c----------------------------------------------------------------------- c p a r t 1 c main storage allocation and program control subroutines c----------------------------------------------------------------------- c subroutine colsys (ncomp, m, aleft, aright, zeta, ipar, ltol, 1 tol, fixpnt, ispace, fspace, iflag, fsub, 2 dfsub, gsub, dgsub, solutn) c c c*********************************************************************** c c purpose c c subroutine colsys solves a multi-point boundary value c problem for a mixed order system of ode-s given by c c (m(i)) c u = f ( x; z(u(x)) ) i = 1, ... ,ncomp c i i c c aleft .lt. x .lt. aright, c c c g ( zeta(j); z(u(zeta(j))) )= 0 j = 1, ... ,mstar c j c mstar=m(1)+m(2)+...+m(ncomp), c c c where t c u = (u , u , ... ,u ) is the exact solution vector c 1 2 ncomp c c (mi) c u is the mi=m(i) th derivative of u c i i c c (1) (m1-1) (mncomp-1) t c z(u(x)) = ( u (x),u (x),...,u (x),...,u (x) ) c 1 1 1 ncomp c c f (x,z(u)) is a (generally) nonlinear function of c i c z(u)=z(u(x)). c c g (zeta(j);z(u)) is a (generally) nonlinear function c j c used to represent a boundary condition. c c the boundary points satisfy c aleft .le. zeta(1) .le. .. .le. zeta(mstar) .le. aright c c the orders mi of the differential equations satisfy c m1 .le. m2 .le. ... .le. mncomp .le. 4. c c c*********************************************************************** c c written by c u. ascher, c department of computer science, c university of british columbia, c vancouver, b. c., canada v6t 1w5 c j. christiansen and c r. d. russell, c mathematics department, c simon fraser university, c burnaby, b. c., canada v5a 1s6 c c*********************************************************************** c c method c c the method used to approximate the solution u is c collocation at gaussian points, using b-splines of c order k+mi and continuity mi-1 in the i-th component, c i = 1, ..., ncomp. here, k is the number of collocation c points per subinterval and is chosen such that k .ge. m(ncomp). c c main references c c (1) u. ascher, j. christiansen and r.d. russell, c c a collocation solver for mixed order c systems of boundary value problems c c tech. rep. 77-13, dept. computer sc., univ. b.c., c vancouver, canada. also in math. comp. 33 (1979), 659-679. c c (2) u. ascher, j. christiansen and r.d. russell, c c colsys - a collocation code for boundary c value problems c c proc. conf. for codes for bvp-s in ode-s, c houston, texas, 1978. c c other references c c (3) u. ascher and r. d. russell c c evaluation of b-splines for solving systems c of boundary value problems c c tech. rep. 77-14, dept. computer sc., univ. b.c., c vancouver, canada. c c (4) c. deboor and r. weiss c c solveblok: a package for solving almost block diagonal c linear systems, with applications to spline approximation c and the numerical solution of ordinary differential equations c c mrc tech report 1625, university of wisconsin - madison c c c (5) r. d. russell and j. christiansen c c adaptive mesh selection strategies for c solving boundary value problems c c siam j. numer. anal. 7(1978), 59-80. c c*********************************************************************** c c *************** input to colsys *************** c c variables c c ncomp - no. of differential equations (ncomp .le. 20) c c m(j) - order of the j-th differential equation ( m(j).le.m(j+1) c and mstar = m(1) + ... + m(ncomp) .le. 40 ) c c aleft - left end of interval c c aright - right end of interval c c zeta(j) - j-th side condition point (boundary point). must c have zeta(j) .le. zeta(j+1) c c ipar - an integer array dimensioned at least 11. c a list of the parameters in ipar and their meaning follows. c some parameters are renamed in colsys; these new names are c given in parentheses. c c ipar(1) ( = nonlin ) c = 0 if the problem is linear c = 1 if the problem is nonlinear c c ipar(2) = no. of collocation points per subinterval (= k ) c where m(ncomp) .lt. k .le. 7 . if ipar(2)=0 then c colsys sets k = max ( m(ncomp)+1, 5-m(ncomp) ) c c ipar(3) = no. of subintervals in the initial mesh ( = n ). c if ipar(3) = 0 then colsys arbitrarily sets n = 5. c c ipar(4) = no. of solution and derivative tolerances. ( = ntol ) c we require 0 .lt. ntol .le. mstar. c c ipar(5) = dimension of fspace. ( = ndimf ) c c ipar(6) = dimension of ispace. ( = ndimi ) c c ipar(7) - output control ( = iprint ) c = -1 for full diagnostic printout c = 0 for selected printout c = 1 for no printout c c ipar(8) ( = iread ) c = 0 causes colsys to generate a uniform initial mesh. c = 1 if the initial mesh is provided by the user. it c is defined in fspace as follows: the mesh c aleft=x(1).lt.x(2).lt. ... .lt.x(n).lt.x(n+1)=aright c will occupy fspace(1), ..., fspace(n+1). the c user needs to supply only the interior mesh c points fspace(j) = x(j), j = 2, ..., n. c = 2 if the initial mesh is supplied by the user c as with ipar(8)=1, and in addition no adaptive c mesh selection is to be done. c c ipar(9) ( = iguess ) c = 0 if no initial guess for the solution is c provided. c = 1 if an initial guess is provided by the user c in subroutine solutn. c = 2 if an initial mesh and approximate solution c coefficients are provided by the user in fspace. c (the former and new mesh are the same). c = 3 if a former mesh and an approximate solution c coefficients are provided by the user in fspace, c and the new mesh is to be taken twice as coarse; c i.e.,every second point from the former mesh. c = 4 if in addition to a former initial mesh and an c approximate solution coefficients, a new mesh c is provided in fspace as well. c (see description of output for further details c on iguess = 2, 3, and 4.) c c ipar(10)= 0 if the problem is regular c = 1 if the first relax factor is =rstart, and the c nonlinear iteration does not rely on past covergence c (use for an extra sensitive nonlinear problem only). c = 2 if we are to return immediately upon (a) two c successive nonconvergences, or (b) after obtaining c error estimate for the first time. c c ipar(11)= no. of fixed points in the mesh other than c aleft and aright. ( = nfxpnt , the dimension of fixpnt) c c ltol - an array of dimension ipar(4). ltol(j) = l specifies c that the j-th tolerance in tol controls the error c in the l-th component of z(u). also require that c 1.le.ltol(1).lt.ltol(2).lt. ... .lt.ltol(ntol).le.mstar c c tol - an array of dimension ipar(4). tol(j) is the c error tolerance on the ltol(j) -th component c of z(u). thus, the code attempts to satisfy c for j=1,...,ntol on each subinterval c abs(z(v)-z(u)) .le. tol(j)*abs(z(u)) +tol(j) c ltol(j) ltol(j) c c if v(x) is the approximate solution vector. c c fixpnt - an array of dimension ipar(11). it contains c the points, other than aleft and aright, which c are to be included in every mesh. c c ispace - an integer work array of dimension ipar(6). c its size provides a constraint on nmax, c the maximum number of subintervals. choose c ipar(6) according to the formula c ipar(6) .ge. nmax*nsizei c where c nsizei = 3 + kdm - nrec c with c kdm = kd + mstar ; kd = k * ncomp ; c nrec = no. of right end boundary conditions. c c c fspace - a real work array of dimension ipar(5). c its size provides a constraint on nmax. c choose ipar(5) according to the formula c ipar(5) .ge. nmax*nsizef c where c nsizef = 4 + k + 2 * kd + (4+2*k) * mstar + c (kdm-nrec) * (kdm+1). c c c iflag - the mode of return from colsys. c = 1 for normal return c = 0 if the collocation matrix is singular. c =-1 if the expected no. of subintervals exceeds storage c specifications. c =-2 if the nonlinear iteration has not converged. c =-3 if there is an input data error. c c c*********************************************************************** c c ************* user supplied subroutines ************* c c c the following subroutines must be declared external in the c main program which calls colsys. c c c fsub - name of subroutine for evaluating f(x,z(u(x))) = c t c (f ,...,f ) at a point x in (aleft,aright). it c 1 ncomp c should have the heading c c subroutine fsub (x , z , f) c c where f is the vector containing the value of fi(x,z(u)) c in the i-th component and t c z(u(x))=(z(1),...,z(mstar)) c is defined as above under purpose . c c c dfsub - name of subroutine for evaluating the jacobian of c f(x,z(u)) at a point x. it should have the heading c c subroutine dfsub (x , z , df) c c where z(u(x)) is defined as for fsub and the (ncomp) by c (mstar) array df should be filled by the partial deriv- c atives of f, viz, for a particular call one calculates c df(i,j) = dfi / dzj, i=1,...,ncomp c j=1,...,mstar. c c c gsub - name of subroutine for evaluating the i-th component of c g(x,z(u(x))) = g (zeta(i),z(u(zeta(i)))) at a point x = c i c zeta(i) where 1.le.i.le.mstar. it should have the heading c c subroutine gsub (i , z , g) c c where z(u) is as for fsub, and i and g=g are as above. c i c note that in contrast to f in fsub , here c only one value per call is returned in g. c c c dgsub - name of subroutine for evaluating the i-th row of c the jacobian of g(x,u(x)). it should have the heading c c subroutine dgsub (i , z , dg) c c where z(u) is as for fsub, i as for gsub and the mstar- c vector dg should be filled with the partial derivatives c of g, viz, for a particular call one calculates c dg(i,j) = dgi / dzj j=1,...,mstar. c c c solutn- name of subroutine to evaluate the initial c approximation for z(u(x)) and for dmval(u(x))= vector c of the mj-th derivatives of u(x). it should have the c heading c c subroutine solutn (x , z , dmval) c c note that this subroutine is needed only if using c ipar(9) = 1, and then all mstar components of z c and ncomp components of dmval should be specified c for any x, aleft .le. x .le. aright . c c c*********************************************************************** c c ************ use of output from colsys ************ c c *** solution evaluation *** c c on return from colsys, the arrays fspace and ispace c contain information specifying the approximate solution. c the user can produce the solution vector z( u(x) ) at c any point x, aleft .le. x .le. aright, by the statement, c c call appsln (x, z, fspace, ispace) c c when saving the coefficients for later reference, only c ispace(1),...,ispace(7+ncomp) and c fspace(1),...,fspace(ispace(7)) need to be saved as c these are the quantities used by appsln. c c c *** simple continuation *** c c c a formerly obtained solution can easily be used as the c first approximation for the nonlinear iteration for a c new problem by setting (iguess =) ipar(9) = 2, 3 or 4. c c if the former solution has just been obtained then the c values needed to define the first approximation are c already in ispace and fspace. c alternatively, if the former solution was obtained in a c previous run and its coefficients were saved then those c coefficients must be put back into c ispace(1),..., ispace(7+ncomp) and c fspace(1),..., fspace(ispace(7)). c c for ipar(9) = 2 or 3 set ipar(3) = ispace(1) ( = the c size of the previous mesh ). c c for ipar(9) = 4 the user specifies a new mesh of n subintervals c as follows. c the values in fspace(1),...,fspace(ispace(7)) have to be c shifted by n+1 locations to fspace(n+2),..,fspace(ispace(7)+n+1) c and the new mesh is then specified in fspace(1),..., fspace(n+1). c also set ipar(3) = n. c c c*********************************************************************** c c *************** package subroutines *************** c c the following description gives a brief overview of how the c procedure is broken down into the subroutines which make up c the package called colsys . for further details the c user should refer to documentation in the various subroutines c and to the references cited above. c c the subroutines fall into four groups: c c part 1 - the main storage allocation and program control subroutines. c c colsys - tests input values, does initialization and breaks up c the work areas, fspace and ispace, into the arrays c used by the program. c c contrl - is the actual driver of the package. this routine c contains the strategy for nonlinear problems. c c c part 2 - mesh selection and error estimation subroutines c c consts - is called once by colsys to initialize constants c which are used for error estimation and mesh selection. c c newmsh - generates meshes. it contains the test to decide c whether or not to redistribute a mesh. c c errchk - produces error estimates and checks against the c tolerances at each subinterval c c c part 3 - collocation system set-up subroutines c c lsyslv - controls the set-up and solution of the linear c algebraic systems of collocation equations which c arise at each newton iteration. c c bldblk - is used by lsyslv to set up the equation(s) associated c with a side condition point or a collocation point. c c c part 4 - b-spline subroutines c c appsln - sets up a standard call to approx . c c approx - evaluates a piecewise polynomial solution. c c bspfix - evaluates the mesh independent b-splines c (i.e. the fixed b-splines) c c bspvar - evaluates the mesh dependent b-splines (i.e. the c varying b-splines) c c bspder - generates values for the derivatives needed to set c up the collocation equations. c c appdif - generates a divided difference table from the b-spline c coefficients for a collocation solution. the table c is used in approx . c c horder - evaluates the highest order derivatives of the c current collocation solution used for mesh refinement. c c c to solve the linear systems of collocation equations c constructed in part 3, colsys uses the package solveblok c of de boor and weiss (to appear in toms). c c c---------------------------------------------------------------------- implicit real*8 (a-h,o-z) common /order/ k,nc,mstar,kd,kdm,mnsum,mt(20) common /appr/ n,nold,nmax,nalpha,mshflg,mshnum,mshlmt,mshalt common /side/ tzeta(40),tleft,tright,izeta,iwr common /nonln/ precis,nonlin,iter,limit,icare,iprint,iguess,ifreez common /eqord/ ind(5), ineq(20), mnd(5), nd, neq common /errors/ ttl(40),wgtmsh(40),tolin(40),root(40), 1 jtol(40),lttol(40),ntol external fsub, dfsub, gsub, dgsub, solutn dimension m(1), zeta(1), ipar(1), ltol(1), tol(1), 1 fixpnt(1), ispace(1), fspace(1) c*********************************************************************** c c the actual subroutine colsys serves as an interface with c the package of subroutines referred to collectively as c colsys. the subroutine serves to test some of the input c parameters, rename some of the parameters (to make under- c standing of the coding easier), to do some initialization, c and to break the work areas fspace and ispace up into the c arrays needed by the program. c c*********************************************************************** c c... specify machine dependent output unit iwr and compute machine c... dependent constant precis = 100 * machine unit roundoff c iwr = 6 precis = 1.d0 10 precis = precis / 2.d0 precp1 = precis + 1.d0 if (precp1 .gt. 1.d0) go to 10 precis = precis * 100.d0 c c... in case incorrect input data is detected, the program returns c... immediately with iflag=-3. c iflag = -3 if (ncomp .lt. 1 .or. ncomp .gt. 20) return if (m(1) .lt. 1 .or. m(ncomp) .gt. 4) return if (ncomp .eq. 1) go to 30 do 20 i=2,ncomp if (m(i-1) .gt. m(i)) return 20 continue 30 continue c c... rename some of the parameters and set default values. c nonlin = ipar(1) k = ipar(2) if (k .eq. 0) k = max0( m(ncomp)+1, 5-m(ncomp) ) n = ipar(3) if (n .eq. 0) n = 5 iread = ipar(8) iguess = ipar(9) if (nonlin .eq. 0 .and. iguess .eq. 1) iguess = 0 if (iguess .ge. 2 .and. iread .eq. 0) iread = 1 icare = ipar(10) ntol = ipar(4) ndimf = ipar(5) ndimi = ipar(6) nfxpnt = ipar(11) iprint = ipar(7) mstar = 0 mnsum = 0 do 40 i=1,ncomp mnsum = mnsum + m(i)**2 40 mstar = mstar + m(i) do 50 i=1,ncomp 50 mt(i) = m(i) do 60 i=1,mstar 60 tzeta(i) = zeta(i) do 70 i=1,ntol lttol(i) = ltol(i) 70 tolin(i) = tol(i) tleft = aleft tright = aright nc = ncomp kd = k * ncomp kdm = kd + mstar c c... print the input data for checking. c if (iprint .gt. (-1)) go to 100 if (nonlin .gt. 0) go to 80 write (iwr,260) ncomp, (m(ip), ip=1,ncomp) go to 90 80 write(iwr,270) ncomp, (m(ip), ip=1,ncomp) 90 write (iwr,280) (zeta(ip), ip=1,mstar) write (iwr,290) k write (iwr,300) (ltol(ip), ip=1,ntol) write (iwr,310) (tol(ip), ip=1,ntol) if (iguess .ge. 2) write (iwr,320) if (iread .eq. 2) write (iwr,330) if (nfxpnt .gt. 0) write (iwr,340) nfxpnt, 1 (fixpnt(ip), ip=1,nfxpnt) 100 continue c c... check for correctness of data c if (k .lt. 0 .or. k .gt. 7) return if (n .lt. 0) return if (iread .lt. 0 .or. iread .gt. 2) return if (iguess .lt. 0 .or. iguess .gt. 4) return if (icare .lt. 0 .or. icare .gt. 2) return if (ntol .lt. 0 .or. ntol .gt. mstar) return if (nfxpnt .lt. 0) return if (iprint .lt. (-1) .or. iprint .gt. 1) return if (mstar .lt. 0 .or. mstar .gt. 40) return c c... set limits on iterations and initialize counters. c... limit = maximum number of newton iterations per mesh. c... see subroutine newmsh for the roles of mshlmt , mshflg , c... mshnum , and mshalt . c mshlmt = 3 mshflg = 0 mshnum = 1 mshalt = 1 limit = 40 c c... compute the maxium possible n for the given sizes of c... ispace and fspace. c nrec = 0 do 110 ii=1,mstar i = mstar + 1 - ii if (zeta(i) .lt. aright) go to 110 nrec = ii 110 continue nfixi = nrec nsizei = 3 + kdm - nrec nfixf = nrec * (kdm+1) + 2 * mnsum + 2 * mstar + 3 nsizef = 4 + k + 2 * kd + (4+2*k) * mstar + 1(kdm-nrec) * (kdm+1) nmaxf = (ndimf - nfixf) / nsizef nmaxi = (ndimi - nfixi) / nsizei if (iprint .lt. 1) write(iwr,350) nmaxf, nmaxi nmax = min0(nmaxf,nmaxi) if (nmax .lt. n) return if (nmax .lt. nfxpnt+1) return if (nmax .lt. 2*nfxpnt+2 .and. iprint .lt. 1) write(iwr,360) c c... generate pointers to break up fspace and ispace . c lxi = 1 la = lxi + nmax + 1 lxiold = la + kdm * (nmax * (kdm-nrec) + nrec) lxij = lxiold + nmax + 1 lalpha = lxij + k * nmax ldlpha = lalpha + nmax * kd + mstar lelpha = ldlpha + nmax * kd + mstar laldif = lelpha + nmax * k * mstar + mnsum lrhs = laldif + nmax * k * mstar + mnsum lvalst = lrhs + nmax * (kdm - nrec) + nrec lslope = lvalst + 4 * mstar * nmax laccum = lslope + nmax lipiv = 1 linteg = lipiv + (lvalst - lrhs) c c... if iguess .ge. 2, move xiold and aldif to their proper c... locations in fspace. c if (iguess .lt. 2) go to 160 nold = n if (iguess .eq. 4) nold = ispace(1) naldif = nold * k * mstar + mnsum np1 = n + 1 if (iguess .eq. 4) np1 = np1 + nold + 1 do 120 i=1,naldif 120 fspace( laldif+i-1 ) = fspace( np1+i ) np1 = nold + 1 if (iguess .eq. 4) go to 140 do 130 i=1,np1 130 fspace( lxiold+i-1 ) = fspace( lxi+i-1 ) go to 160 140 do 150 i=1,np1 150 fspace( lxiold+i-1 ) = fspace( n+1+i ) 160 continue c c... initialize collocation points, constants, mesh. c call consts call newmsh (3+iread, fspace(lxi), fspace(lxiold), 1 fspace(lxij), dum1, dum2, dum3, dum4, 2 nfxpnt, fixpnt) c c... determine which are the different order equations and c... put these orders in mnd , also generate the pointers c... ind and ineq which will be used in bspder . c ind(1) = 1 mnd(1) = m(1) nd = 1 neq = 0 ig = (m(1)+1) * (m(1)+k) + 1 if (ncomp .le. 1) go to 200 do 190 j=2,ncomp mj = m(j) if (mj .eq. m(j-1)) go to 170 nd = nd + 1 ind(nd) = ig mnd(nd) = mj go to 180 170 neq = neq + 1 ineq(neq) = ig 180 ig = ig + (mj+1) * (mj+k) 190 continue ind(nd+1) =ind(nd) + ig 200 continue c c... determine first approximation, if the problem is nonlinear. c if (iguess .ge. 2) go to 230 np1 = n + 1 do 210 i = 1,np1 210 fspace(i + lxiold - 1) = fspace(i + lxi - 1) nold = n if (nonlin .eq. 0 .or. iguess .eq. 1) go to 230 c c... system provides first approximation of the solution. c... choose z(j) = 0 for j=1,..,mstar. c do 220 i = 1,nalpha 220 fspace(i + lalpha - 1) = 0.d0 call appdif (fspace(laldif), fspace(lalpha), fspace(lxi), 1 n, k, nc, mt, mstar) 230 continue if (iguess .ge. 2) iguess = 0 call contrl (fspace(lxi),fspace(lxiold),fspace(lxij), 1 fspace(lalpha),fspace(laldif),fspace(lrhs), 2 fspace(ldlpha), fspace(lelpha), 3 fspace(la),fspace(lvalst),fspace(lslope), 4 fspace(laccum),ispace(lipiv),ispace(linteg), 5 nfxpnt,fixpnt,iflag,fsub,dfsub,gsub,dgsub, 6 solutn) c... prepare output ispace(1) = n ispace(2) = k ispace(3) = ncomp ispace(4) = mstar naldif = n * k * mstar + mnsum ispace(5) = naldif ispace(6) = naldif + n + 2 ispace(7) = ispace(6) + 65 do 240 i=1,ncomp 240 ispace(7+i) = m(i) do 250 i=1,naldif 250 fspace(n+1+i) = fspace(laldif-1+i) return c----------------------------------------------------------------------- 260 format(/// 37h the number of (linear) diff eqns is , i3/ 1x, 1 16htheir orders are, 20i3) 270 format(/// 40h the number of (nonlinear) diff eqns is , i3/ 1x, 1 16htheir orders are, 20i3) 280 format(27h side condition points zeta, 8f10.6, 4( / 27x, 8f10.6)) 290 format(37h number of colloc pts per interval is, i3) 300 format(39h components of z requiring tolerances -,8(7x,i2,1x), 1 4(/38x,8i10)) 310 format(33h corresponding error tolerances -,6x,8d10.2, 1 4(/39x,8d10.2)) 320 format(44h initial mesh(es) and alpha provided by user) 330 format(27h no adaptive mesh selection) 340 format(10h there are ,i5,27h fixed points in the mesh - , 1 10(6d12.4/)) 350 format(44h the maximum number of subintervals is min (, i4, 1 23h (allowed from fspace),,i4, 24h (allowed from ispace) )) 360 format(/53h insufficient space to double mesh for error estimate) end subroutine contrl(xi, xiold, xij, alpha, aldif, rhs, 1 dalpha, ealpha, a, valstr, slope, 2 accum, ipiv, integs, nfxpnt, fixpnt, iflag, 3 fsub, dfsub, gsub, dgsub, solutn) c c********************************************************************** c c purpose c this subroutine is the actual driver. the nonlinear iteration c strategy is controlled here ( see (2) ). upon convergence, errchk c is called to test for satisfaction of the requested tolerances. c c variables c c check - maximum tolerance value, used as part of criteria for c checking for nonlinear iteration convergence c relax - the relaxation factor for damped newton iteration c relmin - minimum allowable value for relax (otherwise the c jacobian is considered singular). c rlxold - previous relax c rstart - initial value for relax when problem is sensitive c ifrz - number of fixed jacobian iterations c lmtfrz - maximum value for ifrz before performing a reinversion c iter - number of iterations (counted only when jacobian c reinversions are performed). c xi - current mesh c xiold - previous mesh c ipred = 0 if relax is determined by a correction c = 1 if relax is determined by a prediction c ifreez = 0 if the jacobian is to be inverted c = 1 if the jacobian is currently fixed (frozen) c icon = 0 if no previous convergence has been obtained c = 1 if convergence on a previous mesh has been obtained c icare =-1 no convergence occurred (used for regular problems) c = 0 a regular problem c = 1 a sensitive problem c = 2 used for continuation (see description of ipar(10) c in colsys). c rnorm - norm of rhs (right hand side) for current iteration c rnold - norm of rhs for previous iteration c anscl - scaled norm of newton correction c anfix - scaled norm of newton correction at next step c anorm - scaled norm of a correction obtained with jacobian fixed c naldif - number of components of aldif (see subroutine approx) c imesh - a control variable for subroutines newmsh and errchk c c*********************************************************************** c implicit real*8 (a-h,o-z) external fsub, dfsub, gsub, dgsub, solutn dimension xi(1), xiold(1), xij(1), alpha(1), aldif(1), rhs(1) dimension a(1), valstr(1), slope(1), accum(1), ipiv(1), integs(1) dimension dalpha(1), ealpha(1) , fixpnt(1) common /order/ k,ncomp,mstar,kd,kdm,mnsum,m(20) common /appr/ n,nold,nmax,nalpha,mshflg,mshnum,mshlmt,mshalt common /side/ zeta(40),aleft,aright,izeta,iwr common /nonln/ precis,nonlin,iter,limit,icare,iprint,iguess,ifreez common /eqord/ ind(5), ineq(20), mnd(5), nd, neq common /errors/ tol(40),wgtmsh(40),tolin(40),root(40), 1 jtol(40),ltol(40),ntol c c... constants for control of nonlinear iteration c relmin = 1.d-3 rstart = 1.d-2 lmtfrz = 4 c c... compute the maximum tolerance c check = 0.d0 do 10 i=1,ntol 10 check = dmax1 (tolin(i), check ) falpha = dfloat(nalpha) imesh = 1 icon = 0 if (nonlin .eq. 0) icon=1 icor = 0 lconv = 0 c c... the main iteration begins here c... loop 20 is executed until error tolerances are satisfied or c... the code fails (due to a singular matrix or storage limitations) c 20 continue c c... initialization for a new mesh c iter = 0 naldif = n * k * mstar + mnsum if (nonlin .gt. 0) go to 60 c c... the linear case. c... set up and solve equations c call lsyslv (iflag, xi, xiold, xij, alpha, aldif, rhs, 1 ealpha, a, ipiv, integs, rnorm, 0, fsub, 2 dfsub, gsub, dgsub, solutn) c c... check for a singular matrix c if (iflag .ne. 0) go to 40 30 if (iprint .lt. 1) write (iwr,490 ) return c c... update the old mesh c 40 np1 = n + 1 do 50 i=1,np1 50 xiold(i) = xi(i) nold = n c c... prepare table of divided differences and call errchk c call appdif (aldif, alpha, xi, n, k, ncomp, m, mstar) go to 450 c c... iteration loop for nonlinear case c... define the initial relaxation parameter (= relax) c 60 relax = 1.d0 c c... check for previous convergence and problem sensitivity c if (icare .eq. 1 .or. icare .eq. (-1)) relax = rstart if (icon .eq. 0) go to 140 c c... convergence on a previous mesh has been obtained. thus c... we have a very good initial approximation for the newton c... process. proceed with one full newton and then iterate c... with a fixed jacobian. c ifreez = 0 c c... evaluate right hand side and its norm c call lsyslv (iflag, xi, xiold, xij, dalpha, aldif, rhs, 1 alpha, a, ipiv, integs, rnorm, 1, fsub, 2 dfsub, gsub, dgsub, solutn) c c... solve for the next iterate . c... the value of ifreez determines whether this is a full c... newton step (=0) or a fixed jacobian iteration (=1). c if (iprint .lt. 0 .and. iter .eq. 0) write(iwr,530) 70 if (iprint .lt. 0) write (iwr,510) iter, rnorm rnold = rnorm call lsyslv (iflag, xi, xiold, xij, dalpha, aldif, rhs, 1 alpha, a, ipiv, integs, rnorm, 2+ifreez , 2 fsub, dfsub, gsub, dgsub, solutn) c c... check for a singular matrix c if (iflag .eq. 0) go to 30 if (ifreez .eq. 1) go to 90 c c... a full newton step c iter = iter + 1 ifrz = 0 c c... update the old mesh. c np1 = n + 1 do 80 i=1,np1 80 xiold(i) = xi(i) nold = n 90 continue c c... update alpha , compute new rhs and its norm c do 100 i=1,nalpha 100 alpha(i) = alpha(i) + dalpha(i) call appdif (aldif, alpha, xi, n, k, ncomp, m, mstar) call lsyslv (iflag, xi, xiold, xij, dalpha, aldif, rhs, 1 alpha, a, ipiv, integs, rnorm, 1, fsub, 2 dfsub, gsub, dgsub, solutn) c c... check monotonicity. if the norm of rhs gets smaller, c... proceed with a fixed jacobian; else proceed cautiously, c... as if convergence has not been obtained before (icon=0). c if (rnorm .lt. precis) go to 405 if (rnorm .le. rnold) go to 120 if (iprint .lt. 0) write (iwr,510) iter, rnorm if (iprint .lt. 0) write (iwr,540) icon = 0 relax = rstart do 110 i=1,nalpha 110 alpha(i) = alpha(i) - dalpha(i) call appdif (aldif, alpha, xi, n, k, ncomp, m, mstar) iter = 0 go to 140 120 if (ifreez .eq. 1) go to 130 ifreez = 1 go to 70 c c... verify that the linear convergence with fixed jacobian c... is fast enough. c 130 ifrz = ifrz + 1 if (ifrz .ge. lmtfrz) ifreez = 0 if (rnold .lt. 4.d0*rnorm) ifreez = 0 go to 300 c c... no previous convergence has been obtained. proceed c... with the modified newton method. c... evaluate rhs. c 140 if(iprint .lt. 0) write (iwr,500) call lsyslv (iflag, xi, xiold, xij, dalpha, aldif, rhs, 1 alpha, a, ipiv, integs, rnorm, 1, fsub, 2 dfsub, gsub, dgsub, solutn) c c... find a newton direction c 150 rnold = rnorm if (iter .ge. limit) go to 420 call lsyslv (iflag, xi, xiold, xij, dalpha, aldif, rhs, 1 alpha, a, ipiv, integs, rnorm, 2, fsub, 2 dfsub, gsub, dgsub, solutn) c c... check for a singular matrix c if (iflag .eq. 0) go to 30 if (iter .gt. 0) go to 170 c c... bookkeeping for first mesh c if ( iguess .eq. 1) iguess = 0 c c... update the old mesh c np1 = n + 1 do 160 i=1,np1 160 xiold(i) = xi(i) nold = n go to 190 170 continue c c... predict relaxation factor for newton step. c andif = 0.d0 do 180 i=1,nalpha 180 andif = andif + (ealpha(i) - dalpha(i))**2 1 / (alpha(i)*alpha(i) + precis) relax = relax * anscl / dmax1( dsqrt(andif/falpha), 1 precis) if (relax .gt. 1.d0) relax = 1.d0 190 rlxold = relax ipred = 1 iter = iter + 1 c c... determine a new alpha and find new rhs and its norm c do 200 i=1,nalpha 200 alpha(i) = alpha(i) + relax * dalpha(i) 210 call appdif (aldif, alpha, xi, n, k, ncomp, m, mstar) call lsyslv (iflag, xi, xiold, xij, dalpha, aldif, rhs, 1 alpha, a, ipiv, integs, rnorm, 1, fsub, 2 dfsub, gsub, dgsub, solutn) c c... compute a fixed jacobian iterate (used to control relax) c call lsyslv (iflag, xi, xiold, xij, ealpha, aldif, rhs, 1 alpha, a, ipiv, integs, rnorm, 3, fsub, 2 dfsub, gsub, dgsub, solutn) c c... find scaled norms of various terms used to correct relax c anorm = 0.d0 anfix = 0.d0 anscl = 0.d0 do 220 i=1,nalpha anscl = anscl + dalpha(i) * dalpha(i) / 1 (alpha(i)*alpha(i) + precis) scale = alpha(i) - relax*dalpha(i) scale = 1.d0 / (scale*scale + precis) anorm = anorm + dalpha(i) * dalpha(i) * scale 220 anfix = anfix + ealpha(i) * ealpha(i) * scale anorm = dsqrt(anorm / falpha) anfix = dsqrt(anfix / falpha) anscl = dsqrt(anscl / falpha) if (icor .eq. 1) go to 230 if (iprint .lt. 0) write (iwr,520) iter, relax, anorm, 1 anfix, rnold, rnorm go to 240 230 if (iprint .lt. 0) write (iwr,550) relax, anorm, anfix, 1 rnold, rnorm 240 icor = 0 c c... check for monotonic decrease in dalpha. c if (anfix.lt.precis .or. rnorm.lt.precis)go to 405 if (anfix .gt. anorm) go to 250 c c... we have a decrease. if ealpha small, check convergence c if (anfix .le. check) go to 290 c c... correct the predicted relax unless the corrected c... value is within 10 percent of the predicted one. c if (ipred .ne. 1) go to 150 250 if (iter .ge. limit) go to 420 c c... correct the relaxation factor. c ipred = 0 arg = (anfix/anorm - 1.d0) / relax + 1.d0 if (arg .lt. 0.d0) go to 150 if (arg .le. .25d0*relax+.125d0*relax**2 ) go to 260 factor = -1.d0 + dsqrt (1.d0+8.d0 * arg) if ( dabs(factor-1.d0) .lt. .1d0*factor ) go to 150 relax = relax / factor go to 270 260 if (relax .ge. .9d0) go to 150 relax = 1.d0 270 icor = 1 if (relax .lt. relmin) go to 430 do 280 i=1,nalpha 280 alpha(i) = alpha(i) + (relax-rlxold) * dalpha(i) rlxold = relax go to 210 c c... check convergence. c... compute divided difference tables for correction terms. c 290 call appdif (a, ealpha, xi, n, k, ncomp, m, mstar) go to 310 c c... if icon = 1 then also save a. c 300 call appdif (ealpha, dalpha, xi, n, k, ncomp, m, mstar) 310 continue inn = 0 jcol = 0 jinit = 1 do 380 i = 1, ntol jend = jtol(i) - 1 if (jend .lt. jinit) go to 330 do 320 j = jinit, jend mj = m(j) nalphj = n * k + mj jcol = jcol + mj inn = inn + mj * nalphj 320 continue 330 jinit = jend + 1 nalphj = n * k + m(jinit) inn1 = inn jcol1 = jcol + 1 340 if (jcol1 .eq. ltol(i)) go to 350 inn1 = inn1 + nalphj jcol1 = jcol1 + 1 go to 340 350 iinit = jcol1 - jcol c c... check that tolerances are satisfied for b-spline coeffs. c do 370 ii = iinit, nalphj in = inn1 + ii if (icon .eq. 1) go to 360 if (dabs(a(in)) .gt. tolin(i) * 1 (dabs(aldif(in)) +1.d0)) go to 410 go to 370 360 if (dabs(ealpha(in)) .gt. tolin(i) * 1 (dabs(aldif(in)) + 1.d0)) go to 410 370 continue 380 continue c c... convergence obtained c if (iprint .lt. 1) write (iwr,560) iter if (icon .eq. 1) go to 450 c c... since convergence obtained, update coeffs with term from c... the fixed jacobian iteration. c do 390 i=1,naldif 390 aldif(i) = aldif(i) + a(i) do 400 i=1,nalpha 400 alpha(i) = alpha(i) + ealpha(i) 405 if ((anfix.lt.precis.or.rnorm.lt.precis).and.iprint.lt.1) 1 write (iwr,560) iter icon = 1 if (icare .eq. (-1)) icare = 0 go to 450 c c... no convergence. repeat c 410 if ( icon .eq. 0) go to 150 go to 70 c c... diagnostics for failure of nonlinear iteration. c 420 if(iprint .lt. 1) write (iwr,570) iter go to 440 430 if(iprint .lt. 1) write(iwr,580) relax, relmin 440 iflag = -2 lconv = lconv + 1 if (icare .eq. 2 .and. lconv .gt. 1) return if (icare .eq. 0) icare = -1 go to 460 c c... check for error tolerance satisfaction c 450 call errchk(imesh,xiold,aldif,valstr,a,mstar,ifin) if (imesh .eq. 1 .or. ifin .eq. 0 .and. 1 icare .ne. 2) go to 460 iflag = 1 return c c... pick a new mesh c... check safeguards for mesh refinement c 460 imesh = 1 if (icon .eq. 0 .or. mshnum .ge. mshlmt 1 .or. mshalt .ge. mshlmt) imesh = 2 if (mshalt .ge. mshlmt .and. mshnum .lt. mshlmt) 1 mshalt = 1 call newmsh(imesh, xi, xiold, xij, aldif, valstr, 1 slope, accum, nfxpnt, fixpnt) c c... exit if expected n is too large (but may try n=nmax once) c if (n .le. nmax) go to 470 n = n / 2 iflag = -1 if (icon .eq. 0 .and. iprint .lt. 1) write (iwr,590) if (icon .eq. 1 .and. iprint .lt. 1) write (iwr,600) return 470 if (icon .eq. 0) imesh = 1 if (icare .eq. 1) icon = 0 go to 20 c --------------------------------------------------------------- 490 format(//24h the matrix is singular ) 500 format(/30h full damped newton iteration,) 510 format(13h iteration = , i3, 15h norm (rhs) = , d10.2) 520 format(13h iteration = ,i3,22h relaxation factor = ,d10.2 1 /33h norm of scaled rhs changes from ,d10.2,3h to,d10.2 2 /33h norm of rhs changes from ,d10.2,3h to,d10.2) 530 format(/27h fixed jacobian iterations,) 540 format(/35h switch to damped newton iteration,) 550 format(40h relaxation factor corrected to relax = , d10.2 1 /33h norm of scaled rhs changes from ,d10.2,3h to,d10.2 2 /33h norm of rhs changes from ,d10.2,3h to,d10.2) 560 format(/18h convergence after , i3,11h iterations /) 570 format(/22h no convergence after , i3, 11h iterations/) 580 format(/37h no convergence. relaxation factor =,d10.3 1 ,24h is too small (less than, d10.3, 1h)/) 590 format(18h (no convergence) ) 600 format(50h (probably tolerances too stringent, or nmax too 1 ,6hsmall) ) end c----------------------------------------------------------------------- c p a r t 2 c mesh selection, error estimation, (and related c constant assignment) routines -- see (1), (2), (5) c----------------------------------------------------------------------- c subroutine newmsh (mode, xi, xiold, xij, aldif, valstr, 1 slope, accum, nfxpnt, fixpnt) c c*********************************************************************** c c purpose c select a mesh on which a collocation solution is to be c determined c c there are 5 possible modes of action: c mode = 5,4,3 - deal mainly with definition of an initial c mesh for the current boundary value problem c = 2,1 - deal with definition of a new mesh, either c by simple mesh halving or by mesh selection c more specifically, for c mode = 5 an initial (generally nonuniform) mesh is c defined by the user and no mesh selection is to c be performed c = 4 an initial (generally nonuniform) mesh is c defined by the user c = 3 a simple uniform mesh (except possibly for some c fixed points) is defined; n= no. of subintervals c = 1 the automatic mesh selection procedure is used c (see (1) and (5) for details) c = 2 a simple mesh halving is performed c c*********************************************************************** c c variables c c n = number of mesh subintervals c nold = number of subintervals for former mesh c xi - mesh point array c xiold - former mesh point array c mshlmt - maximum no. of mesh selections which are permitted c for a given n before mesh halving c mshnum - no. of mesh selections which have actually been c performed for the given n c mshalt - no. of consecutive times ( plus 1 ) the mesh c selection has alternately halved and doubled n. c if mshalt .ge. mshlmt then contrl requires c that the current mesh be halved. c mshflg = 1 the mesh is a halving of its former mesh c (so an error estimate has been calculated) c = 0 otherwise c iguess - ipar(9) in subroutine colsys. it is used c here only for mode=5 and 4, where c = 2 the subroutine sets xi=xiold. this is c used e.g. if continuation is being per- c formed, and a mesh for the old differen- c tial equation is being used c = 3 same as for =2, except xi uses every other c point of xiold (so mesh xiold is mesh xi c halved) c = 4 xi has been defined by the user, and an old c mesh xiold is also available c otherwise, xi has been defined by the user c and we set xiold=xi in this subroutine c slope - an approximate quantity to be equidistributed for c mesh selection (see (1)), viz, c . (k+mj) c slope(i)= max (weight(l) *u (xi(i))) c 1.le.l.le.ntol j c c where j=jtol(l) c slphmx - maximum of slope(i)*(xiold(i+1)-xiold(i)) for c i = 1 ,..., nold. c accum - accum(i) is the integral of slope from aleft c to xiold(i). c valstr - is assigned values needed in errchk for the c error estimate. c*********************************************************************** c implicit real*8 (a-h,o-z) common /order/k,ncomp,mstar,kd,kdm,mnsum,m(20) common /appr/ n,nold,nmax,nalpha,mshflg,mshnum,mshlmt,mshalt common /errors/ tol(40),wgtmsh(40),tolin(40),root(40), 1 jtol(40),ltol(40),ntol common /colloc/ rho(7),wgterr(40) common /side/ zeta(40), aleft, aright, izeta, iwr common /nonln/ precis,nonlin,iter,limit,icare,iprint,iguess,ifreez common /bsplin/ vncol(66,7), vnsave(66,5), vn(66) dimension d1(40), d2(40), zv(40), slope(1), accum(1), valstr(1) dimension xi(1), xiold(1), xij(1), aldif(1), fixpnt(1) c nfxp1 = nfxpnt +1 go to (180, 100, 50, 20, 10), mode c c... mode=5 set mshlmt=1 so that no mesh selection is performed c 10 mshlmt = 1 c c... mode=4 the user-specified initial mesh is already in place. c 20 if (iguess .lt. 2) go to 40 c c... iguess=2, 3 or 4. c noldp1 = nold + 1 if (iprint .lt. 1) write(iwr,360) nold, (xiold(i), i=1,noldp1) if (iguess .ne. 3) go to 40 c c... if iread ( ipar(8) ) .ge. 1 and iguess ( ipar(9) ) c... .eq. 3 then the first mesh is every second point of the c... mesh in xiold . c n = nold /2 i = 0 do 30 j = 1, nold, 2 i = i + 1 30 xi(i) = xiold(j) 40 continue np1 = n + 1 xi(1) = aleft xi(np1) = aright go to 320 c c... mode=3 generate a (piecewise) uniform mesh. if there are c... fixed points then ensure that the n being used is large enough. c 50 if ( n .lt. nfxp1 ) n = nfxp1 np1 = n + 1 xi(1) = aleft ileft = 1 xleft = aleft c c... loop over the subregions between fixed points. c do 90 j = 1,nfxp1 xright = aright iright = np1 if ( j .eq. nfxp1 ) go to 60 xright = fixpnt(j) c c... determine where the j-th fixed point should fall in the c... new mesh - this is xi(iright) and the (j-1)st fixed c... point is in xi(ileft) c nmin = (xright-aleft) / (aright-aleft) * dfloat(n) + 1.5d0 if (nmin .gt. n-nfxpnt+j) nmin = n - nfxpnt + j iright = max0 (ileft+1, nmin) 60 xi(iright) = xright c c... generate equally spaced points between the j-1st and the c... j-th fixed points. c nregn = iright - ileft - 1 if ( nregn .eq. 0 ) go to 80 dx = (xright - xleft) / dfloat(nregn+1) do 70 i = 1,nregn 70 xi(ileft+i) = xleft + dfloat(i) * dx 80 ileft = iright xleft = xright 90 continue go to 320 c c... mode=2 halve the current mesh (i.e. double its size) c 100 n2 = 2 * n c c... check that n does not exceed storage limitations c if (n2 .le. nmax) go to 120 c c... if possible, try with n=nmax. redistribute first. c if (mode .eq. 2) go to 110 n = nmax / 2 go to 220 110 if (iprint .lt. 1) write(iwr,370) n = n2 return c c... calculate the old approximate solution values at c... points to be used in errchk for error estimates. c... if mshflg =1 an error estimate was obtained for c... for the old approximation so half the needed values c... will already be in valstr . c 120 if (mshflg .eq. 0) go to 140 c c... save in valstr the values of the old solution c... at the relative positions 1/6 and 5/6 in each subinterval. c kstore = 1 do 130 i = 1,nold hd6 = (xiold(i+1) - xiold(i)) / 6.d0 x = xiold(i) + hd6 call approx (i, x, valstr(kstore), vnsave(1,2), xiold, 1 nold, aldif, k, ncomp, m, mstar, 3,dumm,0) x = x + 4.d0 * hd6 kstore = kstore + 3 * mstar call approx (i, x, valstr(kstore), vnsave(1,5), 1 xiold, nold, aldif, k, ncomp, m, mstar, 3,dumm,0) kstore = kstore + mstar 130 continue go to 160 c c... save in valstr the values of the old solution c... at the relative positions 1/6, 2/6, 4/6 and 5/6 in c... each subinterval. c 140 kstore = 1 do 150 i = 1,n x = xi(i) hd6 = (xi(i+1) - xi(i)) / 6.d0 do 150 j = 1,4 x = x + hd6 if ( j.eq.3 ) x = x + hd6 call approx (i, x, valstr(kstore), vnsave(1,j+1), 1 xiold, nold, aldif, k, ncomp, m, mstar, 3, 2 dumm,0) kstore = kstore + mstar 150 continue 160 mshflg = 0 mshnum = 1 mode = 2 c c... generate the halved mesh. c j = 2 do 170 i = 1,n xi(j) = (xiold(i) + xiold(i+1)) / 2.d0 xi(j+1) = xiold(i+1) 170 j = j + 2 n = n2 go to 320 c c... mode=1 we do mesh selection if it is deemed worthwhile c 180 if ( nold .eq. 1 ) go to 100 if (nold .le. 2*nfxpnt) go to 100 c c... the first interval has to be treated separately from the c... other intervals (generally the solution on the (i-1)st and ith c... intervals will be used to approximate the needed derivative, but c... here the 1st and second intervals are used.) c i = 1 call horder (1, d1, xiold, aldif) call horder (2, d2, xiold, aldif) call approx (i, xiold(i), zv, vnsave(1,1), xiold, nold, 1 aldif, k, ncomp, m, mstar, 3, dumm, 0) accum(1) = 0.d0 slope(1) = 0.d0 oneovh = 2.d0 / ( xiold(3) - xiold(1) ) do 190 j = 1,ntol jj = jtol(j) jv = ltol(j) 190 slope(1) = dmax1(slope(1),(dabs(d2(jj)-d1(jj))*wgtmsh(j)* 1oneovh / (1.d0 + dabs(zv(jv)))) **root(j)) slphmx = slope(1) * (xiold(2) - xiold(1)) accum(2) = slphmx iflip = 1 c c... go through the remaining intervals generating slope c... and accum . c do 210 i = 2,nold if ( iflip .eq. (-1) ) call horder ( i, d1, xiold, aldif) if ( iflip .eq. 1 ) call horder ( i, d2, xiold, aldif) call approx (i, xiold(i), zv, vnsave(1,1), xiold, nold, 1 aldif, k, ncomp, m, mstar, 3, dumm, 0) oneovh = 2.d0 / ( xiold(i+1) - xiold(i-1) ) slope(i) = 0.d0 c c... evaluate function to be equidistributed c do 200 j = 1,ntol jj = jtol(j) jv = ltol(j) 200 slope(i) = dmax1(slope(i),(dabs(d2(jj)-d1(jj))*wgtmsh(j)* 1 oneovh / (1.d0 + dabs(zv(jv)))) **root(j)) c c... accumulate approximate integral of function to be c... equidistributed c temp = slope(i) * (xiold(i+1)-xiold(i)) slphmx = dmax1(slphmx,temp) accum(i+1) = accum(i) + temp 210 iflip = - iflip avrg = accum(nold+1) / dfloat(nold) degequ = avrg / dmax1(slphmx,precis) c c... naccum=expected n to achieve .1x user requested tolerances c naccum = accum(nold+1) + 1.d0 if (iprint .lt. 0) write(iwr,350) degequ, naccum c c... decide if mesh selection is worthwhile (otherwise, halve) c if (avrg .lt. precis) go to 100 if (degequ .ge. .5d0) go to 100 c c... nmx assures mesh has at least half as many subintervals as the c... previous mesh c nmx = max0 (nold+1, naccum) / 2 c c... this assures that halving will be possible later (for error est) c nmax2 = nmax / 2 c c... the mesh is at most halved c n = min0 (nmax2, nold, nmx) 220 noldp1 = nold + 1 if (n .lt. nfxp1) n=nfxp1 mshnum = mshnum + 1 c c... if the new mesh is smaller than the old mesh set mshnum c... so that the next call to newmsh will produce a halved c... mesh. if n .eq. nold / 2 increment mshalt so there can not c... be an infinite loop alternating between n and n/2 points. c if (n .lt. nold) mshnum = mshlmt if (n .gt. nold/2) mshalt = 1 if (n .eq. nold/2) mshalt = mshalt + 1 mshflg = 0 c c... having decided to generate a new mesh with n subintervals we now c... do so, taking into account that the nfxpnt points in the array c... fixpnt must be included in the new mesh. c in = 1 accl = 0.d0 lold =2 xi(1) = aleft xi(n+1) = aright do 310 i = 1, nfxp1 if (i .eq. nfxp1) go to 250 do 230 j = lold, noldp1 lnew = j if (fixpnt(i) .le. xiold(j)) go to 240 230 continue 240 continue accr = accum(lnew) + (fixpnt(i)-xiold(lnew))*slope(lnew-1) nregn = (accr-accl) / accum(noldp1) * dfloat(n) - .5d0 nregn = min0(nregn, n - in - nfxp1 + i) xi(in + nregn + 1) = fixpnt(i) go to 260 250 accr = accum(noldp1) lnew = noldp1 nregn = n - in 260 if (nregn .eq. 0) go to 300 temp = accl tsum = (accr - accl) / dfloat(nregn+1) do 290 j = 1, nregn in = in + 1 temp = temp + tsum do 270 l = lold, lnew lcarry = l if (temp .le. accum(l)) go to 280 270 continue 280 continue lold = lcarry 290 xi(in) = xiold(lold-1) + (temp - accum(lold-1)) / 1 slope(lold-1) 300 in = in + 1 accl = accr lold = lnew 310 continue mode = 1 320 continue c c... regardless of how the new mesh is chosen, the new collocation c... points xij in (aleft,aright) are generated here c k2 = 1 do 330 i = 1,n h = (xi(i+1) - xi(i)) / 2.d0 xm = (xi(i+1) + xi(i)) / 2.d0 do 330 j = 1,k xij(k2) = rho(j) * h + xm k2 = k2 + 1 330 continue np1 = n + 1 if (iprint .lt. 1) write(iwr,340) n, (xi(i),i=1,np1) nalpha = n * k * ncomp + mstar return c---------------------------------------------------------------- 340 format(/17h the new mesh (of,i5,16h subintervals), ,100(/8f12.6)) 350 format(/21h mesh selection info,/30h degree of equidistribution = 1 , f8.5, 28h prediction for required n = , i8) 360 format(/20h the former mesh (of,i5,15h subintervals),, 1 100(/8f12.6)) 370 format (/23h expected n too large ) end subroutine consts c c*********************************************************************** c c purpose c assign (once) values to various array constants. c c arrays assigned during compilation: c cnsts1 - weights for extrapolation error estimate c cnsts2 - weights for mesh selection c (the above weights come from the theoretical form for c the collocation error -- see (5)) c c arrays assigned during execution: c wgterr - the particular values of cnsts1 used for current run c (depending on k, m) c wgtmsh - gotten from the values of cnsts2 which in turn are c the constants in the theoretical expression for the c errors. the quantities in wgtmsh are 10x the values c in cnsts2 so that the mesh selection algorithm c is aiming for errors .1x as large as the user c requested tolerances. c jtol - components of differential system to which tolerances c refer (viz, if ltol(i) refers to a derivative of u(j), c then jtol(i)=j) c root - reciprocals of expected rates of convergence of compo- c nents of z(j) for which tolerances are specified c rho - the k collocation points on (-1,1) c vncol - the mesh independent b-splines values at collocation c points c c*********************************************************************** c implicit real*8 (a-h,o-z) common /colloc/ rho(7),wgterr(40) common /order/ k,ncomp,mstar,kd,kdm,mnsum,m(20) common /bsplin/ vncol(66,7), vnsave(66,5), vn(66) common /errors/ tol(40),wgtmsh(40),tolin(40),root(40), 1 jtol(40),ltol(40),ntol common /nonln/ precis,nonlin,iter,limit,icare,iprint,iguess,ifreez dimension cnsts1(28), cnsts2(28) data cnsts1 / .25d0, .625d-1, .72169d-1, 1.8342d-2, 1 1.9065d-2, 5.8190d-2, 5.4658d-3, 5.3370d-3, 1.8890d-2, 2 2.7792d-2, 1.6095d-3, 1.4964d-3, 7.5938d-3, 5.7573d-3, 3 1.8342d-2, 4.673d-3, 4.150d-4, 1.919d-3, 1.468d-3, 4 6.371d-3, 4.610d-3, 1.342d-4, 1.138d-4, 4.889d-4, 5 4.177d-4, 1.374d-3, 1.654d-3, 2.863d-3 / data cnsts2 / 1.25d-1, 2.604d-3, 8.019d-3, 2.170d-5, 1 7.453d-5, 5.208d-4, 9.689d-8, 3.689d-7, 3.100d-6, 2.451d-5, 2 2.691d-10, 1.120d-9, 1.076d-8, 9.405d-8, 1.033d-6, 3 5.097d-13, 2.290d-12, 2.446d-11, 2.331d-10, 2.936d-9, 4 3.593d-8, 7.001d-16, 3.363d-15, 3.921d-14, 4.028d-13, 5 5.646d-12, 7.531d-11, 1.129d-9 / c c... assign weights for error estimate c koff = k * ( k + 1 ) / 2 iextra = 1 do 10 j = 1,ncomp lim = m(j) do 10 l = 1,lim wgterr(iextra) = cnsts1(koff - lim + l) iextra = iextra + 1 10 continue c c... assign array values for mesh selection: wgtmsh, jtol, and root c jcomp = 1 mtot = m(1) do 40 i=1,ntol ltoli = ltol(i) 20 continue if (ltoli .le. mtot) go to 30 jcomp = jcomp + 1 mtot = mtot + m(jcomp) go to 20 30 continue jtol(i) = jcomp wgtmsh(i) = 1.d1 * cnsts2(koff+ltoli-mtot) / tolin(i) root(i) = 1.d0 / dfloat(k+mtot-ltoli+1) 40 continue c c... specify collocation points c go to (50,60,70,80,90,100,110),k 50 rho(1) = 0.d0 go to 120 60 rho(2) = .57735026918962576451d0 rho(1) = - rho(2) go to 120 70 rho(3) = .77459666924148337704d0 rho(2) = .0d0 rho(1) = - rho(3) go to 120 80 rho(1) = -.86113631159405257523d0 rho(2) = -.33998104358485626480d0 rho(3) = - rho(2) rho(4) = - rho(1) go to 120 90 rho(5) = .90617984593866399280d0 rho(4) = .53846931010568309104d0 rho(3) = .0d0 rho(2) = - rho(4) rho(1) = - rho(5) go to 120 100 rho(6) = .93246951420315202781d0 rho(5) = .66120938646626451366d0 rho(4) = .23861918608319690863d0 rho(3) = -rho(4) rho(2) = -rho(5) rho(1) = -rho(6) go to 120 110 rho(7) = .949107991234275852452d0 rho(6) = .74153118559939443986d0 rho(5) = .40584515137739716690d0 rho(4) = 0.d0 rho(3) = -rho(5) rho(2) = -rho(6) rho(1) = -rho(7) 120 continue c c... put mesh independent b-splines values at collocation point c... rho(j) into vncol(*,j), j=1,...,k. c do 130 j=1,k arg = .5d0 * (1.d0 - rho(j)) call bspfix (arg, vncol(1,j), k, ncomp, m) 130 continue c c... put mesh independent b-splines values at the points in unit in- c... terval 0, 1/6, 1/3, 2/3, 5/6 into vnsave. these values are to c... be used in newmsh and errchk . c call bspfix (1.d0, vnsave(1,1), k, ncomp, m) call bspfix (5.d0/6.d0, vnsave(1,2), k, ncomp, m) call bspfix (2.d0/3.d0, vnsave(1,3), k, ncomp, m) call bspfix (1.d0/3.d0, vnsave(1,4), k, ncomp, m) call bspfix (1.d0/6.d0, vnsave(1,5), k, ncomp, m) return end subroutine errchk(imesh,xiold,aldif,valstr,work,mstar,ifin) c c*********************************************************************** c c purpose c determine the error estimates and test to see if the c error tolerances are satisfied. c c variables c xiold - current mesh points c valstr - values of the previous solution which are needed c for the extrapolation- like error estimate. c wgterr - weights used in the extrapolation-like error c estimate. the array values are assigned in c subroutine consts. c errest - storage for error estimates c err - temporary storage used for error estimates c work - space to be used to store values of z at the c mesh points for printout. its dimension is c mstar * nmax. c z - approximate solution on mesh xi c ifin - a 0-1 variable. if imesh = 2 then on return it c indicates whether the error tolerances were satisfied. c imesh = 1 the current mesh resulted from mesh selection c or is the initial mesh. c = 2 the current mesh resulted from doubling the c previous mesh c mshflg - is set by errchk to indicate to newmsh whether c any values of the current solution are stored in c the array valstr. (0 for no, 1 for yes) c c********************************************************************** c implicit real*8 (a-h,o-z) dimension err(40),z(40),errest(40) ,dmval(20) common /order/k,ncomp,mstr,kd,kdm,mnsum,m(20) common /appr/ n,nold,nmax,nalpha,mshflg,mshnum,mshlmt,mshalt common /side/ zeta(40), aleft, aright, izeta, iwr common /errors/ tol(40),wgtmsh(40),tolin(40),root(40), 1 jtol(40),ltol(40),ntol common /colloc/ rho(7),wgterr(40) common /nonln/ precis,nonlin,iter,limit,icare,iprint,iguess,ifreez common /bsplin/ vncol(66,7), vnsave(66,5), vn(66) dimension xiold(1), aldif(1), valstr(1), work(mstar,1) c ifin = 1 noldp1 = nold + 1 c c... if full output has been requested, print values of the c... solution components z at the meshpoints. c if ( iprint .ge. 0 ) go to 30 do 10 i = 1, nold 10 call approx(i, xiold(i), work(1,i), vnsave(1,1), xiold, 1 nold, aldif, k, ncomp, m, mstar, 3, dumm, 0) call approx(nold, xiold(noldp1), work(1,noldp1), vn, xiold, 1 nold, aldif, k, ncomp, m, mstar, 2, dumm, 0) do 20 i = 1, mstar write(iwr,140) i 20 write(iwr,150) (work(i,j), j=1, noldp1) 30 continue if (imesh.eq.1) return c c... imesh = 2 so error estimates are to be generated and tested c... to see if the tolerance requirements are satisfied. c do 40 j = 1,mstar 40 errest(j) = 0.d0 do 100 iback = 1,nold i = nold +1 -iback c c... the error estimates are obtained by combining values of c... the numerical solutions for two meshes. c... for each value of iback we will consider the two c... approximations at 2 points in each of c... the new subintervals. we work backwards through c... the subinterval so that new values can be stored c... in valstr in case they prove to be needed later c... for an error estimate. the routine newmsh c... filled in the needed values of the old solution c... in valstr. c mshflg = 1 do 50 j = 1,mstar z(j) = 0.d0 50 err(j) = 0.d0 do 60 j = 1,2 jj = 5 - j knew = ( 4 * (i-1) + 3 - j ) * mstar + 1 kstore = ( 2 * (i-1) + 2 - j ) * mstar + 1 x = xiold(i) + dfloat(3-j)/3.d0*(xiold(i+1)-xiold(i)) call approx (i, x, valstr(knew), vnsave(1,jj), xiold, 1 nold, aldif, k, ncomp, m, mstar, 3,dumm,0) do 60 l = 1,mstar err(l) = err(l) + wgterr(l) * dabs(valstr(knew) - 1 valstr(kstore)) z(l) = z(l) + .5d0 * dabs(valstr(knew)) knew = knew + 1 kstore = kstore + 1 60 continue c c... test whether the tolerance requirements are satisfied c... in the i-th interval. c if (ifin .eq. 0) go to 80 do 70 j = 1, ntol ltolj = ltol(j) 70 if ( err(ltolj) .gt. tolin(j) * (z(ltolj)+1.d0) ) ifin = 0 80 do 90 l = 1,mstar 90 errest(l) = dmax1(errest(l),err(l)) 100 continue if (iprint .lt. 1) write(iwr,130) lj = 1 do 110 j = 1,ncomp mj = lj - 1 + m(j) if (iprint .lt. 1) write(iwr,120) j, (errest(l), l= lj, mj) lj = mj + 1 110 continue return c-------------------------------------------------------------- 120 format (3h u(, i2, 3h) -,4d12.4) 130 format (/26h the estimated errors are,) 140 format( 19h mesh values for z(, i2, 2h), ) 150 format(1h , 8d15.7) end c c----------------------------------------------------------------------- c p a r t 3 c collocation system setup routines -- see (1) c----------------------------------------------------------------------- c subroutine lsyslv (iflag, xi, xiold, xij, alpha, aldif, 1 rhs, alpho, a, ipiv, integs, rnorm, 2 mode, fsub, dfsub, gsub, dgsub, solutn) c********************************************************************* c c purpose c this routine controls the set up and solution of a linear c system of collocation equations. c the matrix a is cast into an almost block diagonal c form by an appropriate ordering of the columns and solved c using the package of de boor-weiss (4). the matrix is composed c of n blocks. the i-th block has the size c integs(1,i) * integs(2,i). c it contains in its last rows the linearized collocation equa- c tions (both bundary conditions and differential equations ) c corresponding to the i-th subinterval. integs(3,i) steps of c gaussian elimination are applied to it to achieve a plu c decomposition. the right hand side vector is put into rhs c and the solution vector is returned in alpha. c c lsyslv operates according to one of 5 modes: c mode = 0 - set up both a and rhs , and solve system c (for linear problems). c mode = 1 - set up rhs only and compute its norm. c mode = 2 - set up a only and solve system. c mode = 3 - perform forward and backward substitution (do not set c up a nor form the rhs). c c for the first iteration on a particular mesh, c integs is computed. also, the initial alpha on c the new mesh is computed. c c variables c c irhs,ia,izeta,ialpho - pointers to rhs,a,zeta,alpho respectively c (necessary to keep track of blocks of a c during matrix manipulations) c alpho - b-spline coeffs for previous solution c dg - partial derivatives of g from dgsub c df - partial derivatives of f from dfsub c rnorm - euclidean norm of rhs c lside - number of side conditions in current and previous blocks c icolc - pointer to current collocation point array xij c id - (another) pointer for rhs c iguess = 1 when current soln is user specified c = 0 otherwise c c********************************************************************* implicit real*8 (a-h,o-z) common /order/ k, ncomp, mstar, kd, kdm, mnsum, m(20) common /side/ zeta(40), aleft, aright, izeta, iwr common /bsplin/ vncol(66,7), vnsave(66,5), vn(66) common /appr/ n,nold,nmax,nalpha,mshflg,mshnum,mshlmt,mshalt common /nonln/ precis,nonlin,iter,limit,icare,iprint,iguess,ifreez common /hi/ dn1, dn2, dn3 external dfsub, dgsub dimension alpho(1), xi(1), xiold(1), xij(1), alpha(1) dimension aldif(1), rhs(1), a(1), ipiv(1), integs(3,1) dimension z(40), f(40), df(800), dmval(20) c m1 = mode + 1 go to (10, 30, 30, 310), m1 c c... linear problem initialization c 10 do 20 i=1,mstar 20 z(i) = 0.d0 c c... initialization c 30 irhs = 0 ia = 1 izeta = 1 lside = 0 rnorm = 0.d0 ialpho = 0 if (iter .ge. 1 .or. mode .eq. 2) go to 80 c c... build integs (describing block structure of matrix) c do 70 i = 1,n integs(2,i) = kdm if (i .lt. n) go to 40 integs(3,i) = kdm lside = mstar go to 60 40 integs(3,i) = kd 50 if( lside .eq. mstar ) go to 60 if ( zeta(lside+1) .ge. xi(i+1) ) go to 60 lside = lside + 1 go to 50 60 nrow = kd + lside 70 integs(1,i) = nrow c c... the do loop 290 sets up the linear system of equations. c 80 do 290 i=1,n xil = xi(1) if (i .gt. 1) xil = xi(i-1) xir = xi(n+1) if (i .lt. n) xir = xi(i+2) dn1 = 1.d0 / (xi(i+1) - xil) dn2 = 1.d0 / (xi(i+1) - xi(i)) dn3 = 1.d0 / (xir - xi(i)) c c... construct a block of a and a corresponding piece of rhs. c nrow = integs(1,i) ii = i icolc = (i-1) * k id = irhs + izeta - 1 c c... go thru the ncomp collocation equations and side conditions c... in the i-th subinterval c do 270 ll=1,k xx = xij (icolc+ll) 100 if ( izeta .gt. mstar ) go to 160 if ( zeta(izeta) .ge. xx ) go to 160 c c... build equation for a side condition. c 110 id = id + 1 ialpho = ialpho + 1 if (mode .eq. 0) go to 130 if (iguess .ne. 1) go to 120 c c... case where user provided current approximation c call solutn (zeta(izeta), z, dmval) go to (130, 140), mode c c... other nonlinear case c 120 call approx (ii, zeta(izeta), z, vn, xiold, nold, 1 aldif, k, ncomp, m, mstar, 1, dummy, 0) if (mode .eq. 2) go to 140 c c... find rhs boundary value. c 130 call gsub (izeta, z, g) rhs(id) = -g rnorm = rnorm + g**2 if (mode .eq. 1) go to 150 c c... build a row of a corresponding to a boundary point c 140 call bldblk (i, zeta(izeta), ll, a(ia), nrow, id-irhs, z, 1 df, ncomp, xi, alpho, ialpho, 1, dfsub, dgsub) 150 izeta = izeta + 1 c c... check for other side conditions. c if (izeta .gt. mstar .and. 1 zeta(mstar) .ge. dmin1(xx,aright)) go to 280 if (xx .gt. xi(n+1)) go to 260 go to 100 c c... this value corresponds to a collocation (interior) c... point. build the corresponding ncomp equations. c 160 if (iguess .ne. 1) go to (210, 170, 230), m1 c c... use initial approximation provided by the user. c call solutn(xx, z, dmval) go to (190, 250), mode c c... find rhs values c 170 if (iter .ge. 1 ) go to 180 call approx (ii, xx, z, vn, xiold, nold, aldif, k, 1 ncomp, m, mstar, 1, dmval, 1) go to 190 180 call approx (i, xx, z, vncol(1,ll), xiold, nold, 1 aldif, k, ncomp, m, mstar, 3, dmval, 1) 190 call fsub (xx, z, f) c c... fill in rhs values (and accumulate its norm). c do 200 j=1,ncomp id = id + 1 value = dmval(j) - f(j) rhs(id) = -value rnorm = rnorm + value**2 if (iter .ge. 1) go to 200 ialpho = ialpho + 1 alpho(ialpho) = dmval(j) 200 continue go to 260 c c... the linear case c 210 call fsub (xx, z, f) do 220 j=1,ncomp id = id + 1 220 rhs(id) = f(j) id = id - ncomp go to 250 c c... evaluate former collocation soln for mode=2 c 230 if (iter .ge. 1 ) go to 240 call approx (ii, xx, z, vn, xiold, nold, aldif, k, 1 ncomp, m, mstar, 1, dummy, 0) go to 250 240 call approx (i, xx, z, vncol(1,ll), xiold, nold, 1 aldif, k, ncomp, m, mstar, 3, dummy, 0) c c... fill in ncomp rows of a c 250 call bldblk (i, xx, ll, a(ia), nrow, id-irhs+1, z, 1 df, ncomp, xi, alpho, ialpho, 2, dfsub, dgsub) id = id + ncomp c c... prepare to set up side conditions for last subinterval c 260 if (ll .lt. k) go to 270 if (i .lt. n .or. izeta .gt. mstar) go to 280 xx = xi(n+1) + 1.d0 go to 110 270 continue c c... update counters -- i-th block completed c 280 irhs = irhs + nrow ia = ia + nrow * kdm 290 continue if (mode .ne. 1) go to 300 rnorm = dsqrt(rnorm / dfloat(nalpha)) return c c... solve the linear system. c c... matrix decomposition c 300 call fcblok (a, integs, n, ipiv, alpha, iflag) c c... check for singular matrix c if(iflag .eq. 0) return c c... perform forward and backward substitution for mode=0,2, or 3. c 310 call sbblok (a, integs, n, ipiv, rhs, alpha) c c... find the coefficients alpha of the initial approx if necessary. c if (iter .ge. 1 .or. mode .ne. 2) return ialpho = 0 irhs = 0 isto = 0 do 320 i=1,n nrow = integs(1,i) irhs = irhs + isto istart = isto + 1 isto = nrow - kd do 320 j=istart,nrow irhs = irhs + 1 ialpho = ialpho + 1 rhs(irhs) = rhs(irhs) + alpho(ialpho) 320 continue call sbblok (a, integs, n, ipiv, rhs, alpho) do 330 i=1,nalpha 330 alpho(i) = alpho(i) - alpha(i) return end subroutine bldblk (i, x, ll, q, nrow, nc, z, df, ncomp, 1 xi, alpho, ialpho, mode, dfsub, dgsub) c c********************************************************************** c c purpose: c c construct collocation matrix rows according to mode: c mode = 1 - a row corresponding to a side condition. c mode = 2 - a group of ncomp rows corresponding c an interior collocation point. c c variables: c c alpho - used only on the first iteration for nonlinear c problems when the first approximation is other c than a b-spline representation on the current mesh. c a right hand side is being built up in alpho which, c when the inverted collocation matrix is applied to it, c will produce a first approximation on the current mesh c in terms of b- splines so the step-length algorithm c in contrl can operate. c x - the collocation or side condition point. c i - the subinterval containing x c ll - if x is a collocation point then it is the ll-th c of k collocation points on the i-th subinterval. c q - the sub-block of the collocation matrix in c which the equations are to be formed. c nrow - no. of rows in q. c nc - the first row in q to be used for equations. c z - z(x) c dg - the derivatives of the side condition. c df - the jacobian at x. c id - the row of q being constructed. c basef - values and derivatives of the b-spline basis c for each of the components. c jcomp - counter for the component being dealt with. c l - counter for the b-splines representing u(jcomp). c j - counter for the lowest m(jcomp) derivatives of c bsplines representing u . c jcomp c c********************************************************************** implicit real*8 (a-h,o-z) common /appr/ n,nold,nmax,nalpha,mshflg,mshnum,mshlmt,mshalt common /order/ k, nd, mstar, kd, kdm, mnsum, m(20) common /side/ zeta(40), aleft, aright, izeta, iwr common /nonln/ precis,nonlin,iter,limit,icare,iprint,iguess,ifreez common /bsplin/ vncol(66,7), vnsave(66,5), vn(66) dimension q(nrow,1), z(1), df(ncomp,1) dimension xi(1), basef(620), alpho(1), dg(40) c nk = nc if (mode .eq. 2) nk = nc + ncomp - 1 do 10 j=nc,nk do 10 l=1,kdm 10 q(j,l)=0.d0 c c... branch according to m o d e c go to (20, 130), mode c c... x is a boundary point c 20 call bspder (vn, xi, n, x, i, basef, 2) c c... provide coefficients of the j-th linearized side condition. c... specifically, at x=zeta(j) the j-th side condition reads c... dg(1)*z(1) + ... +dg(mstar)*z(mstar) + g = 0 c call dgsub (izeta, z, dg) if (iter .ge. 1 .or. nonlin .eq. 0) go to 40 value = 0.d0 do 30 j=1,mstar 30 value = value + dg(j) * z(j) alpho(ialpho) = value 40 iq = 0 iqm = mstar idg = 0 ibasef = 0 id = nc c do 120 jcomp=1,ncomp mj = m(jcomp) mj1 = mj + 1 kmj = k - mj c c... incorporate the values and derivatives for c... the b-splines which are nonzero on the preceeding c... subinterval. c do 60 l=1,mj do 50 j=1,mj 50 q(id, iq+l) = q(id, iq+l) + dg(idg+j) * 1 basef(ibasef + j) 60 ibasef = ibasef + mj1 c c... the b-splines which are nonzero on the current c... subinterval only. c if (kmj .le. 0) go to 90 do 80 l=1,kmj do 70 j=1,mj 70 q(id, iqm+l) = q(id, iqm+l) + dg(idg+j) * 1 basef(ibasef+j) 80 ibasef = ibasef + mj1 c c... the b-splines which are nonzero on the succeeding c... subinterval as well. c 90 do 110 l=1,mj do 100 j=1,mj 100 q(id, iq+kd+l) = q(id, iq+kd+l) + dg(idg+j) * 1 basef(ibasef+j) 110 ibasef = ibasef + mj1 c idg =idg + mj iq = iq + mj iqm = iqm + kmj 120 continue return c c... build ncomp rows for interior collocation point x. c... the linear expressions to be constructed are: c... (m(jj)) c... u - df(jj,1)*z(1) - ... - df(jj,mstar)*z(mstar) c... jj c... for jj = 1 to ncomp. c 130 call bspder (vncol(1,ll), xi, n, x , i, basef, 3) call dfsub (x, z, df) c c... loop over the ncomp expressions to be set up for the c... current collocation point. c do 240 jj=1,ncomp if (iter .ge. 1 .or. nonlin .eq. 0) go to 150 ialpho = ialpho + 1 value = 0.d0 do 140 j=1,mstar 140 value = value + df(jj,j) * z(j) alpho(ialpho) = alpho(ialpho) - value 150 id = jj + nc - 1 iq=0 iqm=mstar idf=0 ibasef=0 c c... note that if jj .eq. jcomp an entry has to be made for the c... m(jcomp)-th derivative of the jcomp-th component. c do 230 jcomp=1,ncomp mj = m(jcomp) mj1 = mj + 1 kmj = k - mj c c... use the b-splines which are nonzero on the preceeding c... subinterval. c do 170 l=1,mj if (jcomp . eq. jj) q(id, iq+l) = basef(ibasef+mj1) do 160 j=1,mj 160 q(id, iq+l) = q(id, iq+l) - df(jj, idf+j) * 1 basef(ibasef+j) 170 ibasef = ibasef + mj1 c c... the b-splines which are nonzero on the current c... subinterval only. c if (kmj .le. 0) go to 200 do 190 l=1,kmj if (jcomp .eq. jj) q(id, iqm+l) = basef(ibasef+mj1) do 180 j=1,mj 180 q(id, iqm+l) = q(id, iqm+l) - 1 df(jj, idf+j) * basef(ibasef+j) 190 ibasef = ibasef + mj1 c c... the b-splines which are nonzero on the succeeding c... subinterval as well. c 200 do 220 l=1,mj if (jcomp .eq. jj) q(id, iq+kd+l) = basef(ibasef+mj1) do 210 j=1,mj 210 q(id, iq+kd+l) = q(id, iq+kd+l) - df(jj, idf+j) * 1 basef(ibasef+j) 220 ibasef = ibasef + mj1 c idf = idf + mj iq = iq + mj iqm = iqm + kmj 230 continue 240 continue return end c c----------------------------------------------------------------------- c p a r t 4 c b-spline routines -- see (3) c----------------------------------------------------------------------- c subroutine appsln (x, z, fspace, ispace) c c***************************************************************** c c purpose c c set up a standard call to approx to evaluate the c approximate solution z = z( u(x) ) at a point x c (it has been computed by a call to colsys ). c the parameters needed for approx are retrieved c from the work arrays ispace and fspace . c c***************************************************************** c implicit real*8 (a-h,o-z) dimension z(1), fspace(1), ispace(1) is6 = ispace(6) + 1 is5 = ispace(1) + 2 i = 1 call approx (i, x, z, fspace(is6), fspace, ispace(1), 1 fspace(is5), ispace(2), ispace(3), ispace(8), 2 ispace(4), 1, dumm, 0) return end subroutine approx (i, x, z, vn, xi, n, aldif, k, ncomp, 1 m, mstar, mode, dmval, modhi) c c*********************************************************************** c c purpose c (1) (m1-1) (mncomp-1) c evaluate z(u(x))=(u (x),u (x),...,u (x),...,u (x) ) c 1 1 1 mncomp c at one point x. c if modhi=1, evaluate mj-th derivatives too. c c variables c vn - triangular array of b-spline values filled in by c routines bspfix and bspvar c xi - the current mesh (having n subintervals) c aldif - the array of divided differences of the current c solution vectors coefficients alpha (and previously c determined in the routine appdif) c mode - determines the amount of initialization needed c = 5 forms z(u(x)) using aldif and vn c = 3 as in =5, but finishes filling in vn using bspvar c = 2 as in =3, but starts filling in vn using bspfix c = 1 as in =2, but determines i such that c xi(i) .le. x .lt. xi(i+1) (unless x=xi(n+1)) c = 4 a special case which only determines i as above c dmval - array of mj-th derivatives of the solution components c uj (evaluated if modhi=1) c c*********************************************************************** c implicit real*8 (a-h,o-z) common /nonln/ precis,nonlin,iter,limit,icare,iprint,iguess,ifreez common /side/ zeta(40), aleft, aright, izeta, iwr dimension z(1), vn(1), xi(1), aldif(1), m(1), dmval(1) c go to (10, 60, 70, 10, 80), mode c c... mode = 1 or 4, locate i so xi(i) .le. x .lt. xi(i+1) c 10 continue if (x .ge. xi(1)-precis .and. x .le. xi(n+1)+precis) 1 go to 20 if (iprint .lt. 1) write(iwr,160) x, xi(1), xi(n+1) if (x .lt. xi(1)) x = xi(1) if (x .gt. xi(n+1)) x = xi(n+1) 20 if (i .gt. n .or. i .lt. 1) i = (n+1) / 2 ileft = i if (x .lt. xi(ileft)) go to 40 do 30 l=ileft,n i = l if (x .lt. xi(l+1)) go to 60 30 continue go to 60 40 iright = ileft - 1 do 50 l=1,iright i = iright + 1 - l if (x .ge. xi(i)) go to 60 50 continue 60 if (mode .eq. 4) return c c... mode = 1 or 2 begin filling in vn using bspfix . c... compute mesh independent splines. c rhox = (xi(i+1) - x) / (xi(i+1) - xi(i)) call bspfix (rhox, vn, k, ncomp, m) c c... mode = 1, 2, or 3 finish filling in vn using bspvar c 70 call bspvar (i, x, vn, xi, n, k, ncomp, m) c c... mode .ne. 4 determine z(u(x)) c 80 do 90 l=1,mstar 90 z(l) = 0.d0 indif = 0 k5 = 1 if (modhi .eq. 0) go to 110 c c... initialize for subsequent evaluation of mj-th derivatives. c ivnhi = k * (k-1) / 2 dnk2 = dfloat(k) / (xi(i+1) - xi(i)) incomp = 0 do 100 j=1,ncomp 100 dmval(j) = 0.d0 c c... evaluate z( u(x) ). c 110 do 150 j = 1, ncomp mj = m(j) nalphj = n * k + mj kmr = k + mj ivn = kmr * (kmr - 1) / 2 do 130 nr = 1, mj left = i * k + mj - kmr do 120 l = 1, kmr leftpl = left + l 120 z(k5) = z(k5) + aldif(indif+leftpl) * vn(ivn+l) kmr = kmr - 1 ivn = ivn - kmr k5 = k5 + 1 130 indif = indif + nalphj if (modhi .eq. 0) go to 150 c c... evaluate dmval(j) = mj-th derivative of uj. c incomp = incomp + (mj-1) * nalphj left = (i-1) * k + mj - 1 do 140 l = 1, k 140 dmval(j) = dmval(j) + dnk2 * (aldif(incomp+left+l+1) - 1 aldif(incomp+left+l)) * vn(ivnhi+l) incomp = incomp + nalphj 150 continue return c-------------------------------------------------------- 160 format(37h ****** domain error in approx ****** 1 /4h x =,d20.10, 10h aleft =,d20.10, 2 11h aright =,d20.10) end subroutine bspfix (rhox, vn, k, ncomp, m) c c********************************************************************** c c purpose c evaluate the mesh independent bsplines at one point c c c variables c vn - triangular array of b-spline values at x for orders c 1 to k+m(ncomp) where xi(i) .le. x .le. xi(i+1) , column c j has length j and contains the j-th order b-spline c values and begins in location i + j*(j-1)/2. values c not computed here are computed in bspvar. c rhox = (xi(i+1)-x)/(xi(i+1)-xi(i)) c c*********************************************************************** c implicit real*8 (a-h,o-z) dimension vn(1), m(1) xrho = 1.d0 - rhox ivn = 0 c c... compute first group of mesh independent b-spline values c vn(1) = 1.d0 do 20 l=1,k ivn = ivn + l vnp = 0.d0 do 10 j=1,l rep = vn(ivn-l+j) vn(ivn+j) = vnp + rep * rhox 10 vnp = rep * xrho 20 vn(ivn+l+1) = vnp c c... compute second group of mesh independent b-spline values c md1 = m(ncomp) - 1 if (md1 .le. 0) return do 40 l=1,md1 ivn = ivn + k + l inc = l + 2 vnp = vn(ivn+1-k) * xrho if (k .lt. inc) return do 30 j=inc,k rep = vn(ivn-l-k+j) vn(ivn+j) = vnp + rep * rhox 30 vnp = rep * xrho 40 vn(ivn+k+1) = vnp return end subroutine bspvar (i, x, vn, xi, n, k, ncomp, m) c c*********************************************************************** c c purpose c evaluate the mesh dependent b-splines at one point x c c variables c vn - triangular array of values of b-splines of orders 1 c to k+m(ncomp) (described in bspfix) c x - satisfies xi(i) .le. x .le. xi(i+1) c c********************************************************************** c implicit real*8 (a-h,o-z) dimension vn(1), xi(1), m(1) md1 = m(ncomp) -1 if(md1 .le. 0) return xil = xi(1) if (i .gt. 1) xil = xi(i-1) xir = xi(n+1) if (i .lt. n) xir = xi(i+2) rho1 = (xi(i+1) - x) / (xi(i+1) - xi(i)) rho2 = (xi(i+1) - x) / (xi(i+1) - xil) rho3 = (xir - x) / (xir - xi(i)) xrho1 = 1.d0 - rho1 xrho2 = 1.d0 - rho2 xrho3 = 1.d0 - rho3 ivn = k * (k+1) / 2 c c... recursively compute b-spline values. c do 30 l=1,md1 ivn = ivn + k + l vnp = 0.d0 do 10 j=1,l rep = vn(ivn-l-k+j) vn(ivn+j) = vnp + rep * rho2 10 vnp = rep * xrho2 vn(ivn+l+1) = vnp + rho1 * vn(ivn-k+1) vnp = vn(ivn-l) * xrho1 do 20 j=1,l rep = vn(ivn+j-l) vn(ivn+k+j) = vnp + rep * rho3 20 vnp = rep * xrho3 30 vn(ivn+k+l+1) = vnp return end subroutine bspder (vn, xmesh, n, x, i, basef, mode) c c*********************************************************************** c c purpose c evaluate the derivatives of the b-splines of appropriate c orders at one point x (used to set up the c collocation equations.) c c variables c c vn - the triangular array of b-spline values calculated in c bspfix and bspvar c basef - b-spline derivatives needed to set up collocation c equations, viz, derivatives of orders 0,1,...,mj of c b-splines of order k+mj (j=1,...,ncomp). these c values are found using vn, alphd, and alphn (see below). c alphd - array of divided differences corresponding to deriva- c tives of b-splines of order k+mncomp c alphn - same as alphd, but for other order b-splines c alphdo - divided differences of one lower order, used to deter- c mine alphd c alphno - divided differences of one lower order, used to deter- c mine alphn c nd - the no. of differential equations of distinct orders c (so no. of other differential equations =neq =ncomp-nd) c mnd - the distinct orders of these nd differential equations c xmesh - current mesh, with xmesh(i) .le. x .lt. xmesh(i+1) (unle c x=xmesh(n+1) c mode - determines the amount of initialization needed c = 4 compute the array basef c = 3 as in =4, but fill in subinterval dependent values c of vn using bspvar c = 2 as in =3, but fill in subinterval independent values c of vn using bspfix c = 1 as in =2, but calculate certain subinterval depen- c dent constants c c*********************************************************************** c implicit real*8 (a-h,o-z) common /order/ k, ncomp, mstar, kd, kdm, mnsum, m(20) common /hi/ dn1, dn2, dn3 common /eqord/ ind(5), ineq(20), mnd(5), nd, neq dimension basef(1), vn(1), xmesh(1) dimension alphd(80), alphdo(80), alphn(280) , alphno(280) c go to (10, 20, 30, 40), mode c c... mode = 1 compute subinterval dependent constants c 10 xil = xmesh(1) if (i .gt. 1) xil = xmesh(i-1) xir = xmesh(n+1) if (i .lt. n) xir = xmesh(i+2) dn1 = 1.d0 / (xmesh(i+1) - xil) dn2 = 1.d0 / (xmesh(i+1) - xmesh(i)) dn3 = 1.d0 / (xir - xmesh(i)) c c... mode = 2 compute subinterval independent b-splines c 20 rhox = (xmesh(i+1) - x) * dn2 call bspfix (rhox, vn, k, ncomp, m) c c... mode = 3 compute subinterval dependent b-splines c 30 call bspvar (i, x, vn, xmesh, n, k, ncomp, m) c c... mode = 4 c 40 md = mnd(nd) kmd = k + md kmd1 = kmd + 1 md1 = md + 1 md2m2 = md * 2 - 2 md2m1 = md2m2 + 1 inl = kmd * 2 c c... initialize arrays alphdo and alphno c do 50 j=1,kmd alphdo (j) = 0.d0 50 alphdo(j+kmd) = 1.d0 kup = kmd * md do 60 j=1,kup 60 alphdo(j+inl) = 0.d0 ndm1 = nd - 1 nrest = md2m2 - k inn = 0 if (nrest .le. 0) go to 100 if (nd .eq. 1) go to 100 inl = 2 * md2m2 do 90 nn = 1,ndm1 mn2 = mnd(nn) + 2 do 70 j = 1,md2m2 alphno(j+inn) = 0.d0 70 alphno(j+inn+md2m2) = 1.d0 kup = md2m2 * mnd(nn) do 80 j=1,kup 80 alphno(j+inn+inl) = 0.d0 90 inn = inn + mn2 * md2m2 100 inns = inn c c... initialize b-spline derivative values basef c do 120 j=1,nd k1 = ind(j) mj = mnd(j) kmj = k + mj mj1 = mj + 1 ivn = kmj * (kmj-1) / 2 do 120 l=1,kmj basef(k1) = vn(ivn+l) do 110 jj=1,mj 110 basef(k1+jj) = 0.d0 120 k1 = k1 + mj1 c c... for each derivative nr do loop 310 c do 310 nr=1,md nr1 = nr + 1 mdr = md - nr k1 = ind(nd) + nr kmdr = k + mdr ivn = kmdr * (kmdr-1) /2 if (mdr .eq. 0) go to 150 c c... first, determine nr(th) derivative of b-splines c... corresponding to the highest order solution component c... (i.e. of order mncomp=md). c do 140 j=1,mdr jr = j + nr jin = jr + nr1 * kmd jink = jin + k do 130 l=j,jr jin1 = jin - kmd1 jink1 = jink - kmd1 alphd(jin) = dn1 * (alphdo(jin) - alphdo(jin1)) alphd(jink) = dn3 * (alphdo(jink) - alphdo(jink1)) in = k1 + (l-1) * md1 basef(in) = basef(in) + alphd(jin) * vn(ivn+j) in = in + k * md1 basef(in) = basef(in) + alphd(jink) * vn(ivn+j+k) jin = jin - kmd 130 jink = jink - kmd 140 continue 150 mdr1 = mdr + 1 if ( mdr1 .gt. k) go to 180 do 170 j = mdr1,k jr = j + nr jin = jr + nr1 * kmd do 160 l = j,jr jin1 = jin - kmd1 alphd(jin) = dn2 * (alphdo(jin) - alphdo(jin1)) in = k1 + (l-1) * md1 basef(in) = basef(in) + alphd(jin) * vn(ivn+j) 160 jin = jin - kmd 170 continue 180 continue if (nd .eq. 1) go to 230 inn = inns c c... now determine nr(th) derivative basef for b-splines c... corresponding to all other solution components (nn) c do 220 nn=1,ndm1 nj = nd - nn mj = mnd(nj) inn = inn - (mj+2) * md2m2 if (nr .gt. mj) go to 230 kmjr = k + mj - nr k1 = ind(nj)+ nr ivn = kmjr * (kmjr-1) / 2 mj1 = mj + 1 jr1 = kmjr - md + 1 jr1 = min0 (jr1, md-1) c c... compute portion of b-spline derivative values (basef) c... using divided differences previously calculated for the c... highest order solution component in alphd. c do 190 j=1,jr1 jr = j + nr jin = jr + nr1 * kmd + md - mj do 190 l=j,jr in = k1 + (l-1) * mj1 basef(in) = basef(in) + alphd(jin) * vn(ivn+j) jin = jin - kmd 190 continue do 200 j=md,kmjr jr = j + nr jin = jr + nr1 * kmd do 200 l=j,jr in = k1 + (l-1) * mj1 basef(in) = basef(in) + alphd(jin) * vn(ivn+j) jin = jin - kmd 200 continue c c... finish computing b-spline derivative values using the c... new nr(th) divided differences alphn c jr2 = md2m2 - kmjr if (jr2 .le. 0) go to 220 do 210 jj=1,jr2 j = jj + jr1 jr = j + nr jin = jr + nr1 * md2m2 + inn do 210 l=j,jr jin1 = jin - md2m1 alphn(jin) = dn2 * (alphno(jin) - alphno(jin1)) in = k1 + (l-1) * mj1 basef(in) = basef(in) + alphn(jin) * vn(ivn+j) jin = jin - md2m2 210 continue 220 continue 230 continue c c... save nr(th) divided difference values, alphd and alphn, c... to be used to determine the next higher order divided c... differences, by storing them in alphdo and alphno c if (nr .eq. md) go to 300 nr2 = nr + 2 inj = nr do 240 l=2,nr2 inj = inj + kmd do 240 j=1,kmdr 240 alphdo(j+inj) = alphd(j+inj) if (nd .eq. 1) go to 300 if (nrest .le. 0) go to 300 inn = 0 do 290 nn = 1,ndm1 mn = mnd(nn) if (mn .le. nr) go to 280 kmnr = k + mn - nr jr1 = min0 (kmnr-md+1, md-1) inj = nr + inn inl = nr + md - mn do 250 l=2,nr2 inj = inj + md2m2 inl = inl + kmd do 250 j=1,jr1 250 alphno(inj+j) = alphd(inl+j) mup = min0 (kmnr, md2m2) inj = nr + inn inl = nr do 260 l=2,nr2 inj = inj + md2m2 inl = inl + kmd do 260 j =md,mup 260 alphno(inj+j) = alphd(inl+j) jr2 = md2m2 - kmnr if (jr2 .le. 0) go to 280 inj = nr + inn do 270 l=2,nr2 inj = inj + md2m2 do 270 jj = 1,jr2 jin = inj + jj + jr1 270 alphno(jin) = alphn(jin) 280 inn = inn + (mn+2) * md2m2 290 continue 300 continue 310 continue c c... properly normalize basef values c do 320 j=1,nd in = ind(j) icons = 1 mj = mnd(j) kmj = k + mj mj1 = mj + 1 do 320 nr = 1,mj icons = icons * (kmj-nr) in = in + 1 do 320 l=1,kmj lbasef = in + (l-1) * mj1 basef(lbasef) = basef(lbasef) * dfloat(icons) 320 continue c c... copy basef values corresponding to equal order solution components c if (neq .eq. 0) return jd = 1 do 360 j=1,neq in1 = ineq(j) 330 if (in1 .lt. ind(jd+1)) go to 340 jd = jd + 1 go to 330 340 mj = mnd(jd) ntot = (k+mj)*(1+mj) in2 = ind(jd) do 350 l=1,ntot 350 basef(in1-1+l) = basef(in2-1+l) 360 continue return end subroutine appdif (aldif, alpha, xi, n, k, ncomp, m, mstar) c c*********************************************************************** c c purpose c compute a divided difference table based upon the vector c of solution components c c variables c alpha - vector of solution coefficients (for all components) c corresponding to the mesh xi(1),...,xi(n+1) c aldif - the divided difference array based upon alpha, viz, c aldif(i,r,j) = (r-1)st divided difference of alpha c corresponding to u (x), for c j c i=r,...,k+n+mj; r=1,...,mj; j=1,...,ncomp c c*********************************************************************** c implicit real*8 (a-h,o-z) dimension aldif(1), alpha(1), xi(1), m(1) kd = k * ncomp incomp = 0 k3 = 0 k4 = 0 c c... construct the difference table for each component. c do 130 j=1,ncomp mj = m(j) kmj = k - mj mjm1 = mj - 1 kmr = k + mj nalphj = n * k + mj inn = incomp k1 = mstar k2 = kd k5 = inn + 1 c c... copy alpha into the first rows (nr=0) of aldif c do 10 l=1,mj aldif(k5) = alpha(k3+l) 10 k5 = k5 + 1 do 50 i = 1, n if (kmj .eq. 0) go to 30 do 20 l = 1, kmj aldif(k5) = alpha(k1+k4+l) 20 k5 = k5 + 1 30 do 40 l = 1, mj aldif(k5) = alpha(k2+k3+l) 40 k5 = k5 + 1 k1 = k1 + kd k2 = k2 + kd 50 continue c c... for each derivative nr compute divided differences c if (mjm1 .eq. 0) go to 120 do 110 nr = 1, mjm1 inn1 = inn + nalphj kmr = kmr - 1 mjr = mj - nr kmjr = k - mjr xip1 = xi(1) dnk2 = dfloat(kmr) / (xi(2) - xip1) c c... for xi(1),xi(2), the divided difference is a special case c do 60 l=1,nr 60 aldif(inn1+l) = 0.d0 do 70 l = nr, mjm1 l1 = l + 1 70 aldif(inn1+l1) = (aldif(inn+l1) - aldif(inn+l)) * 1 dnk2 ibeg1 = mj ibeg2 = k + nr c c... now the divided difference calculations for xi(i),xi(i+1), c... i=1,...,n c do 100 i = 1, n xii = xip1 xip1 = xi(i+1) dnk1 = dfloat(kmr) / (xip1 - xii) if (i .lt. n) dnk2 = dfloat(kmr) / (xi(i+2) - xii) if (i .eq. n) dnk2 = dnk1 c c... the actual calculations involve two loops c do 80 l = 1, kmjr l1 = ibeg1 + l 80 aldif(inn1+l1) = (aldif(inn+l1) - aldif(inn+l1-1)) * 1 dnk1 do 90 l = 1, mjr l1 = ibeg2 + l 90 aldif(inn1+l1) = (aldif(inn+l1) - aldif(inn+l1-1)) * 1 dnk2 ibeg1 = ibeg1 + k ibeg2 = ibeg2 + k 100 continue inn = inn1 110 continue 120 continue k3 = k3 + mj k4 = k4 + kmj incomp = incomp + nalphj * mj 130 continue return end subroutine horder (i, uhigh, xiold, aldif) c c*********************************************************************** c c purpose c determine highest order (piecewise constant) derivatives c of the current collocation solution c c variables c aldif - divided differences of the solution coefficients alpha c uhigh - the array of highest order (piecewise constant) c derivatives of the approximate solution on c (xiold(i),xiold(i+1)), viz, c (k+mj-1) c uhigh(j) = u (x) on (xiold(i),xiold(i+1)) c j c c*********************************************************************** c implicit real*8 (a-h,o-z) common /appr/ n,nold,nmax,nalpha,mshflg,mshnum,mshlmt,mshalt common /order/ k,ncomp,mstar,kd,kdm,mnsum,m(20) dimension uhigh(1) , ar(20), arm1(20) dimension aldif(1), xiold(1) c dn2 = 1.d0 / (xiold(i+1) - xiold(i)) incomp = 0 c c... loop through the ncomp solution components c do 50 j = 1,ncomp mj = m(j) nalphj = k * nold + mj kpmj = k + mj kmr = k + 1 mjm1 = mj - 1 incomp = incomp + mjm1 * nalphj left = i * k + mj - kmr c c... further divided differences of the appropriate aldif c... (viz. of the (mj-1)st divided differences of the alpha) are c... calculated to obtain the (k+mj-1)st divided difference c do 10 l=1,kmr leftpl = left + l 10 arm1(l+mj-1) = aldif(incomp+leftpl) incomp = incomp + nalphj kpmj1 = kpmj - 1 do 40 nr = mj,kpmj1 kmr = kmr - 1 dnk2 = dn2 * dfloat(kmr) do 20 l = 1,kmr 20 ar(l+nr) = dnk2 * (arm1(l+nr) - arm1(l+nr-1)) do 30 l=nr,kpmj1 30 arm1(l+1) = ar(l+1) 40 continue uhigh(j) = ar(kpmj) 50 continue return end c----------------------------------------------------------------------- c for convenience of the user we list here the package c solveblok of de boor - weiss (4), used in colsys. c----------------------------------------------------------------------- c subroutine fcblok ( bloks, integs, nbloks, ipivot, scrtch, iflag ) c c****************************************************************** c c calls subroutines factrb and shiftb . c c fcblok supervises the plu factorization with pivoting of c scaled rows of the almost block diagonal matrix stored in the c arrays bloks and integs . c c factrb = subprogram which carries out steps 1,...,last of gauss c elimination (with pivoting) for an individual block. c shiftb = subprogram which shifts the remaining rows to the top of c the next block c c parameters c bloks an array that initially contains the almost block diagona c matrix a to be factored, and on return contains the com- c puted factorization of a . c integs an integer array describing the block structure of a . c nbloks the number of blocks in a . c ipivot an integer array of dimension sum (integs(1,n) ; n=1, c ...,nbloks) which, on return, contains the pivoting stra- c tegy used. c scrtch work area required, of length max (integs(1,n) ; n=1, c ...,nbloks). c iflag output parameter; c = 0 in case matrix was found to be singular. c otherwise, c = (-1)**(number of row interchanges during factorization) c c*********************************************************************** c integer integs(3,nbloks),ipivot(1),iflag, i,index,indexb,indexn, 1 last,ncol,nrow double precision bloks(1),scrtch(1) iflag = 1 indexb = 1 indexn = 1 i = 1 c c... loop over the blocks. i is loop index c 10 index = indexn nrow = integs(1,i) ncol = integs(2,i) last = integs(3,i) c c... carry out elimination on the i-th block until next block c... enters, i.e., for columns 1,...,last of i-th block. c call factrb ( bloks(index), ipivot(indexb), scrtch, nrow, 1 ncol, last, iflag) c c... check for having reached a singular block or the last block c if (iflag .eq. 0 .or. i .eq. nbloks) return i = i+1 indexn = nrow*ncol + index c c... put the rest of the i-th block onto the next block c call shiftb ( bloks(index), ipivot(indexb), nrow, ncol, 1 last, bloks(indexn), integs(1,i), integs(2,i) ) indexb = indexb + nrow go to 10 end subroutine factrb ( w, ipivot, d, nrow, ncol, last, iflag ) c c*********************************************************************** c c adapted from p.132 of element.numer.analysis by conte-de boor c c constructs a partial plu factorization, corresponding to steps c 1,..., last in gauss elimination, for the matrix w of c order ( nrow , ncol ), using pivoting of scaled rows. c c parameters c w contains the (nrow,ncol) matrix to be partially factored c on input, and the partial factorization on output. c ipivot an integer array of length nrow containing a record of c the pivoting strategy used; row ipivot(i) is used c during the i-th elimination step, i=1,...,last. c d a work array of length nrow used to store row sizes c temporarily. c nrow number of rows of w. c ncol number of columns of w. c last number of elimination steps to be carried out. c iflag on output, equals iflag on input times (-1)**(number of c row interchanges during the factorization process), in c case no zero pivot was encountered. c otherwise, iflag = 0 on output. c c*********************************************************************** c integer ipivot(nrow),ncol,last,iflag, i,ipivi,ipivk,j,k,kp1 double precision w(nrow,ncol),d(nrow), awikdi,colmax,ratio,rowmax double precision dabs,dmax1 c c... initialize ipivot, d c do 20 i=1,nrow ipivot(i) = i rowmax = 0.d0 do 10 j=1,ncol 10 rowmax = dmax1(rowmax, dabs(w(i,j))) if (rowmax .eq. 0.d0) go to 90 20 d(i) = rowmax c c... gauss elimination with pivoting of scaled rows, loop over c... k=1,.,last c k = 1 c c... as pivot row for k-th step, pick among the rows not yet used, c... i.e., from rows ipivot(k),...,ipivot(nrow), the one whose k-th c... entry (compared to the row size) is largest. then, if this row c... does not turn out to be row ipivot(k), redefine ipivot(k) ap- c... propriately and record this interchange by changing the sign c... of iflag . c 30 ipivk = ipivot(k) if (k .eq. nrow) go to 80 j = k kp1 = k+1 colmax = dabs(w(ipivk,k))/d(ipivk) c c... find the (relatively) largest pivot c do 40 i=kp1,nrow ipivi = ipivot(i) awikdi = dabs(w(ipivi,k))/d(ipivi) if (awikdi .le. colmax) go to 40 colmax = awikdi j = i 40 continue if (j .eq. k) go to 50 ipivk = ipivot(j) ipivot(j) = ipivot(k) ipivot(k) = ipivk iflag = -iflag 50 continue c c... if pivot element is too small in absolute value, declare c... matrix to be noninvertible and quit. c if (dabs(w(ipivk,k))+d(ipivk) .le. d(ipivk)) 1 go to 90 c c... otherwise, subtract the appropriate multiple of the pivot c... row from remaining rows, i.e., the rows ipivot(k+1),..., c... ipivot(nrow), to make k-th entry zero. save the multiplier c... in its place. c do 60 i=kp1,nrow ipivi = ipivot(i) w(ipivi,k) = w(ipivi,k)/w(ipivk,k) ratio = -w(ipivi,k) do 60 j=kp1,ncol 60 w(ipivi,j) = ratio*w(ipivk,j) + w(ipivi,j) k = kp1 c c... check for having reached the next block. c if (k .le. last) go to 30 return c c... if last .eq. nrow , check now that pivot element in last row c... is nonzero. c 80 if(dabs(w(ipivk,nrow))+d(ipivk) .gt. d(ipivk)) return c c... singularity flag set c 90 iflag = 0 return end subroutine shiftb ( ai, ipivot, nrowi, ncoli, last, 1 ai1, nrowi1, ncoli1 ) c c********************************************************************** c c shifts the rows in current block, ai, not used as pivot rows, if c any, i.e., rows ipivot(last+1),...,ipivot(nrowi), onto the first c mmax = nrow-last rows of the next block, ai1, with column last+j c of ai going to column j , j=1,...,jmax=ncoli-last. the remaining c columns of these rows of ai1 are zeroed out. c c picture c c original situation after results in a new block i+1 c last = 2 columns have been created and ready to be c done in factrb (assuming no factored by next factrb c interchanges of rows) call. c 1 c x x 1x x x x x x x x c 1 c 0 x 1x x x 0 x x x x c block i 1 --------------- c nrowi = 4 0 0 1x x x 0 0 1x x x 0 01 c ncoli = 5 1 1 1 c last = 2 0 0 1x x x 0 0 1x x x 0 01 c ------------------------------- 1 1 new c 1x x x x x 1x x x x x1 block c 1 1 1 i+1 c block i+1 1x x x x x 1x x x x x1 c nrowi1= 5 1 1 1 c ncoli1= 5 1x x x x x 1x x x x x1 c ------------------------------- 1-------------1 c 1 c c*********************************************************************** c integer ipivot(nrowi),last, ip,j,jmax,jmaxp1,m,mmax double precision ai(nrowi,ncoli),ai1(nrowi1,ncoli1) mmax = nrowi - last jmax = ncoli - last if (mmax .lt. 1 .or. jmax .lt. 1) return c c... put the remainder of block i into ai1 c do 10 m=1,mmax ip = ipivot(last+m) do 10 j=1,jmax 10 ai1(m,j) = ai(ip,last+j) if (jmax .eq. ncoli1) return c c... zero out the upper right corner of ai1 c jmaxp1 = jmax + 1 do 20 j=jmaxp1,ncoli1 do 20 m=1,mmax 20 ai1(m,j) = 0.d0 return end subroutine sbblok ( bloks, integs, nbloks, ipivot, b, x ) c c********************************************************************** c c calls subroutines subfor and subbak . c c supervises the solution (by forward and backward substitution) of c the linear system a*x = b for x, with the plu factorization of c a already generated in fcblok . individual blocks of c equations are solved via subfor and subbak . c c parameters c bloks, integs, nbloks, ipivot are as on return from fcblok. c b the right side, stored corresponding to the storage of c the equations. see comments in s l v b l k for details. c x solution vector c c*********************************************************************** c integer integs(3,nbloks),ipivot(1), i,index,indexb,indexx,j,last, 1 nbp1,ncol,nrow double precision bloks(1),b(1),x(1) c c... forward substitution pass c index = 1 indexb = 1 indexx = 1 do 10 i=1,nbloks nrow = integs(1,i) last = integs(3,i) call subfor ( bloks(index), ipivot(indexb), nrow, last, 1 b(indexb), x(indexx) ) index = nrow*integs(2,i) + index indexb = indexb + nrow 10 indexx = indexx + last c c... back substitution pass c nbp1 = nbloks + 1 do 20 j=1,nbloks i = nbp1 - j nrow = integs(1,i) ncol = integs(2,i) last = integs(3,i) index = index - nrow*ncol indexb = indexb - nrow indexx = indexx - last 20 call subbak ( bloks(index), ipivot(indexb), nrow, ncol, 1 last, x(indexx) ) return end subroutine subfor ( w, ipivot, nrow, last, b, x ) c c*********************************************************************** c c carries out the forward pass of substitution for the current c block, i.e., the action on the right side corresponding to the c elimination carried out in factrb for this block. c at the end, x(j) contains the right side of the transformed c ipivot(j)-th equation in this block, j=1,...,nrow. then, since c for i=1,...,nrow-last, b(nrow+i) is going to be used as the right c side of equation i in the next block (shifted over there from c this block during factorization), it is set equal to x(last+i) c here. c c parameters c w, ipivot, nrow, last are as on return from factrb. c b(j) is expected to contain, on input, the right side of j-th c equation for this block, j=1,...,nrow. c b(nrow+j) contains, on output, the appropriately modified right c side for equation j in next block, j=1,...,nrow-last. c x(j) contains, on output, the appropriately modified right c side of equation ipivot(j) in this block, j=1,...,last c (and even for j=last+1,...,nrow). c c*********************************************************************** c integer ipivot(nrow), ip,jmax,k double precision w(nrow,last),b(1),x(nrow),sum ip = ipivot(1) x(1) = b(ip) if (nrow .eq. 1) go to 40 do 20 k=2,nrow ip = ipivot(k) jmax = min0(k-1,last) sum = 0.d0 do 10 j=1,jmax 10 sum = w(ip,j)*x(j) + sum 20 x(k) = b(ip) - sum c c... transfer modified right sides of equations ipivot(last+1),..., c... ipivot(nrow) to next block. c nrowml = nrow - last if (nrowml .eq. 0) go to 40 lastp1 = last+1 do 30 k=lastp1,nrow 30 b(nrowml+k) = x(k) 40 return end subroutine subbak ( w, ipivot, nrow, ncol, last, x ) c c*********************************************************************** c c carries out backsubstitution for current block. c c parameters c w, ipivot, nrow, ncol, last are as on return from factrb. c x(1),...,x(ncol) contains, on input, the right side for the c equations in this block after backsubstitution has been c carried up to but not including equation ipivot(last). c means that x(j) contains the right side of equation ipi- c vot(j) as modified during elimination, j=1,...,last, c while for j .gt. last, x(j) is already a component of c the solution vector. c x(1),...,x(ncol) contains, on output, the components of the c solution corresponding to the present block. c c********************************************************************** c integer ipivot(nrow),last, ip,j,k,kp1 double precision w(nrow,ncol),x(ncol), sum k = last ip = ipivot(k) sum = 0.d0 if (k .eq. ncol) go to 30 kp1 = k+1 10 do 20 j=kp1,ncol 20 sum = w(ip,j)*x(j) + sum 30 x(k) = (x(k) - sum)/w(ip,k) if (k .eq. 1) return kp1 = k k = k-1 ip = ipivot(k) sum = 0.d0 go to 10 end