! ! Parallel Sparse BLAS version 3.5 ! (C) Copyright 2006-2018 ! Salvatore Filippone ! Alfredo Buttari ! ! Redistribution and use in source and binary forms, with or without ! modification, are permitted provided that the following conditions ! are met: ! 1. Redistributions of source code must retain the above copyright ! notice, this list of conditions and the following disclaimer. ! 2. Redistributions in binary form must reproduce the above copyright ! notice, this list of conditions, and the following disclaimer in the ! documentation and/or other materials provided with the distribution. ! 3. The name of the PSBLAS group or the names of its contributors may ! not be used to endorse or promote products derived from this ! software without specific written permission. ! ! THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ! ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED ! TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR ! PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE PSBLAS GROUP OR ITS CONTRIBUTORS ! BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR ! CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF ! SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS ! INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN ! CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ! ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE ! POSSIBILITY OF SUCH DAMAGE. ! ! ! File: psb_d_pde2d.f90 ! ! Program: psb_d_pde2d ! This sample program solves a linear system obtained by discretizing a ! PDE with Dirichlet BCs. ! ! ! The PDE is a general second order equation in 2d ! ! a1 dd(u) a2 dd(u) b1 d(u) b2 d(u) ! - ------ - ------ ----- + ------ + c u = f ! dxdx dydy dx dy ! ! with Dirichlet boundary conditions ! u = g ! ! on the unit square 0<=x,y<=1. ! ! ! Note that if b1=b2=c=0., the PDE is the Laplace equation. ! ! There are three choices available for data distribution: ! 1. A simple BLOCK distribution ! 2. A ditribution based on arbitrary assignment of indices to processes, ! typically from a graph partitioner ! 3. A 2D distribution in which the unit square is partitioned ! into rectangles, each one assigned to a process. ! module psb_d_pde2d_mod use psb_base_mod, only : psb_dpk_, psb_ipk_, psb_desc_type,& & psb_dspmat_type, psb_d_vect_type, dzero,& & psb_d_base_sparse_mat, psb_d_base_vect_type, psb_i_base_vect_type interface function d_func_2d(x,y) result(val) import :: psb_dpk_ real(psb_dpk_), intent(in) :: x,y real(psb_dpk_) :: val end function d_func_2d end interface interface psb_gen_pde2d module procedure psb_d_gen_pde2d end interface psb_gen_pde2d contains function d_null_func_2d(x,y) result(val) real(psb_dpk_), intent(in) :: x,y real(psb_dpk_) :: val val = dzero end function d_null_func_2d ! ! functions parametrizing the differential equation ! ! ! Note: b1 and b2 are the coefficients of the first ! derivative of the unknown function. The default ! we apply here is to have them zero, so that the resulting ! matrix is symmetric/hermitian and suitable for ! testing with CG and FCG. ! When testing methods for non-hermitian matrices you can ! change the B1/B2 functions to e.g. done/sqrt((2*done)) ! function b1(x,y) use psb_base_mod, only : psb_dpk_, done, dzero implicit none real(psb_dpk_) :: b1 real(psb_dpk_), intent(in) :: x,y b1=dzero end function b1 function b2(x,y) use psb_base_mod, only : psb_dpk_, done, dzero implicit none real(psb_dpk_) :: b2 real(psb_dpk_), intent(in) :: x,y b2=dzero end function b2 function c(x,y) use psb_base_mod, only : psb_dpk_, done, dzero implicit none real(psb_dpk_) :: c real(psb_dpk_), intent(in) :: x,y c=0.d0 end function c function a1(x,y) use psb_base_mod, only : psb_dpk_, done, dzero implicit none real(psb_dpk_) :: a1 real(psb_dpk_), intent(in) :: x,y a1=done/80 end function a1 function a2(x,y) use psb_base_mod, only : psb_dpk_, done, dzero implicit none real(psb_dpk_) :: a2 real(psb_dpk_), intent(in) :: x,y a2=done/80 end function a2 function g(x,y) use psb_base_mod, only : psb_dpk_, done, dzero implicit none real(psb_dpk_) :: g real(psb_dpk_), intent(in) :: x,y g = dzero if (x == done) then g = done else if (x == dzero) then g = exp(-y**2) end if end function g ! ! subroutine to allocate and fill in the coefficient matrix and ! the rhs. ! subroutine psb_d_gen_pde2d(ictxt,idim,a,bv,xv,desc_a,afmt,info,& & f,amold,vmold,imold,partition,nrl,iv) use psb_base_mod use psb_util_mod ! ! Discretizes the partial differential equation ! ! a1 dd(u) a2 dd(u) b1 d(u) b2 d(u) ! - ------ - ------ + ----- + ------ + c u = f ! dxdx dydy dx dy ! ! with Dirichlet boundary conditions ! u = g ! ! on the unit square 0<=x,y<=1. ! ! ! Note that if b1=b2=c=0., the PDE is the Laplace equation. ! implicit none integer(psb_ipk_) :: idim type(psb_dspmat_type) :: a type(psb_d_vect_type) :: xv,bv type(psb_desc_type) :: desc_a integer(psb_ipk_) :: ictxt, info character(len=*) :: afmt procedure(d_func_2d), optional :: f class(psb_d_base_sparse_mat), optional :: amold class(psb_d_base_vect_type), optional :: vmold class(psb_i_base_vect_type), optional :: imold integer(psb_ipk_), optional :: partition, nrl,iv(:) ! Local variables. integer(psb_ipk_), parameter :: nb=20 type(psb_d_csc_sparse_mat) :: acsc type(psb_d_coo_sparse_mat) :: acoo type(psb_d_csr_sparse_mat) :: acsr real(psb_dpk_) :: zt(nb),x,y,z integer(psb_ipk_) :: m,n,nnz,nr,nt,glob_row,nlr,i,j,ii,ib,k, partition_ integer(psb_ipk_) :: ix,iy,iz,ia,indx_owner ! For 2D partition integer(psb_ipk_) :: npx,npy,npdims(2),iamx,iamy,mynx,myny integer(psb_ipk_), allocatable :: bndx(:),bndy(:) ! Process grid integer(psb_ipk_) :: np, iam integer(psb_ipk_) :: icoeff integer(psb_ipk_), allocatable :: irow(:),icol(:),myidx(:) real(psb_dpk_), allocatable :: val(:) ! deltah dimension of each grid cell ! deltat discretization time real(psb_dpk_) :: deltah, sqdeltah, deltah2 real(psb_dpk_), parameter :: rhs=dzero,one=done,zero=dzero real(psb_dpk_) :: t0, t1, t2, t3, tasb, talc, ttot, tgen, tcdasb integer(psb_ipk_) :: err_act procedure(d_func_2d), pointer :: f_ character(len=20) :: name, ch_err,tmpfmt info = psb_success_ name = 'create_matrix' call psb_erractionsave(err_act) call psb_info(ictxt, iam, np) if (present(f)) then f_ => f else f_ => d_null_func_2d end if deltah = done/(idim+2) sqdeltah = deltah*deltah deltah2 = (2*done)* deltah if (present(partition)) then if ((1<= partition).and.(partition <= 3)) then partition_ = partition else write(*,*) 'Invalid partition choice ',partition,' defaulting to 3' partition_ = 3 end if else partition_ = 3 end if ! initialize array descriptor and sparse matrix storage. provide an ! estimate of the number of non zeroes m = idim*idim n = m nnz = ((n*7)/(np)) if(iam == psb_root_) write(psb_out_unit,'("Generating Matrix (size=",i0,")...")')n t0 = psb_wtime() select case(partition_) case(1) ! A BLOCK partition if (present(nrl)) then nr = nrl else ! ! Using a simple BLOCK distribution. ! nt = (m+np-1)/np nr = max(0,min(nt,m-(iam*nt))) end if nt = nr call psb_sum(ictxt,nt) if (nt /= m) then write(psb_err_unit,*) iam, 'Initialization error ',nr,nt,m info = -1 call psb_barrier(ictxt) call psb_abort(ictxt) return end if ! ! First example of use of CDALL: specify for each process a number of ! contiguous rows ! call psb_cdall(ictxt,desc_a,info,nl=nr) myidx = desc_a%get_global_indices() nlr = size(myidx) case(2) ! A partition defined by the user through IV if (present(iv)) then if (size(iv) /= m) then write(psb_err_unit,*) iam, 'Initialization error: wrong IV size',size(iv),m info = -1 call psb_barrier(ictxt) call psb_abort(ictxt) return end if else write(psb_err_unit,*) iam, 'Initialization error: IV not present' info = -1 call psb_barrier(ictxt) call psb_abort(ictxt) return end if ! ! Second example of use of CDALL: specify for each row the ! process that owns it ! call psb_cdall(ictxt,desc_a,info,vg=iv) myidx = desc_a%get_global_indices() nlr = size(myidx) case(3) ! A 2-dimensional partition ! A nifty MPI function will split the process list npdims = 0 call mpi_dims_create(np,2,npdims,info) npx = npdims(1) npy = npdims(2) allocate(bndx(0:npx),bndy(0:npy)) ! We can reuse idx2ijk for process indices as well. call idx2ijk(iamx,iamy,iam,npx,npy,base=0) ! Now let's split the 2D square in rectangles call dist1Didx(bndx,idim,npx) mynx = bndx(iamx+1)-bndx(iamx) call dist1Didx(bndy,idim,npy) myny = bndy(iamy+1)-bndy(iamy) ! How many indices do I own? nlr = mynx*myny allocate(myidx(nlr)) ! Now, let's generate the list of indices I own nr = 0 do i=bndx(iamx),bndx(iamx+1)-1 do j=bndy(iamy),bndy(iamy+1)-1 nr = nr + 1 call ijk2idx(myidx(nr),i,j,idim,idim) end do end do if (nr /= nlr) then write(psb_err_unit,*) iam,iamx,iamy, 'Initialization error: NR vs NLR ',& & nr,nlr,mynx,myny info = -1 call psb_barrier(ictxt) call psb_abort(ictxt) end if ! ! Third example of use of CDALL: specify for each process ! the set of global indices it owns. ! call psb_cdall(ictxt,desc_a,info,vl=myidx) case default write(psb_err_unit,*) iam, 'Initialization error: should not get here' info = -1 call psb_barrier(ictxt) call psb_abort(ictxt) return end select if (info == psb_success_) call psb_spall(a,desc_a,info,nnz=nnz) ! define rhs from boundary conditions; also build initial guess if (info == psb_success_) call psb_geall(xv,desc_a,info) if (info == psb_success_) call psb_geall(bv,desc_a,info) call psb_barrier(ictxt) talc = psb_wtime()-t0 if (info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='allocation rout.' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if ! we build an auxiliary matrix consisting of one row at a ! time; just a small matrix. might be extended to generate ! a bunch of rows per call. ! allocate(val(20*nb),irow(20*nb),& &icol(20*nb),stat=info) if (info /= psb_success_ ) then info=psb_err_alloc_dealloc_ call psb_errpush(info,name) goto 9999 endif ! loop over rows belonging to current process in a block ! distribution. call psb_barrier(ictxt) t1 = psb_wtime() do ii=1, nlr,nb ib = min(nb,nlr-ii+1) icoeff = 1 do k=1,ib i=ii+k-1 ! local matrix pointer glob_row=myidx(i) ! compute gridpoint coordinates call idx2ijk(ix,iy,glob_row,idim,idim) ! x, y coordinates x = (ix-1)*deltah y = (iy-1)*deltah zt(k) = f_(x,y) ! internal point: build discretization ! ! term depending on (x-1,y) ! val(icoeff) = -a1(x,y)/sqdeltah-b1(x,y)/deltah2 if (ix == 1) then zt(k) = g(dzero,y)*(-val(icoeff)) + zt(k) else call ijk2idx(icol(icoeff),ix-1,iy,idim,idim) irow(icoeff) = glob_row icoeff = icoeff+1 endif ! term depending on (x,y-1) val(icoeff) = -a2(x,y)/sqdeltah-b2(x,y)/deltah2 if (iy == 1) then zt(k) = g(x,dzero)*(-val(icoeff)) + zt(k) else call ijk2idx(icol(icoeff),ix,iy-1,idim,idim) irow(icoeff) = glob_row icoeff = icoeff+1 endif ! term depending on (x,y) val(icoeff)=(2*done)*(a1(x,y) + a2(x,y))/sqdeltah + c(x,y) call ijk2idx(icol(icoeff),ix,iy,idim,idim) irow(icoeff) = glob_row icoeff = icoeff+1 ! term depending on (x,y+1) val(icoeff)=-a2(x,y)/sqdeltah+b2(x,y)/deltah2 if (iy == idim) then zt(k) = g(x,done)*(-val(icoeff)) + zt(k) else call ijk2idx(icol(icoeff),ix,iy+1,idim,idim) irow(icoeff) = glob_row icoeff = icoeff+1 endif ! term depending on (x+1,y) val(icoeff)=-a1(x,y)/sqdeltah+b1(x,y)/deltah2 if (ix==idim) then zt(k) = g(done,y)*(-val(icoeff)) + zt(k) else call ijk2idx(icol(icoeff),ix+1,iy,idim,idim) irow(icoeff) = glob_row icoeff = icoeff+1 endif end do call psb_spins(icoeff-1,irow,icol,val,a,desc_a,info) if(info /= psb_success_) exit call psb_geins(ib,myidx(ii:ii+ib-1),zt(1:ib),bv,desc_a,info) if(info /= psb_success_) exit zt(:)=dzero call psb_geins(ib,myidx(ii:ii+ib-1),zt(1:ib),xv,desc_a,info) if(info /= psb_success_) exit end do tgen = psb_wtime()-t1 if(info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='insert rout.' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if deallocate(val,irow,icol) call psb_barrier(ictxt) t1 = psb_wtime() call psb_cdasb(desc_a,info,mold=imold) tcdasb = psb_wtime()-t1 call psb_barrier(ictxt) t1 = psb_wtime() if (info == psb_success_) then if (present(amold)) then call psb_spasb(a,desc_a,info,dupl=psb_dupl_err_,mold=amold) else call psb_spasb(a,desc_a,info,dupl=psb_dupl_err_,afmt=afmt) end if end if call psb_barrier(ictxt) if(info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='asb rout.' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if if (info == psb_success_) call psb_geasb(xv,desc_a,info,mold=vmold) if (info == psb_success_) call psb_geasb(bv,desc_a,info,mold=vmold) if(info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='asb rout.' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if tasb = psb_wtime()-t1 call psb_barrier(ictxt) ttot = psb_wtime() - t0 call psb_amx(ictxt,talc) call psb_amx(ictxt,tgen) call psb_amx(ictxt,tasb) call psb_amx(ictxt,ttot) if(iam == psb_root_) then tmpfmt = a%get_fmt() write(psb_out_unit,'("The matrix has been generated and assembled in ",a3," format.")')& & tmpfmt write(psb_out_unit,'("-allocation time : ",es12.5)') talc write(psb_out_unit,'("-coeff. gen. time : ",es12.5)') tgen write(psb_out_unit,'("-desc asbly time : ",es12.5)') tcdasb write(psb_out_unit,'("- mat asbly time : ",es12.5)') tasb write(psb_out_unit,'("-total time : ",es12.5)') ttot end if call psb_erractionrestore(err_act) return 9999 call psb_error_handler(ictxt,err_act) return end subroutine psb_d_gen_pde2d end module psb_d_pde2d_mod program psb_d_pde2d use psb_base_mod use psb_prec_mod use psb_krylov_mod use psb_util_mod use psb_d_pde2d_mod implicit none ! input parameters character(len=20) :: kmethd, ptype character(len=5) :: afmt integer(psb_ipk_) :: idim ! miscellaneous real(psb_dpk_), parameter :: one = done real(psb_dpk_) :: t1, t2, tprec ! sparse matrix and preconditioner type(psb_dspmat_type) :: a type(psb_dprec_type) :: prec ! descriptor type(psb_desc_type) :: desc_a ! dense vectors type(psb_d_vect_type) :: xxv,bv ! parallel environment integer(psb_ipk_) :: ictxt, iam, np ! solver parameters integer(psb_ipk_) :: iter, itmax,itrace, istopc, irst integer(psb_long_int_k_) :: amatsize, precsize, descsize, d2size real(psb_dpk_) :: err, eps ! other variables integer(psb_ipk_) :: info, i character(len=20) :: name,ch_err character(len=40) :: fname info=psb_success_ call psb_init(ictxt) call psb_info(ictxt,iam,np) if (iam < 0) then ! This should not happen, but just in case call psb_exit(ictxt) stop endif if(psb_get_errstatus() /= 0) goto 9999 name='pde2d90' call psb_set_errverbosity(itwo) ! ! Hello world ! if (iam == psb_root_) then write(*,*) 'Welcome to PSBLAS version: ',psb_version_string_ write(*,*) 'This is the ',trim(name),' sample program' end if ! ! get parameters ! call get_parms(ictxt,kmethd,ptype,afmt,idim,istopc,itmax,itrace,irst) ! ! allocate and fill in the coefficient matrix, rhs and initial guess ! call psb_barrier(ictxt) t1 = psb_wtime() call psb_gen_pde2d(ictxt,idim,a,bv,xxv,desc_a,afmt,info) call psb_barrier(ictxt) t2 = psb_wtime() - t1 if(info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='psb_gen_pde2d' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if if (iam == psb_root_) write(psb_out_unit,'("Overall matrix creation time : ",es12.5)')t2 if (iam == psb_root_) write(psb_out_unit,'(" ")') ! ! prepare the preconditioner. ! if(iam == psb_root_) write(psb_out_unit,'("Setting preconditioner to : ",a)')ptype call prec%init(ictxt,ptype,info) call psb_barrier(ictxt) t1 = psb_wtime() call prec%build(a,desc_a,info) if(info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='psb_precbld' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if tprec = psb_wtime()-t1 call psb_amx(ictxt,tprec) if (iam == psb_root_) write(psb_out_unit,'("Preconditioner time : ",es12.5)')tprec if (iam == psb_root_) write(psb_out_unit,'(" ")') call prec%descr() ! ! iterative method parameters ! if(iam == psb_root_) write(psb_out_unit,'("Calling iterative method ",a)')kmethd call psb_barrier(ictxt) t1 = psb_wtime() eps = 1.d-6 call psb_krylov(kmethd,a,prec,bv,xxv,eps,desc_a,info,& & itmax=itmax,iter=iter,err=err,itrace=itrace,istop=istopc,irst=irst) if(info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='solver routine' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if call psb_barrier(ictxt) t2 = psb_wtime() - t1 call psb_amx(ictxt,t2) amatsize = a%sizeof() descsize = desc_a%sizeof() precsize = prec%sizeof() call psb_sum(ictxt,amatsize) call psb_sum(ictxt,descsize) call psb_sum(ictxt,precsize) if (iam == psb_root_) then write(psb_out_unit,'(" ")') write(psb_out_unit,'("Number of processes : ",i0)')np write(psb_out_unit,'("Time to solve system : ",es12.5)')t2 write(psb_out_unit,'("Time per iteration : ",es12.5)')t2/iter write(psb_out_unit,'("Number of iterations : ",i0)')iter write(psb_out_unit,'("Convergence indicator on exit : ",es12.5)')err write(psb_out_unit,'("Info on exit : ",i0)')info write(psb_out_unit,'("Total memory occupation for A: ",i12)')amatsize write(psb_out_unit,'("Total memory occupation for PREC: ",i12)')precsize write(psb_out_unit,'("Total memory occupation for DESC_A: ",i12)')descsize write(psb_out_unit,'("Storage format for A: ",a)') a%get_fmt() write(psb_out_unit,'("Storage format for DESC_A: ",a)') desc_a%get_fmt() end if ! ! cleanup storage and exit ! call psb_gefree(bv,desc_a,info) call psb_gefree(xxv,desc_a,info) call psb_spfree(a,desc_a,info) call prec%free(info) call psb_cdfree(desc_a,info) if(info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='free routine' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if call psb_exit(ictxt) stop 9999 call psb_error(ictxt) stop contains ! ! get iteration parameters from standard input ! subroutine get_parms(ictxt,kmethd,ptype,afmt,idim,istopc,itmax,itrace,irst) integer(psb_ipk_) :: ictxt character(len=*) :: kmethd, ptype, afmt integer(psb_ipk_) :: idim, istopc,itmax,itrace,irst integer(psb_ipk_) :: np, iam integer(psb_ipk_) :: ip, inp_unit character(len=1024) :: filename call psb_info(ictxt, iam, np) if (iam == 0) then if (command_argument_count()>0) then call get_command_argument(1,filename) inp_unit = 30 open(inp_unit,file=filename,action='read',iostat=info) if (info /= 0) then write(psb_err_unit,*) 'Could not open file ',filename,' for input' call psb_abort(ictxt) stop else write(psb_err_unit,*) 'Opened file ',trim(filename),' for input' end if else inp_unit=psb_inp_unit end if read(inp_unit,*) ip if (ip >= 3) then read(inp_unit,*) kmethd read(inp_unit,*) ptype read(inp_unit,*) afmt read(inp_unit,*) idim if (ip >= 4) then read(inp_unit,*) istopc else istopc=1 endif if (ip >= 5) then read(inp_unit,*) itmax else itmax=500 endif if (ip >= 6) then read(inp_unit,*) itrace else itrace=-1 endif if (ip >= 7) then read(inp_unit,*) irst else irst=1 endif write(psb_out_unit,'("Solving matrix : ell1")') write(psb_out_unit,'("Grid dimensions : ",i5," x ",i5)')idim,idim write(psb_out_unit,'("Number of processors : ",i0)')np write(psb_out_unit,'("Data distribution : BLOCK")') write(psb_out_unit,'("Preconditioner : ",a)') ptype write(psb_out_unit,'("Iterative method : ",a)') kmethd write(psb_out_unit,'(" ")') else ! wrong number of parameter, print an error message and exit call pr_usage(izero) call psb_abort(ictxt) stop 1 endif if (inp_unit /= psb_inp_unit) then close(inp_unit) end if end if ! broadcast parameters to all processors call psb_bcast(ictxt,kmethd) call psb_bcast(ictxt,afmt) call psb_bcast(ictxt,ptype) call psb_bcast(ictxt,idim) call psb_bcast(ictxt,istopc) call psb_bcast(ictxt,itmax) call psb_bcast(ictxt,itrace) call psb_bcast(ictxt,irst) return end subroutine get_parms ! ! print an error message ! subroutine pr_usage(iout) integer(psb_ipk_) :: iout write(iout,*)'incorrect parameter(s) found' write(iout,*)' usage: pde2d90 methd prec dim & &[istop itmax itrace]' write(iout,*)' where:' write(iout,*)' methd: cgstab cgs rgmres bicgstabl' write(iout,*)' prec : bjac diag none' write(iout,*)' dim number of points along each axis' write(iout,*)' the size of the resulting linear ' write(iout,*)' system is dim**3' write(iout,*)' istop stopping criterion 1, 2 ' write(iout,*)' itmax maximum number of iterations [500] ' write(iout,*)' itrace <=0 (no tracing, default) or ' write(iout,*)' >= 1 do tracing every itrace' write(iout,*)' iterations ' end subroutine pr_usage end program psb_d_pde2d