!******************************************************************************* !> ! Module for the refactored DVODE. ! !### See also ! * The original DVODE.f (1989-2002) can be found ! [here](https://computing.llnl.gov/projects/odepack/software) ! !### History ! * Translated into modern Fortran by Jacob Williams. ! !@todo Replace GOTO statements module dvode_module #ifndef HAS_BLAS use dvode_blas_module #endif use dvode_linpack_module use dvode_kinds_module, only: dvode_wp use iso_fortran_env, only: output_unit implicit none private integer,parameter :: wp = dvode_wp !! real kind to use public :: dvode_wp #ifdef HAS_BLAS ! user is linking against an external BLAS library external :: daxpy,dcopy,dscal #endif real(wp),parameter :: zero = 0.0_wp real(wp),parameter :: epmach = epsilon(1.0_wp) !! machine epsilon integer,parameter :: iumach = output_unit !! standard output unit number integer,dimension(2),parameter :: mord = [12,5] type,public :: dvode_data_t !! Internal variables for DVODE. !! variables which are communicated between subroutines in the !! dvode package, or which are to be saved between calls to dvode. private ! type declarations formerly in common block dvod01 real(wp) :: acnrm = zero !! weighted r.m.s. norm of accumulated correction vectors. real(wp) :: ccmxj = zero !! threshhold on drc for updating the jacobian. (see drc.) real(wp) :: conp = zero !! the saved value of tq(5). real(wp) :: crate = zero !! estimated corrector convergence rate constant. real(wp) :: drc = zero !! relative change in h*rl1 since last dvjac call. real(wp) :: el(13) = zero !! real array of integration coefficients. see dvset. real(wp) :: eta = zero !! saved tentative ratio of new to old h. real(wp) :: etamax = zero !! saved maximum value of eta to be allowed. real(wp) :: h = zero !! the step size. real(wp) :: hmin = zero !! the minimum absolute value of the step size h to be used. real(wp) :: hmxi = zero !! inverse of the maximum absolute value of h to be used. !! hmxi = 0.0 is allowed and corresponds to an infinite hmax. real(wp) :: hnew = zero !! the step size to be attempted on the next step. real(wp) :: hscal = zero !! step size `h` used in scaling of nordsieck array `yh` in [[dvjust]]. !! (if `iord = +1`, [[dvjust]] assumes that `hscal = tau(1)`.) !! see references 1 and 2 for details. real(wp) :: prl1 = zero !! the saved value of rl1. real(wp) :: rc = zero !! ratio of current h*rl1 to value on last dvjac call. real(wp) :: rl1 = zero !! the reciprocal of the coefficient el(1). real(wp) :: tau(13) = zero !! real vector of past nq step sizes, length 13. real(wp) :: tq(5) = zero !! a real vector of length 5 in which dvset stores constants !! used for the convergence test, the error test, and the !! selection of h at a new order. real(wp) :: tn = zero !! the independent variable, updated on each step taken. real(wp) :: uround = zero !! the machine unit roundoff. the smallest positive real number !! such that 1.0 + uround /= 1.0 integer :: icf = 0 !! integer flag for convergence failure in dvnlsd: !! !! * 0 means no failures. !! * 1 means convergence failure with out of date jacobian !! (recoverable error). !! * 2 means convergence failure with current jacobian or !! singular matrix (unrecoverable error). integer :: init = 0 !! saved integer flag indicating whether initialization of the !! problem has been done (init = 1) or not. integer :: ipup = 0 !! saved flag to signal updating of newton matrix. !! !! In [[dvnlsd]]: own variable flag with values and meanings as follows: !! !! * 0, do not update the newton matrix. !! * `miter /= 0`, update newton matrix, because it is the !! initial step, order was changed, the error !! test failed, or an update is indicated by !! the scalar `rc` or step counter `nst`. integer :: jcur = 0 !! output flag from [[dvjac]] showing jacobian status: !! !! * jcur = 0 means j is not current. !! * jcur = 1 means j is current. integer :: jstart = 0 !! integer flag used as input to dvstep: !! !! * 0 means perform the first step. !! * 1 means take a new step continuing from the last. !! * -1 means take the next step with a new value of maxord, !! hmin, hmxi, n, meth, miter, and/or matrix parameters. !! !! on return, dvstep sets jstart = 1. integer :: jsv = 0 !! integer flag for jacobian saving, = sign(mf). integer :: kflag = 0 !! a completion code from dvstep with the following meanings: !! !! * 0 the step was succesful. !! * -1 the requested error could not be achieved. !! * -2 corrector convergence could not be achieved. !! * -3, -4 fatal error in vnls (can not occur here). integer :: kuth = 0 !! input flag to dvstep showing whether h was reduced by the !! driver. kuth = 1 if h was reduced, = 0 otherwise. integer :: l = 0 !! integer variable, nq + 1, current order plus one. integer :: lmax = 0 !! maxord + 1 (used for dimensioning). integer :: lyh = 0 !! saved integer pointers to segments of rwork and iwork. integer :: lewt = 0 !! saved integer pointers to segments of rwork and iwork. integer :: lacor = 0 !! saved integer pointers to segments of rwork and iwork. integer :: lsavf = 0 !! saved integer pointers to segments of rwork and iwork. integer :: lwm = 0 !! saved integer pointers to segments of rwork and iwork. integer :: liwm = 0 !! saved integer pointers to segments of rwork and iwork. integer :: locjs = 0 !! a pointer to the saved jacobian, whose storage starts at !! wm(locjs), if jsv = 1. integer :: maxord = 0 !! the maximum order of integration method to be allowed. integer :: meth = 0 !! the method flags. see mf. integer :: miter = 0 !! the method flags. see mf. integer :: msbj = 0 !! the maximum number of steps between j evaluations, = 50. integer :: mxhnil = 0 !! saved value of optional input mxhnil. integer :: mxstep = 0 !! saved value of optional input mxstep. integer :: n = 0 !! the number of first-order odes, = neq. integer :: newh = 0 !! saved integer to flag change of h. integer :: newq = 0 !! the method order to be used on the next step. integer :: nhnil = 0 !! saved counter for occurrences of t + h = t. integer :: nq = 0 !! integer variable, the current integration method order. integer :: nqnyh = 0 !! saved value of nq*nyh. integer :: nqwait = 0 !! a counter controlling the frequency of order changes. !! an order change is about to be considered if nqwait = 1. integer :: nslj = 0 !! the number of steps taken as of the last jacobian update. integer :: nslp = 0 !! saved value of nst as of last newton matrix update. integer :: nyh = 0 !! saved value of the initial value of neq. ! type declarations formerly in common block dvod02 real(wp) :: hu = zero !! the step size in t last used. integer :: ncfn = 0 !! number of nonlinear convergence failures so far. integer :: netf = 0 !! the number of error test failures of the integrator so far. integer :: nfe = 0 !! the number of f evaluations for the problem so far. integer :: nje = 0 !! the number of jacobian evaluations so far. integer :: nlu = 0 !! the number of matrix lu decompositions so far. integer :: nni = 0 !! number of nonlinear iterations so far. integer :: nqu = 0 !! the method order last used. integer :: nst = 0 !! the number of steps taken for the problem so far. end type dvode_data_t !***************************************************************************************** !> ! Main DVODE class. ! !### Summary of usage. ! ! communication between the user and the dvode package, for normal ! situations, is summarized here. this summary describes only a subset ! of the full set of options available. see the full description under ! [[dvode]] for details, including optional communication, nonstandard ! options, and instructions for special situations. ! ! 1. first provide a subroutine `f` of the form [[f_func]]: ! which supplies the vector function f by loading ydot(i) with f(i). ! ! 2. next determine (or guess) whether or not the problem is stiff. ! stiffness occurs when the jacobian matrix df/dy has an eigenvalue ! whose real part is negative and large in magnitude, compared to the ! reciprocal of the t span of interest. if the problem is nonstiff, ! use a method flag mf = 10. if it is stiff, there are four standard ! choices for mf (21, 22, 24, 25), and dvode requires the jacobian ! matrix in some form. in these cases (mf > 0), dvode will use a ! saved copy of the jacobian matrix. if this is undesirable because of ! storage limitations, set mf to the corresponding negative value ! (-21, -22, -24, -25). (see full description of mf below.) ! the jacobian matrix is regarded either as full (mf = 21 or 22), ! or banded (mf = 24 or 25). in the banded case, dvode requires two ! half-bandwidth parameters ml and mu. these are, respectively, the ! widths of the lower and upper parts of the band, excluding the main ! diagonal. thus the band consists of the locations (i,j) with ! i-ml <= j <= i+mu, and the full bandwidth is ml+mu+1. ! ! 3. if the problem is stiff, you are encouraged to supply the jacobian ! directly (mf = 21 or 24), but if this is not feasible, dvode will ! compute it internally by difference quotients (mf = 22 or 25). ! if you are supplying the jacobian, provide a subroutine of the form [[f_jac]] ! which supplies df/dy by loading pd as follows (in either case, only nonzero elements need be loaded): ! ! * for a full jacobian (mf = 21), load pd(i,j) with df(i)/dy(j), ! the partial derivative of f(i) with respect to y(j). (ignore the ! ml and mu arguments in this case.) ! * for a banded jacobian (mf = 24), load pd(i-j+mu+1,j) with ! df(i)/dy(j), i.e. load the diagonal lines of df/dy into the rows of ! pd from the top down. ! ! 4. write a main program which calls subroutine dvode once for ! each point at which answers are desired. this should also provide ! for possible use of logical unit 6 for output of error messages ! by dvode. on the first call to dvode, supply arguments as follows: ! ! * f = name of subroutine for right-hand side vector f. ! * jac = name of subroutine for jacobian matrix (mf = 21 or 24). ! * neq = number of first order odes. ! * y = array of initial values, of length neq. ! * t = the initial value of the independent variable. ! * tout = first point where output is desired (/= t). ! * itol = 1 or 2 according as atol (below) is a scalar or array. ! * rtol = relative tolerance parameter. ! * atol = absolute tolerance parameter. ! * itask = 1 for normal computation of output values of y at t = tout. ! * istate = integer flag (input and output). set istate = 1. ! * iopt = 0 to indicate no optional input used. ! * rwork = real work array ! * lrw = declared length of rwork. ! * iwork = integer work array. ! * liw = declared length of iwork (in user's dimension statement). ! * mf = method flag. ! ! 5. the output from the first call (or any call) is: ! * y = array of computed values of y(t) vector. ! * t = corresponding value of independent variable (normally tout). ! * istate = 2 if dvode was successful, negative otherwise. ! * -1 means excess work done on this call. (perhaps wrong mf.) ! * -2 means excess accuracy requested. (tolerances too small.) ! * -3 means illegal input detected. (see printed message.) ! * -4 means repeated error test failures. (check all input.) ! * -5 means repeated convergence failures. (perhaps bad ! jacobian supplied or wrong choice of mf or tolerances.) ! * -6 means error weight became zero during problem. (solution ! component i vanished, and atol or atol(i) = 0.) ! ! 6. to continue the integration after a successful return, simply ! reset tout and call [[dvode]] again. no other parameters need be reset. ! !### Other routines callable ! ! the following are optional calls which the user may make to ! gain additional capabilities in conjunction with dvode. ! (the routines [[xsetun]] and [[xsetf]] are designed to conform to the ! slatec error handling package.) ! ! * [[xsetun]] ! * [[xsetf]] ! * [[dvsrco]] ! * [[dvindy]] type,public :: dvode_t private type(dvode_data_t) :: dat !! internal data (formerly in common blocks) ! formerly save variables real(wp) :: etaq = zero real(wp) :: etaqm1 = zero integer :: lunit = -1 !! logical unit number for messages. the default is obtained !! by a call to iumach (may be machine-dependent). integer :: mesflg = 1 !! print control flag: !! !! * 1 means print all messages (the default). !! * 0 means no printing. procedure(f_func),pointer :: f => null() procedure(f_jac),pointer :: jac => null() procedure(f_dewset),pointer :: dewset => null() procedure(f_dvnorm),pointer :: dvnorm => null() contains private procedure,public :: initialize !! this routine must be called first before using the solver. procedure,public :: solve => dvode !! main solver routine. procedure,public :: xsetun !! set the logical unit number, `lun`, for !! output of messages from [[dvode]], if !! the default is not desired. !! the default value of `lun` is `output_unit`. !! this call may be made at any time and !! will take effect immediately. procedure,public :: xsetf !! set a flag to control the printing of !! messages by [[dvode]]: !! !! * `mflag = 0` means do not print. (danger: !! this risks losing valuable information.) !! * `mflag = 1` means print (the default). !! !! this call may be made at any time and !! will take effect immediately. procedure,public :: dvsrco !! saves and restores the contents of !! the internal variables used by [[dvode]]. !! [[dvsrco]] is useful if one is !! interrupting a run and restarting !! later, or alternating between two or !! more problems solved with [[dvode]]. procedure,public :: dvindy !! provide derivatives of y, of various !! orders, at a specified point `t`, if !! desired. it may be called only after !! a successful return from [[dvode]]. procedure :: dvhin procedure :: dvstep procedure :: dvset procedure :: dvjust procedure :: dvnlsd procedure :: dvjac procedure :: dvsol procedure :: ixsav procedure :: xerrwd end type dvode_t abstract interface subroutine f_func (me, neq, t, y, ydot) import :: dvode_t, wp implicit none class(dvode_t),intent(inout) :: me integer :: neq real(wp) :: t real(wp) :: y(neq) real(wp) :: ydot(neq) end subroutine f_func subroutine f_jac (me, neq, t, y, ml, mu, pd, nrowpd) import :: dvode_t, wp implicit none class(dvode_t),intent(inout) :: me integer :: neq integer :: nrowpd real(wp) :: t, y(neq), pd(nrowpd,neq) integer :: ml, mu end subroutine f_jac subroutine f_dewset(me,n,itol,rtol,atol,ycur,ewt) import :: dvode_t, wp implicit none class(dvode_t),intent(inout) :: me integer,intent(in) :: n integer,intent(in) :: itol real(wp),intent(in) :: rtol(*) real(wp),intent(in) :: atol(*) real(wp),intent(in) :: ycur(n) real(wp),intent(out) :: ewt(n) end subroutine f_dewset real(wp) function f_dvnorm(me,n,v,w) import :: dvode_t, wp implicit none class(dvode_t),intent(inout) :: me integer,intent(in) :: n real(wp),intent(in) :: v(n) real(wp),intent(in) :: w(n) end function f_dvnorm end interface public :: dvode contains !***************************************************************************************** !> ! Set the function pointers. This must be called before [[dvode]] is called. ! !### optionally replaceable solver routines ! ! The two routines (`dewset` and `dvnorm`) relate to the measurement of errors. ! either routine can be replaced by a user-supplied version, if desired. ! however, since such a replacement may have a major impact on performance, ! it should be done only when absolutely necessary, and only with great caution. ! (note: the means by which the package version of a routine is ! superseded by the user's version may be system-dependent.) ! If not changed by the users, the defaults are used: ! ! * [[dewset_default]] ! * [[dvnorm_default]] subroutine initialize(me,f,jac,dewset,dvnorm) implicit none class(dvode_t),intent(inout) :: me procedure(f_func) :: f !! the name of the user-supplied subroutine defining the !! ode system. the system must be put in the first-order !! form dy/dt = f(t,y), where f is a vector-valued function !! of the scalar t and the vector y. subroutine f is to !! compute the function f. it is to have the form !! subroutine f (neq, t, y, ydot) !! real(wp) t, y(neq), ydot(neq) !! where neq, t, and y are input, and the array ydot = f(t,y) !! is output. y and ydot are arrays of length neq. !! subroutine f should not alter y(1),...,y(neq). !! f must be declared external in the calling program. !! !! if quantities computed in the f routine are needed !! externally to dvode, an extra call to f should be made !! for this purpose, for consistent and accurate results. !! if only the derivative dy/dt is needed, use [[dvindy]] instead. procedure(f_jac),optional :: jac !! the name of the user-supplied routine (miter = 1 or 4) to !! compute the jacobian matrix, df/dy, as a function of !! the scalar t and the vector y. it is to have the form [[f_jac]], !! where neq, t, y, ml, mu, and nrowpd are input and the array !! pd is to be loaded with partial derivatives (elements of the !! jacobian matrix) in the output. pd must be given a first !! dimension of nrowpd. t and y have the same meaning as in !! subroutine f. !! !! * in the full matrix case (miter = 1), ml and mu are !! ignored, and the jacobian is to be loaded into pd in !! columnwise manner, with df(i)/dy(j) loaded into pd(i,j). !! !! * in the band matrix case (miter = 4), the elements !! within the band are to be loaded into pd in columnwise !! manner, with diagonal lines of df/dy loaded into the rows !! of pd. thus df(i)/dy(j) is to be loaded into pd(i-j+mu+1,j). !! ml and mu are the half-bandwidth parameters. (see iwork). !! the locations in pd in the two triangular areas which !! correspond to nonexistent matrix elements can be ignored !! or loaded arbitrarily, as they are overwritten by dvode. !! !! jac need not provide df/dy exactly. a crude !! approximation (possibly with a smaller bandwidth) will do. !! !! in either case, pd is preset to zero by the solver, !! so that only the nonzero elements need be loaded by jac. !! each call to jac is preceded by a call to f with the same !! arguments neq, t, and y. thus to gain some efficiency, !! intermediate quantities shared by both calculations may be !! saved in a user common block by f and not recomputed by jac, !! if desired. also, jac may alter the y array, if desired. !! jac must be declared external in the calling program. procedure(f_dewset),optional :: dewset !! the following subroutine is called just before each internal !! integration step, and sets the array of error weights, ewt, as !! described under itol/rtol/atol above: !! `subroutine dewset (neq, itol, rtol, atol, ycur, ewt)` !! where neq, itol, rtol, and atol are as in the [[dvode]] call sequence, !! ycur contains the current dependent variable vector, and !! ewt is the array of weights set by dewset. !! !! if the user supplies this subroutine, it must return in ewt(i) !! (i = 1,...,neq) a positive quantity suitable for comparison with !! errors in y(i). the ewt array returned by dewset is passed to the !! dvnorm routine (see below.), and also used by dvode in the computation !! of the optional output imxer, the diagonal jacobian approximation, !! and the increments for difference quotient jacobians. !! !! in the user-supplied version of dewset, it may be desirable to use !! the current values of derivatives of y. derivatives up to order nq !! are available from the history array yh, described above under !! optional output. in dewset, yh is identical to the ycur array, !! extended to nq + 1 columns with a column length of nyh and scale !! factors of h**j/factorial(j). on the first call for the problem, !! given by nst = 0, nq is 1 and h is temporarily set to 1.0. !! nyh is the initial value of neq. the quantities nq, h, and nst !! can be obtained by including in dewset the statements: !!```fortran !! type(dvode_data_t) :: sav !! call me%dvsrco(sav,job=1) !! nq = sav%nq !! h = sav%h !! nst = sav%nst !!``` !! thus, for example, the current value of dy/dt can be obtained as !! `ycur(nyh+i)/h (i=1,...,neq)` (and the division by h is !! unnecessary when nst = 0). procedure(f_dvnorm),optional :: dvnorm !! the following is a real function routine which computes the weighted !! root-mean-square norm of a vector v: !! `d = dvnorm (n, v, w)` !! where: !! !! * n = the length of the vector, !! * v = real array of length n containing the vector, !! * w = real array of length n containing weights, !! * d = `sqrt( (1/n) * sum(v(i)*w(i))**2 )`. !! !! dvnorm is called with n = neq and with w(i) = 1.0/ewt(i), where !! ewt is as set by subroutine dewset. !! !! if the user supplies this function, it should return a non-negative !! value of dvnorm suitable for use in the error control in dvode. !! none of the arguments should be altered by dvnorm. !! for example, a user-supplied dvnorm routine might: !! !! * substitute a max-norm of (v(i)*w(i)) for the rms-norm, or !! * ignore some components of v in the norm, with the effect of !! suppressing the error control on those components of y. me%f => f if (present(jac)) then me%jac => jac else me%jac => null() end if if (present(dewset)) then me%dewset => dewset else me%dewset => dewset_default end if if (present(dvnorm)) then me%dvnorm => dvnorm else me%dvnorm => dvnorm_default end if end subroutine initialize !***************************************************************************************** !***************************************************************************************** !> ! dvode: variable-coefficient ordinary differential equation solver, ! with fixed-leading-coefficient implementation. ! ! dvode solves the initial value problem for stiff or nonstiff ! systems of first order odes: ! ! $$ \frac{d}{dt} \mathbf{y} = \mathbf{f}(t,\mathbf{y}) $$ ! ! or, in component form: ! ! $$ \frac{d}{dt} \left[ \begin{array}{c} y_1 \\ y_2 \\ \vdots \\ y_{n_{eq}} \end{array} \right] = \left[ \begin{array}{c} f_1 \\ f_2 \\ \vdots \\ f_{n_{eq}} \end{array} \right] = \left[ \begin{array}{c} f(t,y_1) \\ f(t,y_2) \\ \vdots \\ f(t,y_{n_{eq}}) \end{array} \right] $$ ! ! dvode is a package based on the `episode` and `episodeb` packages, and ! on the `odepack` user interface standard, with minor modifications. ! !### Interrupting and restarting ! ! if the integration of a given problem by dvode is to be ! interrrupted and then later continued, such as when restarting ! an interrupted run or alternating between two or more ode problems, ! the user should save, following the return from the last dvode call ! prior to the interruption, the contents of the call sequence ! variables and internal variables, and later restore these ! values before the next dvode call for that problem. to save ! and restore the variables, use subroutine [[dvsrco]]. ! ! in addition, if non-default values for either `lun` or `mflag` are ! desired, an extra call to [[xsetun]] and/or [[xsetf]] should be made just ! before continuing the integration. ! !### Authors: ! * peter n. brown and alan c. hindmarsh, ! center for applied scientific computing, l-561, ! lawrence livermore national laboratory, ! livermore, ca 94551 ! * george d. byrne, ! illinois institute of technology, ! chicago, il 60616 ! !### References ! ! 1. p. n. brown, g. d. byrne, and a. c. hindmarsh, "vode: a variable ! coefficient ode solver," siam j. sci. stat. comput., 10 (1989), ! pp. 1038-1051. also, llnl report ucrl-98412, june 1988. ! 2. g. d. byrne and a. c. hindmarsh, "a polyalgorithm for the ! numerical solution of ordinary differential equations," ! acm trans. math. software, 1 (1975), pp. 71-96. ! 3. a. c. hindmarsh and g. d. byrne, "episode: an effective package ! for the integration of systems of ordinary differential ! equations," llnl report ucid-30112, rev. 1, april 1977. ! 4. g. d. byrne and a. c. hindmarsh, "episodeb: an experimental ! package for the integration of systems of ordinary differential ! equations with banded jacobians," llnl report ucid-30132, april ! 1976. ! 5. a. c. hindmarsh, "odepack, a systematized collection of ode ! solvers," in scientific computing, r. s. stepleman et al., eds., ! north-holland, amsterdam, 1983, pp. 55-64. ! 6. k. r. jackson and r. sacks-davis, "an alternative implementation ! of variable step-size multistep formulas for stiff odes," acm ! trans. math. software, 6 (1980), pp. 295-318. ! !### Revision history ! * 19890615 date written. initial release. ! * 19890922 added interrupt/restart ability, minor changes throughout. ! * 19910228 minor revisions in line format, prologue, etc. ! * 19920227 modifications by d. pang: ! (1) applied subgennam to get generic intrinsic names. ! (2) changed intrinsic names to generic in comments. ! (3) added *deck lines before each routine. ! * 19920721 names of routines and labeled common blocks changed, so as ! to be unique in combined single/real(wp) code (ach). ! * 19920722 minor revisions to prologue (ach). ! * 19920831 conversion to real(wp) done (ach). ! * 19921106 fixed minor bug: etaq,etaqm1 in dvstep save statement (ach). ! * 19921118 changed lunsav/mflgsv to ixsav (ach). ! * 19941222 removed mf overwrite; attached sign to h in estimated second ! deriv. in dvhin; misc. comment changes throughout (ach). ! * 19970515 minor corrections to comments in prologue, dvjac (ach). ! * 19981111 corrected block b by adding final line, go to 200 (ach). ! * 20020430 various upgrades (ach): use odepack error handler package. ! replaced d1mach by dumach. various changes to main ! prologue and other routine prologues. ! !@note the legality of input parameters will be thoroughly checked on the ! initial call for the problem, but not checked thereafter unless a ! change in input parameters is flagged by `istate = 3` in the input. ! !@note the work arrays must not be altered between calls to [[dvode]] ! for the same problem, except possibly for the conditional and ! optional input, and except for the last `3*neq` words of `rwork`. ! the latter space is used for internal scratch space, and so is ! available for use by the user outside [[dvode]] between calls, if ! desired (but not for use by `f` or `jac`). subroutine dvode(me,neq,y,t,tout,itol,rtol,atol,itask,istate,iopt, & rwork,lrw,iwork,liw,mf) class(dvode_t),intent(inout) :: me real(wp),intent(inout) :: y(*) !! a real array for the vector of dependent variables, of !! length `neq` or more. used for both input and output on the !! first call (`istate = 1`), and only for output on other calls. !! on the first call, `y` must contain the vector of initial !! values. in the output, `y` contains the computed solution !! evaluated at `t`. if desired, the `y` array may be used !! for other purposes between calls to the solver. !! !! this array is passed as the `y` argument in all calls to !! `f` and `jac`. real(wp),intent(inout) :: t !! the independent variable. in the input, `t` is used only on !! the first call, as the initial point of the integration. !! in the output, after each call, `t` is the value at which a !! computed solution `y` is evaluated (usually the same as `tout`). !! on an error return, `t` is the farthest point reached. real(wp),intent(in) :: tout !! the next value of `t` at which a computed solution is desired. !! !! when starting the problem (`istate = 1`), `tout` may be equal !! to `t` for one call, then should `/= t` for the next call. !! for the initial `t`, an input value of `tout /= t` is used !! in order to determine the direction of the integration !! (i.e. the algebraic sign of the step sizes) and the rough !! scale of the problem. integration in either direction !! (forward or backward in `t`) is permitted. !! !! if `itask = 2 or 5` (one-step modes), `tout` is ignored after !! the first call (i.e. the first call with `tout /= t`). !! otherwise, `tout` is required on every call. !! !! if `itask = 1, 3, or 4`, the values of `tout` need not be !! monotone, but a value of `tout` which backs up is limited !! to the current internal `t` interval, whose endpoints are !! `tcur - hu` and `tcur`. (see optional output, below, for !! `tcur` and `hu`.) real(wp),intent(in) :: rtol(*) !! a relative error tolerance parameter, either a scalar or !! an array of length `neq`. see description under `atol`. real(wp),intent(in) :: atol(*) !! an absolute error tolerance parameter, either a scalar or !! an array of length `neq`. !! !! the input parameters `itol`, `rtol`, and `atol` determine !! the error control performed by the solver. the solver will !! control the vector `e = (e(i))` of estimated local errors !! in `y`, according to an inequality of the form !!``` !! rms-norm of ( e(i)/ewt(i) ) <= 1, !! where ewt(i) = rtol(i)*abs(y(i)) + atol(i), !!``` !! and the rms-norm (root-mean-square norm) here is !! `rms-norm(v) = sqrt(sum v(i)**2 / neq)`. here `ewt = (ewt(i))` !! is a vector of weights which must always be positive, and !! the values of `rtol` and `atol` should all be non-negative. !! the following table gives the types (scalar/array) of !! `rtol` and `atol`, and the corresponding form of `ewt(i)`. !! !!``` !! itol rtol atol ewt(i) !! 1 scalar scalar rtol*abs(y(i)) + atol !! 2 scalar array rtol*abs(y(i)) + atol(i) !! 3 array scalar rtol(i)*abs(y(i)) + atol !! 4 array array rtol(i)*abs(y(i)) + atol(i) !!``` !! !! when either of these parameters is a scalar, it need not !! be dimensioned in the user's calling program. !! !! if none of the above choices (with `itol`, `rtol`, and `atol` !! fixed throughout the problem) is suitable, more general !! error controls can be obtained by substituting !! user-supplied routines for the setting of `ewt` and/or for !! the norm calculation. !! !! if global errors are to be estimated by making a repeated !! run on the same problem with smaller tolerances, then all !! components of `rtol` and `atol` (i.e. of `ewt`) should be scaled !! down uniformly. !! !! use `rtol = 0.0` for pure absolute error control, and !! use `atol = 0.0` (or `atol(i) = 0.0`) for pure relative error !! control. caution: actual (global) errors may exceed these !! local tolerances, so choose them conservatively. integer,intent(in) :: lrw !! the length of the array rwork, as declared by the user. !! (this will be checked by the solver.) real(wp) :: rwork(lrw) !! a real working array. !! the length of `rwork` must be at least !! `20 + nyh*(maxord + 1) + 3*neq + lwm` where: !! !! * `nyh` = the initial value of neq, !! * `maxord` = 12 (if `meth` = 1) or 5 (if `meth` = 2) (unless a !! smaller value is given as an optional input), !! * `lwm` = length of work space for matrix-related data: !! !! * `lwm = 0 ` if `miter` = 0, !! * `lwm = 2*neq**2 + 2 ` if `miter` = 1 or 2, and `mf`>0, !! * `lwm = neq**2 + 2 ` if `miter` = 1 or 2, and `mf`<0, !! * `lwm = neq + 2 ` if `miter` = 3, !! * `lwm = (3*ml+2*mu+2)*neq + 2` if `miter` = 4 or 5, and `mf`>0, !! * `lwm = (2*ml+mu+1)*neq + 2 ` if `miter` = 4 or 5, and `mf`<0. !! !! (see the `mf` description for `meth` and `miter`.) !! thus if maxord has its default value and `neq` is constant, !! this length is: !! !! * `20 + 16*neq ` for `mf` = 10, !! * `22 + 16*neq + 2*neq**2 ` for `mf` = 11 or 12, !! * `22 + 16*neq + neq**2 ` for `mf` = -11 or -12, !! * `22 + 17*neq ` for `mf` = 13, !! * `22 + 18*neq + (3*ml+2*mu)*neq` for `mf` = 14 or 15, !! * `22 + 17*neq + (2*ml+mu)*neq ` for `mf` = -14 or -15, !! * `20 + 9*neq ` for `mf` = 20, !! * `22 + 9*neq + 2*neq**2 ` for `mf` = 21 or 22, !! * `22 + 9*neq + neq**2 ` for `mf` = -21 or -22, !! * `22 + 10*neq ` for `mf` = 23, !! * `22 + 11*neq + (3*ml+2*mu)*neq` for `mf` = 24 or 25. !! * `22 + 10*neq + (2*ml+mu)*neq ` for `mf` = -24 or -25. !! !! the first 20 words of `rwork` are reserved for conditional !! and optional input and optional output. !! !! `rwork` can also used for conditional and !! optional input and optional output. !! !! the following word in `rwork` is a conditional input: !! !! * `rwork(1) = tcrit` = critical value of `t` which the solver !! is not to overshoot. required if `itask` is !! 4 or 5, and ignored otherwise. (see `itask`.) !! !! The following optional input requires `iopt = 1`, and in that !! case all of this input is examined. a value of zero for any !! of these optional input variables will cause the default value to be !! used. thus to use a subset of the optional input, simply preload !! locations 5 to 10 in `rwork` to 0.0, and !! then set those of interest to nonzero values: !! !! * `rwork(5) = h0` = the step size to be attempted on the first step. !! the default value is determined by the solver. !! !! * `rwork(6) = hmax` = the maximum absolute step size allowed. !! the default value is infinite. !! !! * `rwork(7) = hmin` = the minimum absolute step size allowed. !! the default value is 0. (this lower bound is not !! enforced on the final step before reaching `tcrit` !! when `itask = 4 or 5`.) !! !! the following optional outputs !! below are quantities related to the performance of [[dvode]] !! which are available to the user. !! except where stated otherwise, all of this output is defined !! on any successful return from [[dvode]], and on any return with !! istate = -1, -2, -4, -5, or -6. on an illegal input return !! (istate = -3), they will be unchanged from their existing values !! (if any), except possibly for `tolsf`. !! on any error return, output relevant to the error will be defined, !! as noted below: !! !! * `rwork(11) = hu` = the step size in t last used (successfully). !! !! * `rwork(12) = hcur` = the step size to be attempted on the next step. !! !! * `rwork(13) = tcur` = the current value of the independent variable !! which the solver has actually reached, i.e. the !! current internal mesh point in t. in the output, !! tcur will always be at least as far from the !! initial value of t as the current argument t, !! but may be farther (if interpolation was done). !! !! * `rwork(14) = tolsf` = a tolerance scale factor, greater than 1.0, !! computed when a request for too much accuracy was !! detected (istate = -3 if detected at the start of !! the problem, istate = -2 otherwise). if itol is !! left unaltered but rtol and atol are uniformly !! scaled up by a factor of tolsf for the next call, !! then the solver is deemed likely to succeed. !! (the user may also ignore tolsf and alter the !! tolerance parameters in any other way appropriate.) !! !! the following two arrays are segments of the `rwork` array which !! may also be of interest to the user as optional output: !! !! * `rwork(21:) = yh` = the nordsieck history array, of size `nyh` by !! `(nqcur + 1)`, where `nyh` is the initial value !! of `neq`. for `j = 0,1,...,nqcur`, column `j+1` !! of `yh` contains `hcur**j/factorial(j)` times !! the `j-th` derivative of the interpolating !! polynomial currently representing the !! solution, evaluated at `t = tcur`. !! !! * `rwork(lenrw-neq+1:) = acor` = `lenrw-neq+1` array !! of size `neq` used for the accumulated !! corrections on each step, scaled in the output !! to represent the estimated local error in `y` !! on the last step. this is the vector `e` in !! the description of the error control. it is !! defined only on a successful return from [[dvode]]. integer,intent(in) :: neq !! the size of the ode system (number of first order !! ordinary differential equations). !! `neq` may not be increased during the problem, but !! can be decreased (with `istate = 3` in the input). integer,intent(in) :: itol !! an indicator for the type of error control. !! 1 or 2 according as `atol` is a scalar or array. !! see description under `atol`. integer,intent(in) :: itask !! an index specifying the task to be performed. !! input only. itask has the following values and meanings: !! !! * 1 means normal computation of output values of y(t) at !! t = tout (by overshooting and interpolating). !! * 2 means take one step only and return. !! * 3 means stop at the first internal mesh point at or !! beyond t = tout and return. !! * 4 means normal computation of output values of y(t) at !! t = tout but without overshooting t = tcrit. !! tcrit must be input as rwork(1). tcrit may be equal to !! or beyond tout, but not behind it in the direction of !! integration. this option is useful if the problem !! has a singularity at or beyond t = tcrit. !! * 5 means take one step, without passing tcrit, and return. !! tcrit must be input as rwork(1). !! !! note: if itask = 4 or 5 and the solver reaches tcrit !! (within roundoff), it will return t = tcrit (exactly) to !! indicate this (unless itask = 4 and tout comes before tcrit, !! in which case answers at t = tout are returned first). integer,intent(inout) :: istate !! an index used for input and output to specify the !! the state of the calculation. !! !! in the input, the values of istate are as follows: !! !! 1. means this is the first call for the problem !! (initializations will be done). see note below. !! 2. means this is not the first call, and the calculation !! is to continue normally, with no change in any input !! parameters except possibly tout and itask. !! (if itol, rtol, and/or atol are changed between calls !! with istate = 2, the new values will be used but not !! tested for legality.) !! 3. means this is not the first call, and the !! calculation is to continue normally, but with !! a change in input parameters other than !! tout and itask. changes are allowed in !! neq, itol, rtol, atol, iopt, lrw, liw, mf, ml, mu, !! and any of the optional input except h0. !! (see iwork description for ml and mu.) !! !! note: a preliminary call with tout = t is not counted !! as a first call here, as no initialization or checking of !! input is done. (such a call is sometimes useful to include !! the initial conditions in the output.) !! thus the first call for which tout /= t requires !! istate = 1 in the input. !! !! in the output, istate has the following values and meanings: !! !! * 1 means nothing was done, as tout was equal to t with !! istate = 1 in the input. !! * 2 means the integration was performed successfully. !! * -1 means an excessive amount of work (more than mxstep !! steps) was done on this call, before completing the !! requested task, but the integration was otherwise !! successful as far as t. (mxstep is an optional input !! and is normally 500.) to continue, the user may !! simply reset istate to a value > 1 and call again. !! (the excess work step counter will be reset to 0.) !! in addition, the user may increase mxstep to avoid !! this error return. (see optional input below.) !! * -2 means too much accuracy was requested for the precision !! of the machine being used. this was detected before !! completing the requested task, but the integration !! was successful as far as t. to continue, the tolerance !! parameters must be reset, and istate must be set !! to 3. the optional output tolsf may be used for this !! purpose. (note: if this condition is detected before !! taking any steps, then an illegal input return !! (istate = -3) occurs instead.) !! * -3 means illegal input was detected, before taking any !! integration steps. see written message for details. !! note: if the solver detects an infinite loop of calls !! to the solver with illegal input, it will cause !! the run to stop. !! * -4 means there were repeated error test failures on !! one attempted step, before completing the requested !! task, but the integration was successful as far as t. !! the problem may have a singularity, or the input !! may be inappropriate. !! * -5 means there were repeated convergence test failures on !! one attempted step, before completing the requested !! task, but the integration was successful as far as t. !! this may be caused by an inaccurate jacobian matrix, !! if one is being used. !! * -6 means ewt(i) became zero for some i during the !! integration. pure relative error control (atol(i)=0.0) !! was requested on a variable which has now vanished. !! the integration was successful as far as t. !! !! note: since the normal output value of istate is 2, !! it does not need to be reset for normal continuation. !! also, since a negative input value of istate will be !! regarded as illegal, a negative output value requires the !! user to change it, and possibly other input, before !! calling the solver again. integer,intent(in) :: iopt !! an integer flag to specify whether or not any optional !! input is being used on this call. !! the optional input is listed separately below. !! !! * iopt = 0 means no optional input is being used. !! default values will be used in all cases. !! * iopt = 1 means optional input is being used. integer,intent(in) :: liw !! the length of the array iwork, as declared by the user. !! (this will be checked by the solver.) integer :: iwork(liw) !! an integer work array. the length of iwork must be at least: !! !! * `30` if miter = 0 or 3 (mf = 10, 13, 20, 23), or !! * `30 + neq` otherwise (abs(mf) = 11,12,14,15,21,22,24,25). !! !! the first 30 words of `iwork` are reserved for conditional and !! optional input and optional output. !! !! the following 2 words in `iwork` are conditional input: !! !! * `iwork(1) = ml` !! * `iwork(2) = mu` !! !! these are the lower and upper !! half-bandwidths, respectively, of the !! banded jacobian, excluding the main diagonal. !! the band is defined by the matrix locations !! `(i,j)` with `i-ml <= j <= i+mu`. `ml` and `mu` !! must satisfy `0 <= ml,mu <= neq-1`. !! these are required if `miter` is 4 or 5, and !! ignored otherwise. `ml` and `mu` may in fact be !! the band parameters for a matrix to which !! `df/dy` is only approximately equal. !! !! The following optional input requires `iopt = 1`, and in that !! case all of this input is examined. a value of zero for any !! of these optional input variables will cause the default value to be !! used. thus to use a subset of the optional input, simply preload !! locations 5 to 10 in `iwork` to 0, and !! then set those of interest to nonzero values. !! !! * `iwork(5) = maxord` = the maximum order to be allowed. the default !! value is 12 if `meth = 1`, and 5 if `meth = 2`. !! if `maxord` exceeds the default value, it will !! be reduced to the default value. !! if `maxord` is changed during the problem, it may !! cause the current order to be reduced. !! !! * `iwork(6) = mxstep` = maximum number of (internally defined) steps !! allowed during one call to the solver. !! the default value is 500. !! !! * `iwork(7) = mxhnil` = maximum number of messages printed (per problem) !! warning that `t + h = t` on a step (`h` = step size). !! this must be positive to result in a non-default !! value. the default value is 10. !! !! as optional additional output from dvode, the variables listed !! below are quantities related to the performance of dvode !! which are available to the user. !! except where stated otherwise, all of this output is defined !! on any successful return from dvode, and on any return with !! istate = -1, -2, -4, -5, or -6. on an illegal input return !! (istate = -3), they will be unchanged from their existing values !! (if any), except possibly for `lenrw`, and `leniw`. !! on any error return, output relevant to the error will be defined, !! as noted below: !! !! * `iwork(11) = nst` = the number of steps taken for the problem so far. !! * `iwork(12) = nfe` = the number of f evaluations for the problem so far. !! * `iwork(13) = nje` = the number of jacobian evaluations so far. !! * `iwork(14) = nqu` = the method order last used (successfully). !! * `iwork(15) = nqcur` = the order to be attempted on the next step. !! * `iwork(16) = imxer` = the index of the component of largest magnitude in !! the weighted local error vector ( e(i)/ewt(i) ), !! on an error return with istate = -4 or -5. !! * `iwork(17) = lenrw` = the length of rwork actually required. !! this is defined on normal returns and on an illegal !! input return for insufficient storage. !! * `iwork(18) = leniw` = the length of iwork actually required. !! this is defined on normal returns and on an illegal !! input return for insufficient storage. !! * `iwork(19) = nlu` = the number of matrix lu decompositions so far. !! * `iwork(20) = nni` = the number of nonlinear (newton) iterations so far. !! * `iwork(21) = ncfn` = the number of convergence failures of the nonlinear !! solver so far. !! * `iwork(22) = netf` = the number of error test failures of the integrator !! so far. integer,intent(in) :: mf !! method flag. standard values are: !! !! * 10 for nonstiff (adams) method, no jacobian used. !! * 21 for stiff (bdf) method, user-supplied full jacobian. !! * 22 for stiff method, internally generated full jacobian. !! * 24 for stiff method, user-supplied banded jacobian. !! * 25 for stiff method, internally generated banded jacobian. !! !! the complete set of legal values of !! mf are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, !! -11, -12, -14, -15, -21, -22, -24, -25. !! !! `mf` is a signed two-digit integer, `mf = jsv*(10*meth + miter)` !! !! `jsv = sign(mf)` indicates the jacobian-saving strategy: !! !! * jsv = 1 means a copy of the jacobian is saved for reuse !! in the corrector iteration algorithm. !! * jsv = -1 means a copy of the jacobian is not saved !! (valid only for miter = 1, 2, 4, or 5). !! !! `meth` indicates the basic linear multistep method: !! !! * meth = 1 means the implicit adams method. !! * meth = 2 means the method based on backward !! differentiation formulas (bdf-s). !! !! `miter` indicates the corrector iteration method: !! !! * miter = 0 means functional iteration (no jacobian matrix !! is involved). !! * miter = 1 means chord iteration with a user-supplied !! full (neq by neq) jacobian. !! * miter = 2 means chord iteration with an internally !! generated (difference quotient) full jacobian !! (using neq extra calls to f per df/dy value). !! * miter = 3 means chord iteration with an internally !! generated diagonal jacobian approximation !! (using 1 extra call to f per df/dy evaluation). !! * miter = 4 means chord iteration with a user-supplied !! banded jacobian. !! * miter = 5 means chord iteration with an internally !! generated banded jacobian (using ml+mu+1 extra !! calls to f per df/dy evaluation). !! !! if miter = 1 or 4, the user must supply a subroutine `jac` !! (the name is arbitrary) as described above under `jac`. !! for other values of miter, a dummy argument can be used. logical :: ihit real(wp) :: atoli , big , ewti , h0 , hmax , hmx , & rh , rtoli , size , tcrit , & tnext , tolsf , tp integer :: i , ier , iflag , imxer , jco , kgo , leniw , lenj , & lenp , lenrw , lenwm , lf0 , mband , mfa , ml , & mu , niter , nslast character(len=80) :: msg integer,parameter :: mxhnl0 = 10 integer,parameter :: mxstp0 = 500 real(wp),parameter :: one = 1.0_wp real(wp),parameter :: two = 2.0_wp real(wp),parameter :: four = 4.0_wp real(wp),parameter :: pt2 = 0.2_wp real(wp),parameter :: hun = 100.0_wp !----------------------------------------------------------------------- ! block a. ! this code block is executed on every call. ! it tests istate and itask for legality and branches appropriately. ! if istate > 1 but the flag init shows that initialization has ! not yet been done, an error return occurs. ! if istate = 1 and tout = t, return immediately. !----------------------------------------------------------------------- if ( istate<1 .or. istate>3 ) then !----------------------------------------------------------------------- ! block i. ! the following block handles all error returns due to illegal input ! (istate = -3), as detected before calling the core integrator. ! first the error message routine is called. if the illegal input ! is a negative istate, the run is aborted (apparent infinite loop). !----------------------------------------------------------------------- msg = 'dvode-- istate (=i1) illegal ' call me%xerrwd(msg,30,1,1,1,istate,0,0,zero,zero) if ( istate>=0 ) goto 1500 msg = 'dvode-- run aborted: apparent infinite loop ' call me%xerrwd(msg,50,303,2,0,0,0,0,zero,zero) return else if ( itask<1 .or. itask>5 ) then msg = 'dvode-- itask (=i1) illegal ' call me%xerrwd(msg,30,2,1,1,itask,0,0,zero,zero) goto 1500 else if ( istate==1 ) then me%dat%init = 0 if ( tout==t ) return else if ( me%dat%init/=1 ) then msg = 'dvode-- istate (=i1) > 1 but dvode not initialized ' call me%xerrwd(msg,60,3,1,1,istate,0,0,zero,zero) goto 1500 else if ( istate==2 ) then goto 50 endif !----------------------------------------------------------------------- ! block b. ! the next code block is executed for the initial call (istate = 1), ! or for a continuation call with parameter changes (istate = 3). ! it contains checking of all input and various initializations. ! ! first check legality of the non-optional input neq, itol, iopt, ! mf, ml, and mu. !----------------------------------------------------------------------- if ( neq<=0 ) then msg = 'dvode-- neq (=i1) < 1 ' call me%xerrwd(msg,30,4,1,1,neq,0,0,zero,zero) goto 1500 else if ( istate/=1 ) then if ( neq>me%dat%n ) then msg = 'dvode-- istate = 3 and neq increased (i1 to i2) ' call me%xerrwd(msg,50,5,1,2,me%dat%n,neq,0,zero,zero) goto 1500 endif endif me%dat%n = neq if ( itol<1 .or. itol>4 ) then msg = 'dvode-- itol (=i1) illegal ' call me%xerrwd(msg,30,6,1,1,itol,0,0,zero,zero) goto 1500 else if ( iopt<0 .or. iopt>1 ) then msg = 'dvode-- iopt (=i1) illegal ' call me%xerrwd(msg,30,7,1,1,iopt,0,0,zero,zero) goto 1500 else me%dat%jsv = sign(1,mf) mfa = abs(mf) me%dat%meth = mfa/10 me%dat%miter = mfa - 10*me%dat%meth if ( me%dat%meth<1 .or. me%dat%meth>2 ) goto 800 if ( me%dat%miter<0 .or. me%dat%miter>5 ) goto 800 if ( me%dat%miter>3 ) then ml = iwork(1) mu = iwork(2) if ( ml<0 .or. ml>=me%dat%n ) then msg = 'dvode-- ml (=i1) illegal: <0 or >=neq (=i2)' call me%xerrwd(msg,50,9,1,2,ml,neq,0,zero,zero) goto 1500 else if ( mu<0 .or. mu>=me%dat%n ) then msg = 'dvode-- mu (=i1) illegal: <0 or >=neq (=i2)' call me%xerrwd(msg,50,10,1,2,mu,neq,0,zero,zero) goto 1500 endif endif ! next process and check the optional input. --------------------------- if ( iopt==1 ) then me%dat%maxord = iwork(5) if ( me%dat%maxord<0 ) then msg = 'dvode-- maxord (=i1) < 0 ' call me%xerrwd(msg,30,11,1,1,me%dat%maxord,0,0,zero,zero) goto 1500 else if ( me%dat%maxord==0 ) me%dat%maxord = 100 me%dat%maxord = min(me%dat%maxord,mord(me%dat%meth)) me%dat%mxstep = iwork(6) if ( me%dat%mxstep<0 ) then msg = 'dvode-- mxstep (=i1) < 0 ' call me%xerrwd(msg,30,12,1,1,me%dat%mxstep,0,0,zero,zero) goto 1500 else if ( me%dat%mxstep==0 ) me%dat%mxstep = mxstp0 me%dat%mxhnil = iwork(7) if ( me%dat%mxhnil<0 ) then msg = 'dvode-- mxhnil (=i1) < 0 ' call me%xerrwd(msg,30,13,1,1,me%dat%mxhnil,0,0,zero,zero) goto 1500 else if ( me%dat%mxhnil==0 ) me%dat%mxhnil = mxhnl0 if ( istate==1 ) then h0 = rwork(5) if ( (tout-t)*h0<zero ) then msg = 'dvode-- tout (=r1) behind t (=r2) ' call me%xerrwd(msg,40,14,1,0,0,0,2,tout,t) msg = ' integration direction is given by h0 (=r1) ' call me%xerrwd(msg,50,14,1,0,0,0,1,h0,zero) goto 1500 endif endif hmax = rwork(6) if ( hmax<zero ) then msg = 'dvode-- hmax (=r1) < 0.0 ' call me%xerrwd(msg,30,15,1,0,0,0,1,hmax,zero) goto 1500 else me%dat%hmxi = zero if ( hmax>zero ) me%dat%hmxi = one/hmax me%dat%hmin = rwork(7) if ( me%dat%hmin<zero ) then msg = 'dvode-- hmin (=r1) < 0.0 ' call me%xerrwd(msg,30,16,1,0,0,0,1,me%dat%hmin,zero) goto 1500 endif endif endif endif endif else me%dat%maxord = mord(me%dat%meth) me%dat%mxstep = mxstp0 me%dat%mxhnil = mxhnl0 if ( istate==1 ) h0 = zero me%dat%hmxi = zero me%dat%hmin = zero endif !----------------------------------------------------------------------- ! set work array pointers and check lengths lrw and liw. ! pointers to segments of rwork and iwork are named by prefixing l to ! the name of the segment. e.g., the segment yh starts at rwork(lyh). ! segments of rwork (in order) are denoted yh, wm, ewt, savf, acor. ! within wm, locjs is the location of the saved jacobian (jsv > 0). !----------------------------------------------------------------------- me%dat%lyh = 21 if ( istate==1 ) me%dat%nyh = me%dat%n me%dat%lwm = me%dat%lyh + (me%dat%maxord+1)*me%dat%nyh jco = max(0,me%dat%jsv) select case (me%dat%miter) case(0) lenwm = 0 case(1:2) lenwm = 2 + (1+jco)*me%dat%n*me%dat%n me%dat%locjs = me%dat%n*me%dat%n + 3 case(3) lenwm = 2 + me%dat%n case(4:5) mband = ml + mu + 1 lenp = (mband+ml)*me%dat%n lenj = mband*me%dat%n lenwm = 2 + lenp + jco*lenj me%dat%locjs = lenp + 3 end select me%dat%lewt = me%dat%lwm + lenwm me%dat%lsavf = me%dat%lewt + me%dat%n me%dat%lacor = me%dat%lsavf + me%dat%n lenrw = me%dat%lacor + me%dat%n - 1 iwork(17) = lenrw me%dat%liwm = 1 leniw = 30 + me%dat%n if ( me%dat%miter==0 .or. me%dat%miter==3 ) leniw = 30 iwork(18) = leniw if ( lenrw>lrw ) then msg = 'dvode-- rwork length needed, lenrw (=i1), exceeds lrw (=i2)' call me%xerrwd(msg,60,17,1,2,lenrw,lrw,0,zero,zero) goto 1500 else if ( leniw>liw ) then msg = 'dvode-- iwork length needed, leniw (=i1), exceeds liw (=i2)' call me%xerrwd(msg,60,18,1,2,leniw,liw,0,zero,zero) goto 1500 else ! check rtol and atol for legality. ------------------------------------ rtoli = rtol(1) atoli = atol(1) do i = 1 , me%dat%n if ( itol>=3 ) rtoli = rtol(i) if ( itol==2 .or. itol==4 ) atoli = atol(i) if ( rtoli<zero ) goto 900 if ( atoli<zero ) goto 1000 enddo if ( istate==1 ) then !----------------------------------------------------------------------- ! block c. ! the next block is for the initial call only (istate = 1). ! it contains all remaining initializations, the initial call to f, ! and the calculation of the initial step size. ! the error weights in ewt are inverted after being loaded. !----------------------------------------------------------------------- me%dat%uround = epmach me%dat%tn = t if ( itask==4 .or. itask==5 ) then tcrit = rwork(1) if ( (tcrit-tout)*(tout-t)<zero ) goto 1300 if ( h0/=zero .and. (t+h0-tcrit)*h0>zero ) & h0 = tcrit - t endif me%dat%jstart = 0 if ( me%dat%miter>0 ) rwork(me%dat%lwm) = sqrt(me%dat%uround) me%dat%ccmxj = pt2 me%dat%msbj = 50 me%dat%nhnil = 0 me%dat%nst = 0 me%dat%nje = 0 me%dat%nni = 0 me%dat%ncfn = 0 me%dat%netf = 0 me%dat%nlu = 0 me%dat%nslj = 0 nslast = 0 me%dat%hu = zero me%dat%nqu = 0 ! initial call to f. (lf0 points to yh(*,2).) ------------------------- lf0 = me%dat%lyh + me%dat%nyh call me%f(me%dat%n,t,y(1:me%dat%n),rwork(lf0)) me%dat%nfe = 1 ! load the initial value vector in yh. --------------------------------- call dcopy(me%dat%n,y,1,rwork(me%dat%lyh),1) ! load and invert the ewt array. (h is temporarily set to 1.0.) ------- me%dat%nq = 1 me%dat%h = one call me%dewset(me%dat%n,itol,rtol(1:me%dat%n),atol(1:me%dat%n),& rwork(me%dat%lyh), rwork(me%dat%lewt)) do i = 1 , me%dat%n if ( rwork(i+me%dat%lewt-1)<=zero ) goto 1100 rwork(i+me%dat%lewt-1) = one/rwork(i+me%dat%lewt-1) enddo if ( h0==zero ) then ! call dvhin to set initial step size h0 to be attempted. -------------- call me%dvhin(me%dat%n,t,rwork(me%dat%lyh),rwork(lf0), & tout,me%dat%uround,rwork(me%dat%lewt),itol,& atol,y,rwork(me%dat%lacor),h0,niter,ier) me%dat%nfe = me%dat%nfe + niter if ( ier/=0 ) then msg = & 'dvode-- tout (=r1) too close to t(=r2) to start integration' call me%xerrwd(msg,60,22,1,0,0,0,2,tout,t) goto 1500 endif endif ! adjust h0 if necessary to meet hmax bound. --------------------------- rh = abs(h0)*me%dat%hmxi if ( rh>one ) h0 = h0/rh ! load h with h0 and scale yh(*,2) by h0. ------------------------------ me%dat%h = h0 call dscal(me%dat%n,h0,rwork(lf0),1) goto 200 else ! if istate = 3, set flag to signal parameter changes to dvstep. ------- me%dat%jstart = -1 ! maxord was reduced below nq. copy yh(*,maxord+2) into savf. --------- if ( me%dat%nq>me%dat%maxord ) & call dcopy(me%dat%n,rwork(me%dat%lwm),1,rwork(me%dat%lsavf),1) ! reload wm(1) = rwork(lwm), since lwm may have changed. --------------- if ( me%dat%miter>0 ) rwork(me%dat%lwm) = sqrt(me%dat%uround) endif endif endif endif endif !----------------------------------------------------------------------- ! block d. ! the next code block is for continuation calls only (istate = 2 or 3) ! and is to check stop conditions before taking a step. !----------------------------------------------------------------------- 50 nslast = me%dat%nst me%dat%kuth = 0 select case (itask) case (2) goto 100 case (3) tp = me%dat%tn - me%dat%hu*(one+hun*me%dat%uround) if ( (tp-tout)*me%dat%h>zero ) then msg = 'dvode-- itask = i1 and tout (=r1) behind tcur - hu (= r2) ' call me%xerrwd(msg,60,23,1,1,itask,0,2,tout,tp) goto 1500 else if ( (me%dat%tn-tout)*me%dat%h>=zero ) goto 300 goto 100 endif case (4) tcrit = rwork(1) if ( (me%dat%tn-tcrit)*me%dat%h>zero ) goto 1200 if ( (tcrit-tout)*me%dat%h<zero ) goto 1300 if ( (me%dat%tn-tout)*me%dat%h>=zero ) then call me%dvindy(tout,0,rwork(me%dat%lyh),me%dat%nyh,y,iflag) if ( iflag/=0 ) goto 1400 t = tout goto 400 endif case (5) tcrit = rwork(1) if ( (me%dat%tn-tcrit)*me%dat%h>zero ) goto 1200 case default if ( (me%dat%tn-tout)*me%dat%h<zero ) goto 100 call me%dvindy(tout,0,rwork(me%dat%lyh),me%dat%nyh,y,iflag) if ( iflag/=0 ) goto 1400 t = tout goto 400 end select hmx = abs(me%dat%tn) + abs(me%dat%h) ihit = abs(me%dat%tn-tcrit)<=hun*me%dat%uround*hmx if ( ihit ) goto 300 tnext = me%dat%tn + me%dat%hnew*(one+four*me%dat%uround) if ( (tnext-tcrit)*me%dat%h>zero ) then me%dat%h = (tcrit-me%dat%tn)*(one-four*me%dat%uround) me%dat%kuth = 1 endif endif !----------------------------------------------------------------------- ! block e. ! the next block is normally executed for all calls and contains ! the call to the one-step core integrator dvstep. ! ! this is a looping point for the integration steps. ! ! first check for too many steps being taken, update ewt (if not at ! start of problem), check for too much accuracy being requested, and ! check for h below the roundoff level in t. !----------------------------------------------------------------------- 100 if ( (me%dat%nst-nslast)>=me%dat%mxstep ) then !----------------------------------------------------------------------- ! block h. ! the following block handles all unsuccessful returns other than ! those for illegal input. first the error message routine is called. ! if there was an error test or convergence test failure, imxer is set. ! then y is loaded from yh, and t is set to tn. ! the optional output is loaded into the work arrays before returning. !----------------------------------------------------------------------- ! the maximum number of steps was taken before reaching tout. ---------- msg = 'dvode-- at current t (=r1), mxstep (=i1) steps ' call me%xerrwd(msg,50,201,1,0,0,0,0,zero,zero) msg = ' taken on this call before reaching tout ' call me%xerrwd(msg,50,201,1,1,me%dat%mxstep,0,1,me%dat%tn,zero) istate = -1 goto 700 else call me%dewset(me%dat%n,itol,rtol(1:me%dat%n),atol(1:me%dat%n),& rwork(me%dat%lyh),rwork(me%dat%lewt)) do i = 1 , me%dat%n if ( rwork(i+me%dat%lewt-1)<=zero ) goto 500 rwork(i+me%dat%lewt-1) = one/rwork(i+me%dat%lewt-1) enddo endif 200 tolsf = me%dat%uround*me%dvnorm(me%dat%n,rwork(me%dat%lyh),rwork(me%dat%lewt)) if ( tolsf<=one ) then if ( (me%dat%tn+me%dat%h)==me%dat%tn ) then me%dat%nhnil = me%dat%nhnil + 1 if ( me%dat%nhnil<=me%dat%mxhnil ) then msg = 'dvode-- warning: internal t (=r1) and h (=r2) are' call me%xerrwd(msg,50,101,1,0,0,0,0,zero,zero) msg = ' such that in the machine, t + h = t on the next step ' call me%xerrwd(msg,60,101,1,0,0,0,0,zero,zero) msg = ' (h = step size). solver will continue anyway' call me%xerrwd(msg,50,101,1,0,0,0,2,me%dat%tn,me%dat%h) if ( me%dat%nhnil>=me%dat%mxhnil ) then msg = 'dvode-- above warning has been issued i1 times. ' call me%xerrwd(msg,50,102,1,0,0,0,0,zero,zero) msg = ' it will not be issued again for this problem' call me%xerrwd(msg,50,102,1,1,me%dat%mxhnil,0,0,zero,zero) endif endif endif call me%dvstep(y,& ! y rwork(me%dat%lyh),& ! yh me%dat%nyh,& ! nyh rwork(me%dat%lyh),& ! yh rwork(me%dat%lewt), & ! ewt rwork(me%dat%lsavf),& ! savf y,& ! vsav rwork(me%dat%lacor),& ! acor rwork(me%dat%lwm),& ! wm iwork(me%dat%liwm)) ! iwm kgo = 1 - me%dat%kflag ! branch on kflag. note: in this version, kflag can not be set to -3. ! kflag == 0, -1, -2 select case (kgo) case (2) ! kflag = -1. error test failed repeatedly or with abs(h) = hmin. ----- msg = 'dvode-- at t(=r1) and step size h(=r2), the error' call me%xerrwd(msg,50,204,1,0,0,0,0,zero,zero) msg = ' test failed repeatedly or with abs(h) = hmin' call me%xerrwd(msg,50,204,1,0,0,0,2,me%dat%tn,me%dat%h) istate = -4 goto 600 case (3) ! kflag = -2. convergence failed repeatedly or with abs(h) = hmin. ---- msg = 'dvode-- at t (=r1) and step size h (=r2), the ' call me%xerrwd(msg,50,205,1,0,0,0,0,zero,zero) msg = ' corrector convergence failed repeatedly ' call me%xerrwd(msg,50,205,1,0,0,0,0,zero,zero) msg = ' or with abs(h) = hmin ' call me%xerrwd(msg,30,205,1,0,0,0,2,me%dat%tn,me%dat%h) istate = -5 goto 600 case default !----------------------------------------------------------------------- ! block f. ! the following block handles the case of a successful return from the ! core integrator (kflag = 0). test for stop conditions. !----------------------------------------------------------------------- me%dat%init = 1 me%dat%kuth = 0 select case (itask) case (2) case (3) ! itask = 3. jump to exit if tout was reached. ------------------------ if ( (me%dat%tn-tout)*me%dat%h<zero ) goto 100 case (4) ! itask = 4. see if tout or tcrit was reached. adjust h if necessary. if ( (me%dat%tn-tout)*me%dat%h<zero ) then hmx = abs(me%dat%tn) + abs(me%dat%h) ihit = abs(me%dat%tn-tcrit)<=hun*me%dat%uround*hmx if ( .not.(ihit) ) then tnext = me%dat%tn + me%dat%hnew*(one+four*me%dat%uround) if ( (tnext-tcrit)*me%dat%h>zero ) then me%dat%h = (tcrit-me%dat%tn)*(one-four*me%dat%uround) me%dat%kuth = 1 endif goto 100 endif else call me%dvindy(tout,0,rwork(me%dat%lyh),me%dat%nyh,y,iflag) t = tout goto 400 endif case (5) ! itask = 5. see if tcrit was reached and jump to exit. --------------- hmx = abs(me%dat%tn) + abs(me%dat%h) ihit = abs(me%dat%tn-tcrit)<=hun*me%dat%uround*hmx case default ! itask = 1. if tout has been reached, interpolate. ------------------- if ( (me%dat%tn-tout)*me%dat%h<zero ) goto 100 call me%dvindy(tout,0,rwork(me%dat%lyh),me%dat%nyh,y,iflag) t = tout goto 400 end select end select else tolsf = tolsf*two if ( me%dat%nst==0 ) then msg = 'dvode-- at start of problem, too much accuracy ' call me%xerrwd(msg,50,26,1,0,0,0,0,zero,zero) msg = ' requested for precision of machine: see tolsf (=r1) ' call me%xerrwd(msg,60,26,1,0,0,0,1,tolsf,zero) rwork(14) = tolsf goto 1500 else ! too much accuracy requested for machine precision. ------------------- msg = 'dvode-- at t (=r1), too much accuracy requested ' call me%xerrwd(msg,50,203,1,0,0,0,0,zero,zero) msg = ' for precision of machine: see tolsf (=r2) ' call me%xerrwd(msg,50,203,1,0,0,0,2,me%dat%tn,tolsf) rwork(14) = tolsf istate = -2 goto 700 endif endif !----------------------------------------------------------------------- ! block g. ! the following block handles all successful returns from dvode. ! if itask /= 1, y is loaded from yh and t is set accordingly. ! istate is set to 2, and the optional output is loaded into the work ! arrays before returning. !----------------------------------------------------------------------- 300 call dcopy(me%dat%n,rwork(me%dat%lyh),1,y,1) t = me%dat%tn if ( itask==4 .or. itask==5 ) then if ( ihit ) t = tcrit endif 400 istate = 2 rwork(11) = me%dat%hu rwork(12) = me%dat%hnew rwork(13) = me%dat%tn iwork(11) = me%dat%nst iwork(12) = me%dat%nfe iwork(13) = me%dat%nje iwork(14) = me%dat%nqu iwork(15) = me%dat%newq iwork(19) = me%dat%nlu iwork(20) = me%dat%nni iwork(21) = me%dat%ncfn iwork(22) = me%dat%netf return ! ewt(i) <= 0.0 for some i (not at start of problem). ---------------- 500 ewti = rwork(me%dat%lewt+i-1) msg = 'dvode-- at t (=r1), ewt(i1) has become r2 <= 0.' call me%xerrwd(msg,50,202,1,1,i,0,2,me%dat%tn,ewti) istate = -6 goto 700 ! compute imxer if relevant. ------------------------------------------- 600 big = zero imxer = 1 do i = 1 , me%dat%n size = abs(rwork(i+me%dat%lacor-1)*rwork(i+me%dat%lewt-1)) if ( big<size ) then big = size imxer = i endif enddo iwork(16) = imxer ! set y vector, t, and optional output. -------------------------------- 700 call dcopy(me%dat%n,rwork(me%dat%lyh),1,y,1) t = me%dat%tn rwork(11) = me%dat%hu rwork(12) = me%dat%h rwork(13) = me%dat%tn iwork(11) = me%dat%nst iwork(12) = me%dat%nfe iwork(13) = me%dat%nje iwork(14) = me%dat%nqu iwork(15) = me%dat%nq iwork(19) = me%dat%nlu iwork(20) = me%dat%nni iwork(21) = me%dat%ncfn iwork(22) = me%dat%netf return 800 msg = 'dvode-- mf (=i1) illegal ' call me%xerrwd(msg,30,8,1,1,mf,0,0,zero,zero) goto 1500 900 msg = 'dvode-- rtol(i1) is r1 < 0.0 ' call me%xerrwd(msg,40,19,1,1,i,0,1,rtoli,zero) goto 1500 1000 msg = 'dvode-- atol(i1) is r1 < 0.0 ' call me%xerrwd(msg,40,20,1,1,i,0,1,atoli,zero) goto 1500 1100 ewti = rwork(me%dat%lewt+i-1) msg = 'dvode-- ewt(i1) is r1 <= 0.0 ' call me%xerrwd(msg,40,21,1,1,i,0,1,ewti,zero) goto 1500 1200 msg = 'dvode-- itask = 4 or 5 and tcrit (=r1) behind tcur (=r2) ' call me%xerrwd(msg,60,24,1,0,0,0,2,tcrit,me%dat%tn) goto 1500 1300 msg = 'dvode-- itask = 4 or 5 and tcrit (=r1) behind tout (=r2) ' call me%xerrwd(msg,60,25,1,0,0,0,2,tcrit,tout) goto 1500 1400 msg = 'dvode-- trouble from dvindy. itask = i1, tout = r1. ' call me%xerrwd(msg,60,27,1,1,itask,0,1,tout,zero) 1500 istate = -3 end subroutine dvode !***************************************************************************************** !> ! this routine computes the step size, h0, to be attempted on the ! first step, when the user has not supplied a value for this. ! ! first we check that tout - t0 differs significantly from zero. then ! an iteration is done to approximate the initial second derivative ! and this is used to define h from w.r.m.s.norm(h**2 * yddot / 2) = 1. ! a bias factor of 1/2 is applied to the resulting h. ! the sign of h0 is inferred from the initial values of tout and t0. subroutine dvhin(me,n,t0,y0,ydot,tout,uround,ewt,itol,& atol,y,temp,h0,niter,ier) class(dvode_t),intent(inout) :: me real(wp),intent(in) :: t0 !! initial value of independent variable real(wp),intent(in) :: y0(*) !! vector of initial conditions real(wp),intent(in) :: ydot(*) !! vector of initial first derivatives real(wp),intent(in) :: tout !! first output value of independent variable real(wp),intent(in) :: uround !! machine unit roundoff real(wp),intent(in) :: ewt(*) !! error weights and tolerance parameters as described in the driver routine real(wp),intent(in) :: atol(*) !! error weights and tolerance parameters as described in the driver routine real(wp),intent(inout) :: y(n) !! work array of length n real(wp),intent(inout) :: temp(n) !! work array of length n integer,intent(in) :: n !! size of ode system integer,intent(in) :: itol !! error weights and tolerance parameters as described in the driver routine real(wp),intent(out) :: h0 !! step size to be attempted integer,intent(out) :: niter !! number of iterations (and of f evaluations) to compute h0 integer,intent(out) :: ier !! the error flag, returned with the value: !! !! * ier = 0 if no trouble occurred, or !! * ier = -1 if tout and t0 are considered too close to proceed. real(wp) :: afi , atoli , delyi , h , hg , hlb , & hnew , hrat , hub , t1 , tdist , & tround , yddnrm integer :: i , iter real(wp),parameter :: half = 0.5_wp real(wp),parameter :: hun = 100.0_wp real(wp),parameter :: pt1 = 0.1_wp real(wp),parameter :: two = 2.0_wp main : block niter = 0 tdist = abs(tout-t0) tround = uround*max(abs(t0),abs(tout)) if ( tdist<two*tround ) then ! error return for tout - t0 too small. -------------------------------- ier = -1 return else ! set a lower bound on h based on the roundoff level in t0 and tout. --- hlb = hun*tround ! set an upper bound on h based on tout-t0 and the initial y and ydot. - hub = pt1*tdist atoli = atol(1) do i = 1 , n if ( itol==2 .or. itol==4 ) atoli = atol(i) delyi = pt1*abs(y0(i)) + atoli afi = abs(ydot(i)) if ( afi*hub>delyi ) hub = delyi/afi enddo ! set initial guess for h as geometric mean of upper and lower bounds. - iter = 0 hg = sqrt(hlb*hub) ! if the bounds have crossed, exit with the mean value. ---------------- if ( hub<hlb ) then h0 = hg exit main endif do ! looping point for iteration. ----------------------------------------- ! estimate the second derivative as a difference quotient in f. -------- h = sign(hg,tout-t0) t1 = t0 + h do i = 1 , n y(i) = y0(i) + h*ydot(i) enddo call me%f(n,t1,y,temp) do i = 1 , n temp(i) = (temp(i)-ydot(i))/h enddo yddnrm = me%dvnorm(n,temp,ewt(1:n)) ! get the corresponding new value of h. -------------------------------- if ( yddnrm*hub*hub>two ) then hnew = sqrt(two/yddnrm) else hnew = sqrt(hg*hub) endif iter = iter + 1 !----------------------------------------------------------------------- ! test the stopping conditions. ! stop if the new and previous h values differ by a factor of < 2. ! stop if four iterations have been done. also, stop with previous h ! if hnew/hg > 2 after first iteration, as this probably means that ! the second derivative value is bad because of cancellation error. !----------------------------------------------------------------------- if ( iter<4 ) then hrat = hnew/hg if ( (hrat<=half) .or. (hrat>=two) ) then if ( (iter>=2) .and. (hnew>two*hg) ) then hnew = hg exit endif hg = hnew cycle endif endif exit end do ! iteration done. apply bounds, bias factor, and sign. then exit. ---- h0 = hnew*half if ( h0<hlb ) h0 = hlb if ( h0>hub ) h0 = hub endif end block main h0 = sign(h0,tout-t0) niter = iter ier = 0 end subroutine dvhin !***************************************************************************************** !> ! dvindy computes interpolated values of the k-th derivative of the ! dependent variable vector y, and stores it in dky. this routine ! is called within the package with k = 0 and t = tout, but may ! also be called by the user for any k up to the current order. ! (see detailed instructions in the usage documentation.) ! !``` ! the computed values in dky are gotten by interpolation using the ! nordsieck history array yh. this array corresponds uniquely to a ! vector-valued polynomial of degree nqcur or less, and dky is set ! to the k-th derivative of this polynomial at t. ! the formula for dky is: ! q ! dky(i) = sum c(j,k) * (t - tn)**(j-k) * h**(-j) * yh(i,j+1) ! j=k ! where c(j,k) = j*(j-1)*...*(j-k+1), q = nqcur, tn = tcur, h = hcur. ! the quantities nq = nqcur, l = nq+1, n, tn, and h are ! communicated by common. the above sum is done in reverse order. ! iflag is returned negative if either k or t is out of bounds. ! ! discussion above and comments in driver explain all variables. !``` subroutine dvindy(me,t,k,yh,ldyh,dky,iflag) class(dvode_t),intent(inout) :: me real(wp),intent(in) :: t !! value of independent variable where answers are desired !! (normally the same as the t last returned by [[dvode]]). !! for valid results, t must lie between tcur - hu and tcur. !! (see optional output for tcur and hu.) integer,intent(in) :: k !! integer order of the derivative desired. k must satisfy !! 0 <= k <= nqcur, where nqcur is the current order !! (see optional output). the capability corresponding !! to k = 0, i.e. computing y(t), is already provided !! by [[dvode]] directly. since nqcur >= 1, the first !! derivative dy/dt is always available with [[dvindy]]. integer,intent(in) :: ldyh !! column length of yh, equal to the initial value of neq. real(wp),intent(in) :: yh(ldyh,*) !! the history array yh real(wp),intent(out) :: dky(*) !! a real array of length neq containing the computed value !! of the k-th derivative of y(t). integer,intent(out) :: iflag !! integer flag, returned as 0 if k and t were legal, !! -1 if k was illegal, and -2 if t was illegal. !! on an error return, a message is also written. real(wp) :: c , r , s , tfuzz , tn1 , tp integer :: i , ic , j , jb , jb2 , jj , jj1 , jp1 character(len=80) :: msg real(wp),parameter :: hun = 100.0_wp iflag = 0 if ( k<0 .or. k>me%dat%nq ) then msg = 'dvindy-- k (=i1) illegal ' call me%xerrwd(msg,30,51,1,1,k,0,0,zero,zero) iflag = -1 return else tfuzz = hun*me%dat%uround*sign(abs(me%dat%tn)+abs(me%dat%hu),me%dat%hu) tp = me%dat%tn - me%dat%hu - tfuzz tn1 = me%dat%tn + tfuzz if ( (t-tp)*(t-tn1)>zero ) then msg = 'dvindy-- t (=r1) illegal ' call me%xerrwd(msg,30,52,1,0,0,0,1,t,zero) msg = ' t not in interval tcur - hu (= r1) to tcur (=r2) ' call me%xerrwd(msg,60,52,1,0,0,0,2,tp,me%dat%tn) iflag = -2 return else s = (t-me%dat%tn)/me%dat%h ic = 1 if ( k/=0 ) then jj1 = me%dat%l - k do jj = jj1 , me%dat%nq ic = ic*jj enddo endif c = real(ic,wp) do i = 1 , me%dat%n dky(i) = c*yh(i,me%dat%l) enddo if ( k/=me%dat%nq ) then jb2 = me%dat%nq - k do jb = 1 , jb2 j = me%dat%nq - jb jp1 = j + 1 ic = 1 if ( k/=0 ) then jj1 = jp1 - k do jj = jj1 , j ic = ic*jj enddo endif c = real(ic,wp) do i = 1 , me%dat%n dky(i) = c*yh(i,jp1) + s*dky(i) enddo enddo if ( k==0 ) return endif endif endif r = me%dat%h**(-k) call dscal(me%dat%n,r,dky,1) end subroutine dvindy !***************************************************************************************** !> ! dvstep performs one step of the integration of an initial value ! problem for a system of ordinary differential equations. ! ! dvstep calls subroutine [[dvnlsd]] for the solution of the nonlinear system ! arising in the time step. thus it is independent of the problem ! jacobian structure and the type of nonlinear system solution method. ! dvstep returns a completion flag kflag (in common). ! a return with kflag = -1 or -2 means either abs(h) = hmin or 10 ! consecutive failures occurred. on a return with kflag negative, ! the values of tn and the yh array are as of the beginning of the last ! step, and h is the last step size attempted. subroutine dvstep(me,y,yh,ldyh,yh1,ewt,savf,vsav,acor,wm,iwm) class(dvode_t),intent(inout) :: me integer,intent(in) :: ldyh !! a constant integer >= n, the first dimension of yh. !! n is the number of odes in the system. real(wp),intent(inout) :: y(*) !! an array of length n used for the dependent variable vector. real(wp),intent(inout) :: yh(ldyh,*) !! an ldyh by lmax array containing the dependent variables !! and their approximate scaled derivatives, where !! lmax = maxord + 1. yh(i,j+1) contains the approximate !! j-th derivative of y(i), scaled by h**j/factorial(j) !! (j = 0,1,...,nq). on entry for the first step, the first !! two columns of yh must be set from the initial values. real(wp),intent(inout) :: yh1(*) !! a one-dimensional array occupying the same space as yh. real(wp),intent(in) :: ewt(*) !! an array of length n containing multiplicative weights !! for local error measurements. local errors in y(i) are !! compared to 1.0/ewt(i) in various error tests. real(wp) :: savf(*) !! an array of working storage, of length n. !! also used for input of yh(*,maxord+2) when jstart = -1 !! and maxord < the current order nq. real(wp) :: vsav(*) !! a work array of length n passed to subroutine [[dvnlsd]]. real(wp) :: acor(*) !! a work array of length n, used for the accumulated !! corrections. on a successful return, acor(i) contains !! the estimated one-step local error in y(i). real(wp) :: wm(*) !! real work array associated with matrix !! operations in [[dvnlsd]]. integer :: iwm(*) !! integer work array associated with matrix !! operations in [[dvnlsd]]. real(wp) :: cnquot , ddn , dsm , dup , etaqp1 , & flotl , r , told integer :: i , i1 , i2 , iback , j , jb , ncf , nflag integer,parameter :: kfc = -3 integer,parameter :: kfh = -7 integer,parameter :: mxncf = 10 real(wp),parameter :: one = 1.0_wp real(wp),parameter :: addon = 1.0e-6_wp real(wp),parameter :: bias1 = 6.0_wp real(wp),parameter :: bias2 = 6.0_wp real(wp),parameter :: bias3 = 10.0_wp real(wp),parameter :: etacf = 0.25_wp real(wp),parameter :: etamin = 0.1_wp real(wp),parameter :: etamxf = 0.2_wp real(wp),parameter :: etamx1 = 10000.0_wp real(wp),parameter :: etamx2 = 10.0_wp real(wp),parameter :: etamx3 = 10.0_wp real(wp),parameter :: onepsm = 1.00001_wp real(wp),parameter :: thresh = 1.5_wp me%dat%kflag = 0 told = me%dat%tn ncf = 0 me%dat%jcur = 0 nflag = 0 if ( me%dat%jstart<=0 ) then if ( me%dat%jstart==-1 ) then !----------------------------------------------------------------------- ! the following block handles preliminaries needed when jstart = -1. ! if n was reduced, zero out part of yh to avoid undefined references. ! if maxord was reduced to a value less than the tentative order newq, ! then nq is set to maxord, and a new h ratio eta is chosen. ! otherwise, we take the same preliminary actions as for jstart > 0. ! in any case, nqwait is reset to l = nq + 1 to prevent further ! changes in order for that many steps. ! the new h ratio eta is limited by the input h if kuth = 1, ! by hmin if kuth = 0, and by hmxi in any case. ! finally, the history array yh is rescaled. !----------------------------------------------------------------------- me%dat%lmax = me%dat%maxord + 1 if ( me%dat%n/=ldyh ) then i1 = 1 + (me%dat%newq+1)*ldyh i2 = (me%dat%maxord+1)*ldyh if ( i1<=i2 ) then do i = i1 , i2 yh1(i) = zero enddo endif endif if ( me%dat%newq>me%dat%maxord ) then flotl = real(me%dat%lmax,wp) if ( me%dat%maxord<me%dat%nq-1 ) then ddn = me%dvnorm(me%dat%n,savf(1:me%dat%n),ewt(1:me%dat%n))/me%dat%tq(1) me%dat%eta = one/((bias1*ddn)**(one/flotl)+addon) endif if ( me%dat%maxord==me%dat%nq .and. me%dat%newq==me%dat%nq+1 ) me%dat%eta = me%etaq if ( me%dat%maxord==me%dat%nq-1 .and. me%dat%newq==me%dat%nq+1 ) then me%dat%eta = me%etaqm1 call me%dvjust(yh,ldyh,-1) endif if ( me%dat%maxord==me%dat%nq-1 .and. me%dat%newq==me%dat%nq ) then ddn = me%dvnorm(me%dat%n,savf(1:me%dat%n),ewt(1:me%dat%n))/me%dat%tq(1) me%dat%eta = one/((bias1*ddn)**(one/flotl)+addon) call me%dvjust(yh,ldyh,-1) endif me%dat%eta = min(me%dat%eta,one) me%dat%nq = me%dat%maxord me%dat%l = me%dat%lmax endif if ( me%dat%kuth==1 ) me%dat%eta = min(me%dat%eta,abs(me%dat%h/me%dat%hscal)) if ( me%dat%kuth==0 ) me%dat%eta = max(me%dat%eta,me%dat%hmin/abs(me%dat%hscal)) me%dat%eta = me%dat%eta/max(one,abs(me%dat%hscal)*me%dat%hmxi*me%dat%eta) me%dat%newh = 1 me%dat%nqwait = me%dat%l if ( me%dat%newq>me%dat%maxord ) goto 300 else !----------------------------------------------------------------------- ! on the first call, the order is set to 1, and other variables are ! initialized. etamax is the maximum ratio by which h can be increased ! in a single step. it is normally 10, but is larger during the ! first step to compensate for the small initial h. if a failure ! occurs (in corrector convergence or error test), etamax is set to 1 ! for the next increase. !----------------------------------------------------------------------- me%dat%lmax = me%dat%maxord + 1 me%dat%nq = 1 me%dat%l = 2 me%dat%nqnyh = me%dat%nq*ldyh me%dat%tau(1) = me%dat%h me%dat%prl1 = one me%dat%rc = zero me%dat%etamax = etamx1 me%dat%nqwait = 2 me%dat%hscal = me%dat%h goto 400 end if else if ( me%dat%kuth==1 ) then !----------------------------------------------------------------------- ! take preliminary actions on a normal continuation step (jstart>0). ! if the driver changed h, then eta must be reset and newh set to 1. ! if a change of order was dictated on the previous step, then ! it is done here and appropriate adjustments in the history are made. ! on an order decrease, the history array is adjusted by dvjust. ! on an order increase, the history array is augmented by a column. ! on a change of step size h, the history array yh is rescaled. !----------------------------------------------------------------------- me%dat%eta = min(me%dat%eta,me%dat%h/me%dat%hscal) me%dat%newh = 1 endif if ( me%dat%newh==0 ) goto 400 if ( me%dat%newq<me%dat%nq ) then call me%dvjust(yh,ldyh,-1) me%dat%nq = me%dat%newq me%dat%l = me%dat%nq + 1 me%dat%nqwait = me%dat%l else if ( me%dat%newq>me%dat%nq ) then call me%dvjust(yh,ldyh,1) me%dat%nq = me%dat%newq me%dat%l = me%dat%nq + 1 me%dat%nqwait = me%dat%l endif ! rescale the history array for a change in h by a factor of eta. ------ 300 r = one do j = 2 , me%dat%l r = r*me%dat%eta call dscal(me%dat%n,r,yh(1,j),1) enddo me%dat%h = me%dat%hscal*me%dat%eta me%dat%hscal = me%dat%h me%dat%rc = me%dat%rc*me%dat%eta me%dat%nqnyh = me%dat%nq*ldyh !----------------------------------------------------------------------- ! this section computes the predicted values by effectively ! multiplying the yh array by the pascal triangle matrix. ! dvset is called to calculate all integration coefficients. ! rc is the ratio of new to old values of the coefficient h/el(2)=h/l1. !----------------------------------------------------------------------- 400 me%dat%tn = me%dat%tn + me%dat%h i1 = me%dat%nqnyh + 1 do jb = 1 , me%dat%nq i1 = i1 - ldyh do i = i1 , me%dat%nqnyh yh1(i) = yh1(i) + yh1(i+ldyh) enddo enddo call me%dvset() me%dat%rl1 = one/me%dat%el(2) me%dat%rc = me%dat%rc*(me%dat%rl1/me%dat%prl1) me%dat%prl1 = me%dat%rl1 ! call the nonlinear system solver. ------------------------------------ call me%dvnlsd(y,yh,ldyh,vsav,savf,ewt,acor,iwm,wm,nflag) if ( nflag==0 ) then !----------------------------------------------------------------------- ! the corrector has converged (nflag = 0). the local error test is ! made and control passes to statement 500 if it fails. !----------------------------------------------------------------------- dsm = me%dat%acnrm/me%dat%tq(2) if ( dsm>one ) then !----------------------------------------------------------------------- ! the error test failed. kflag keeps track of multiple failures. ! restore tn and the yh array to their previous values, and prepare ! to try the step again. compute the optimum step size for the ! same order. after repeated failures, h is forced to decrease ! more rapidly. !----------------------------------------------------------------------- me%dat%kflag = me%dat%kflag - 1 me%dat%netf = me%dat%netf + 1 nflag = -2 me%dat%tn = told i1 = me%dat%nqnyh + 1 do jb = 1 , me%dat%nq i1 = i1 - ldyh do i = i1 , me%dat%nqnyh yh1(i) = yh1(i) - yh1(i+ldyh) enddo enddo if ( abs(me%dat%h)<=me%dat%hmin*onepsm ) then !----------------------------------------------------------------------- ! all returns are made through this section. ! on a successful return, etamax is reset and acor is scaled. !----------------------------------------------------------------------- me%dat%kflag = -1 me%dat%jstart = 1 return else me%dat%etamax = one if ( me%dat%kflag>kfc ) then ! compute ratio of new h to current h at the current order. ------------ flotl = real(me%dat%l,wp) me%dat%eta = one/((bias2*dsm)**(one/flotl)+addon) me%dat%eta = max(me%dat%eta,me%dat%hmin/abs(me%dat%h),etamin) if ( (me%dat%kflag<=-2) .and. (me%dat%eta>etamxf) ) me%dat%eta = etamxf goto 300 !----------------------------------------------------------------------- ! control reaches this section if 3 or more consecutive failures ! have occurred. it is assumed that the elements of the yh array ! have accumulated errors of the wrong order. the order is reduced ! by one, if possible. then h is reduced by a factor of 0.1 and ! the step is retried. after a total of 7 consecutive failures, ! an exit is taken with kflag = -1. !----------------------------------------------------------------------- else if ( me%dat%kflag==kfh ) then me%dat%kflag = -1 me%dat%jstart = 1 return else if ( me%dat%nq==1 ) then me%dat%eta = max(etamin,me%dat%hmin/abs(me%dat%h)) me%dat%h = me%dat%h*me%dat%eta me%dat%hscal = me%dat%h me%dat%tau(1) = me%dat%h call me%f(me%dat%n,me%dat%tn,y(1:me%dat%n),savf(1:me%dat%n)) me%dat%nfe = me%dat%nfe + 1 do i = 1 , me%dat%n yh(i,2) = me%dat%h*savf(i) enddo me%dat%nqwait = 10 goto 400 else me%dat%eta = max(etamin,me%dat%hmin/abs(me%dat%h)) call me%dvjust(yh,ldyh,-1) me%dat%l = me%dat%nq me%dat%nq = me%dat%nq - 1 me%dat%nqwait = me%dat%l goto 300 endif endif else !----------------------------------------------------------------------- ! after a successful step, update the yh and tau arrays and decrement ! nqwait. if nqwait is then 1 and nq < maxord, then acor is saved ! for use in a possible order increase on the next step. ! if etamax = 1 (a failure occurred this step), keep nqwait >= 2. !----------------------------------------------------------------------- me%dat%kflag = 0 me%dat%nst = me%dat%nst + 1 me%dat%hu = me%dat%h me%dat%nqu = me%dat%nq do iback = 1 , me%dat%nq i = me%dat%l - iback me%dat%tau(i+1) = me%dat%tau(i) enddo me%dat%tau(1) = me%dat%h do j = 1 , me%dat%l call daxpy(me%dat%n,me%dat%el(j),acor,1,yh(1,j),1) enddo me%dat%nqwait = me%dat%nqwait - 1 if ( (me%dat%l/=me%dat%lmax) .and. (me%dat%nqwait==1) ) then call dcopy(me%dat%n,acor,1,yh(1,me%dat%lmax),1) me%dat%conp = me%dat%tq(5) endif if ( me%dat%etamax/=one ) then !----------------------------------------------------------------------- ! if nqwait = 0, an increase or decrease in order by one is considered. ! factors etaq, etaqm1, etaqp1 are computed by which h could ! be multiplied at order q, q-1, or q+1, respectively. ! the largest of these is determined, and the new order and ! step size set accordingly. ! a change of h or nq is made only if h increases by at least a ! factor of thresh. if an order change is considered and rejected, ! then nqwait is set to 2 (reconsider it after 2 steps). !----------------------------------------------------------------------- ! compute ratio of new h to current h at the current order. ------------ flotl = real(me%dat%l,wp) me%etaq = one/((bias2*dsm)**(one/flotl)+addon) if ( me%dat%nqwait==0 ) then me%dat%nqwait = 2 me%etaqm1 = zero if ( me%dat%nq/=1 ) then ! compute ratio of new h to current h at the current order less one. --- ddn = me%dvnorm(me%dat%n,yh(1,me%dat%l),ewt(1:me%dat%n))/me%dat%tq(1) me%etaqm1 = one/((bias1*ddn)**(one/(flotl-one))+addon) endif etaqp1 = zero if ( me%dat%l/=me%dat%lmax ) then ! compute ratio of new h to current h at current order plus one. ------- cnquot = (me%dat%tq(5)/me%dat%conp)*(me%dat%h/me%dat%tau(2))**me%dat%l do i = 1 , me%dat%n savf(i) = acor(i) - cnquot*yh(i,me%dat%lmax) enddo dup = me%dvnorm(me%dat%n,savf(1:me%dat%n),ewt(1:me%dat%n))/me%dat%tq(3) etaqp1 = one/((bias3*dup)**(one/(flotl+one))+addon) endif if ( me%etaq<etaqp1 ) then if ( etaqp1<=me%etaqm1 ) goto 420 me%dat%eta = etaqp1 me%dat%newq = me%dat%nq + 1 call dcopy(me%dat%n,acor,1,yh(1,me%dat%lmax),1) goto 450 else if ( me%etaq<me%etaqm1 ) then goto 420 endif endif me%dat%eta = me%etaq me%dat%newq = me%dat%nq goto 450 else if ( me%dat%nqwait<2 ) me%dat%nqwait = 2 me%dat%newq = me%dat%nq me%dat%newh = 0 me%dat%eta = one me%dat%hnew = me%dat%h goto 500 endif 420 me%dat%eta = me%etaqm1 me%dat%newq = me%dat%nq - 1 endif ! test tentative new h against thresh, etamax, and hmxi, then exit. ---- 450 if ( me%dat%eta<thresh .or. me%dat%etamax==one ) then me%dat%newq = me%dat%nq me%dat%newh = 0 me%dat%eta = one me%dat%hnew = me%dat%h else me%dat%eta = min(me%dat%eta,me%dat%etamax) me%dat%eta = me%dat%eta/max(one,abs(me%dat%h)*me%dat%hmxi*me%dat%eta) me%dat%newh = 1 me%dat%hnew = me%dat%h*me%dat%eta endif else !----------------------------------------------------------------------- ! the vnls routine failed to achieve convergence (nflag /= 0). ! the yh array is retracted to its values before prediction. ! the step size h is reduced and the step is retried, if possible. ! otherwise, an error exit is taken. !----------------------------------------------------------------------- ncf = ncf + 1 me%dat%ncfn = me%dat%ncfn + 1 me%dat%etamax = one me%dat%tn = told i1 = me%dat%nqnyh + 1 do jb = 1 , me%dat%nq i1 = i1 - ldyh do i = i1 , me%dat%nqnyh yh1(i) = yh1(i) - yh1(i+ldyh) enddo enddo if ( nflag<-1 ) then if ( nflag==-2 ) me%dat%kflag = -3 if ( nflag==-3 ) me%dat%kflag = -4 me%dat%jstart = 1 return else if ( abs(me%dat%h)<=me%dat%hmin*onepsm ) then me%dat%kflag = -2 me%dat%jstart = 1 return else if ( ncf==mxncf ) then me%dat%kflag = -2 me%dat%jstart = 1 return else me%dat%eta = etacf me%dat%eta = max(me%dat%eta,me%dat%hmin/abs(me%dat%h)) nflag = -1 goto 300 endif endif 500 me%dat%etamax = etamx3 if ( me%dat%nst<=10 ) me%dat%etamax = etamx2 r = one/me%dat%tq(2) call dscal(me%dat%n,r,acor,1) me%dat%jstart = 1 end subroutine dvstep !***************************************************************************************** !> ! dvset is called by [[dvstep]] and sets coefficients for use there. ! ! for each order `nq`, the coefficients in `el` are calculated by use of ! the generating polynomial `lambda(x)`, with coefficients `el(i)`. ! ! $$ \lambda(x) = \mathrm{el}(1) + \mathrm{el}(2) x + \cdots + \mathrm{el}(n_q+1)(x^n_q) $$ ! ! for the backward differentiation formulas, ! ! $$ \lambda(x) = \left(1 + x/x_i \cdot (n_q) \right) \prod_{i=1}^{n_q-1} \left(1 + x/x_i(i) \right) $$ ! ! for the adams formulas, ! ! $$ \frac{d}{dx} \lambda(x) = c \prod_{i=1}^{n_q-1} \left(1 + x/x_i(i) \right) $$ ! $$ \lambda(-1) = 0 , \lambda(0) = 1 $$ ! ! where `c` is a normalization constant. ! in both cases, \(x_i(i)\) is defined by ! ! $$ \begin{align} ! h x_i(i) &= t_n - t_{n-i} \\ ! &= h + \tau(1) + \tau(2) + \cdots + \tau(i-1) ! \end{align} $$ ! ! in addition to variables described previously, communication ! with [[dvset]] uses the following class variables: ! ! * `tau` = a vector of length 13 containing the past `nq` values ! of h. ! * `el` = a vector of length 13 in which vset stores the ! coefficients for the corrector formula. ! * `tq` = a vector of length 5 in which vset stores constants ! used for the convergence test, the error test, and the ! selection of `h` at a new order. ! * `meth` = the basic method indicator. ! * `nq` = the current order. ! * `l` = `nq + 1`, the length of the vector stored in el, and ! the number of columns of the `yh` array being used. ! * `nqwait` = a counter controlling the frequency of order changes. ! an order change is about to be considered if `nqwait = 1`. subroutine dvset(me) class(dvode_t),intent(inout) :: me real(wp) :: ahatn0 , alph0 , cnqm1 , csum , elp , & em(13) , em0 , floti , flotl , flotnq , hsum , & rxi , rxis , s , t1 , t2 , t3 , t4 , t5 , & t6 , xi integer :: i , iback , j , jp1 , nqm1 , nqm2 real(wp),parameter :: cortes = 0.1_wp real(wp),parameter :: one = 1.0_wp real(wp),parameter :: six = 6.0_wp real(wp),parameter :: two = 2.0_wp flotl = real(me%dat%l,wp) nqm1 = me%dat%nq - 1 nqm2 = me%dat%nq - 2 if ( me%dat%meth==2 ) then ! set coefficients for bdf methods. ------------------------------------ do i = 3 , me%dat%l me%dat%el(i) = zero enddo me%dat%el(1) = one me%dat%el(2) = one alph0 = -one ahatn0 = -one hsum = me%dat%h rxi = one rxis = one if ( me%dat%nq/=1 ) then do j = 1 , nqm2 ! in el, construct coefficients of (1+x/xi(1))*...*(1+x/xi(j+1)). ------ hsum = hsum + me%dat%tau(j) rxi = me%dat%h/hsum jp1 = j + 1 alph0 = alph0 - one/real(jp1,wp) do iback = 1 , jp1 i = (j+3) - iback me%dat%el(i) = me%dat%el(i) + me%dat%el(i-1)*rxi enddo enddo alph0 = alph0 - one/real(me%dat%nq,wp) rxis = -me%dat%el(2) - alph0 hsum = hsum + me%dat%tau(nqm1) rxi = me%dat%h/hsum ahatn0 = -me%dat%el(2) - rxi do iback = 1 , me%dat%nq i = (me%dat%nq+2) - iback me%dat%el(i) = me%dat%el(i) + me%dat%el(i-1)*rxis enddo endif t1 = one - ahatn0 + alph0 t2 = one + real(me%dat%nq,wp)*t1 me%dat%tq(2) = abs(alph0*t2/t1) me%dat%tq(5) = abs(t2/(me%dat%el(me%dat%l)*rxi/rxis)) if ( me%dat%nqwait==1 ) then cnqm1 = rxis/me%dat%el(me%dat%l) t3 = alph0 + one/real(me%dat%nq,wp) t4 = ahatn0 + rxi elp = t3/(one-t4+t3) me%dat%tq(1) = abs(elp/cnqm1) hsum = hsum + me%dat%tau(me%dat%nq) rxi = me%dat%h/hsum t5 = alph0 - one/real(me%dat%nq+1,wp) t6 = ahatn0 - rxi elp = t2/(one-t6+t5) me%dat%tq(3) = abs(elp*rxi*(flotl+one)*t5) endif ! set coefficients for adams methods. ---------------------------------- else if ( me%dat%nq/=1 ) then hsum = me%dat%h em(1) = one flotnq = flotl - one do i = 2 , me%dat%l em(i) = zero enddo do j = 1 , nqm1 if ( (j==nqm1) .and. (me%dat%nqwait==1) ) then s = one csum = zero do i = 1 , nqm1 csum = csum + s*em(i)/real(i+1,wp) s = -s enddo me%dat%tq(1) = em(nqm1)/(flotnq*csum) endif rxi = me%dat%h/hsum do iback = 1 , j i = (j+2) - iback em(i) = em(i) + em(i-1)*rxi enddo hsum = hsum + me%dat%tau(j) enddo ! compute integral from -1 to 0 of polynomial and of x times it. ------- s = one em0 = zero csum = zero do i = 1 , me%dat%nq floti = real(i,wp) em0 = em0 + s*em(i)/floti csum = csum + s*em(i)/(floti+one) s = -s enddo ! in el, form coefficients of normalized integrated polynomial. -------- s = one/em0 me%dat%el(1) = one do i = 1 , me%dat%nq me%dat%el(i+1) = s*em(i)/real(i,wp) enddo xi = hsum/me%dat%h me%dat%tq(2) = xi*em0/csum me%dat%tq(5) = xi/me%dat%el(me%dat%l) if ( me%dat%nqwait==1 ) then ! for higher order control constant, multiply polynomial by 1+x/xi(q). - rxi = one/xi do iback = 1 , me%dat%nq i = (me%dat%l+1) - iback em(i) = em(i) + em(i-1)*rxi enddo ! compute integral of polynomial. -------------------------------------- s = one csum = zero do i = 1 , me%dat%l csum = csum + s*em(i)/real(i+1,wp) s = -s enddo me%dat%tq(3) = flotl*em0/csum endif else me%dat%el(1) = one me%dat%el(2) = one me%dat%tq(1) = one me%dat%tq(2) = two me%dat%tq(3) = six*me%dat%tq(2) me%dat%tq(5) = one endif me%dat%tq(4) = cortes*me%dat%tq(2) end subroutine dvset !***************************************************************************************** !> ! this subroutine adjusts the `yh` array on reduction of order, ! and also when the order is increased for the stiff option (meth = 2). subroutine dvjust(me,yh,ldyh,iord) class(dvode_t),intent(inout) :: me integer,intent(in) :: ldyh !! leading dimension of `yh` integer,intent(in) :: iord !! an integer flag used when meth = 2 to indicate an order !! increase (iord = +1) or an order decrease (iord = -1). real(wp),intent(inout) :: yh(ldyh,*) real(wp) :: alph0 , alph1 , hsum , prod , t1 , xi , xiold integer :: i , iback , j , jp1 , lp1 , nqm1 , nqm2 , nqp1 real(wp),parameter :: one = 1.0_wp if ( (me%dat%nq==2) .and. (iord/=1) ) return nqm1 = me%dat%nq - 1 nqm2 = me%dat%nq - 2 if ( me%dat%meth==2 ) then !----------------------------------------------------------------------- ! stiff option... ! check to see if the order is being increased or decreased. !----------------------------------------------------------------------- if ( iord==1 ) then ! order increase. ------------------------------------------------------ do j = 1 , me%dat%lmax me%dat%el(j) = zero enddo me%dat%el(3) = one alph0 = -one alph1 = one prod = one xiold = one hsum = me%dat%hscal if ( me%dat%nq/=1 ) then do j = 1 , nqm1 ! construct coefficients of x*x*(x+xi(1))*...*(x+xi(j)). --------------- jp1 = j + 1 hsum = hsum + me%dat%tau(jp1) xi = hsum/me%dat%hscal prod = prod*xi alph0 = alph0 - one/real(jp1,wp) alph1 = alph1 + one/xi do iback = 1 , jp1 i = (j+4) - iback me%dat%el(i) = me%dat%el(i)*xiold + me%dat%el(i-1) enddo xiold = xi enddo endif t1 = (-alph0-alph1)/prod ! load column l + 1 in yh array. --------------------------------------- lp1 = me%dat%l + 1 do i = 1 , me%dat%n yh(i,lp1) = t1*yh(i,me%dat%lmax) enddo ! add correction terms to yh array. ------------------------------------ nqp1 = me%dat%nq + 1 do j = 3 , nqp1 call daxpy(me%dat%n,me%dat%el(j),yh(1,lp1),1,yh(1,j),1) enddo else ! order decrease. ------------------------------------------------------ do j = 1 , me%dat%lmax me%dat%el(j) = zero enddo me%dat%el(3) = one hsum = zero do j = 1 , nqm2 ! construct coefficients of x*x*(x+xi(1))*...*(x+xi(j)). --------------- hsum = hsum + me%dat%tau(j) xi = hsum/me%dat%hscal jp1 = j + 1 do iback = 1 , jp1 i = (j+4) - iback me%dat%el(i) = me%dat%el(i)*xi + me%dat%el(i-1) enddo enddo ! subtract correction terms from yh array. ----------------------------- do j = 3 , me%dat%nq do i = 1 , me%dat%n yh(i,j) = yh(i,j) - yh(i,me%dat%l)*me%dat%el(j) enddo enddo return endif !----------------------------------------------------------------------- ! nonstiff option... ! check to see if the order is being increased or decreased. !----------------------------------------------------------------------- else if ( iord==1 ) then ! order increase. ------------------------------------------------------ ! zero out next column in yh array. ------------------------------------ lp1 = me%dat%l + 1 do i = 1 , me%dat%n yh(i,lp1) = zero enddo return else ! order decrease. ------------------------------------------------------ do j = 1 , me%dat%lmax me%dat%el(j) = zero enddo me%dat%el(2) = one hsum = zero do j = 1 , nqm2 ! construct coefficients of x*(x+xi(1))*...*(x+xi(j)). ----------------- hsum = hsum + me%dat%tau(j) xi = hsum/me%dat%hscal jp1 = j + 1 do iback = 1 , jp1 i = (j+3) - iback me%dat%el(i) = me%dat%el(i)*xi + me%dat%el(i-1) enddo enddo ! construct coefficients of integrated polynomial. --------------------- do j = 2 , nqm1 me%dat%el(j+1) = real(me%dat%nq,wp)*me%dat%el(j)/real(j,wp) enddo ! subtract correction terms from yh array. ----------------------------- do j = 3 , me%dat%nq do i = 1 , me%dat%n yh(i,j) = yh(i,j) - yh(i,me%dat%l)*me%dat%el(j) enddo enddo return endif end subroutine dvjust !***************************************************************************************** !> ! subroutine dvnlsd is a nonlinear system solver, which uses functional ! iteration or a chord (modified newton) method. for the chord method ! direct linear algebraic system solvers are used. subroutine dvnlsd ! then handles the corrector phase of this integration package. subroutine dvnlsd(me,y,yh,ldyh,vsav,savf,ewt,acor,iwm,wm,nflag) class(dvode_t),intent(inout) :: me real(wp),intent(inout) :: y(*) !! the dependent variable, a vector of length n integer,intent(in) :: ldyh !! a constant >= n, the first dimension of yh real(wp),intent(inout) :: yh(ldyh,*) !! the nordsieck (taylor) array, ldyh by lmax, input !! and output. on input, it contains predicted values. real(wp) :: vsav(*) !! unused work array. real(wp) :: savf(*) !! a work array of length n. real(wp),intent(in) :: ewt(*) !! an error weight vector of length n real(wp),intent(inout) :: acor(*) !! a work array of length n, used for the accumulated !! corrections to the predicted y vector. real(wp),intent(inout) :: wm(*) !! real work array associated with matrix !! operations in chord iteration (miter /= 0). integer,intent(inout) :: iwm(*) !! integer work array associated with matrix !! operations in chord iteration (miter /= 0). integer,intent(inout) :: nflag !! input/output flag, with values and meanings as follows: !! !! * **input:** !! * 0 first call for this time step. !! * -1 convergence failure in previous call to dvnlsd. !! * -2 error test failure in dvstep. !! !! * **output:** !! * 0 successful completion of nonlinear solver. !! * -1 convergence failure or singular matrix. !! * -2 unrecoverable error in matrix preprocessing (cannot occur here). !! * -3 unrecoverable error in solution (cannot occur here). real(wp) :: cscale , dcon , del , delp integer :: i , ierpj , iersl , m integer,parameter :: maxcor = 3 integer,parameter :: msbp = 20 real(wp),parameter :: ccmax = 0.3_wp real(wp),parameter :: crdown = 0.3_wp real(wp),parameter :: rdiv = 2.0_wp real(wp),parameter :: one = 1.0_wp real(wp),parameter :: two = 2.0_wp !----------------------------------------------------------------------- ! on the first step, on a change of method order, or after a ! nonlinear convergence failure with nflag = -2, set ipup = miter ! to force a jacobian update when miter /= 0. !----------------------------------------------------------------------- if ( me%dat%jstart==0 ) me%dat%nslp = 0 if ( nflag==0 ) me%dat%icf = 0 if ( nflag==-2 ) me%dat%ipup = me%dat%miter if ( (me%dat%jstart==0) .or. (me%dat%jstart==-1) ) me%dat%ipup = me%dat%miter ! if this is functional iteration, set crate == 1 and drop to 220 if ( me%dat%miter==0 ) then me%dat%crate = one else !----------------------------------------------------------------------- ! rc is the ratio of new to old values of the coefficient h/el(2)=h/l1. ! when rc differs from 1 by more than ccmax, ipup is set to miter ! to force dvjac to be called, if a jacobian is involved. ! in any case, dvjac is called at least every msbp steps. !----------------------------------------------------------------------- me%dat%drc = abs(me%dat%rc-one) if ( me%dat%drc>ccmax .or. me%dat%nst>=me%dat%nslp+msbp ) me%dat%ipup = me%dat%miter end if corrector : do !----------------------------------------------------------------------- ! up to maxcor corrector iterations are taken. a convergence test is ! made on the r.m.s. norm of each correction, weighted by the error ! weight vector ewt. the sum of the corrections is accumulated in the ! vector acor(i). the yh array is not altered in the corrector loop. !----------------------------------------------------------------------- m = 0 delp = zero call dcopy(me%dat%n,yh(1,1),1,y,1) call me%f(me%dat%n,me%dat%tn,y(1:me%dat%n),savf(1:me%dat%n)) me%dat%nfe = me%dat%nfe + 1 if ( me%dat%ipup>0 ) then !----------------------------------------------------------------------- ! if indicated, the matrix p = i - h*rl1*j is reevaluated and ! preprocessed before starting the corrector iteration. ipup is set ! to 0 as an indicator that this has been done. !----------------------------------------------------------------------- call me%dvjac(y,yh,ldyh,ewt,acor,savf,wm,iwm,ierpj) me%dat%ipup = 0 me%dat%rc = one me%dat%drc = zero me%dat%crate = one me%dat%nslp = me%dat%nst ! if matrix is singular, take error return to force cut in step size. -- if ( ierpj/=0 ) exit corrector endif do i = 1 , me%dat%n acor(i) = zero enddo do ! this is a looping point for the corrector iteration. ----------------- if ( me%dat%miter/=0 ) then !----------------------------------------------------------------------- ! in the case of the chord method, compute the corrector error, ! and solve the linear system with that as right-hand side and ! p as coefficient matrix. the correction is scaled by the factor ! 2/(1+rc) to account for changes in h*rl1 since the last dvjac call. !----------------------------------------------------------------------- do i = 1 , me%dat%n y(i) = (me%dat%rl1*me%dat%h)*savf(i) - (me%dat%rl1*yh(i,2)+acor(i)) enddo call me%dvsol(wm,iwm,y,iersl) me%dat%nni = me%dat%nni + 1 if ( iersl>0 ) exit if ( me%dat%meth==2 .and. me%dat%rc/=one ) then cscale = two/(one+me%dat%rc) call dscal(me%dat%n,cscale,y,1) endif del = me%dvnorm(me%dat%n,y(1:me%dat%n),ewt(1:me%dat%n)) call daxpy(me%dat%n,one,y,1,acor,1) do i = 1 , me%dat%n y(i) = yh(i,1) + acor(i) enddo else !----------------------------------------------------------------------- ! in the case of functional iteration, update y directly from ! the result of the last function evaluation. !----------------------------------------------------------------------- do i = 1 , me%dat%n savf(i) = me%dat%rl1*(me%dat%h*savf(i)-yh(i,2)) enddo do i = 1 , me%dat%n y(i) = savf(i) - acor(i) enddo del = me%dvnorm(me%dat%n,y(1:me%dat%n),ewt(1:me%dat%n)) do i = 1 , me%dat%n y(i) = yh(i,1) + savf(i) enddo call dcopy(me%dat%n,savf,1,acor,1) endif !----------------------------------------------------------------------- ! test for convergence. if m > 0, an estimate of the convergence ! rate constant is stored in crate, and this is used in the test. !----------------------------------------------------------------------- if ( m/=0 ) me%dat%crate = max(crdown*me%dat%crate,del/delp) dcon = del*min(one,me%dat%crate)/me%dat%tq(4) if ( dcon<=one ) then ! return for successful step. ------------------------------------------ nflag = 0 me%dat%jcur = 0 me%dat%icf = 0 if ( m==0 ) me%dat%acnrm = del if ( m>0 ) me%dat%acnrm = me%dvnorm(me%dat%n,acor(1:me%dat%n),ewt(1:me%dat%n)) return else m = m + 1 if ( m/=maxcor ) then if ( m<2 .or. del<=rdiv*delp ) then delp = del call me%f(me%dat%n,me%dat%tn,y(1:me%dat%n),savf(1:me%dat%n)) me%dat%nfe = me%dat%nfe + 1 cycle endif endif exit endif end do if ( me%dat%miter/=0 .and. me%dat%jcur/=1 ) then me%dat%icf = 1 me%dat%ipup = me%dat%miter else exit corrector endif end do corrector nflag = -1 me%dat%icf = 2 me%dat%ipup = me%dat%miter end subroutine dvnlsd !***************************************************************************************** !> ! dvjac is called by [[dvnlsd]] to compute and process the matrix ! `p = i - h*rl1*j` , where j is an approximation to the jacobian. ! ! here j is computed by the user-supplied routine `jac` if ! miter = 1 or 4, or by finite differencing if miter = 2, 3, or 5. ! if miter = 3, a diagonal approximation to j is used. ! if jsv = -1, j is computed from scratch in all cases. ! if jsv = 1 and miter = 1, 2, 4, or 5, and if the saved value of j is ! considered acceptable, then p is constructed from the saved j. ! j is stored in wm and replaced by p. if miter /= 3, p is then ! subjected to lu decomposition in preparation for later solution ! of linear systems with p as coefficient matrix. this is done ! by [[dgefa]] if miter = 1 or 2, and by dgbfa if miter = 4 or 5. subroutine dvjac(me,y,yh,ldyh,ewt,ftem,savf,wm,iwm,ierpj) class(dvode_t),intent(inout) :: me real(wp),intent(inout) :: y(*) !! vector containing predicted values on entry. integer,intent(in) :: ldyh !! a constant >= n, the first dimension of `yh`. real(wp),intent(in) :: yh(ldyh,*) !! the nordsieck array, an `ldyh` by `lmax` array. real(wp),intent(in) :: ewt(*) !! an error weight vector of length `n`. real(wp),intent(in) :: ftem(*) real(wp),intent(in) :: savf(*) !! array containing f evaluated at predicted y. real(wp),intent(inout) :: wm(*) !! real work space for matrices. in the output, it contains !! the inverse diagonal matrix if miter = 3 and the lu !! decomposition of p if miter is 1, 2 , 4, or 5. !! storage of matrix elements starts at wm(3). !! storage of the saved jacobian starts at wm(locjs). !! wm also contains the following matrix-related data: !! wm(1) = sqrt(uround), used in numerical jacobian step. !! wm(2) = h*rl1, saved for later use if miter = 3. integer,intent(inout) :: iwm(*) !! integer work space containing pivot information, !! starting at iwm(31), if miter is 1, 2, 4, or 5. !! iwm also contains band parameters ml = iwm(1) and !! mu = iwm(2) if miter is 4 or 5. integer,intent(out) :: ierpj !! output error flag, = 0 if no trouble, 1 if the p !! matrix is found to be singular. real(wp) :: con , di , fac , hrl1 , r , r0 , & srur , yi , yj , yjj integer :: i , i1 , i2 , ier , ii , j , j1 , jj , jok , lenp , mba , & mband , meb1 , meband , ml , ml3 , mu , np1 real(wp),parameter :: one = 1.0_wp real(wp),parameter :: thou = 1000.0_wp real(wp),parameter :: pt1 = 0.1_wp ierpj = 0 hrl1 = me%dat%h*me%dat%rl1 ! rl1 = 1/el(2) ! see whether j should be evaluated (jok = -1) or not (jok = 1). ------- jok = me%dat%jsv if ( me%dat%jsv==1 ) then if ( me%dat%nst==0 .or. me%dat%nst>me%dat%nslj+me%dat%msbj ) jok = -1 if ( me%dat%icf==1 .and. me%dat%drc<me%dat%ccmxj ) jok = -1 if ( me%dat%icf==2 ) jok = -1 endif ! end of setting jok. -------------------------------------------------- if ( jok==-1 .and. me%dat%miter==1 ) then ! if jok = -1 and miter = 1, call me%jac to evaluate jacobian. ------------ me%dat%nje = me%dat%nje + 1 me%dat%nslj = me%dat%nst me%dat%jcur = 1 lenp = me%dat%n*me%dat%n do i = 1 , lenp wm(i+2) = zero enddo call me%jac(me%dat%n,me%dat%tn,y(1:me%dat%n),0,0,wm(3),me%dat%n) if ( me%dat%jsv==1 ) call dcopy(lenp,wm(3),1,wm(me%dat%locjs),1) endif if ( jok==-1 .and. me%dat%miter==2 ) then ! if miter = 2, make n calls to f to approximate the jacobian. --------- me%dat%nje = me%dat%nje + 1 me%dat%nslj = me%dat%nst me%dat%jcur = 1 fac = me%dvnorm(me%dat%n,savf(1:me%dat%n),ewt(1:me%dat%n)) r0 = thou*abs(me%dat%h)*me%dat%uround*real(me%dat%n,wp)*fac if ( r0==zero ) r0 = one srur = wm(1) j1 = 2 do j = 1 , me%dat%n yj = y(j) r = max(srur*abs(yj),r0/ewt(j)) y(j) = y(j) + r fac = one/r call me%f(me%dat%n,me%dat%tn,y(1:me%dat%n),ftem(1:me%dat%n)) do i = 1 , me%dat%n wm(i+j1) = (ftem(i)-savf(i))*fac enddo y(j) = yj j1 = j1 + me%dat%n enddo me%dat%nfe = me%dat%nfe + me%dat%n lenp = me%dat%n*me%dat%n if ( me%dat%jsv==1 ) call dcopy(lenp,wm(3),1,wm(me%dat%locjs),1) endif if ( jok==1 .and. (me%dat%miter==1 .or. me%dat%miter==2) ) then me%dat%jcur = 0 lenp = me%dat%n*me%dat%n call dcopy(lenp,wm(me%dat%locjs),1,wm(3),1) endif if ( me%dat%miter==1 .or. me%dat%miter==2 ) then ! multiply jacobian by scalar, add identity, and do lu decomposition. -- con = -hrl1 call dscal(lenp,con,wm(3),1) j = 3 np1 = me%dat%n + 1 do i = 1 , me%dat%n wm(j) = wm(j) + one j = j + np1 enddo me%dat%nlu = me%dat%nlu + 1 call dgefa(wm(3),me%dat%n,me%dat%n,iwm(31),ier) if ( ier/=0 ) ierpj = 1 return endif ! end of code block for miter = 1 or 2. -------------------------------- if ( me%dat%miter==3 ) then ! if miter = 3, construct a diagonal approximation to j and p. --------- me%dat%nje = me%dat%nje + 1 me%dat%jcur = 1 wm(2) = hrl1 r = me%dat%rl1*pt1 do i = 1 , me%dat%n y(i) = y(i) + r*(me%dat%h*savf(i)-yh(i,2)) enddo call me%f(me%dat%n,me%dat%tn,y(1:me%dat%n),wm(3)) me%dat%nfe = me%dat%nfe + 1 do i = 1 , me%dat%n r0 = me%dat%h*savf(i) - yh(i,2) di = pt1*r0 - me%dat%h*(wm(i+2)-savf(i)) wm(i+2) = one if ( abs(r0)>=me%dat%uround/ewt(i) ) then if ( abs(di)==zero ) then ierpj = 1 return end if wm(i+2) = pt1*r0/di endif enddo return endif ! end of code block for miter = 3. ------------------------------------- ! set constants for miter = 4 or 5. ------------------------------------ ml = iwm(1) mu = iwm(2) ml3 = ml + 3 mband = ml + mu + 1 meband = mband + ml lenp = meband*me%dat%n if ( jok==-1 .and. me%dat%miter==4 ) then ! if jok = -1 and miter = 4, call me%jac to evaluate jacobian. ------------ me%dat%nje = me%dat%nje + 1 me%dat%nslj = me%dat%nst me%dat%jcur = 1 do i = 1 , lenp wm(i+2) = zero enddo call me%jac(me%dat%n,me%dat%tn,y(1:me%dat%n),ml,mu,wm(ml3),meband) if ( me%dat%jsv==1 ) call dacopy(mband,me%dat%n,wm(ml3),meband,wm(me%dat%locjs),mband) endif if ( jok==-1 .and. me%dat%miter==5 ) then ! if miter = 5, make ml+mu+1 calls to f to approximate the jacobian. --- me%dat%nje = me%dat%nje + 1 me%dat%nslj = me%dat%nst me%dat%jcur = 1 mba = min(mband,me%dat%n) meb1 = meband - 1 srur = wm(1) fac = me%dvnorm(me%dat%n,savf(1:me%dat%n),ewt(1:me%dat%n)) r0 = thou*abs(me%dat%h)*me%dat%uround*real(me%dat%n,wp)*fac if ( r0==zero ) r0 = one do j = 1 , mba do i = j , me%dat%n , mband yi = y(i) r = max(srur*abs(yi),r0/ewt(i)) y(i) = y(i) + r enddo call me%f(me%dat%n,me%dat%tn,y(1:me%dat%n),ftem(1:me%dat%n)) do jj = j , me%dat%n , mband y(jj) = yh(jj,1) yjj = y(jj) r = max(srur*abs(yjj),r0/ewt(jj)) fac = one/r i1 = max(jj-mu,1) i2 = min(jj+ml,me%dat%n) ii = jj*meb1 - ml + 2 do i = i1 , i2 wm(ii+i) = (ftem(i)-savf(i))*fac enddo enddo enddo me%dat%nfe = me%dat%nfe + mba if ( me%dat%jsv==1 ) call dacopy(mband,me%dat%n,wm(ml3),meband,wm(me%dat%locjs),mband) endif if ( jok==1 ) then me%dat%jcur = 0 call dacopy(mband,me%dat%n,wm(me%dat%locjs),mband,wm(ml3),meband) endif ! multiply jacobian by scalar, add identity, and do lu decomposition. con = -hrl1 call dscal(lenp,con,wm(3),1) ii = mband + 2 do i = 1 , me%dat%n wm(ii) = wm(ii) + one ii = ii + meband enddo me%dat%nlu = me%dat%nlu + 1 call dgbfa(wm(3),meband,me%dat%n,ml,mu,iwm(31),ier) if ( ier/=0 ) ierpj = 1 ! end of code block for miter = 4 or 5. -------------------------------- end subroutine dvjac !***************************************************************************************** !> ! this routine copies one rectangular array, `a`, to another, `b`, ! where `a` and `b` may have different row dimensions, `nrowa` and `nrowb`. ! the data copied consists of `nrow` rows and `ncol` columns. subroutine dacopy(nrow,ncol,a,nrowa,b,nrowb) integer,intent(in) :: nrow integer,intent(in) :: ncol integer,intent(in) :: nrowa integer,intent(in) :: nrowb real(wp),intent(in) :: a(nrowa,ncol) real(wp),intent(out) :: b(nrowb,ncol) integer :: ic do ic = 1 , ncol call dcopy(nrow,a(1,ic),1,b(1,ic),1) enddo end subroutine dacopy !***************************************************************************************** !> ! This routine manages the solution of the linear system arising from ! a chord iteration. it is called if `miter /= 0`: ! ! * if miter is 1 or 2, it calls [[dgesl]] to accomplish this. ! * if miter = 3 it updates the coefficient `h*rl1` in the diagonal ! matrix, and then computes the solution. ! * if miter is 4 or 5, it calls [[dgbsl]]. subroutine dvsol(me,wm,iwm,x,iersl) class(dvode_t),intent(inout) :: me real(wp),intent(inout) :: wm(*) !! real work space containing the inverse diagonal matrix if !! miter = 3 and the lu decomposition of the matrix otherwise. !! storage of matrix elements starts at wm(3). !! wm also contains the following matrix-related data: !! wm(1) = sqrt(uround) (not used here), !! wm(2) = hrl1, the previous value of h*rl1, used if miter = 3. real(wp),intent(inout) :: x(*) !! the right-hand side vector on input, !! and the solution vector !! on output, of length n. integer :: iwm(*) !! integer work space containing pivot information, starting at !! iwm(31), if miter is 1, 2, 4, or 5. iwm also contains band !! parameters ml = iwm(1) and mu = iwm(2) if miter is 4 or 5. integer,intent(out) :: iersl !! output flag. iersl = 0 if no trouble occurred. !! iersl = 1 if a singular matrix arose with miter = 3. integer :: i , meband , ml , mu real(wp) :: di , hrl1 , phrl1 , r real(wp),parameter :: one = 1.0_wp iersl = 0 select case (me%dat%miter) case (3) phrl1 = wm(2) hrl1 = me%dat%h*me%dat%rl1 wm(2) = hrl1 if ( hrl1/=phrl1 ) then r = hrl1/phrl1 do i = 1 , me%dat%n di = one - r*(one-one/wm(i+2)) if ( abs(di)==zero ) then iersl = 1 return end if wm(i+2) = one/di enddo endif do i = 1 , me%dat%n x(i) = wm(i+2)*x(i) enddo case (4,5) ml = iwm(1) mu = iwm(2) meband = 2*ml + mu + 1 call dgbsl(wm(3),meband,me%dat%n,ml,mu,iwm(31),x,0) case default call dgesl(wm(3),me%dat%n,me%dat%n,iwm(31),x,0) end select end subroutine dvsol !***************************************************************************************** !> ! this routine saves or restores (depending on `job`) the contents of the ! dvode internal variables. subroutine dvsrco(me,sav,job) class(dvode_t),intent(inout) :: me type(dvode_data_t),intent(inout) :: sav integer,intent(in) :: job !! flag indicating to save or restore the data: !! !! * `job = 1` if common is to be saved (written to `sav`). !! * `job = 2` if common is to be restored (read from `sav`). !! !! a call with job = 2 presumes a prior call with job = 1. select case (job) case(1); sav = me%dat case(2); me%dat = sav case default error stop 'invalid input to dvsrco' end select end subroutine dvsrco !***************************************************************************************** !> ! Set error weight vector. ! ! this subroutine sets the error weight vector ewt according to !``` ! ewt(i) = rtol(i)*abs(ycur(i)) + atol(i), i = 1,...,n, !``` ! with the subscript on rtol and/or atol possibly replaced by 1 above, ! depending on the value of itol. ! !### Author ! * hindmarsh, alan c., (llnl) ! !### Revision history ! * 791129 date written ! * 890501 modified prologue to slatec/ldoc format. (fnf) ! * 890503 minor cosmetic changes. (fnf) ! * 930809 renamed to allow single/real(wp) versions. (ach) subroutine dewset_default(me,n,itol,rtol,atol,ycur,ewt) class(dvode_t),intent(inout) :: me integer,intent(in) :: n integer,intent(in) :: itol real(wp),intent(in) :: rtol(*) real(wp),intent(in) :: atol(*) real(wp),intent(in) :: ycur(n) real(wp),intent(out) :: ewt(n) integer :: i select case (itol) case (2) do i = 1 , n ewt(i) = rtol(1)*abs(ycur(i)) + atol(i) enddo return case (3) do i = 1 , n ewt(i) = rtol(i)*abs(ycur(i)) + atol(1) enddo return case (4) do i = 1 , n ewt(i) = rtol(i)*abs(ycur(i)) + atol(i) enddo return case default end select do i = 1 , n ewt(i) = rtol(1)*abs(ycur(i)) + atol(1) enddo end subroutine dewset_default !***************************************************************************************** !> ! weighted root-mean-square vector norm. ! ! this function routine computes the weighted root-mean-square norm ! of the vector of length n contained in the array v, with weights ! contained in the array w of length n: !``` ! dvnorm = sqrt( (1/n) * sum( v(i)*w(i) )**2 ) !``` ! !### Author ! * hindmarsh, alan c., (llnl) ! !### Revision history ! * 791129 date written ! * 890501 modified prologue to slatec/ldoc format. (fnf) ! * 890503 minor cosmetic changes. (fnf) ! * 930809 renamed to allow single/real(wp) versions. (ach) real(wp) function dvnorm_default(me,n,v,w) class(dvode_t),intent(inout) :: me integer,intent(in) :: n real(wp),intent(in) :: v(n) real(wp),intent(in) :: w(n) integer :: i real(wp) :: sum sum = zero do i = 1 , n sum = sum + (v(i)*w(i))**2 enddo dvnorm_default = sqrt(sum/n) end function dvnorm_default !***************************************************************************************** !> ! write error message with values. ! ! subroutines [[xerrwd]], [[xsetf]], [[xsetun]], and the function routine [[ixsav]], ! as given here, constitute a simplified version of the slatec error ! handling package. ! !### Note ! ! This routine is machine-dependent and specialized for use ! in limited context, in the following ways: ! ! 1. the argument msg is assumed to be of type character, and ! the message is printed with a format of (1x,a). ! 2. the message is assumed to take only one line. ! multi-line messages are generated by repeated calls. ! 3. if level = 2, control passes to the statement stop ! to abort the run. this statement may be machine-dependent. ! 4. r1 and r2 are assumed to be in real(wp) and are printed ! in d21.13 format. ! !### Author ! * hindmarsh, alan c., (llnl) ! !### Revision history ! * 920831 date written ! * 921118 replaced mflgsv/lunsav by ixsav. (ach) ! * 930329 modified prologue to slatec format. (fnf) ! * 930407 changed msg from character*1 array to variable. (fnf) ! * 930922 minor cosmetic change. (fnf) ! !### internal notes: ! ! for a different default logical unit number, [[ixsav]] (or a subsidiary ! routine that it calls) will need to be modified. ! for a different run-abort command, change the statement at the end. subroutine xerrwd(me,msg,nmes,nerr,level,ni,i1,i2,nr,r1,r2) class(dvode_t),intent(inout) :: me character(len=*),intent(in) :: msg !! the message (character array). integer,intent(in) :: nmes !! the length of msg (number of characters). integer,intent(in) :: nerr !! the error number (not used). integer,intent(in) :: level !! the error level: !! !! * 0 or 1 means recoverable (control returns to caller). !! * 2 means fatal (run is aborted). integer,intent(in) :: ni !! number of integers (0, 1, or 2) to be printed with message. integer,intent(in) :: i1 !! integer to be printed, depending on `ni`. integer,intent(in) :: i2 !! integer to be printed, depending on `ni`. integer,intent(in) :: nr !! number of reals (0, 1, or 2) to be printed with message. real(wp),intent(in) :: r1 !! real to be printed, depending on `nr`. real(wp),intent(in) :: r2 !! real to be printed, depending on `nr`. integer :: lunit , mesflg ! get logical unit number and message print flag. lunit = me%ixsav(1,0,.false.) mesflg = me%ixsav(2,0,.false.) if ( mesflg/=0 ) then ! write the message. write (lunit,'(A)') trim(msg) select case (ni) case(1) write (lunit,'(6X,A,I10)') 'in above message, i1 =', i1 case(2) write (lunit,'(6X,A,I10,3X,A,I10)') 'in above message, i1 =', i1, 'i2 =', i2 end select select case (nr) case(1) write (lunit,'(6X,A,D21.13)') 'in above message, r1 =', r1 case (2) write (lunit,'(6X,A,D21.13,3X,A,D21.13)') 'in above message, r1 =', r1, 'r2 =', r2 end select endif ! abort the run if level = 2. if ( level==2 ) stop end subroutine xerrwd !***************************************************************************************** !> ! reset the error print control flag. ! ! xsetf sets the error print control flag to mflag: ! ! * mflag=1 means print all messages (the default). ! * mflag=0 means no printing. ! !### Author ! * hindmarsh, alan c., (llnl) ! !### Revision history ! * 921118 date written ! * 930329 added slatec format prologue. (fnf) ! * 930407 corrected see also section. (fnf) ! * 930922 made user-callable, and other cosmetic changes. (fnf) subroutine xsetf(me,mflag) class(dvode_t),intent(inout) :: me integer :: mflag integer :: junk if ( mflag==0 .or. mflag==1 ) junk = me%ixsav(2,mflag,.true.) end subroutine xsetf !***************************************************************************************** !> ! reset the logical unit number for error messages. ! ! xsetun sets the logical unit number for error messages to lun. ! !### Author ! * hindmarsh, alan c., (llnl) ! !### Revision history ! * 921118 date written ! * 930329 added slatec format prologue. (fnf) ! * 930407 corrected see also section. (fnf) ! * 930922 made user-callable, and other cosmetic changes. (fnf) subroutine xsetun(me,lun) class(dvode_t),intent(inout) :: me integer :: lun integer :: junk if ( lun>0 ) junk = me%ixsav(1,lun,.true.) end subroutine xsetun !***************************************************************************************** !> ! save and recall error message control parameters. ! ! [[ixsav]] saves and recalls one of two error message parameters: ! ! * `lunit`, the logical unit number to which messages are printed, and ! * `mesflg`, the message print flag. ! ! this is a modification of the slatec library routine `j4save`. ! ! on return: ! ! * `ixsav` = the (old) value of the parameter. ! !### See also ! * [[xerrwd]] ! * [[xerrwv]] ! !### Author ! * hindmarsh, alan c., (llnl) ! !### Revision history ! * 921118 date written ! * 930329 modified prologue to slatec format. (fnf) ! * 930915 added iumach call to get default output unit. (ach) ! * 930922 minor cosmetic changes. (fnf) ! * 010425 type declaration for iumach added. (ach) integer function ixsav(me,ipar,ivalue,iset) class(dvode_t),intent(inout) :: me integer,intent(in) :: ipar !! parameter indicator (1 for `lunit`, 2 for `mesflg`). integer :: ivalue !! the value to be set for the parameter, if iset = .true. logical,intent(in) :: iset !! logical flag to indicate whether to read or write. !! if `iset = .true.`, the parameter will be given !! the value `ivalue`. if `iset = .false.`, the parameter !! will be unchanged, and `ivalue` is a dummy argument. select case (ipar) case ( 1 ) if ( me%lunit==-1 ) me%lunit = iumach ixsav = me%lunit if ( iset ) me%lunit = ivalue case ( 2 ) ixsav = me%mesflg if ( iset ) me%mesflg = ivalue end select end function ixsav end module dvode_module