocate any storage. When
C this subroutine is called, the first N components of Y are
C intermediate approximations to the solution components.
C The user should not alter these values. Here YDOT is a
C vector of length N. The user should only compute YDOT(I)
C for I from 1 to N. Normally a return from F passes
C control back to SDRIV3. However, if the user would like
C to abort the calculation, i.e., return control to the
C program which calls SDRIV3, he should set N to zero.
C SDRIV3 will signal this by returning a value of NSTATE
C equal to 6 . Altering the value of N in F has no effect
C on the value of N in the call sequence of SDRIV3.
C
C NSTATE = An integer describing the status of integration. The
C meaning of NSTATE is as follows:
C 1 (Input) Means the first call to the routine. This
C value must be set by the user. On all subsequent
C calls the value of NSTATE should be tested by the
C user, but must not be altered. (As a convenience to
C the user who may wish to put out the initial
C conditions, SDRIV3 can be called with NSTATE=1, and
C TOUT=T. In this case the program will return with
C NSTATE unchanged, i.e., NSTATE=1.)
C 2 (Output) Means a successful integration. If a normal
C continuation is desired (i.e., a further integration
C in the same direction), simply advance TOUT and call
C again. All other parameters are automatically set.
C 3 (Output)(Unsuccessful) Means the integrator has taken
C MXSTEP steps without reaching TOUT. The user can
C continue the integration by simply calling SDRIV3
C again.
C 4 (Output)(Unsuccessful) Means too much accuracy has
C been requested. EPS has been increased to a value
C the program estimates is appropriate. The user can
C continue the integration by simply calling SDRIV3
C again.
C 5 (Output) A root was found at a point less than TOUT.
C The user can continue the integration toward TOUT by
C simply calling SDRIV3 again.
C 6 (Output)(Unsuccessful) N has been set to zero in
C SUBROUTINE F.
C 7 (Output)(Unsuccessful) N has been set to zero in
C FUNCTION G. See description of G below.
C 8 (Output)(Unsuccessful) N has been set to zero in
C SUBROUTINE JACOBN. See description of JACOBN below.
C 9 (Output)(Unsuccessful) N has been set to zero in
C SUBROUTINE FA. See description of FA below.
C 10 (Output)(Unsuccessful) N has been set to zero in
C SUBROUTINE USERS. See description of USERS below.
C 11 (Output)(Successful) For NTASK = 2 or 3, T is beyond
C TOUT. The solution was obtained by interpolation.
C The user can continue the integration by simply
C advancing TOUT and calling SDRIV3 again.
C 12 (Output)(Unsuccessful) The solution could not be
C obtained. The value of IERFLG (see description
C below) for a "Recoverable" situation indicates the
C type of difficulty encountered: either an illegal
C value for a parameter or an inability to continue the
C solution. For this condition the user should take
C corrective action and reset NSTATE to 1 before
C calling SDRIV3 again. Otherwise the program will
C terminate the run.
C
C TOUT = (Input) The point at which the solution is desired. The
C position of TOUT relative to T on the first call
C determines the direction of integration.
C
C NTASK = (Input) An index specifying the manner of returning the
C solution, according to the following:
C NTASK = 1 Means SDRIV3 will integrate past TOUT and
C interpolate the solution. This is the most
C efficient mode.
C NTASK = 2 Means SDRIV3 will return the solution after
C each internal integration step, or at TOUT,
C whichever comes first. In the latter case,
C the program integrates exactly to TOUT.
C NTASK = 3 Means SDRIV3 will adjust its internal step to
C reach TOUT exactly (useful if a singularity
C exists beyond TOUT.)
C
C NROOT = (Input) The number of equations whose roots are desired.
C If NROOT is zero, the root search is not active. This
C option is useful for obtaining output at points which are
C not known in advance, but depend upon the solution, e.g.,
C when some solution component takes on a specified value.
C The root search is carried out using the user-written
C function G (see description of G below.) SDRIV3 attempts
C to find the value of T at which one of the equations
C changes sign. SDRIV3 can find at most one root per
C equation per internal integration step, and will then
C return the solution either at TOUT or at a root, whichever
C occurs first in the direction of integration. The initial
C point is never reported as a root. The index of the
C equation whose root is being reported is stored in the
C sixth element of IWORK.
C NOTE: NROOT is never altered by this program.
C
C EPS = On input, the requested relative accuracy in all solution
C components. EPS = 0 is allowed. On output, the adjusted
C relative accuracy if the input value was too small. The
C value of EPS should be set as large as is reasonable,
C because the amount of work done by SDRIV3 increases as EPS
C decreases.
C
C EWT = (Input) Problem zero, i.e., the smallest, nonzero,
C physically meaningful value for the solution. (Array,
C possibly of length one. See following description of
C IERROR.) Setting EWT smaller than necessary can adversely
C affect the running time.
C
C IERROR = (Input) Error control indicator. A value of 3 is
C suggested for most problems. Other choices and detailed
C explanations of EWT and IERROR are given below for those
C who may need extra flexibility.
C
C These last three input quantities EPS, EWT and IERROR
C control the accuracy of the computed solution. EWT and
C IERROR are used internally to compute an array YWT. One
C step error estimates divided by YWT(I) are kept less than
C EPS in root mean square norm.
C IERROR (Set by the user) =
C 1 Means YWT(I) = 1. (Absolute error control)
C EWT is ignored.
C 2 Means YWT(I) = ABS(Y(I)), (Relative error control)
C EWT is ignored.
C 3 Means YWT(I) = MAX(ABS(Y(I)), EWT(1)).
C 4 Means YWT(I) = MAX(ABS(Y(I)), EWT(I)).
C This choice is useful when the solution components
C have differing scales.
C 5 Means YWT(I) = EWT(I).
C If IERROR is 3, EWT need only be dimensioned one.
C If IERROR is 4 or 5, the user must dimension EWT at least
C N, and set its values.
C
C MINT = (Input) The integration method indicator.
C MINT = 1 Means the Adams methods, and is used for
C non-stiff problems.
C MINT = 2 Means the stiff methods of Gear (i.e., the
C backward differentiation formulas), and is
C used for stiff problems.
C MINT = 3 Means the program dynamically selects the
C Adams methods when the problem is non-stiff
C and the Gear methods when the problem is
C stiff. When using the Adams methods, the
C program uses a value of MITER=0; when using
C the Gear methods, the program uses the value
C of MITER provided by the user. Only a value
C of IMPL = 0 and a value of MITER = 1, 2, 4, or
C 5 is allowed for this option. The user may
C not alter the value of MINT or MITER without
C restarting, i.e., setting NSTATE to 1.
C
C MITER = (Input) The iteration method indicator.
C MITER = 0 Means functional iteration. This value is
C suggested for non-stiff problems.
C MITER = 1 Means chord method with analytic Jacobian.
C In this case, the user supplies subroutine
C JACOBN (see description below).
C MITER = 2 Means chord method with Jacobian calculated
C internally by finite differences.
C MITER = 3 Means chord method with corrections computed
C by the user-written routine USERS (see
C description of USERS below.) This option
C allows all matrix algebra and storage
C decisions to be made by the user. When using
C a value of MITER = 3, the subroutine FA is
C not required, even if IMPL is not 0. For
C further information on using this option, see
C Section IV-E below.
C MITER = 4 Means the same as MITER = 1 but the A and
C Jacobian matrices are assumed to be banded.
C MITER = 5 Means the same as MITER = 2 but the A and
C Jacobian matrices are assumed to be banded.
C
C IMPL = (Input) The implicit method indicator.
C IMPL = 0 Means solving dY(I)/dT = F(Y(I),T).
C IMPL = 1 Means solving A*dY(I)/dT = F(Y(I),T), non-
C singular A (see description of FA below.)
C Only MINT = 1 or 2, and MITER = 1, 2, 3, 4,
C or 5 are allowed for this option.
C IMPL = 2,3 Means solving certain systems of hybrid
C differential/algebraic equations (see
C description of FA below.) Only MINT = 2 and
C MITER = 1, 2, 3, 4, or 5, are allowed for
C this option.
C The value of IMPL must not be changed during a problem.
C
C ML = (Input) The lower half-bandwidth in the case of a banded
C A or Jacobian matrix. (I.e., maximum(R-C) for nonzero
C A(R,C).)
C
C MU = (Input) The upper half-bandwidth in the case of a banded
C A or Jacobian matrix. (I.e., maximum(C-R).)
C
C MXORD = (Input) The maximum order desired. This is .LE. 12 for
C the Adams methods and .LE. 5 for the Gear methods. Normal
C value is 12 and 5, respectively. If MINT is 3, the
C maximum order used will be MIN(MXORD, 12) when using the
C Adams methods, and MIN(MXORD, 5) when using the Gear
C methods. MXORD must not be altered during a problem.
C
C HMAX = (Input) The maximum magnitude of the step size that will
C be used for the problem. This is useful for ensuring that
C important details are not missed. If this is not the
C case, a large value, such as the interval length, is
C suggested.
C
C WORK
C LENW = (Input)
C WORK is an array of LENW real words used
C internally for temporary storage. The user must allocate
C space for this array in the calling program by a statement
C such as
C REAL WORK(...)
C The following table gives the required minimum value for
C the length of WORK, depending on the value of IMPL and
C MITER. LENW should be set to the value used. The
C contents of WORK should not be disturbed between calls to
C SDRIV3.
C
C IMPL = 0 1 2 3
C ---------------------------------------------------------
C MITER = 0 (MXORD+4)*N Not allowed Not allowed Not allowed
C + 2*NROOT
C + 250
C
C 1,2 N*N + 2*N*N + N*N + N*(N + NDE)
C (MXORD+5)*N (MXORD+5)*N (MXORD+6)*N + (MXORD+5)*N
C + 2*NROOT + 2*NROOT + 2*NROOT + 2*NROOT
C + 250 + 250 + 250 + 250
C
C 3 (MXORD+4)*N (MXORD+4)*N (MXORD+4)*N (MXORD+4)*N
C + 2*NROOT + 2*NROOT + 2*NROOT + 2*NROOT
C + 250 + 250 + 250 + 250
C
C 4,5 (2*ML+MU+1) 2*(2*ML+MU+1) (2*ML+MU+1) (2*ML+MU+1)*
C *N + *N + *N + (N+NDE) +
C (MXORD+5)*N (MXORD+5)*N (MXORD+6)*N + (MXORD+5)*N
C + 2*NROOT + 2*NROOT + 2*NROOT + 2*NROOT
C + 250 + 250 + 250 + 250
C ---------------------------------------------------------
C
C IWORK
C LENIW = (Input)
C IWORK is an integer array of length LENIW used internally
C for temporary storage. The user must allocate space for
C this array in the calling program by a statement such as
C INTEGER IWORK(...)
C The length of IWORK should be at least
C 50 if MITER is 0 or 3, or
C N+50 if MITER is 1, 2, 4, or 5, or MINT is 3,
C and LENIW should be set to the value used. The contents
C of IWORK should not be disturbed between calls to SDRIV3.
C
C JACOBN = A subroutine supplied by the user, if MITER is 1 or 4.
C If this is the case, the name must be declared EXTERNAL in
C the user's calling program. Given a system of N
C differential equations, it is meaningful to speak about
C the partial derivative of the I-th right hand side with
C respect to the J-th dependent variable. In general there
C are N*N such quantities. Often however the equations can
C be ordered so that the I-th differential equation only
C involves dependent variables with index near I, e.g., I+1,
C I-2. Such a system is called banded. If, for all I, the
C I-th equation depends on at most the variables
C Y(I-ML), Y(I-ML+1), ... , Y(I), Y(I+1), ... , Y(I+MU)
C then we call ML+MU+1 the bandwidth of the system. In a
C banded system many of the partial derivatives above are
C automatically zero. For the cases MITER = 1, 2, 4, and 5,
C some of these partials are needed. For the cases
C MITER = 2 and 5 the necessary derivatives are
C approximated numerically by SDRIV3, and we only ask the
C user to tell SDRIV3 the value of ML and MU if the system
C is banded. For the cases MITER = 1 and 4 the user must
C derive these partials algebraically and encode them in
C subroutine JACOBN. By computing these derivatives the
C user can often save 20-30 per cent of the computing time.
C Usually, however, the accuracy is not much affected and
C most users will probably forego this option. The optional
C user-written subroutine JACOBN has the form:
C SUBROUTINE JACOBN (N, T, Y, DFDY, MATDIM, ML, MU)
C REAL Y(*), DFDY(MATDIM,*)
C .
C .
C Calculate values of DFDY
C .
C .
C END (Sample)
C Here Y is a vector of length at least N. The actual
C length of Y is determined by the user's declaration in the
C program which calls SDRIV3. Thus the dimensioning of Y in
C JACOBN, while required by FORTRAN convention, does not
C actually allocate any storage. When this subroutine is
C called, the first N components of Y are intermediate
C approximations to the solution components. The user
C should not alter these values. If the system is not
C banded (MITER=1), the partials of the I-th equation with
C respect to the J-th dependent function are to be stored in
C DFDY(I,J). Thus partials of the I-th equation are stored
C in the I-th row of DFDY. If the system is banded
C (MITER=4), then the partials of the I-th equation with
C respect to Y(J) are to be stored in DFDY(K,J), where
C K=I-J+MU+1 . Normally a return from JACOBN passes control
C back to SDRIV3. However, if the user would like to abort
C the calculation, i.e., return control to the program which
C calls SDRIV3, he should set N to zero. SDRIV3 will signal
C this by returning a value of NSTATE equal to +8(-8).
C Altering the value of N in JACOBN has no effect on the
C value of N in the call sequence of SDRIV3.
C
C FA = A subroutine supplied by the user if IMPL is not zero, and
C MITER is not 3. If so, the name must be declared EXTERNAL
C in the user's calling program. This subroutine computes
C the array A, where A*dY(I)/dT = F(Y(I),T).
C There are three cases:
C
C IMPL=1.
C Subroutine FA is of the form:
C SUBROUTINE FA (N, T, Y, A, MATDIM, ML, MU, NDE)
C REAL Y(*), A(MATDIM,*)
C .
C .
C Calculate ALL values of A
C .
C .
C END (Sample)
C In this case A is assumed to be a nonsingular matrix,
C with the same structure as DFDY (see JACOBN description
C above). Programming considerations prevent complete
C generality. If MITER is 1 or 2, A is assumed to be full
C and the user must compute and store all values of
C A(I,J), I,J=1, ... ,N. If MITER is 4 or 5, A is assumed
C to be banded with lower and upper half bandwidth ML and
C MU. The left hand side of the I-th equation is a linear
C combination of dY(I-ML)/dT, dY(I-ML+1)/dT, ... ,
C dY(I)/dT, ... , dY(I+MU-1)/dT, dY(I+MU)/dT. Thus in the
C I-th equation, the coefficient of dY(J)/dT is to be
C stored in A(K,J), where K=I-J+MU+1.
C NOTE: The array A will be altered between calls to FA.
C
C IMPL=2.
C Subroutine FA is of the form:
C SUBROUTINE FA (N, T, Y, A, MATDIM, ML, MU, NDE)
C REAL Y(*), A(*)
C .
C .
C Calculate non-zero values of A(1),...,A(NDE)
C .
C .
C END (Sample)
C In this case it is assumed that the system is ordered by
C the user so that the differential equations appear
C first, and the algebraic equations appear last. The
C algebraic equations must be written in the form:
C 0 = F(Y(I),T). When using this option it is up to the
C user to provide initial values for the Y(I) that satisfy
C the algebraic equations as well as possible. It is
C further assumed that A is a vector of length NDE. All
C of the components of A, which may depend on T, Y(I),
C etc., must be set by the user to non-zero values.
C
C IMPL=3.
C Subroutine FA is of the form:
C SUBROUTINE FA (N, T, Y, A, MATDIM, ML, MU, NDE)
C REAL Y(*), A(MATDIM,*)
C .
C .
C Calculate ALL values of A
C .
C .
C END (Sample)
C In this case A is assumed to be a nonsingular NDE by NDE
C matrix with the same structure as DFDY (see JACOBN
C description above). Programming considerations prevent
C complete generality. If MITER is 1 or 2, A is assumed
C to be full and the user must compute and store all
C values of A(I,J), I,J=1, ... ,NDE. If MITER is 4 or 5,
C A is assumed to be banded with lower and upper half
C bandwidths ML and MU. The left hand side of the I-th
C equation is a linear combination of dY(I-ML)/dT,
C dY(I-ML+1)/dT, ... , dY(I)/dT, ... , dY(I+MU-1)/dT,
C dY(I+MU)/dT. Thus in the I-th equation, the coefficient
C of dY(J)/dT is to be stored in A(K,J), where K=I-J+MU+1.
C It is assumed that the system is ordered by the user so
C that the differential equations appear first, and the
C algebraic equations appear last. The algebraic
C equations must be written in the form 0 = F(Y(I),T).
C When using this option it is up to the user to provide
C initial values for the Y(I) that satisfy the algebraic
C equations as well as possible.
C NOTE: For IMPL = 3, the array A will be altered between
C calls to FA.
C Here Y is a vector of length at least N. The actual
C length of Y is determined by the user's declaration in the
C program which calls SDRIV3. Thus the dimensioning of Y in
C FA, while required by FORTRAN convention, does not
C actually allocate any storage. When this subroutine is
C called, the first N components of Y are intermediate
C approximations to the solution components. The user
C should not alter these values. FA is always called
C immediately after calling F, with the same values of T
C and Y. Normally a return from FA passes control back to
C SDRIV3. However, if the user would like to abort the
C calculation, i.e., return control to the program which
C calls SDRIV3, he should set N to zero. SDRIV3 will signal
C this by returning a value of NSTATE equal to +9(-9).
C Altering the value of N in FA has no effect on the value
C of N in the call sequence of SDRIV3.
C
C NDE = (Input) The number of differential equations. This is
C required only for IMPL = 2 or 3, with NDE .LT. N.
C
C MXSTEP = (Input) The maximum number of internal steps allowed on
C one call to SDRIV3.
C
C G = A real FORTRAN function supplied by the user
C if NROOT is not 0. In this case, the name must be
C declared EXTERNAL in the user's calling program. G is
C repeatedly called with different values of IROOT to obtain
C the value of each of the NROOT equations for which a root
C is desired. G is of the form:
C REAL FUNCTION G (N, T, Y, IROOT)
C REAL Y(*)
C GO TO (10, ...), IROOT
C 10 G = ...
C .
C .
C END (Sample)
C Here, Y is a vector of length at least N, whose first N
C components are the solution components at the point T.
C The user should not alter these values. The actual length
C of Y is determined by the user's declaration in the
C program which calls SDRIV3. Thus the dimensioning of Y in
C G, while required by FORTRAN convention, does not actually
C allocate any storage. Normally a return from G passes
C control back to SDRIV3. However, if the user would like
C to abort the calculation, i.e., return control to the
C program which calls SDRIV3, he should set N to zero.
C SDRIV3 will signal this by returning a value of NSTATE
C equal to +7(-7). In this case, the index of the equation
C being evaluated is stored in the sixth element of IWORK.
C Altering the value of N in G has no effect on the value of
C N in the call sequence of SDRIV3.
C
C USERS = A subroutine supplied by the user, if MITER is 3.
C If this is the case, the name must be declared EXTERNAL in
C the user's calling program. The routine USERS is called
C by SDRIV3 when certain linear systems must be solved. The
C user may choose any method to form, store and solve these
C systems in order to obtain the solution result that is
C returned to SDRIV3. In particular, this allows sparse
C matrix methods to be used. The call sequence for this
C routine is:
C
C SUBROUTINE USERS (Y, YH, YWT, SAVE1, SAVE2, T, H, EL,
C 8 IMPL, N, NDE, IFLAG)
C REAL Y(*), YH(*), YWT(*), SAVE1(*),
C 8 SAVE2(*), T, H, EL
C
C The input variable IFLAG indicates what action is to be
C taken. Subroutine USERS should perform the following
C operations, depending on the value of IFLAG and IMPL.
C
C IFLAG = 0
C IMPL = 0. USERS is not called.
C IMPL = 1, 2 or 3. Solve the system A*X = SAVE2,
C returning the result in SAVE2. The array SAVE1 can
C be used as a work array. For IMPL = 1, there are N
C components to the system, and for IMPL = 2 or 3,
C there are NDE components to the system.
C
C IFLAG = 1
C IMPL = 0. Compute, decompose and store the matrix
C (I - H*EL*J), where I is the identity matrix and J
C is the Jacobian matrix of the right hand side. The
C array SAVE1 can be used as a work array.
C IMPL = 1, 2 or 3. Compute, decompose and store the
C matrix (A - H*EL*J). The array SAVE1 can be used as
C a work array.
C
C IFLAG = 2
C IMPL = 0. Solve the system
C (I - H*EL*J)*X = H*SAVE2 - YH - SAVE1,
C returning the result in SAVE2.
C IMPL = 1, 2 or 3. Solve the system
C (A - H*EL*J)*X = H*SAVE2 - A*(YH + SAVE1)
C returning the result in SAVE2.
C The array SAVE1 should not be altered.
C If IFLAG is 0 and IMPL is 1 or 2 and the matrix A is
C singular, or if IFLAG is 1 and one of the matrices
C (I - H*EL*J), (A - H*EL*J) is singular, the INTEGER
C variable IFLAG is to be set to -1 before RETURNing.
C Normally a return from USERS passes control back to
C SDRIV3. However, if the user would like to abort the
C calculation, i.e., return control to the program which
C calls SDRIV3, he should set N to zero. SDRIV3 will signal
C this by returning a value of NSTATE equal to +10(-10).
C Altering the value of N in USERS has no effect on the
C value of N in the call sequence of SDRIV3.
C
C IERFLG = An error flag. The error number associated with a
C diagnostic message (see Section III-A below) is the same
C as the corresponding value of IERFLG. The meaning of
C IERFLG:
C 0 The routine completed successfully. (No message is
C issued.)
C 3 (Warning) The number of steps required to reach TOUT
C exceeds MXSTEP.
C 4 (Warning) The value of EPS is too small.
C 11 (Warning) For NTASK = 2 or 3, T is beyond TOUT.
C The solution was obtained by interpolation.
C 15 (Warning) The integration step size is below the
C roundoff level of T. (The program issues this
C message as a warning but does not return control to
C the user.)
C 22 (Recoverable) N is not positive.
C 23 (Recoverable) MINT is less than 1 or greater than 3 .
C 24 (Recoverable) MITER is less than 0 or greater than
C 5 .
C 25 (Recoverable) IMPL is less than 0 or greater than 3 .
C 26 (Recoverable) The value of NSTATE is less than 1 or
C greater than 12 .
C 27 (Recoverable) EPS is less than zero.
C 28 (Recoverable) MXORD is not positive.
C 29 (Recoverable) For MINT = 3, either MITER = 0 or 3, or
C IMPL = 0 .
C 30 (Recoverable) For MITER = 0, IMPL is not 0 .
C 31 (Recoverable) For MINT = 1, IMPL is 2 or 3 .
C 32 (Recoverable) Insufficient storage has been allocated
C for the WORK array.
C 33 (Recoverable) Insufficient storage has been allocated
C for the IWORK array.
C 41 (Recoverable) The integration step size has gone
C to zero.
C 42 (Recoverable) The integration step size has been
C reduced about 50 times without advancing the
C solution. The problem setup may not be correct.
C 43 (Recoverable) For IMPL greater than 0, the matrix A
C is singular.
C 999 (Fatal) The value of NSTATE is 12 .
C
C III. OTHER COMMUNICATION TO THE USER ..............................
C
C A. The solver communicates to the user through the parameters
C above. In addition it writes diagnostic messages through the
C standard error handling program XERMSG. A complete description
C of XERMSG is given in "Guide to the SLATEC Common Mathematical
C Library" by Kirby W. Fong et al.. At installations which do not
C have this error handling package the short but serviceable
C routine, XERMSG, available with this package, can be used. That
C program uses the file named OUTPUT to transmit messages.
C
C B. The first three elements of WORK and the first five elements of
C IWORK will contain the following statistical data:
C AVGH The average step size used.
C HUSED The step size last used (successfully).
C AVGORD The average order used.
C IMXERR The index of the element of the solution vector that
C contributed most to the last error test.
C NQUSED The order last used (successfully).
C NSTEP The number of steps taken since last initialization.
C NFE The number of evaluations of the right hand side.
C NJE The number of evaluations of the Jacobian matrix.
C
C IV. REMARKS .......................................................
C
C A. Other routines used:
C SDNTP, SDZRO, SDSTP, SDNTL, SDPST, SDCOR, SDCST,
C SDPSC, and SDSCL;
C SGEFA, SGESL, SGBFA, SGBSL, and SNRM2 (from LINPACK)
C R1MACH (from the Bell Laboratories Machine Constants Package)
C XERMSG (from the SLATEC Common Math Library)
C The last seven routines above, not having been written by the
C present authors, are not explicitly part of this package.
C
C B. On any return from SDRIV3 all information necessary to continue
C the calculation is contained in the call sequence parameters,
C including the work arrays. Thus it is possible to suspend one
C problem, integrate another, and then return to the first.
C
C C. If this package is to be used in an overlay situation, the user
C must declare in the primary overlay the variables in the call
C sequence to SDRIV3.
C
C D. Changing parameters during an integration.
C The value of NROOT, EPS, EWT, IERROR, MINT, MITER, or HMAX may
C be altered by the user between calls to SDRIV3. For example, if
C too much accuracy has been requested (the program returns with
C NSTATE = 4 and an increased value of EPS) the user may wish to
C increase EPS further. In general, prudence is necessary when
C making changes in parameters since such changes are not
C implemented until the next integration step, which is not
C necessarily the next call to SDRIV3. This can happen if the
C program has already integrated to a point which is beyond the
C new point TOUT.
C
C E. As the price for complete control of matrix algebra, the SDRIV3
C USERS option puts all responsibility for Jacobian matrix
C evaluation on the user. It is often useful to approximate
C numerically all or part of the Jacobian matrix. However this
C must be done carefully. The FORTRAN sequence below illustrates
C the method we recommend. It can be inserted directly into
C subroutine USERS to approximate Jacobian elements in rows I1
C to I2 and columns J1 to J2.
C REAL DFDY(N,N), EPSJ, H, R, R1MACH,
C 8 SAVE1(N), SAVE2(N), T, UROUND, Y(N), YJ, YWT(N)
C UROUND = R1MACH(4)
C EPSJ = SQRT(UROUND)
C DO 30 J = J1,J2
C R = EPSJ*MAX(ABS(YWT(J)), ABS(Y(J)))
C IF (R .EQ. 0.E0) R = YWT(J)
C YJ = Y(J)
C Y(J) = Y(J) + R
C CALL F (N, T, Y, SAVE1)
C IF (N .EQ. 0) RETURN
C Y(J) = YJ
C DO 20 I = I1,I2
C 20 DFDY(I,J) = (SAVE1(I) - SAVE2(I))/R
C 30 CONTINUE
C Many problems give rise to structured sparse Jacobians, e.g.,
C block banded. It is possible to approximate them with fewer
C function evaluations than the above procedure uses; see Curtis,
C Powell and Reid, J. Inst. Maths Applics, (1974), Vol. 13,
C pp. 117-119.
C
C F. When any of the routines JACOBN, FA, G, or USERS, is not
C required, difficulties associated with unsatisfied externals can
C be avoided by using the name of the routine which calculates the
C right hand side of the differential equations in place of the
C corresponding name in the call sequence of SDRIV3.
C
C***REFERENCES C. W. Gear, Numerical Initial Value Problems in
C Ordinary Differential Equations, Prentice-Hall, 1971.
C***ROUTINES CALLED R1MACH, SDNTP, SDSTP, SDZRO, SGBFA, SGBSL, SGEFA,
C SGESL, SNRM2, XERMSG
C***REVISION HISTORY (YYMMDD)
C 790601 DATE WRITTEN
C 900329 Initial submission to SLATEC.
C***END PROLOGUE SDRIV3