C********************************************************************** C C Copyright (C) 1992 Roland W. Freund and Noel M. Nachtigal C All rights reserved. C C This code is part of a copyrighted package. For details, see the C file "cpyrit.doc" in the top-level directory. C C ***************************************************************** C ANY USE OF THIS CODE CONSTITUTES ACCEPTANCE OF THE TERMS OF THE C COPYRIGHT NOTICE C ***************************************************************** C C********************************************************************** C C This file contains the routines for the QMR algorithm for C symmetric matrices, using the coupled two-term recurrence variant C of the look-ahead Lanczos algorithm. C C********************************************************************** C SUBROUTINE SSCPL (NDIM,NLEN,NLIM,MAXPQ,MAXVW,M,MVEC,NORMS, $DWK,IDX,IWK,VECS,TOL,INFO) C C Purpose: C This subroutine uses the QMR algorithm based on the coupled two- C term variant of the look-ahead Lanczos process to solve linear C systems. It runs the algorithm to convergence or until a user- C specified limit on the number of iterations is reached. It is set C up to solve symmetric systems starting with identical starting C vectors. C C The code is set up to solve the system A x = b with initial C guess x_0 = 0. Here A x = b denotes the preconditioned system, C and it is connected with the original system as follows. Let C B y = c be the original unpreconditioned system to be solved, and C let y_0 be an arbitrary initial guess for its solution. Then: C A x = b, where A = M_1^{-1} B M_2^{-1}, C x = M_2 (y - y_0), b = M_1^{-1} (c - B y_0). C Here M = M_1 M_2 is the preconditioner. C C To recover the final iterate y_n for the original system B y = c C from the final iterate x_n for the preconditioned system A x = b, C set C y_n = y_0 + M_2^{-1} x_n. C C The algorithm was first described in the RIACS Technical Report C 92.15, An Implementation of the QMR Method Based on Coupled Two- C Term Recurrences, June 1992. A good reference for the details C of this implementation is the RIACS Technical Report 92.19, C Implementation details of the coupled QMR algorithm, by R.W. C Freund and N.M. Nachtigal, October 1992. C C Parameters: C For a description of the parameters, see the file "sscpl.doc" in C the current directory. C C External routines used: C subroutine saxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C single precision sdot(n,x,incx,y,incy) C BLAS-1 routine, computes y^H * x. C single precision slamch(ch) C LAPACK routine, computes machine-related constants. C single precision snrm2(n,x,incx) C BLAS-1 routine, computes the 2-norm of x. C subroutine sqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine sqrsl(x,ldx,n,k,qraux,y,qy,qty,b,rsd,xb,job,info) C LINPACK routine, applies the QR factorization of x. C subroutine srandn(n,x,seed) C Library routine, fill x with random numbers. C subroutine srotg(a,b,cos,sin) C BLAS-1 routine, computes the Givens rotation which rotates the C vector [a; b] into [ sqrt(a**2 + b**2); 0 ]. C subroutine ssvdc(x,ldx,n,p,s,e,u,ldu,v,ldv,work,job,info) C LINPACK routine, computes the SVD of x. C subroutine sscpl1(ndim,nlen,m,n,k,kstar,l,lstar,mk,mkstar,nl, C nlstar,vf,ierr,adjust,norm1,norm2,dwk,iwk,idx,vecs) C Low-level routine, rebuilds vectors PQ in CPL. C subroutine sscpl2(ndim,nlen,m,n,k,kstar,l,lstar,mk,mkstar,nl, C nlstar,vf,ierr,adjust,norm1,norm2,dwk,iwk,idx,vecs) C Low-level routine, rebuilds vectors VW in CPL. C single precision function sscpll(i,n) C User-supplied routine, computes inner recurrence coefficients. C single precision sscplom(n) C User-supplied routine, specifies the QMR scaling factors. C single precision function sscplu(i,n) C User-supplied routine, computes inner recurrence coefficients. C C Noel M. Nachtigal C March 1, 1992 C C********************************************************************** C INTRINSIC ABS, FLOAT, AMAX1, SQRT, MAX0, MIN0, MOD EXTERNAL SAXPBY, SDOT, SLAMCH, SNRM2, SQRDC, SQRSL, SRANDN, SROTG EXTERNAL SSVDC, SSCPL1, SSCPL2, SSCPLL, SSCPLO, SSCPLU REAL SDOT, SLAMCH, SNRM2, SSCPLL, SSCPLO, SSCPLU C INTEGER INFO(4), M, MAXPQ, MAXVW, MVEC, NDIM, NLEN, NLIM INTEGER IDX(4,NLIM+2), IWK(M,13) REAL DWK(M,8*M+14), NORMS(2), TOL REAL VECS(NDIM,3*MVEC+3-1) C C Common block variables. C C C Common block SSCPLX. C REAL NORMA COMMON /SSCPLX/NORMA C C Miscellaneous parameters. C REAL DHUN, DONE, DTEN, SZERO PARAMETER (DHUN = 1.0E2,DONE = 1.0E0,DTEN = 1.0E1,SZERO = 0.0E0) C C Local variables, permanent. C LOGICAL IBUILT, INNER, RERUN SAVE IBUILT, INNER, RERUN INTEGER IEND, IERR, K, KBLKSZ, KSTAR, L, LBLKSZ, LSTAR, MK SAVE IEND, IERR, K, KBLKSZ, KSTAR, L, LBLKSZ, LSTAR, MK INTEGER MKMAX, MKSTAR, MPQBLT, MVWBLT, N, NL, NLMAX, NLSTAR, NMAX SAVE MKMAX, MKSTAR, MPQBLT, MVWBLT, N, NL, NLMAX, NLSTAR, NMAX INTEGER NP1, NQMR, NUMCHK, PQBEG, RETLBL, TF, TRES, VF, VWBEG SAVE NP1, NQMR, NUMCHK, PQBEG, RETLBL, TF, TRES, VF, VWBEG REAL ADJUST, MAXOMG, NORM1, NORM2, TMAX, TMIN, TNRM SAVE ADJUST, MAXOMG, NORM1, NORM2, TMAX, TMIN, TNRM REAL R0, RESN, UCHK, UNRM SAVE R0, RESN, UCHK, UNRM C C Local variables, transient. C INTEGER I, IBASE, IBLKSZ, IJ, J, KEND, LEND, MI, MIP1 INTEGER NI, NIP1, REVCOM REAL STMP, STMP1, STMP2, STMP3, STMP4 REAL ISTMP1, ISTMP3 C C Initialize some of the permanent variables. C DATA RETLBL /0/ C C Check the reverse communication flag to see where to branch. C REVCOM RETLBL Comment C 0 0 first call, go to label 10 C 1 200 returning from AXB, go to label 200 C 1 350 returning from AXB, go to label 350 C 1 720 returning from AXB, go to label 720 C REVCOM = INFO(2) INFO(2) = 0 IF (REVCOM.EQ.0) THEN N = 0 NQMR = 0 MPQBLT = 0 MVWBLT = 0 IF (RETLBL.EQ.0) GO TO 10 ELSE IF (REVCOM.EQ.1) THEN IF (RETLBL.EQ.200) THEN GO TO 200 ELSE IF (RETLBL.EQ.350) THEN GO TO 350 ELSE IF (RETLBL.EQ.720) THEN GO TO 720 END IF END IF IERR = 1 GO TO 750 C C Check whether the inputs are valid. C 10 IERR = 0 IF (NDIM.LT.1) IERR = 2 IF (NLEN.LT.1) IERR = 2 IF (NLIM.LT.1) IERR = 2 IF (MAXPQ.LT.1) IERR = 2 IF (MAXVW.LT.1) IERR = 2 IF (NLEN.GT.NDIM) IERR = 2 IF (MVEC.LT.MAXPQ+1) IERR = 2 IF (MVEC.LT.MAXVW+1) IERR = 2 IF (M.LT.MAXVW+MAXPQ+2) IERR = 2 IF (IERR.NE.0) GO TO 750 C C Extract from INFO the output units TF and VF, the true residual C flag TRES. C VF = MAX0(INFO(1),0) I = VF / 100000 VF = VF - I * 100000 TRES = VF / 10000 VF = VF - TRES * 10000 TF = VF / 100 VF = VF - TF * 100 C C Extract the norms. C NORMA = SZERO NORM1 = ABS(NORMS(1)) NORM2 = ABS(NORMS(2)) C C Set the adjustment parameters. C NUMCHK = 25 ADJUST = DTEN C C Extract and check the various tolerances and norm estimates. C TNRM = SLAMCH('E') * DTEN TMIN = SQRT(SQRT(SLAMCH('S'))) TMAX = DONE / TMIN IF (TOL.LE.SZERO) TOL = SQRT(SLAMCH('E')) C C Start the trace messages and convergence history. C IF (VF.NE.0) WRITE (VF,'(I8,2E11.4)') 0, DONE, DONE IF (TF.NE.0) WRITE (TF,'(I8,2E11.4)') 0, DONE, DONE C C Set up wrapped indices. The following indices are used: C IDX(1,I) = indices used for the work arrays row-dimensioned M; C IDX(2,I) = indices used for v_i; C IDX(3,I) = indices used for p_i; C IDX(4,I) = indices used for s_i. C DO 20 I = 1, NLIM+2 IDX(1,I) = MOD(I-1,M) + 1 IDX(2,I) = 3 + 1 + 0*MVEC + MOD(I-1,MVEC) IDX(3,I) = 3 + 1 + 1*MVEC + MOD(I-1,MVEC) IDX(4,I) = 3 + 1 + 2*MVEC + MOD(I-1,MVEC-1) 20 CONTINUE C C Set x_0 = 0 and compute the norm of the initial residual. C CALL SAXPBY (NLEN,VECS(1,IDX(2,1)),DONE,VECS(1,2),SZERO,VECS(1,$IDX(2,1))) CALL SAXPBY (NLEN,VECS(1,1),SZERO,VECS(1,1),SZERO,VECS(1,1)) R0 = SNRM2(NLEN,VECS(1,IDX(2,1)),1) IF ((TOL.GE.DONE).OR.(R0.EQ.SZERO)) GO TO 750 C C Scale the first pair of Lanczos vectors and check for invariant C subspaces. C STMP1 = R0 IF (STMP1.LT.TNRM) IERR = IERR + 16 IF (IERR.NE.0) GO TO 750 DWK(IDX(1,1),IDX(1,1)) = SDOT(NLEN,VECS(1,IDX(2,1)),1,VECS(1,IDX(2 $,1)),1) / ( STMP1**2 ) IF ((STMP1.GE.TMAX).OR.(STMP1.LE.TMIN)) THEN STMP = DONE / STMP1 CALL SAXPBY (NLEN,VECS(1,IDX(2,1)),STMP,VECS(1,IDX(2,1)),SZERO,$VECS(1,IDX(2,1))) STMP1 = DONE END IF DWK(IDX(1,1),8*M+10) = DONE / STMP1 C C Initialize the counters. C K = 0 L = 1 N = 1 NMAX = 0 KSTAR = 1 LSTAR = 1 MKMAX = 0 NLMAX = 1 PQBEG = 1 VWBEG = 1 IWK(IDX(1,0+1),2) = 1 IWK(IDX(1,1+1),2) = 1 IWK(IDX(1,0+1),1) = 1 IWK(IDX(1,1+1),1) = 1 MPQBLT = 1 MVWBLT = 1 MK = IWK(IDX(1,K+1),1) NL = IWK(IDX(1,L+1),2) IWK(IDX(1,N),3) = NL NLSTAR = IWK(IDX(1,LSTAR+1),2) IWK(IDX(1,N),5) = NLSTAR C C Set up the QMR iteration. C RESN = DONE DWK(IDX(1,1),8*M+8) = SSCPLO(1) DWK(IDX(1,1),8*M+4) = DWK(IDX(1,1),8*M+8) * R0 MAXOMG = DONE / DWK(IDX(1,1),8*M+8) C C This is one step of the coupled two-term Lanczos algorithm. C V_n, W_n, P_{n-1}, Q_{n-1}, D_{n-1}, E_{n-1}, F_{n-1}, L_{n-1}, C U_{n-1}, and D_{nn} are given. C C Note that the previous step was not necessarily step N-1, as it C could have been a restart. C Except at the first step, the following hold: C NL.LE.N, MK.LE.N-1, N-NL.LE.MAXVW-1, N-MK.LE.MAXPQ. C C The references in the comments to step numbers correspond to the C description in Section 4 of the report Implementation details of C the coupled QMR algorithm, by Freund and Nachtigal, RIACS report C 92.19, October 1992. C 30 IWK(IDX(1,N),9) = L IWK(IDX(1,N),7) = N IWK(IDX(1,N),8) = K IWK(IDX(1,N),12) = K IWK(IDX(1,N),10) = KSTAR IWK(IDX(1,N),13) = KSTAR IWK(IDX(1,N),11) = LSTAR C C Build p_n. C There are three possible cases: C N > NMAX : the PQ sequence is being built new. C N > MKMAX : the PQ sequence is being rebuilt, build as if new. C N <= MKMAX : the VW sequence is being rebuilt, just rerun PQ. C The first two are identical in terms of code. C The code is inlined because of the matrix multiplications. C C Initialize Lanczos step variables. C NP1 = N + 1 DWK(IDX(1,N),8*M+14) = SZERO KBLKSZ = N - MK IBUILT = .FALSE. IERR = 0 C C Clear current column of U. C DO 40 I = 1, M DWK(IDX(1,I),6*M+IDX(1,N)) = SZERO 40 CONTINUE DWK(IDX(1,N),6*M+IDX(1,N)) = DONE C C The first step is different, deal with it here. C IF (N.EQ.1) THEN NP1 = 2 MKSTAR = 1 DWK(IDX(1,1),8*M+13) = DONE DWK(IDX(1,1),8*M+12) = DWK(IDX(1,1),8*M+10) INNER = .FALSE. CALL SAXPBY (NLEN,VECS(1,IDX(3,1)),DONE,VECS(1,IDX(2,1)),SZERO, $VECS(1,IDX(3,1))) GO TO 340 END IF C C Step 1: C Update D^{(n-1)} to D^{(n)}. C DO 60 I = NL, N-1 STMP = SZERO DO 50 J = MAX0(NLSTAR,IWK(IDX(1,I),3)), N-1 STMP = STMP + DWK(IDX(1,I),IDX(1,J)) * DWK(IDX(1,J),2*M+$IDX(1,N-1)) 50 CONTINUE STMP = ( DWK(IDX(1,I),M+IDX(1,N-1)) - STMP ) / DWK(IDX(1,N) $,2*M+IDX(1,N-1)) DWK(IDX(1,I),IDX(1,N)) = STMP DWK(IDX(1,N),IDX(1,I)) = STMP 60 CONTINUE C C Step 2: C Compute k^\star. C I = KSTAR DO 70 J = I+1, K IF (IWK(IDX(1,J+1),1).LE.NL-1) KSTAR = J 70 CONTINUE MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Check memory allocation (MKSTAR.GE.PQBEG). Update the range of C valid indices: the new vector p_n will overwrite the vector at C location N-MVEC. C IF (MKSTAR.LT.PQBEG) THEN IERR = 64 GO TO 750 END IF PQBEG = MAX0(PQBEG,N-MVEC+1) C C Step 3: C Compute F_{n,1:n-1} = F_{n,m_{k^\star}:n-1}. C DO 90 I = MKSTAR, N-1 STMP = SZERO DO 80 J = MAX0(NL,IWK(IDX(1,I),5)), I+1 STMP = STMP + DWK(IDX(1,N),IDX(1,J)) * DWK(IDX(1,J),2*M+$IDX(1,I)) 80 CONTINUE DWK(IDX(1,N),M+IDX(1,I)) = STMP STMP = ABS(STMP) / DWK(IDX(1,I),8*M+13) NORMA = AMAX1(NORMA,STMP) NORM1 = AMAX1(ADJUST*NORMA,NORM1) NORM2 = AMAX1(ADJUST*NORMA,NORM2) 90 CONTINUE C C Step 4: C Check whether E_k is nonsingular. C IF (KBLKSZ.EQ.1) THEN INNER = ABS(DWK(IDX(1,MK),5*M+IDX(1,MK))).EQ.SZERO ELSE DO 110 I = MK, N-1 DO 100 J = MK, N-1 DWK(I-MK+1,4*M+J-MK+1) = DWK(IDX(1,I),5*M+IDX(1,J)) 100 CONTINUE 110 CONTINUE CALL SSVDC (DWK(1,4*M+1),M,KBLKSZ,KBLKSZ,DWK(1,8*M+1),DWK(1,8*M $+3),SZERO,0,SZERO,0,DWK(1,8*M+2),0,I) IF (I.NE.0) THEN IERR = -I GO TO 750 END IF STMP = SZERO IF (DWK(1,8*M+1).NE.SZERO) STMP = DWK(KBLKSZ,8*M+1) / DWK(1,8*M$+1) INNER = STMP.LT.(FLOAT(NLEN) * SLAMCH('E')) END IF KEND = K IF (INNER) KEND = K - 1 C C Step 5: C Compute U_{m_i:m_{i+1}-1,n}, for i = k^\star, ..., kend, kend = k C or kend = k-1. C IF (KEND.EQ.K) IWK(IDX(1,K+1+1),1) = N DO 150 I = KSTAR, KEND MI = IWK(IDX(1,I+1),1) MIP1 = IWK(IDX(1,I+1+1),1) IBLKSZ = MIP1 - MI IF (IBLKSZ.EQ.1) THEN DWK(IDX(1,MI),6*M+IDX(1,N)) = DWK(IDX(1,N),M+IDX(1,MI)) / $DWK(IDX(1,MI),5*M+IDX(1,MI)) ELSE DO 130 J = MI, MIP1-1 DWK(J-MI+1,8*M+1) = DWK(IDX(1,N),M+IDX(1,J)) DO 120 IJ = MI, MIP1-1 DWK(J-MI+1,4*M+IJ-MI+1) = DWK(IDX(1,J),5*M+IDX(1,IJ)) 120 CONTINUE 130 CONTINUE CALL SQRDC (DWK(1,4*M+1),M,IBLKSZ,IBLKSZ,DWK(1,8*M+3),0,$SZERO,0) CALL SQRSL (DWK(1,4*M+1),M,IBLKSZ,IBLKSZ,DWK(1,8*M+3),DWK(1, $8*M+1),$ SZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1), $SZERO,100,J) DO 140 J = MI, MIP1-1 DWK(IDX(1,J),6*M+IDX(1,N)) = DWK(J-MI+1,8*M+2) 140 CONTINUE END IF 150 CONTINUE C C Step 6: C Build the part common to both inner and regular vectors. C DWK(IDX(1,N),8*M+12) = DWK(IDX(1,N),8*M+10) CALL SAXPBY (NLEN,VECS(1,3),DONE,VECS(1,IDX(2,N)),SZERO,VECS(1,3)) DO 160 I = MKSTAR, MK-1 STMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+12) /$DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,3),DONE,VECS(1,3),-STMP,VECS(1,IDX(3,I $))) 160 CONTINUE C C Check whether PQ is being rerun. C RERUN = (N.LE.NMAX).AND.(N.LE.MKMAX) IF (RERUN) THEN C C Check whether E_k has become singular (this should not happen). C IF ((IWK(IDX(1,K+1+1),1).EQ.N).AND.INNER) THEN IERR = 128 GO TO 750 END IF C C Determine whether this was an inner vector or not. C INNER = IWK(IDX(1,K+1+1),1).NE.N IF (VF.NE.0) THEN IF (INNER) THEN WRITE (VF,'(A17,I5,A14)') 'Rerunning vector ',N,$' (PQ) as inner' ELSE WRITE (VF,'(A17,I5,A16)') 'Rerunning vector ',N, $' (PQ) as regular' END IF END IF C C If building an inner vector, set the coefficients U_{m_k:n-1,n}. C IF (INNER) THEN DO 170 I = MK, N-1 DWK(IDX(1,I),6*M+IDX(1,N)) = SSCPLU(I,N) 170 CONTINUE END IF ELSE C C PQ is not being rerun, determine whether it is being rebuilt. C IF ((VF.NE.0).AND.(N.LE.NMAX)) THEN WRITE (VF,'(A15,I8)') 'Rebuilding P&Q:', N END IF C C If E_k is singular, build an inner vector. C IF (INNER) THEN IF (VF.NE.0) WRITE (VF,'(A31)')$'... moment matrix E is singular' GO TO 300 END IF END IF C C Either E_k is nonsingular, or PQ is being rerun. For the latter, C just finish building the vectors. For the former, the check in C Step 7 (G_{m_{k-1}:n-1,n-1}) could be done. However, the look- C ahead strategy requires the smaller of the checks in Step 7 and C Step 9, in case the norm estimates need updating. Hence, Step 7 C and Step 9 are switched, and their checks are later performed C together. C C Leave the common part of the vectors in the temporary vectors. C CALL SAXPBY (NLEN,VECS(1,IDX(3,N)),DONE,VECS(1,3),SZERO,VECS(1, $IDX(3,N))) C C Step 8: C Build regular vectors and compute A p_n and p_n^T A p_n. C DO 180 I = MK, N-1 STMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+12) /$DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,IDX(3,N)),DONE,VECS(1,IDX(3,N)),-STMP, $VECS(1,IDX(3,I))) 180 CONTINUE C C Have the caller carry out AXB, then return here. C CALL AXB (VECS(1,IDX(3,N)),VECS(1,IDX(2,NP1))) C 190 INFO(2) = 1 INFO(3) = IDX(3,N) INFO(4) = IDX(2,NP1) RETLBL = 200 RETURN 200 DWK(IDX(1,N),5*M+IDX(1,N)) = SDOT(NLEN,VECS(1,IDX(3,N)),1,VECS(1,$IDX(2,NP1)),1) * ( DWK(IDX(1,N),8*M+12)**2 ) C C Step 9: C Compute the last column of E_k^{-1}. C IF (KBLKSZ.EQ.1) THEN DWK(IDX(1,MK),7*M+IDX(1,K)) = DONE / DWK(IDX(1,MK),5*M+IDX(1,MK $)) ELSE DO 210 I = 1, KBLKSZ-1 DWK(I,8*M+1) = SZERO 210 CONTINUE DWK(KBLKSZ,8*M+1) = DONE CALL SQRSL (DWK(1,4*M+1),M,KBLKSZ,KBLKSZ,DWK(1,8*M+3),DWK(1,8*M$+1), $SZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1),SZERO,$100,J) DO 220 I = MK, N-1 DWK(IDX(1,I),7*M+IDX(1,K)) = DWK(I-MK+1,8*M+2) 220 CONTINUE END IF C C Check for zero norms. C DWK(IDX(1,N),8*M+13) = SNRM2(NLEN,VECS(1,IDX(3,N)),1) * DWK(IDX(1, $N),8*M+12) IF (DWK(IDX(1,N),8*M+13).EQ.SZERO) GO TO 360 C C If PQ is being rerun, then skip to the end. C IF (RERUN) GO TO 360 C C Step 7: C Build the full column G_{m_{k-1}:n-1,n-1} and compute its norm. C STMP1 = SZERO DO 240 I = IWK(IDX(1,K-1+1),1), N-1 STMP = SZERO DO 230 J = MAX0(I,NLSTAR), N STMP = STMP + DWK(IDX(1,I),6*M+IDX(1,J)) * DWK(IDX(1,J),2*M+$IDX(1,N-1)) 230 CONTINUE STMP = ABS(STMP) STMP1 = STMP1 + STMP * DWK(IDX(1,I),8*M+13) 240 CONTINUE STMP1 = STMP1 / DWK(IDX(1,N-1),8*M+13) C C Step 9: C Build the 2nd term for the next step, G_{m_k:n-1,n}, regular. C Compute the norm of G_{m_k:n-1,n}. C STMP3 = SZERO STMP4 = DWK(IDX(1,N),5*M+IDX(1,N)) * DWK(IDX(1,N),2*M+IDX(1,N-1)) DO 250 I = MK, N-1 STMP = ABS(STMP4 * DWK(IDX(1,I),7*M+IDX(1,K))) STMP3 = STMP3 + STMP * DWK(IDX(1,I),8*M+13) 250 CONTINUE STMP3 = STMP3 / DWK(IDX(1,N),8*M+13) C C Steps 7 and 9: C Check G_{m_{k-1}:n-1,n-1} and G_{m_k:n-1,n}. C STMP = AMAX1(STMP1,STMP3) INNER = STMP.GT.NORM1 IF (.NOT.INNER) GO TO 360 DWK(IDX(1,N),8*M+14) = STMP C C If G_{m_{k-1}:n-1,n-1} is bad, build inner vectors. C IF (STMP1.GT.NORM1) GO TO 300 C C If G_{m_k:n-1,n} is bad, check the inner vectors. C This only applies if m_k > 1. C IF (MK.LE.1) GO TO 300 C C Build the inner vectors to compute the 2nd term at the next step. C Get the coefficients U_{m_k:n-1,n} in a temporary location. C Build the inner vectors, their norms are needed. C IBUILT = .TRUE. DO 260 I = MK, N-1 DWK(IDX(1,I),6*M+IDX(1,NP1)) = SSCPLU(I,N) STMP = DWK(IDX(1,I),6*M+IDX(1,NP1)) * DWK(IDX(1,I),8*M+12 $) / DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,3),DONE,VECS(1,3),-STMP,VECS(1,IDX(3,I$))) 260 CONTINUE DWK(IDX(1,NP1),8*M+13) = SNRM2(NLEN,VECS(1,3),1) * DWK(IDX(1,N), $8*M+12) IF (DWK(IDX(1,NP1),8*M+13).EQ.SZERO) GO TO 310 C C Build the 2nd term for the next step, G_{m_{k-1}:m_k-1,n}, inner. C Compute the norm of G_{m_{k-1}:m_k-1,n}. C STMP = SZERO DO 270 J = MKSTAR, MK-1 STMP = STMP + DWK(IDX(1,J),6*M+IDX(1,N)) * DWK(IDX(1,J),5*M+$IDX(1,MK)) 270 CONTINUE DO 280 J = MK, N-1 STMP = STMP + DWK(IDX(1,J),6*M+IDX(1,NP1)) * DWK(IDX(1,J),5*M+ $IDX(1,MK)) 280 CONTINUE STMP1 = SZERO STMP = DWK(IDX(1,N),M+IDX(1,MK)) - STMP STMP4 = STMP * DWK(IDX(1,MK),2*M+IDX(1,MK-1)) DO 290 I = IWK(IDX(1,K-1+1),1), MK-1 STMP = ABS(DWK(IDX(1,I),7*M+IDX(1,K-1)) * STMP4) STMP1 = STMP1 + STMP * DWK(IDX(1,I),8*M+13) 290 CONTINUE STMP1 = STMP1 / DWK(IDX(1,NP1),8*M+13) C C Compare the inner and regular versions of the 2nd term at the next C step. Build the vector corresponding to the smaller term. C INNER = STMP3.GT.STMP1 IF (.NOT.INNER) GO TO 360 C C Build inner vectors. C Check whether the P&Q block has to be forced to close. C 300 IF (VF.NE.0) WRITE (VF,'(A7,I5,A14)') 'Vector ',N,' (PQ) is inner' IF (N-MK.EQ.MAXPQ) THEN CALL SSCPL1 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL,NLSTAR,$ VF,IERR,ADJUST,NORM1,NORM2,DWK,IWK,IDX,VECS) IF (IERR.NE.0) GO TO 750 INNER = .FALSE. KBLKSZ = N - MK RERUN = .TRUE. NP1 = N + 1 GO TO 190 END IF C C The temporary vectors contain either just partial inner vectors, C or the completed ones, depending on whether IBUILT is TRUE or C FALSE. In either case, replace the regular vectors. C 310 CALL SAXPBY (NLEN,VECS(1,IDX(3,N)),DONE,VECS(1,3),SZERO,VECS(1, $IDX(3,N))) C C Step 11: C Get the coefficients U_{m_k:n-1,n} and build inner vectors. C IF (IBUILT) THEN DO 320 I = MK, N-1 DWK(IDX(1,I),6*M+IDX(1,N)) = DWK(IDX(1,I),6*M+IDX(1,NP1)) 320 CONTINUE DWK(IDX(1,N),8*M+13) = DWK(IDX(1,NP1),8*M+13) ELSE DO 330 I = MK, N-1 DWK(IDX(1,I),6*M+IDX(1,N)) = SSCPLU(I,N) STMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+12)$ / DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,IDX(3,N)),DONE,VECS(1,IDX(3,N)),- $STMP,VECS(1,IDX(3,I))) 330 CONTINUE DWK(IDX(1,N),8*M+13) = SNRM2(NLEN,VECS(1,IDX(3,N)),1) *$DWK(IDX(1,N),8*M+12) END IF C C Step 11: C Compute A p_n and p_n^T A p_n. C Have the caller carry out AXB, then return here. C CALL AXB (VECS(1,IDX(3,N)),VECS(1,IDX(2,NP1))) C 340 INFO(2) = 1 INFO(3) = IDX(3,N) INFO(4) = IDX(2,NP1) RETLBL = 350 RETURN 350 DWK(IDX(1,N),5*M+IDX(1,N)) = SDOT(NLEN,VECS(1,IDX(3,N)),1,VECS(1, $IDX(2,NP1)),1) * ( DWK(IDX(1,N),8*M+12)**2 ) C C Step 10: C If regular vectors were built, update the counters. C 360 IF (.NOT.INNER) THEN K = K + 1 MK = IWK(IDX(1,K+1),1) END IF C C Update counters. C MPQBLT = MAX0(MPQBLT,MK-IWK(IDX(1,K-1+1),1)) MKMAX = MAX0(MK,MKMAX) IWK(IDX(1,N),6) = MKSTAR IWK(IDX(1,N),4) = MK C C Check for invariant subspaces. C IF (DWK(IDX(1,N),8*M+13).LT.TNRM) IERR = IERR + 16 IF (IERR.NE.0) GO TO 750 C C Update the norm estimates. C 370 IF (N.LE.NUMCHK) THEN STMP1 = SNRM2(NLEN,VECS(1,IDX(2,NP1)),1) * DWK(IDX(1,N),8*M+12)$ / DWK(IDX(1,N),8*M+13) NORMA = AMAX1(STMP1,NORMA) END IF STMP = ABS(DWK(IDX(1,N),5*M+IDX(1,N))) / ( DWK(IDX(1,N),8*M+13) $**2 ) NORMA = AMAX1(STMP,NORMA) NORM1 = AMAX1(ADJUST*NORMA,NORM1) NORM2 = AMAX1(ADJUST*NORMA,NORM2) C C Build v_{n+1} and w_{n+1}. C There are three cases: C N > NMAX : the VW sequence is being built new. C N >= NLMAX : the VW sequence is being rebuilt, build as if new. C N < NLMAX : the PQ sequence is being rebuilt, just rerun VW. C The first two are identical in terms of code. C IWK(IDX(1,N),12) = K IWK(IDX(1,N),13) = KSTAR DWK(IDX(1,N),8*M+11) = SZERO IBUILT = .FALSE. LBLKSZ = N - NL + 1 C C Clear current column of L. C DO 380 I = 1, M DWK(IDX(1,I),2*M+IDX(1,N)) = SZERO 380 CONTINUE C C Step 14: C Update E^{(n-1)} to E^{(n)}. C DO 400 I = MK, N-1 STMP = SZERO DO 390 J = MAX0(MKSTAR,IWK(IDX(1,I),4)), N-1 STMP = STMP + DWK(IDX(1,J),6*M+IDX(1,N)) * DWK(IDX(1,J),5*M+$IDX(1,I)) 390 CONTINUE STMP = DWK(IDX(1,N),M+IDX(1,I)) - STMP DWK(IDX(1,N),5*M+IDX(1,I)) = STMP DWK(IDX(1,I),5*M+IDX(1,N)) = STMP STMP1 = ABS(DWK(IDX(1,N),5*M+IDX(1,I))) / ( DWK(IDX(1,N),8*M $+13)**2 ) NORMA = AMAX1(NORMA,STMP1) NORM1 = AMAX1(ADJUST*NORMA,NORM1) NORM2 = AMAX1(ADJUST*NORMA,NORM2) 400 CONTINUE C C Step 15: C Compute l^\star. C I = LSTAR DO 410 J = I+1, L IF (IWK(IDX(1,J+1),2).LE.MK) LSTAR = J 410 CONTINUE NLSTAR = IWK(IDX(1,LSTAR+1),2) C C Check memory allocation (NLSTAR.GE.VWBEG). Update the range of C valid indices: the new vectors v_{n+1} and w_{n+1} will overwrite C the vectors at location N+1-MVEC. C IF (NLSTAR.LT.VWBEG) THEN IERR = 64 GO TO 750 END IF VWBEG = MAX0(VWBEG,NP1-MVEC+1) C C Step 16: C Compute F_{1:n,n} = F_{n_{l^\star}:n,n}. C DO 430 I = NLSTAR, N STMP = SZERO DO 420 J = MAX0(MK,IWK(IDX(1,I),6)), I STMP = STMP + DWK(IDX(1,J),6*M+IDX(1,I)) * DWK(IDX(1,J),5*M+$IDX(1,N)) 420 CONTINUE DWK(IDX(1,I),M+IDX(1,N)) = STMP STMP = ABS(STMP) / DWK(IDX(1,N),8*M+13) NORMA = AMAX1(NORMA,STMP) NORM1 = AMAX1(ADJUST*NORMA,NORM1) NORM2 = AMAX1(ADJUST*NORMA,NORM2) 430 CONTINUE C C Step 17: C Check whether D_l is nonsingular. C IF (LBLKSZ.EQ.1) THEN INNER = ABS(DWK(IDX(1,NL),IDX(1,NL))).EQ.SZERO ELSE DO 450 I = NL, N DO 440 J = NL, N DWK(I-NL+1,4*M+J-NL+1) = DWK(IDX(1,I),IDX(1,J)) 440 CONTINUE 450 CONTINUE CALL SSVDC (DWK(1,4*M+1),M,LBLKSZ,LBLKSZ,DWK(1,8*M+1),DWK(1,8*M $+3),SZERO,0,SZERO,0,DWK(1,8*M+2),0,I) IF (I.NE.0) THEN IERR = -I GO TO 750 END IF STMP = SZERO IF (DWK(1,8*M+1).NE.SZERO) STMP = DWK(LBLKSZ,8*M+1) / DWK(1,8*M$+1) INNER = STMP.LT.(FLOAT(NLEN) * SLAMCH('E')) END IF LEND = L IF (INNER) LEND = L - 1 C C Step 18: C Compute L_{n_i:n_{i+1}-1,n}, for i = l^\star, ..., lend, lend = l C or lend = l-1. C IF (LEND.EQ.L) IWK(IDX(1,L+1+1),2) = NP1 DO 490 I = LSTAR, LEND NI = IWK(IDX(1,I+1),2) NIP1 = IWK(IDX(1,I+1+1),2) IBLKSZ = NIP1 - NI IF (IBLKSZ.EQ.1) THEN DWK(IDX(1,NI),2*M+IDX(1,N)) = DWK(IDX(1,NI),M+IDX(1,N)) / $DWK(IDX(1,NI),IDX(1,NI)) ELSE DO 470 J = NI, NIP1-1 DWK(J-NI+1,8*M+1) = DWK(IDX(1,J),M+IDX(1,N)) DO 460 IJ = NI, NIP1-1 DWK(J-NI+1,4*M+IJ-NI+1) = DWK(IDX(1,J),IDX(1,IJ)) 460 CONTINUE 470 CONTINUE CALL SQRDC (DWK(1,4*M+1),M,IBLKSZ,IBLKSZ,DWK(1,8*M+3),0,$SZERO,0) CALL SQRSL (DWK(1,4*M+1),M,IBLKSZ,IBLKSZ,DWK(1,8*M+3),DWK(1, $8*M+1),$ SZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1), $SZERO,100,J) DO 480 J = NI, NIP1-1 DWK(IDX(1,J),2*M+IDX(1,N)) = DWK(J-NI+1,8*M+2) 480 CONTINUE END IF 490 CONTINUE C C Step 19: C Build the part common to both inner and regular vectors. C Assume that V(NP1) = A p_n. C DWK(IDX(1,NP1),2*M+IDX(1,N)) = SZERO DO 500 I = NLSTAR, NL-1 STMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+10) /$DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,IDX(2,NP1)),- $STMP,VECS(1,IDX(2,I))) 500 CONTINUE C C Check whether VW is being rerun. C RERUN = (N.LE.NMAX).AND.(N.LT.NLMAX) IF (RERUN) THEN C C Check whether D_l has become singular (this should not happen). C IF ((IWK(IDX(1,L+1+1),2).EQ.NP1).AND.INNER) THEN IERR = 128 GO TO 750 END IF C C Determine whether this was an inner vector or not. C INNER = IWK(IDX(1,L+1+1),2).NE.NP1 IF (VF.NE.0) THEN IF (INNER) THEN WRITE (VF,'(A17,I5,A14)') 'Rerunning vector ',NP1,$' (VW) as inner' ELSE WRITE (VF,'(A17,I5,A16)') 'Rerunning vector ',NP1, $' (VW) as regular' END IF END IF C C If building an inner vector, set the coefficients L_{n_l:n,n}. C IF (INNER) THEN DO 510 I = NL, N DWK(IDX(1,I),2*M+IDX(1,N)) = SSCPLL(I,N) 510 CONTINUE END IF ELSE C C VW is not being rerun, determine whether it is being rebuilt. C IF ((VF.NE.0).AND.(N.LE.NMAX)) THEN WRITE (VF,'(A15,I8)') 'Rebuilding V&W:', NP1 END IF C C If E_k is singular, build an inner vector. C IF (INNER) THEN IF (VF.NE.0) WRITE (VF,'(A31)')$'... moment matrix D is singular' GO TO 630 END IF END IF C C Either D_l is nonsingular, or VW is being rerun. For the latter, C just finish building the vectors. For the former, the check in C Step 20 (H_{n_{l-1}:n,n}) could be done. However, the look-ahead C strategy requires the smaller of the checks in Step 20 and Step C 22, in case the norm estimates need updating. Hence, Step 20 and C Step 22 are switched, and their checks are later performed C together. C C Save the common part of the vectors. C CALL SAXPBY (NLEN,VECS(1,3),DONE,VECS(1,IDX(2,NP1)),SZERO,VECS(1,3 $)) C C Step 21. C Build regular vectors. C DO 520 I = NL, N STMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+10) /$DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,IDX(2,NP1)),- $STMP,VECS(1,IDX(2,I))) 520 CONTINUE C C Compute scale factors for the new vectors. C 530 STMP3 = SNRM2(NLEN,VECS(1,IDX(2,NP1)),1) STMP1 = DWK(IDX(1,N),8*M+12) * STMP3 DWK(IDX(1,NP1),2*M+IDX(1,N)) = STMP1 IF (STMP1.LT.TNRM) IERR = IERR + 16 IF (IERR.NE.0) GO TO 670 DWK(IDX(1,NP1),IDX(1,NP1)) = SDOT(NLEN,VECS(1,IDX(2,NP1)),1,VECS(1$,IDX(2,NP1)),1) / ( STMP3**2 ) IF ((STMP3.GE.TMAX).OR.(STMP3.LE.TMIN)) THEN STMP = DONE / STMP3 CALL SAXPBY (NLEN,VECS(1,IDX(2,NP1)),STMP,VECS(1,IDX(2,NP1)), $SZERO,VECS(1,IDX(2,NP1))) STMP3 = DONE END IF DWK(IDX(1,NP1),8*M+10) = DONE / STMP3 C C Step 22: C Compute the last column of D_l^{-1}. C IF (LBLKSZ.EQ.1) THEN DWK(IDX(1,NL),3*M+IDX(1,L)) = DONE / DWK(IDX(1,NL),IDX(1,NL)) ELSE DO 540 I = 1, LBLKSZ-1 DWK(I,8*M+1) = SZERO 540 CONTINUE DWK(LBLKSZ,8*M+1) = DONE CALL SQRSL (DWK(1,4*M+1),M,LBLKSZ,LBLKSZ,DWK(1,8*M+3),DWK(1,8*M$+1), $SZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1),SZERO,$100,J) DO 550 I = NL, N DWK(IDX(1,I),3*M+IDX(1,L)) = DWK(I-NL+1,8*M+2) 550 CONTINUE END IF C C If VW is being rerun, then skip to the end. C IF (RERUN) GO TO 670 C C Step 20: C Build the full column H_{n_{l-1}:n,n} and compute its norm. C STMP1 = SZERO DO 570 I = IWK(IDX(1,L-1+1),2), N STMP = SZERO DO 560 J = MAX0(1,I-1), N STMP = STMP + DWK(IDX(1,I),2*M+IDX(1,J)) * DWK(IDX(1,J),6*M+ $IDX(1,N)) 560 CONTINUE STMP = ABS(STMP) STMP1 = STMP1 + STMP 570 CONTINUE C C Step 22: C Build the 2nd term for the next step, H_{n_l:n,n+1}, regular. C Compute the norm of H_{n_l:n,n+1}. C STMP3 = SZERO STMP4 = DWK(IDX(1,NP1),IDX(1,NP1)) * DWK(IDX(1,NP1),2*M+IDX(1,N)) DO 580 I = NL, N STMP = ABS(STMP4 * DWK(IDX(1,I),3*M+IDX(1,L))) STMP3 = STMP3 + STMP 580 CONTINUE C C Steps 20 and 22: C Check H_{n_{l-1}:n,n} and H_{n_l:n-1,n+1}. C STMP = AMAX1(STMP1,STMP3) INNER = STMP.GT.NORM2 IF (.NOT.INNER) GO TO 670 DWK(IDX(1,N),8*M+11) = STMP C C If H_{n_{l-1}:n,n} is bad, build inner vectors. C IF (STMP1.GT.NORM2) GO TO 630 C C If H_{n_l:n-1,n+1} is bad, check the inner vectors. C This only applies if n_l > 1. C IF (NL.LE.1) GO TO 630 C C Build the inner vectors to compute the 2nd term at the next step. C Get the coefficients L_{n_l:n+1,n} in a temporary location. C Build inner vector in VECS(1,3). C IBUILT = .TRUE. DO 590 I = NL, N DWK(IDX(1,I),2*M+IDX(1,NP1)) = SSCPLL(I,N) STMP = DWK(IDX(1,I),2*M+IDX(1,NP1)) * DWK(IDX(1,I),8*M+10$) / DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,3),DONE,VECS(1,3),-STMP,VECS(1,IDX(2,I $))) 590 CONTINUE ISTMP3 = SNRM2(NLEN,VECS(1,3),1) ISTMP1 = DWK(IDX(1,N),8*M+12) * ISTMP3 DWK(IDX(1,NP1),2*M+IDX(1,NP1)) = ISTMP1 IF (ISTMP1.LT.TNRM) IERR = IERR + 16 IF (IERR.NE.0) GO TO 640 C C Build the 2nd term for the next step, H_{n_{l-1}:n_l-1,n+1}, C inner. Compute the norm of H_{n_{l-1}:n_l-1,n+1}. C STMP = SZERO DO 600 J = NLSTAR, NL-1 STMP = STMP + DWK(IDX(1,NL),IDX(1,J)) * DWK(IDX(1,I),2*M+IDX(1,$N)) 600 CONTINUE DO 610 J = NL, N STMP = STMP + DWK(IDX(1,NL),IDX(1,J)) * DWK(IDX(1,I),2*M+IDX(1, $NP1)) 610 CONTINUE STMP1 = SZERO STMP = ( DWK(IDX(1,NL),M+IDX(1,N)) - STMP ) / DWK(IDX(1,NP1),2*M+$IDX(1,NP1)) STMP4 = STMP * DWK(IDX(1,NL),2*M+IDX(1,NL-1)) DO 620 I = IWK(IDX(1,L-1+1),2), NL-1 STMP = ABS(DWK(IDX(1,I),3*M+IDX(1,L-1)) * STMP4) STMP1 = STMP1 + STMP 620 CONTINUE C C Compare the inner and regular versions of the 2nd term at the next C step. Build the vector corresponding to the smaller term. C INNER = STMP3.GT.STMP1 IF (.NOT.INNER) GO TO 670 C C Build inner vectors. C Check whether the V&W block has to be forced to close. C 630 IF (VF.NE.0) WRITE (VF,'(A7,I5,A14)') 'Vector ',NP1, $' (VW) is inner' IF (NP1-NL.EQ.MAXVW) THEN CALL SSCPL2 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL,NLSTAR,$ VF,IERR,ADJUST,NORM1,NORM2,DWK,IWK,IDX,VECS) IF (IERR.NE.0) GO TO 750 LBLKSZ = N - NL + 1 INNER = .FALSE. RERUN = .TRUE. NP1 = N + 1 GO TO 530 END IF C C The temporary vectors contain either just partial inner vectors, C or the completed ones, depending on whether IBUILT is TRUE or C FALSE. In either case, replace the regular vectors. C 640 CALL SAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,3),SZERO,VECS(1, $IDX(2,NP1))) C C Step 24: C Get the coefficients L_{n_l:n+1,n} and build inner vectors. C IF (IBUILT) THEN DO 650 I = NL, NP1 DWK(IDX(1,I),2*M+IDX(1,N)) = DWK(IDX(1,I),2*M+IDX(1,NP1)) 650 CONTINUE STMP1 = ISTMP1 STMP3 = ISTMP3 ELSE DO 660 I = NL, N DWK(IDX(1,I),2*M+IDX(1,N)) = SSCPLL(I,N) STMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+10$) / DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,IDX(2,NP1)) $,-STMP,VECS(1,IDX(2,I))) 660 CONTINUE STMP3 = SNRM2(NLEN,VECS(1,IDX(2,NP1)),1) STMP1 = DWK(IDX(1,N),8*M+12) * STMP3 DWK(IDX(1,NP1),2*M+IDX(1,N)) = STMP1 IF (STMP1.LT.TNRM) IERR = IERR + 16 END IF IF (IERR.NE.0) GO TO 670 DWK(IDX(1,NP1),IDX(1,NP1)) = SDOT(NLEN,VECS(1,IDX(2,NP1)),1,VECS(1$,IDX(2,NP1)),1) / ( STMP3**2 ) IF ((STMP3.GE.TMAX).OR.(STMP3.LE.TMIN)) THEN STMP = DONE / STMP3 CALL SAXPBY (NLEN,VECS(1,IDX(2,NP1)),STMP,VECS(1,IDX(2,NP1)), $SZERO,VECS(1,IDX(2,NP1))) STMP3 = DONE END IF DWK(IDX(1,NP1),8*M+10) = DONE / STMP3 C C Step 23: C If regular vectors were built, update the counters. C 670 IF (.NOT.INNER) THEN L = L + 1 NL = IWK(IDX(1,L+1),2) END IF C C Update counters. C MVWBLT = MAX0(MVWBLT,NL-IWK(IDX(1,L-1+1),2)) NLMAX = MAX0(NL,NLMAX) IWK(IDX(1,N),5) = NLSTAR IWK(IDX(1,N),3) = NL C C Update the counter for steps taken. C NMAX = MAX0(NMAX,N) IF ((N.LT.MKMAX).OR.(N.LT.NLMAX-1)) GO TO 740 C C The QMR code starts here. C At this point, (N.GE.MKMAX).AND.(N.GE.NLMAX-1), so that steps up C to MIN(MKMAX,NLMAX-1) are guaranteed not to be rebuilt. Also, no C errors are allowed in IERR, other than possibly having found one C or both invariant subspaces, in which case all remaining iterates C are computed. C IEND = MIN0(MKMAX,NLMAX-1) IF (IERR.NE.0) IEND = N 680 IF (NQMR.GT.IEND-1) GO TO 740 C C Update the QMR iteration counter. C NQMR = NQMR + 1 C C Get the next scaling factor omega(i) and update MAXOMG. C DWK(IDX(1,NQMR+1),8*M+8) = SSCPLO(NQMR+1) MAXOMG = AMAX1(MAXOMG,DONE/DWK(IDX(1,NQMR+1),8*M+8)) C C Compute the starting index IBASE for the column of \hat{R}. C IBASE = MAX0(1,IWK(IDX(1,NQMR),5)-1) DWK(IDX(1,IBASE),8*M+5) = SZERO C C Multiply the new column by the previous omegas. C DO 690 J = IWK(IDX(1,NQMR),5), NQMR+1 DWK(IDX(1,J),8*M+5) = DWK(IDX(1,J),8*M+8) * DWK(IDX(1,J),2*M+$IDX(1,NQMR)) 690 CONTINUE C C Apply the previous rotations. C The loop below explicitly implements a call to SROT: C CALL SROT (1,DWK(IDX(1,J-1),8*M+5),1,DWK(IDX(1,J),8*M+5),1,DWK(IDX(1,J),8*M+9),DWK(IDX(1,J),8*M+6)) C DO 700 J = IBASE+1, NQMR STMP1 = DWK(IDX(1,J),8*M+5) STMP2 = DWK(IDX(1,J-1),8*M+5) DWK(IDX(1,J-1),8*M+5) = DWK(IDX(1,J),8*M+9) * STMP2 + DWK(IDX(1 $,J),8*M+6) * STMP1 DWK(IDX(1,J),8*M+5) = DWK(IDX(1,J),8*M+9) * STMP1 - DWK(IDX(1$,J),8*M+6) * STMP2 700 CONTINUE C C Compute the rotation for the last element (this also applies it). C CALL SROTG (DWK(IDX(1,NQMR),8*M+5),DWK(IDX(1,NQMR+1),8*M+5), $DWK(IDX(1,NQMR+1),8*M+9),DWK(IDX(1,NQMR+1),8*M+6)) C C Apply the new rotation to the right-hand side vector. C Could be replaced with: C DWK(IDX(1,NQMR+1),8*M+4) = SZERO C CALL SROT (1,DWK(IDX(1,NQMR),8*M+4),1,DWK(IDX(1,NQMR+1),8*M+4),1,DWK(IDX(1,NQMR+1),8*M+9),DWK(IDX(1,NQMR+1),8*M+6)) C DWK(IDX(1,NQMR+1),8*M+4) = -DWK(IDX(1,NQMR+1),8*M+6) * DWK(IDX(1,$NQMR),8*M+4) DWK(IDX(1,NQMR),8*M+4) = DWK(IDX(1,NQMR+1),8*M+9) * DWK(IDX(1, $NQMR),8*M+4) C C Compute the next search direction s_i. C This is more complicated than it might have to be because storage C for the vectors VECS(1,IDX(4,NQMR)) is minimized. C STMP2 = SZERO STMP = DWK(IDX(1,NQMR),8*M+12) DO 710 J = IBASE, NQMR-1 STMP1 = DWK(IDX(1,J),8*M+5) * DWK(IDX(1,J),8*M+7) / STMP IF (STMP1.EQ.SZERO) GO TO 710 CALL SAXPBY (NLEN,VECS(1,IDX(4,NQMR)),STMP2,VECS(1,IDX(4,NQMR))$,-STMP1,VECS(1,IDX(4,J))) STMP2 = DONE 710 CONTINUE CALL SAXPBY (NLEN,VECS(1,IDX(4,NQMR)),STMP2,VECS(1,IDX(4,NQMR)), $DONE,VECS(1,IDX(3,NQMR))) STMP = STMP / DWK(IDX(1,NQMR),8*M+5) C C Compute the new QMR iterate, then scale the search direction. C STMP1 = STMP * DWK(IDX(1,NQMR),8*M+4) CALL SAXPBY (NLEN,VECS(1,1),DONE,VECS(1,1),STMP1,VECS(1,IDX(4,NQMR$))) DWK(IDX(1,NQMR),8*M+7) = STMP STMP = ABS(STMP) IF ((STMP.GE.TMAX).OR.(STMP.LE.TMIN)) THEN DWK(IDX(1,NQMR),8*M+7) = DONE CALL SAXPBY (NLEN,VECS(1,IDX(4,NQMR)),STMP,VECS(1,IDX(4,NQMR)), $SZERO,VECS(1,IDX(4,NQMR))) END IF C C Compute the residual norm upper bound. C If the scaled upper bound is within one order of magnitude of the C target convergence norm, compute the true residual norm. C UNRM = SQRT(FLOAT(NQMR+1)) * MAXOMG * ABS(DWK(IDX(1,NQMR+1),8*M+4)$) / R0 UCHK = UNRM IF ((TRES.EQ.0).AND.(UNRM/TOL.GT.DTEN).AND.(N.LT.NLIM)) GO TO 730 C C Have the caller carry out AXB, then return here. C CALL AXB (VECS(1,1),VECS(1,3)) C INFO(2) = 1 INFO(3) = 1 INFO(4) = 3 RETLBL = 720 RETURN 720 CALL SAXPBY (NLEN,VECS(1,3),DONE,VECS(1,2),-DONE,VECS(1,3)) RESN = SNRM2(NLEN,VECS(1,3),1) / R0 UCHK = RESN C C Output the convergence history. C 730 IF (VF.NE.0) WRITE (VF,'(I8,2E11.4)') NQMR, UNRM, RESN IF (TF.NE.0) WRITE (TF,'(I8,2E11.4)') NQMR, UNRM, RESN C C Check for convergence or termination. Stop if: C 1. algorithm converged; C 2. there is an error condition; C 3. the residual norm upper bound is smaller than the computed C residual norm by a factor of at least 100; C 4. algorithm exceeded the iterations limit. C IF (RESN.LE.TOL) THEN IERR = 0 GO TO 750 ELSE IF (IERR.NE.0) THEN GO TO 750 ELSE IF (UNRM.LT.UCHK/DHUN) THEN IERR = 4 GO TO 750 ELSE IF (NQMR.GE.NLIM) THEN IERR = 4 GO TO 750 END IF GO TO 680 C C Update the running counter. C 740 IF (IERR.NE.0) GO TO 750 IERR = 4 IF (N.GE.NLIM) GO TO 750 N = N + 1 GO TO 30 C C Done. C 750 RETLBL = 0 NLIM = NQMR INFO(1) = IERR MAXPQ = MPQBLT MAXVW = MVWBLT NORMS(1) = NORM1 NORMS(2) = NORM2 C RETURN END C C********************************************************************** C SUBROUTINE SSCPL1 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL, $NLSTAR,VF,IERR,ADJUST,NORM1,NORM2,DWK,IWK,$ IDX,VECS) C C Purpose: C This subroutine rebuilds the data for the vector p_n at the point C where a block was forced to close. The routine is called C internally by the coupled Lanczos code, and is not meant to be C called directly by the user code. C C Parameters: C See descriptions in the main routine SSCPL. C C External routines used: C subroutine saxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C single precision sdot(n,x,incx,y,incy) C BLAS-1 routine, computes y^H * x. C single precision snrm2(n,x,incx) C BLAS-1 routine, computes the 2-norm of x. C subroutine sqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine sqrsl(x,ldx,n,k,qraux,y,qy,qty,b,rsd,xb,job,info) C LINPACK routine, applies the QR factorization of x. C C Noel M. Nachtigal C May 31, 1993 C C********************************************************************** C INTRINSIC AMAX1, MAX0 EXTERNAL SAXPBY, SQRDC, SQRSL C INTEGER IERR, K, KSTAR, L, LSTAR, M, MK, MKSTAR, NL, NLSTAR, N INTEGER IDX(4,*), IWK(M,13), NDIM, NLEN, VF REAL ADJUST, DWK(M,8*M+14), NORM1, NORM2 REAL VECS(NDIM,*) C C Miscellaneous parameters. C REAL DONE, SZERO PARAMETER (DONE = 1.0E0,SZERO = 0.0E0) C C Local variables. C INTEGER I, IJ, J, KBLKSZ, NP1 REAL STMP1, STMP2 C C Find the index of the vector pair with the smallest pass value. C IERR = 0 IF (VF.NE.0) WRITE (VF,'(A23)') 'PQ block did not close:' J = MK + 1 STMP1 = DWK(IDX(1,J),8*M+14) DO 100 I = MK+2, N STMP2 = DWK(IDX(1,I),8*M+14) IF (STMP2.GT.SZERO) THEN IF ((STMP1.EQ.SZERO).OR.(STMP2.LT.STMP1)) THEN J = I STMP1 = STMP2 END IF END IF 100 CONTINUE IF (STMP1.EQ.SZERO) THEN IF (VF.NE.0) WRITE (VF,'(A47)') $'... no new norm estimates available (aborting).' IERR = 8 RETURN END IF NORM1 = ADJUST * STMP1 NORM2 = AMAX1(NORM1,NORM2) IF (VF.NE.0) WRITE (VF,'(A40,I5,2E11.4)')$ '... updated norms, restarting from step:', IWK(IDX(1,J),7), $NORM1, NORM2 IF (IWK(IDX(1,J),7).EQ.N) RETURN N = IWK(IDX(1,J),7) L = IWK(IDX(1,N),9) NL = IWK(IDX(1,L+1),2) K = IWK(IDX(1,N),8) MK = IWK(IDX(1,K+1),1) LSTAR = IWK(IDX(1,N),11) NLSTAR = IWK(IDX(1,LSTAR+1),2) KSTAR = IWK(IDX(1,N),10) MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Initialize local variables. C DWK(IDX(1,N),6*M+IDX(1,N)) = DONE NP1 = N + 1 DWK(IDX(1,N),8*M+14) = SZERO KBLKSZ = N - MK C C Step 2: C Compute k^\star. C I = KSTAR DO 110 J = I+1, K IF (IWK(IDX(1,J+1),1).LE.NL-1) KSTAR = J 110 CONTINUE MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Compute U_{m_k:n-1,n}. Save the old coefficients. C IWK(IDX(1,K+1+1),1) = N IF (KBLKSZ.EQ.1) THEN DWK(IDX(1,MK),6*M+IDX(1,NP1)) = DWK(IDX(1,MK),6*M+IDX(1,N)) DWK(IDX(1,MK),6*M+IDX(1,N)) = DWK(IDX(1,N),M+IDX(1,MK)) /$DWK(IDX(1,MK),5*M+IDX(1,MK)) ELSE DO 130 J = MK, N-1 DWK(J-MK+1,8*M+1) = DWK(IDX(1,N),M+IDX(1,J)) DO 120 IJ = MK, N-1 DWK(J-MK+1,4*M+IJ-MK+1) = DWK(IDX(1,J),5*M+IDX(1,IJ)) 120 CONTINUE 130 CONTINUE CALL SQRDC (DWK(1,4*M+1),M,KBLKSZ,KBLKSZ,DWK(1,8*M+3),0,SZERO,0 $) CALL SQRSL (DWK(1,4*M+1),M,KBLKSZ,KBLKSZ,DWK(1,8*M+3),DWK(1,8*M$+1), $SZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1),SZERO,$100,J) DO 140 J = MK, N-1 DWK(IDX(1,J),6*M+IDX(1,NP1)) = DWK(IDX(1,J),6*M+IDX(1,N)) DWK(IDX(1,J),6*M+IDX(1,N)) = DWK(J-MK+1,8*M+2) 140 CONTINUE END IF C C Convert inner vectors to regular vectors. C DO 150 I = MK, N-1 STMP1 = DWK(IDX(1,I),6*M+IDX(1,N)) - DWK(IDX(1,I),6*M+IDX(1,NP1 $)) STMP2 = STMP1 * DWK(IDX(1,I),8*M+12) / DWK(IDX(1,N),8*M+12) CALL SAXPBY (NLEN,VECS(1,IDX(3,N)),DONE,VECS(1,IDX(3,N)),-STMP2$,VECS(1,IDX(3,I))) 150 CONTINUE C RETURN END C C********************************************************************** C SUBROUTINE SSCPL2 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL, $NLSTAR,VF,IERR,ADJUST,NORM1,NORM2,DWK,IWK,$ IDX,VECS) C C Purpose: C This subroutine rebuilds the data for the vector v_{n+1} at the C point where a block was forced to close. The routine is called C internally by the coupled Lanczos code, and is not meant to be C called directly by the user code. C C Parameters: C See descriptions in the main routine SSCPL. C C External routines used: C subroutine saxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C subroutine sqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine sqrsl(x,ldx,n,k,qraux,y,qy,qty,b,rsd,xb,job,info) C LINPACK routine, applies the QR factorization of x. C C Noel M. Nachtigal C May 31, 1993 C C********************************************************************** C INTRINSIC ABS, AMAX1, MAX0 EXTERNAL SAXPBY, SQRDC, SQRSL C INTEGER IERR, K, KSTAR, L, LSTAR, M, MK, MKSTAR, NL, NLSTAR, N INTEGER IDX(4,*), IWK(M,13), NDIM, NLEN, VF REAL ADJUST, DWK(M,8*M+14), NORM1, NORM2 REAL VECS(NDIM,*) C C Miscellaneous parameters. C REAL DONE, SZERO PARAMETER (DONE = 1.0E0,SZERO = 0.0E0) C C Local variables. C INTEGER I, IJ, J, LBLKSZ, NP1 REAL STMP1, STMP2, SCALV C C Find the index of the vector pair with the smallest pass value. C IERR = 0 IF (VF.NE.0) WRITE (VF,'(A23)') 'VW block did not close:' J = NL STMP1 = DWK(IDX(1,J),8*M+11) DO 100 I = NL+1, N STMP2 = DWK(IDX(1,I),8*M+11) IF (STMP2.GT.SZERO) THEN IF ((STMP1.EQ.SZERO).OR.(STMP2.LT.STMP1)) THEN J = I STMP1 = STMP2 END IF END IF 100 CONTINUE IF (STMP1.EQ.SZERO) THEN IF (VF.NE.0) WRITE (VF,'(A47)') $'... no new norm estimates available (aborting).' IERR = 8 RETURN END IF NORM2 = ADJUST * STMP1 NORM1 = AMAX1(NORM1,NORM2) IF (VF.NE.0) WRITE (VF,'(A40,I5,2E11.4)')$ '... updated norms, restarting from step:', IWK(IDX(1,J),7), $NORM1, NORM2 IF (IWK(IDX(1,J),7).EQ.N) RETURN N = IWK(IDX(1,J),7) L = IWK(IDX(1,N),9) NL = IWK(IDX(1,L+1),2) K = IWK(IDX(1,N),12) MK = IWK(IDX(1,K+1),1) LSTAR = IWK(IDX(1,N),11) NLSTAR = IWK(IDX(1,LSTAR+1),2) KSTAR = IWK(IDX(1,N),13) MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Initialize local variables. C NP1 = N + 1 DWK(IDX(1,N),8*M+11) = SZERO LBLKSZ = N - NL + 1 C C Step 15: C Compute l^\star. C I = LSTAR DO 110 J = I+1, L IF (IWK(IDX(1,J+1),2).LE.MK) LSTAR = J 110 CONTINUE NLSTAR = IWK(IDX(1,LSTAR+1),2) C C Compute L_{n_l:n,n}. Save the old coefficients. C IWK(IDX(1,L+1+1),2) = NP1 IF (LBLKSZ.EQ.1) THEN DWK(IDX(1,NL),2*M+IDX(1,NP1)) = DWK(IDX(1,NL),2*M+IDX(1,N)) DWK(IDX(1,NL),2*M+IDX(1,N)) = DWK(IDX(1,NL),M+IDX(1,N)) /$DWK(IDX(1,NL),IDX(1,NL)) ELSE DO 130 J = NL, N DWK(J-NL+1,8*M+1) = DWK(IDX(1,J),M+IDX(1,N)) DO 120 IJ = NL, N DWK(J-NL+1,4*M+IJ-NL+1) = DWK(IDX(1,J),IDX(1,IJ)) 120 CONTINUE 130 CONTINUE CALL SQRDC (DWK(1,4*M+1),M,LBLKSZ,LBLKSZ,DWK(1,8*M+3),0,SZERO,0 $) CALL SQRSL (DWK(1,4*M+1),M,LBLKSZ,LBLKSZ,DWK(1,8*M+3),DWK(1,8*M$+1), $SZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1),SZERO,$100,J) DO 140 J = NL, N DWK(IDX(1,J),2*M+IDX(1,NP1)) = DWK(IDX(1,J),2*M+IDX(1,N)) DWK(IDX(1,J),2*M+IDX(1,N)) = DWK(J-NL+1,8*M+2) 140 CONTINUE END IF C C Convert inner vectors to regular vectors. C SCALV = DWK(IDX(1,NP1),2*M+IDX(1,N)) * DWK(IDX(1,NP1),8*M+10) DWK(IDX(1,NP1),2*M+IDX(1,N)) = SZERO DO 150 I = NL, N STMP1 = DWK(IDX(1,I),2*M+IDX(1,N)) - DWK(IDX(1,I),2*M+IDX(1,NP1 $)) STMP2 = STMP1 * DWK(IDX(1,I),8*M+10) / SCALV CALL SAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,IDX(2,NP1)),-$STMP2,VECS(1,IDX(2,I))) 150 CONTINUE C RETURN END C C********************************************************************** C REAL FUNCTION SSCPLL(I,J) C C Purpose: C Return the recurrence coefficients for inner vectors VW. C C Parameters: C I = row index of the coefficient (input). C J = column index of the coefficient (input). C C Noel M. Nachtigal C April 17, 1993 C C********************************************************************** C C C Common block SSCPLX. C REAL NORMA COMMON /SSCPLX/NORMA C C INTEGER I, J C IF ((I.LT.1).OR.(J.LT.1)) THEN SSCPLL = 0.0E0 ELSE IF (I.EQ.J) THEN SSCPLL = NORMA ELSE SSCPLL = 0.0E0 END IF C RETURN END C C********************************************************************** C REAL FUNCTION SSCPLO (I) C C Purpose: C Return the scaling parameter OMEGA(I). C C Parameters: C I = the index of the parameter OMEGA (input). C C Noel M. Nachtigal C June 1, 1992 C C********************************************************************** C INTEGER I C SSCPLO = 1.0E0 C RETURN END C C********************************************************************** C REAL FUNCTION SSCPLU(I,J) C C Purpose: C Return the recurrence coefficients for inner vectors PQ. C C Parameters: C I = row index of the coefficient (input). C J = column index of the coefficient (input). C C Noel M. Nachtigal C April 17, 1993 C C********************************************************************** C INTEGER I, J C IF ((I.LT.1).OR.(J.LT.1)) THEN SSCPLU = 0.0E0 ELSE SSCPLU = 0.0E0 END IF C RETURN END C C**********************************************************************