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C/MEMBR ADD NAME=STODE,SSI=0
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subroutine stode (neq, y, yh, nyh, yh1, ewt, savf, acor,
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1 wm, iwm, f, jac, pjac, slvs)
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external f, jac, pjac, slvs
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integer iownd, ialth, ipup, lmax, meo, nqnyh, nslp,
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1 icf, ierpj, iersl, jcur, jstart, kflag, l, meth, miter,
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2 maxord, maxcor, msbp, mxncf, n, nq, nst, nfe, nje, nqu
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integer i, i1, iredo, iret, j, jb, m, ncf, newq
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double precision y, yh, yh1, ewt, savf, acor, wm
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double precision rownd,
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1 conit, crate, el, elco, hold, rmax, tesco,
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2 ccmax, el0, h, hmin, hmxi, hu, rc, tn, uround
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double precision dcon, ddn, del, delp, dsm, dup, exdn, exsm, exup,
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1 r, rh, rhdn, rhsm, rhup, told, vnorm
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dimension neq(*), y(*), yh(nyh,*), yh1(*), ewt(*), savf(*),
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1 acor(*), wm(*), iwm(*)
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common /ls0001/ rownd, conit, crate, el(13), elco(13,12),
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1 hold, rmax, tesco(3,12),
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2 ccmax, el0, h, hmin, hmxi, hu, rc, tn, uround, iownd(14),
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3 ialth, ipup, lmax, meo, nqnyh, nslp,
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4 icf, ierpj, iersl, jcur, jstart, kflag, l, meth, miter,
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5 maxord, maxcor, msbp, mxncf, n, nq, nst, nfe, nje, nqu
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c-----------------------------------------------------------------------
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c stode performs one step of the integration of an initial value
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c problem for a system of ordinary differential equations.
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c note.. stode is independent of the value of the iteration method
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c indicator miter, when this is .ne. 0, and hence is independent
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c of the type of chord method used, or the jacobian structure.
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c communication with stode is done with the following variables..
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c neq = integer array containing problem size in neq(1), and
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c passed as the neq argument in all calls to f and jac.
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c y = an array of length .ge. n used as the y argument in
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c all calls to f and jac.
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c yh = an nyh by lmax array containing the dependent variables
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c and their approximate scaled derivatives, where
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c lmax = maxord + 1. yh(i,j+1) contains the approximate
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c j-th derivative of y(i), scaled by h**j/factorial(j)
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c (j = 0,1,...,nq). on entry for the first step, the first
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c two columns of yh must be set from the initial values.
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c nyh = a constant integer .ge. n, the first dimension of yh.
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c yh1 = a one-dimensional array occupying the same space as yh.
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c ewt = an array of length n containing multiplicative weights
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c for local error measurements. local errors in y(i) are
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c compared to 1.0/ewt(i) in various error tests.
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c savf = an array of working storage, of length n.
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c also used for input of yh(*,maxord+2) when jstart = -1
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c and maxord .lt. the current order nq.
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c acor = a work array of length n, used for the accumulated
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c corrections. on a successful return, acor(i) contains
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c the estimated one-step local error in y(i).
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c wm,iwm = real and integer work arrays associated with matrix
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c operations in chord iteration (miter .ne. 0).
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c pjac = name of routine to evaluate and preprocess jacobian matrix
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c and p = i - h*el0*jac, if a chord method is being used.
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c slvs = name of routine to solve linear system in chord iteration.
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c ccmax = maximum relative change in h*el0 before pjac is called.
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c h = the step size to be attempted on the next step.
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c h is altered by the error control algorithm during the
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c problem. h can be either positive or negative, but its
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c sign must remain constant throughout the problem.
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c hmin = the minimum absolute value of the step size h to be used.
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c hmxi = inverse of the maximum absolute value of h to be used.
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c hmxi = 0.0 is allowed and corresponds to an infinite hmax.
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c hmin and hmxi may be changed at any time, but will not
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c take effect until the next change of h is considered.
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c tn = the independent variable. tn is updated on each step taken.
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c jstart = an integer used for input only, with the following
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c values and meanings..
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c 0 perform the first step.
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c .gt.0 take a new step continuing from the last.
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c -1 take the next step with a new value of h, maxord,
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c n, meth, miter, and/or matrix parameters.
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c -2 take the next step with a new value of h,
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c but with other inputs unchanged.
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c on return, jstart is set to 1 to facilitate continuation.
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c kflag = a completion code with the following meanings..
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c 0 the step was succesful.
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c -1 the requested error could not be achieved.
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c -2 corrector convergence could not be achieved.
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c -3 fatal error in pjac or slvs.
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c a return with kflag = -1 or -2 means either
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c abs(h) = hmin or 10 consecutive failures occurred.
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c on a return with kflag negative, the values of tn and
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c the yh array are as of the beginning of the last
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c step, and h is the last step size attempted.
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c maxord = the maximum order of integration method to be allowed.
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c maxcor = the maximum number of corrector iterations allowed.
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c msbp = maximum number of steps between pjac calls (miter .gt. 0).
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c mxncf = maximum number of convergence failures allowed.
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c meth/miter = the method flags. see description in driver.
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c n = the number of first-order differential equations.
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c-----------------------------------------------------------------------
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if (jstart .gt. 0) go to 200
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if (jstart .eq. -1) go to 100
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if (jstart .eq. -2) go to 160
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c-----------------------------------------------------------------------
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c on the first call, the order is set to 1, and other variables are
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c initialized. rmax is the maximum ratio by which h can be increased
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c in a single step. it is initially 1.e4 to compensate for the small
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c initial h, but then is normally equal to 10. if a failure
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c occurs (in corrector convergence or error test), rmax is set at 2
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c for the next increase.
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c-----------------------------------------------------------------------
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c-----------------------------------------------------------------------
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c the following block handles preliminaries needed when jstart = -1.
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c ipup is set to miter to force a matrix update.
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c if an order increase is about to be considered (ialth = 1),
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c ialth is reset to 2 to postpone consideration one more step.
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c if the caller has changed meth, cfode is called to reset
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c the coefficients of the method.
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c if the caller has changed maxord to a value less than the current
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c order nq, nq is reduced to maxord, and a new h chosen accordingly.
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c if h is to be changed, yh must be rescaled.
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c if h or meth is being changed, ialth is reset to l = nq + 1
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c to prevent further changes in h for that many steps.
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c-----------------------------------------------------------------------
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if (ialth .eq. 1) ialth = 2
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if (meth .eq. meo) go to 110
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call cfode (meth, elco(1,1), tesco(1,1))
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if (nq .gt. maxord) go to 120
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110 if (nq .le. maxord) go to 160
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125 el(i) = elco(i,nq)
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conit = 0.50d+0/dble(nq+2)
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ddn = vnorm (n, savf, ewt)/tesco(1,l)
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exdn = 1.0d+0/dble(l)
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rhdn = 1.0d+0/(1.30d+0*ddn**exdn + 0.00000130d+0)
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rh = min(rhdn,1.0d+0)
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if (h .eq. hold) go to 170
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rh = min(rh,abs(h/hold))
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c-----------------------------------------------------------------------
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c cfode is called to get all the integration coefficients for the
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c current meth. then the el vector and related constants are reset
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c whenever the order nq is changed, or at the start of the problem.
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c-----------------------------------------------------------------------
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140 call cfode (meth, elco(1,1), tesco(1,1))
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155 el(i) = elco(i,nq)
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conit = 0.50d+0/dble(nq+2)
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go to (160, 170, 200), iret
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c-----------------------------------------------------------------------
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c if h is being changed, the h ratio rh is checked against
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c rmax, hmin, and hmxi, and the yh array rescaled. ialth is set to
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c l = nq + 1 to prevent a change of h for that many steps, unless
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c forced by a convergence or error test failure.
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c-----------------------------------------------------------------------
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160 if (h .eq. hold) go to 200
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170 rh = max(rh,hmin/abs(h))
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175 rh = min(rh,rmax)
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rh = rh/max(1.0d+0,abs(h)*hmxi*rh)
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180 yh(i,j) = yh(i,j)*r
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if (iredo .eq. 0) go to 690
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c-----------------------------------------------------------------------
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c this section computes the predicted values by effectively
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c multiplying the yh array by the pascal triangle matrix.
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c rc is the ratio of new to old values of the coefficient h*el(1).
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c when rc differs from 1 by more than ccmax, ipup is set to miter
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c to force pjac to be called, if a jacobian is involved.
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c in any case, pjac is called at least every msbp steps.
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c-----------------------------------------------------------------------
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200 if (abs(rc-1.0d+0) .gt. ccmax) ipup = miter
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if (nst .ge. nslp+msbp) ipup = miter
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210 yh1(i) = yh1(i) + yh1(i+nyh)
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c-----------------------------------------------------------------------
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c up to maxcor corrector iterations are taken. a convergence test is
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c made on the r.m.s. norm of each correction, weighted by the error
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c weight vector ewt. the sum of the corrections is accumulated in the
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c vector acor(i). the yh array is not altered in the corrector loop.
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c-----------------------------------------------------------------------
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call f (neq, tn, y, savf)
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if (ipup .le. 0) go to 250
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c-----------------------------------------------------------------------
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c if indicated, the matrix p = i - h*el(1)*j is reevaluated and
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c preprocessed before starting the corrector iteration. ipup is set
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c to 0 as an indicator that this has been done.
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c-----------------------------------------------------------------------
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call pjac (neq, y, yh, nyh, ewt, acor, savf, wm, iwm, f, jac)
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if (ierpj .ne. 0) go to 430
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270 if (miter .ne. 0) go to 350
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c-----------------------------------------------------------------------
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c in the case of functional iteration, update y directly from
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c the result of the last function evaluation.
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c-----------------------------------------------------------------------
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savf(i) = h*savf(i) - yh(i,2)
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290 y(i) = savf(i) - acor(i)
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del = vnorm (n, y, ewt)
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y(i) = yh(i,1) + el(1)*savf(i)
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300 acor(i) = savf(i)
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c-----------------------------------------------------------------------
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c in the case of the chord method, compute the corrector error,
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c and solve the linear system with that as right-hand side and
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c p as coefficient matrix.
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c-----------------------------------------------------------------------
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360 y(i) = h*savf(i) - (yh(i,2) + acor(i))
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call slvs (wm, iwm, y, savf)
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if (iersl .lt. 0) go to 430
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if (iersl .gt. 0) go to 410
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del = vnorm (n, y, ewt)
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acor(i) = acor(i) + y(i)
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380 y(i) = yh(i,1) + el(1)*acor(i)
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c-----------------------------------------------------------------------
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c test for convergence. if m.gt.0, an estimate of the convergence
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c rate constant is stored in crate, and this is used in the test.
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c-----------------------------------------------------------------------
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400 if (m .ne. 0) crate = max(0.20d+0*crate,del/delp)
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dcon = del*min(1.0d+0,1.50d+0*crate)/(tesco(2,nq)*conit)
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if (dcon .le. 1.0d+0) go to 450
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if (m .eq. maxcor) go to 410
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if (m .ge. 2 .and. del .gt. 2.0d+0*delp) go to 410
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call f (neq, tn, y, savf)
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c-----------------------------------------------------------------------
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c the corrector iteration failed to converge in maxcor tries.
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c if miter .ne. 0 and the jacobian is out of date, pjac is called for
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c the next try. otherwise the yh array is retracted to its values
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c before prediction, and h is reduced, if possible. if h cannot be
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c reduced or mxncf failures have occurred, exit with kflag = -2.
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c-----------------------------------------------------------------------
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410 if (miter .eq. 0 .or. jcur .eq. 1) go to 430
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440 yh1(i) = yh1(i) - yh1(i+nyh)
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if (ierpj .lt. 0 .or. iersl .lt. 0) go to 680
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if (abs(h) .le. hmin*1.000010d+0) go to 670
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if (ncf .eq. mxncf) go to 670
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c-----------------------------------------------------------------------
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c the corrector has converged. jcur is set to 0
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c to signal that the jacobian involved may need updating later.
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c the local error test is made and control passes to statement 500
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c-----------------------------------------------------------------------
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if (m .eq. 0) dsm = del/tesco(2,nq)
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if (m .gt. 0) dsm = vnorm (n, acor, ewt)/tesco(2,nq)
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if (dsm .gt. 1.0d+0) go to 500
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c-----------------------------------------------------------------------
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c after a successful step, update the yh array.
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c consider changing h if ialth = 1. otherwise decrease ialth by 1.
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c if ialth is then 1 and nq .lt. maxord, then acor is saved for
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c use in a possible order increase on the next step.
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c if a change in h is considered, an increase or decrease in order
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c by one is considered also. a change in h is made only if it is by a
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c factor of at least 1.1. if not, ialth is set to 3 to prevent
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c testing for that many steps.
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c-----------------------------------------------------------------------
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470 yh(i,j) = yh(i,j) + el(j)*acor(i)
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if (ialth .eq. 0) go to 520
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if (ialth .gt. 1) go to 700
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if (l .eq. lmax) go to 700
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490 yh(i,lmax) = acor(i)
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c-----------------------------------------------------------------------
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c the error test failed. kflag keeps track of multiple failures.
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c restore tn and the yh array to their previous values, and prepare
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c to try the step again. compute the optimum step size for this or
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c one lower order. after 2 or more failures, h is forced to decrease
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c by a factor of 0.2 or less.
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c-----------------------------------------------------------------------
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500 kflag = kflag - 1
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510 yh1(i) = yh1(i) - yh1(i+nyh)
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if (abs(h) .le. hmin*1.000010d+0) go to 660
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if (kflag .le. -3) go to 640
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c-----------------------------------------------------------------------
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c regardless of the success or failure of the step, factors
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c rhdn, rhsm, and rhup are computed, by which h could be multiplied
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c at order nq - 1, order nq, or order nq + 1, respectively.
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c in the case of failure, rhup = 0.0 to avoid an order increase.
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c the largest of these is determined and the new order chosen
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c accordingly. if the order is to be increased, we compute one
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c additional scaled derivative.
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c-----------------------------------------------------------------------
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if (l .eq. lmax) go to 540
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530 savf(i) = acor(i) - yh(i,lmax)
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dup = vnorm (n, savf, ewt)/tesco(3,nq)
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exup = 1.0d+0/dble(l+1)
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rhup = 1.0d+0/(1.40d+0*dup**exup + 0.00000140d+0)
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540 exsm = 1.0d+0/dble(l)
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rhsm = 1.0d+0/(1.20d+0*dsm**exsm + 0.00000120d+0)
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if (nq .eq. 1) go to 560
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ddn = vnorm (n, yh(1,l), ewt)/tesco(1,nq)
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exdn = 1.0d+0/dble(nq)
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rhdn = 1.0d+0/(1.30d+0*ddn**exdn + 0.00000130d+0)
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560 if (rhsm .ge. rhup) go to 570
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if (rhup .gt. rhdn) go to 590
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570 if (rhsm .lt. rhdn) go to 580
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if (kflag .lt. 0 .and. rh .gt. 1.0d+0) rh = 1.0d+0
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if (rh .lt. 1.10d+0) go to 610
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600 yh(i,newq+1) = acor(i)*r
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620 if ((kflag .eq. 0) .and. (rh .lt. 1.10d+0)) go to 610
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if (kflag .le. -2) rh = min(rh,0.20d+0)
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c-----------------------------------------------------------------------
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c if there is a change of order, reset nq, l, and the coefficients.
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c in any case h is reset according to rh and the yh array is rescaled.
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c then exit from 690 if the step was ok, or redo the step otherwise.
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c-----------------------------------------------------------------------
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if (newq .eq. nq) go to 170
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c-----------------------------------------------------------------------
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c control reaches this section if 3 or more failures have occured.
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c if 10 failures have occurred, exit with kflag = -1.
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c it is assumed that the derivatives that have accumulated in the
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c yh array have errors of the wrong order. hence the first
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c derivative is recomputed, and the order is set to 1. then
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c h is reduced by a factor of 10, and the step is retried,
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c until it succeeds or h reaches hmin.
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c-----------------------------------------------------------------------
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640 if (kflag .eq. -10) go to 660
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rh = max(hmin/abs(h),rh)
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call f (neq, tn, y, savf)
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650 yh(i,2) = h*savf(i)
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if (nq .eq. 1) go to 200
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c-----------------------------------------------------------------------
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c all returns are made through this section. h is saved in hold
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c to allow the caller to change h on the next step.
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c-----------------------------------------------------------------------
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700 r = 1.0d+0/tesco(2,nqu)
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710 acor(i) = acor(i)*r
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c----------------------- end of subroutine stode -----------------------
added
removed
routines/integ/stode.f
