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 unsymmetric matrices, using the coupled two-term recurrence C variant of the look-ahead Lanczos algorithm. C C********************************************************************** C SUBROUTINE DUCPL (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. 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 "ducpl.doc" in C the current directory. C C External routines used: C subroutine daxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C double precision ddot(n,x,incx,y,incy) C BLAS-1 routine, computes y^H * x. C double precision dlamch(ch) C LAPACK routine, computes machine-related constants. C double precision dnrm2(n,x,incx) C BLAS-1 routine, computes the 2-norm of x. C subroutine dqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine dqrsl(x,ldx,n,k,qraux,y,qy,qty,b,rsd,xb,job,info) C LINPACK routine, applies the QR factorization of x. C subroutine drandn(n,x,seed) C Library routine, fill x with random numbers. C subroutine drotg(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 dsvdc(x,ldx,n,p,s,e,u,ldu,v,ldv,work,job,info) C LINPACK routine, computes the SVD of x. C subroutine ducpl1(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 ducpl2(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 double precision function ducpll(i,n) C User-supplied routine, computes inner recurrence coefficients. C double precision ducplom(n) C User-supplied routine, specifies the QMR scaling factors. C double precision function ducplu(i,n) C User-supplied routine, computes inner recurrence coefficients. C C Noel M. Nachtigal C March 1, 1992 C C********************************************************************** C INTRINSIC DABS, DBLE, DMAX1, DSQRT, MAX0, MIN0, MOD EXTERNAL DAXPBY, DDOT, DLAMCH, DNRM2, DQRDC, DQRSL, DRANDN, DROTG EXTERNAL DSVDC, DUCPL1, DUCPL2, DUCPLL, DUCPLO, DUCPLU DOUBLE PRECISION DDOT, DLAMCH, DNRM2, DUCPLL, DUCPLO, DUCPLU C INTEGER INFO(4), M, MAXPQ, MAXVW, MVEC, NDIM, NLEN, NLIM INTEGER IDX(6,NLIM+2), IWK(M,13) DOUBLE PRECISION DWK(M,8*M+18), NORMS(2), TOL DOUBLE PRECISION VECS(NDIM,5*MVEC+4-1) C C Common block variables. C C C Common block DUCPLX. C DOUBLE PRECISION NORMA COMMON /DUCPLX/NORMA C C Miscellaneous parameters. C DOUBLE PRECISION DHUN, DONE, DTEN, DZERO PARAMETER (DHUN = 1.0D2,DONE = 1.0D0,DTEN = 1.0D1,DZERO = 0.0D0) 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 DOUBLE PRECISION ADJUST, MAXOMG, NORM1, NORM2, TMAX, TMIN, TNRM SAVE ADJUST, MAXOMG, NORM1, NORM2, TMAX, TMIN, TNRM DOUBLE PRECISION R0, RESN, UCHK, UNRM SAVE R0, RESN, UCHK, UNRM C C Local variables, transient. C INTEGER I, IBASE, IBLKSZ, IJ, INIT, J, KEND, LEND, MI, MIP1 INTEGER NI, NIP1, REVCOM DOUBLE PRECISION DTMP, DTMP1, DTMP2, DTMP3, DTMP4, DTMP5 DOUBLE PRECISION IDTMP1, IDTMP2, IDTMP3, IDTMP4 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 2 370 returning from ATXB, go to label 370 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 ELSE IF (REVCOM.EQ.2) THEN IF (RETLBL.EQ.370) GO TO 370 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, and the left starting vector flag INIT. C VF = MAX0(INFO(1),0) INIT = VF / 100000 VF = VF - INIT * 100000 TRES = VF / 10000 VF = VF - TRES * 10000 TF = VF / 100 VF = VF - TF * 100 C C Extract the norms. C NORMA = DZERO NORM1 = DABS(NORMS(1)) NORM2 = DABS(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 = DLAMCH('E') * DTEN TMIN = DSQRT(DSQRT(DLAMCH('S'))) TMAX = DONE / TMIN IF (TOL.LE.DZERO) TOL = DSQRT(DLAMCH('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 w_i; C IDX(4,I) = indices used for p_i; C IDX(5,I) = indices used for q_i; C IDX(6,I) = indices used for s_i. C DO 20 I = 1, NLIM+2 IDX(1,I) = MOD(I-1,M) + 1 IDX(2,I) = 4 + 1 + 0*MVEC + MOD(I-1,MVEC) IDX(3,I) = 4 + 1 + 1*MVEC + MOD(I-1,MVEC) IDX(4,I) = 4 + 1 + 2*MVEC + MOD(I-1,MVEC) IDX(5,I) = 4 + 1 + 3*MVEC + MOD(I-1,MVEC) IDX(6,I) = 4 + 1 + 4*MVEC + MOD(I-1,MVEC-1) 20 CONTINUE C C Set x_0 = 0 and compute the norm of the initial residual. C CALL DAXPBY (NLEN,VECS(1,IDX(2,1)),DONE,VECS(1,2),DZERO,VECS(1,$IDX(2,1))) CALL DAXPBY (NLEN,VECS(1,1),DZERO,VECS(1,1),DZERO,VECS(1,1)) R0 = DNRM2(NLEN,VECS(1,IDX(2,1)),1) IF ((TOL.GE.DONE).OR.(R0.EQ.DZERO)) GO TO 750 C C Check whether the auxiliary vector must be supplied. C IF (INIT.EQ.0) CALL DRANDN (NLEN,VECS(1,3),1) CALL DAXPBY (NLEN,VECS(1,IDX(3,1)),DONE,VECS(1,3),DZERO,VECS(1, $IDX(3,1))) C C Scale the first pair of Lanczos vectors and check for invariant C subspaces. C DTMP1 = R0 DTMP2 = DNRM2(NLEN,VECS(1,IDX(3,1)),1) IF (DTMP1.LT.TNRM) IERR = IERR + 16 IF (DTMP2.LT.TNRM) IERR = IERR + 32 IF (IERR.NE.0) GO TO 750 DWK(IDX(1,1),8*M+10) = DONE DWK(IDX(1,1),IDX(1,1)) = DDOT(NLEN,VECS(1,IDX(2,1)),1,VECS(1,IDX(3$,1)),1) / ( DTMP1 * DTMP2 ) IF ((DTMP1.GE.TMAX).OR.(DTMP1.LE.TMIN)) THEN DTMP = DONE / DTMP1 CALL DAXPBY (NLEN,VECS(1,IDX(2,1)),DTMP,VECS(1,IDX(2,1)),DZERO, $VECS(1,IDX(2,1))) DTMP1 = DONE END IF IF ((DTMP2.GE.TMAX).OR.(DTMP2.LE.TMIN)) THEN DTMP = DONE / DTMP2 CALL DAXPBY (NLEN,VECS(1,IDX(3,1)),DTMP,VECS(1,IDX(3,1)),DZERO,$VECS(1,IDX(3,1))) DTMP2 = DONE END IF DWK(IDX(1,1),8*M+11) = DONE / DTMP1 DWK(IDX(1,1),8*M+12) = DONE / DTMP2 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) = DUCPLO(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 and q_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+18) = DZERO 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)) = DZERO 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+16) = DONE DWK(IDX(1,1),8*M+17) = DONE DWK(IDX(1,1),8*M+14) = DWK(IDX(1,1),8*M+11) DWK(IDX(1,1),8*M+15) = DWK(IDX(1,1),8*M+12) INNER = .FALSE. CALL DAXPBY (NLEN,VECS(1,IDX(4,1)),DONE,VECS(1,IDX(2,1)),DZERO, $VECS(1,IDX(4,1))) CALL DAXPBY (NLEN,VECS(1,IDX(5,1)),DONE,VECS(1,IDX(3,1)),DZERO,$VECS(1,IDX(5,1))) GO TO 340 END IF C C Step 1: C Update D^{(n-1)} to D^{(n)}. C DO 60 I = NL, N-1 DTMP = DZERO DO 50 J = MAX0(NLSTAR,IWK(IDX(1,I),3)), N-1 DTMP = DTMP + DWK(IDX(1,I),IDX(1,J)) * DWK(IDX(1,J),2*M+ $IDX(1,N-1)) 50 CONTINUE DTMP = ( DWK(IDX(1,I),M+IDX(1,N-1)) - DTMP ) / DWK(IDX(1,N)$,2*M+IDX(1,N-1)) DWK(IDX(1,I),IDX(1,N)) = DTMP DWK(IDX(1,N),IDX(1,I)) = DTMP * DWK(IDX(1,N),8*M+10) / $DWK(IDX(1,I),8*M+10) 60 CONTINUE C C Step 2: C Compute k^\star. C I = KSTAR DO 70 J = I+1, K IF (IWK(IDX(1,J+1),1).LE.NL-1) KSTAR = J 70 CONTINUE MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Check memory allocation (MKSTAR.GE.PQBEG). Update the range of C valid indices: the new vectors p_n and q_n will overwrite the C vectors at location N-MVEC. C IF (MKSTAR.LT.PQBEG) THEN IERR = 64 GO TO 750 END IF PQBEG = MAX0(PQBEG,N-MVEC+1) C C Step 3: C Compute F_{n,1:n-1} = F_{n,m_{k^\star}:n-1}. C DO 90 I = MKSTAR, N-1 DTMP = DZERO DO 80 J = MAX0(NL,IWK(IDX(1,I),5)), I+1 DTMP = DTMP + DWK(IDX(1,N),IDX(1,J)) * DWK(IDX(1,J),2*M+$IDX(1,I)) 80 CONTINUE DWK(IDX(1,N),M+IDX(1,I)) = DTMP DTMP = DABS(DTMP) / DWK(IDX(1,I),8*M+16) NORMA = DMAX1(NORMA,DTMP) NORM1 = DMAX1(ADJUST*NORMA,NORM1) NORM2 = DMAX1(ADJUST*NORMA,NORM2) 90 CONTINUE C C Step 4: C Check whether E_k is nonsingular. C IF (KBLKSZ.EQ.1) THEN INNER = DABS(DWK(IDX(1,MK),5*M+IDX(1,MK))).EQ.DZERO 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 DSVDC (DWK(1,4*M+1),M,KBLKSZ,KBLKSZ,DWK(1,8*M+1),DWK(1,8*M $+3),DZERO,0,DZERO,0,DWK(1,8*M+2),0,I) IF (I.NE.0) THEN IERR = -I GO TO 750 END IF DTMP = DZERO IF (DWK(1,8*M+1).NE.DZERO) DTMP = DWK(KBLKSZ,8*M+1) / DWK(1,8*M$+1) INNER = DTMP.LT.(DBLE(NLEN) * DLAMCH('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)) / $DWK(IDX(1,MI),5*M+IDX(1,MI)) * DWK(IDX(1,MI),8*M+10) / DWK(IDX(1,N$),8*M+10) ELSE DO 130 J = MI, MIP1-1 DWK(J-MI+1,8*M+1) = DWK(IDX(1,N),M+IDX(1,J)) * DWK(IDX(1, $J),8*M+10) / DWK(IDX(1,N),8*M+10) DO 120 IJ = MI, MIP1-1 DWK(J-MI+1,4*M+IJ-MI+1) = DWK(IDX(1,J),5*M+IDX(1,IJ)) 120 CONTINUE 130 CONTINUE CALL DQRDC (DWK(1,4*M+1),M,IBLKSZ,IBLKSZ,DWK(1,8*M+3),0,$DZERO,0) CALL DQRSL (DWK(1,4*M+1),M,IBLKSZ,IBLKSZ,DWK(1,8*M+3),DWK(1, $8*M+1),$ DZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1), $DZERO,100,J) DO 140 J = MI, MIP1-1 DWK(IDX(1,J),6*M+IDX(1,N)) = DWK(J-MI+1,8*M+2) 140 CONTINUE END IF 150 CONTINUE C C Step 6: C Build the part common to both inner and regular vectors. C DWK(IDX(1,N),8*M+14) = DWK(IDX(1,N),8*M+11) DWK(IDX(1,N),8*M+15) = DWK(IDX(1,N),8*M+12) CALL DAXPBY (NLEN,VECS(1,3),DONE,VECS(1,IDX(2,N)),DZERO,VECS(1,3)) CALL DAXPBY (NLEN,VECS(1,4),DONE,VECS(1,IDX(3,N)),DZERO,VECS(1,4)) DO 160 I = MKSTAR, MK-1 DTMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+14) /$DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,3),DONE,VECS(1,3),-DTMP,VECS(1,IDX(4,I $))) DTMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+15) /$DWK(IDX(1,N),8*M+15) * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M+10) CALL DAXPBY (NLEN,VECS(1,4),DONE,VECS(1,4),-DTMP,VECS(1,IDX(5,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, $' (PQ) as regular' END IF END IF C C If building an inner vector, set the coefficients U_{m_k:n-1,n}. C IF (INNER) THEN DO 170 I = MK, N-1 DWK(IDX(1,I),6*M+IDX(1,N)) = DUCPLU(I,N) 170 CONTINUE END IF ELSE C C PQ is not being rerun, determine whether it is being rebuilt. C IF ((VF.NE.0).AND.(N.LE.NMAX)) THEN WRITE (VF,'(A15,I8)') 'Rebuilding P&Q:', N END IF C C If E_k is singular, build an inner vector. C IF (INNER) THEN IF (VF.NE.0) WRITE (VF,'(A31)')$'... 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 DAXPBY (NLEN,VECS(1,IDX(4,N)),DONE,VECS(1,3),DZERO,VECS(1, $IDX(4,N))) CALL DAXPBY (NLEN,VECS(1,IDX(5,N)),DONE,VECS(1,4),DZERO,VECS(1,$IDX(5,N))) C C Step 8: C Build regular vectors and compute A p_n and q_n^T A p_n. C DO 180 I = MK, N-1 DTMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+14) / $DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,IDX(4,N)),DONE,VECS(1,IDX(4,N)),-DTMP,$VECS(1,IDX(4,I))) DTMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+15) / $DWK(IDX(1,N),8*M+15) * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M+10) CALL DAXPBY (NLEN,VECS(1,IDX(5,N)),DONE,VECS(1,IDX(5,N)),-DTMP,$VECS(1,IDX(5,I))) 180 CONTINUE C C Have the caller carry out AXB, then return here. C CALL AXB (VECS(1,IDX(4,N)),VECS(1,IDX(2,NP1))) C 190 INFO(2) = 1 INFO(3) = IDX(4,N) INFO(4) = IDX(2,NP1) RETLBL = 200 RETURN 200 DWK(IDX(1,N),5*M+IDX(1,N)) = DDOT(NLEN,VECS(1,IDX(5,N)),1,VECS(1, $IDX(2,NP1)),1) * DWK(IDX(1,N),8*M+14) * DWK(IDX(1,N),8*M+15) C C Step 9: C Compute the last column of E_k^{-1}. C IF (KBLKSZ.EQ.1) THEN DWK(IDX(1,MK),7*M+IDX(1,K)) = DONE / DWK(IDX(1,MK),5*M+IDX(1,MK$)) ELSE DO 210 I = 1, KBLKSZ-1 DWK(I,8*M+1) = DZERO 210 CONTINUE DWK(KBLKSZ,8*M+1) = DONE CALL DQRSL (DWK(1,4*M+1),M,KBLKSZ,KBLKSZ,DWK(1,8*M+3),DWK(1,8*M $+1),$ DZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1),DZERO, $100,J) DO 220 I = MK, N-1 DWK(IDX(1,I),7*M+IDX(1,K)) = DWK(I-MK+1,8*M+2) 220 CONTINUE END IF C C Check for zero norms. C DWK(IDX(1,N),8*M+16) = DNRM2(NLEN,VECS(1,IDX(4,N)),1) * DWK(IDX(1,$N),8*M+14) DWK(IDX(1,N),8*M+17) = DNRM2(NLEN,VECS(1,IDX(5,N)),1) * DWK(IDX(1, $N),8*M+15) IF (DWK(IDX(1,N),8*M+16).EQ.DZERO) GO TO 360 IF (DWK(IDX(1,N),8*M+17).EQ.DZERO) GO TO 360 C C If PQ is being rerun, then skip to the end. C IF (RERUN) GO TO 360 C C Step 7: C Build the full column G_{m_{k-1}:n-1,n-1} and compute its norm. C DTMP1 = DZERO DTMP2 = DZERO DO 240 I = IWK(IDX(1,K-1+1),1), N-1 DTMP = DZERO DO 230 J = MAX0(I,NLSTAR), N DTMP = DTMP + DWK(IDX(1,I),6*M+IDX(1,J)) * DWK(IDX(1,J),2*M+$IDX(1,N-1)) 230 CONTINUE DTMP = DABS(DTMP) DTMP1 = DTMP1 + DTMP * DWK(IDX(1,I),8*M+16) DTMP2 = DTMP2 + DTMP * DWK(IDX(1,I),8*M+17) / DWK(IDX(1,I),8*M+ $10) 240 CONTINUE DTMP1 = DTMP1 / DWK(IDX(1,N-1),8*M+16) DTMP2 = DTMP2 * DWK(IDX(1,N-1),8*M+10) / DWK(IDX(1,N-1),8*M+17) C C Step 9: C Build the 2nd term for the next step, G_{m_k:n-1,n}, regular. C Compute the norm of G_{m_k:n-1,n}. C DTMP3 = DZERO DTMP4 = DZERO DTMP5 = DWK(IDX(1,N),5*M+IDX(1,N)) * DWK(IDX(1,N),2*M+IDX(1,N-1))$* DWK(IDX(1,N-1),8*M+10) / DWK(IDX(1,N),8*M+10) DO 250 I = MK, N-1 DTMP = DABS(DTMP5 * DWK(IDX(1,I),7*M+IDX(1,K))) DTMP3 = DTMP3 + DTMP * DWK(IDX(1,I),8*M+16) DTMP4 = DTMP4 + DTMP * DWK(IDX(1,I),8*M+17) / DWK(IDX(1,I),8*M+ $10) 250 CONTINUE DTMP3 = DTMP3 / DWK(IDX(1,N),8*M+16) DTMP4 = DTMP4 * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,N),8*M+17) C C Steps 7 and 9: C Check G_{m_{k-1}:n-1,n-1} and G_{m_k:n-1,n}. C DTMP = DMAX1(DTMP1,DTMP2,DTMP3,DTMP4) INNER = DTMP.GT.NORM1 IF (.NOT.INNER) GO TO 360 DWK(IDX(1,N),8*M+18) = DTMP C C If G_{m_{k-1}:n-1,n-1} is bad, build inner vectors. C IF (DMAX1(DTMP1,DTMP2).GT.NORM1) GO TO 300 C C If G_{m_k:n-1,n} is bad, check the inner vectors. C This only applies if m_k > 1. C IF (MK.LE.1) GO TO 300 C C Build the inner vectors to compute the 2nd term at the next step. C Get the coefficients U_{m_k:n-1,n} in a temporary location. C Build the inner vectors, their norms are needed. C IBUILT = .TRUE. DO 260 I = MK, N-1 DWK(IDX(1,I),6*M+IDX(1,NP1)) = DUCPLU(I,N) DTMP = DWK(IDX(1,I),6*M+IDX(1,NP1)) * DWK(IDX(1,I),8*M+14$) / DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,3),DONE,VECS(1,3),-DTMP,VECS(1,IDX(4,I $))) DTMP = DWK(IDX(1,I),6*M+IDX(1,NP1)) * DWK(IDX(1,I),8*M+15$) / DWK(IDX(1,N),8*M+15) * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M $+10) CALL DAXPBY (NLEN,VECS(1,4),DONE,VECS(1,4),-DTMP,VECS(1,IDX(5,I$))) 260 CONTINUE DWK(IDX(1,NP1),8*M+16) = DNRM2(NLEN,VECS(1,3),1) * DWK(IDX(1,N), $8*M+14) DWK(IDX(1,NP1),8*M+17) = DNRM2(NLEN,VECS(1,4),1) * DWK(IDX(1,N),$8*M+15) IF (DWK(IDX(1,NP1),8*M+16).EQ.DZERO) GO TO 310 IF (DWK(IDX(1,NP1),8*M+17).EQ.DZERO) 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 DTMP = DZERO DO 270 J = MKSTAR, MK-1 DTMP = DTMP + DWK(IDX(1,J),6*M+IDX(1,N)) * DWK(IDX(1,J),5*M+ $IDX(1,MK)) / DWK(IDX(1,J),8*M+10) 270 CONTINUE DO 280 J = MK, N-1 DTMP = DTMP + DWK(IDX(1,J),6*M+IDX(1,NP1)) * DWK(IDX(1,J),5*M+$IDX(1,MK)) / DWK(IDX(1,J),8*M+10) 280 CONTINUE DTMP1 = DZERO DTMP2 = DZERO DTMP3 = DMAX1(DTMP3,DTMP4) DTMP = DWK(IDX(1,N),M+IDX(1,MK)) - DTMP * DWK(IDX(1,N),8*M+10) DTMP5 = DTMP * DWK(IDX(1,MK),2*M+IDX(1,MK-1)) * DWK(IDX(1,MK-1), $8*M+10) / DWK(IDX(1,MK),8*M+10) DO 290 I = IWK(IDX(1,K-1+1),1), MK-1 DTMP = DABS(DTMP5 * DWK(IDX(1,I),7*M+IDX(1,K-1))) DTMP1 = DTMP1 + DTMP * DWK(IDX(1,I),8*M+16) DTMP2 = DTMP2 + DTMP * DWK(IDX(1,I),8*M+17) / DWK(IDX(1,I),8*M+$10) 290 CONTINUE DTMP1 = DTMP1 / DWK(IDX(1,NP1),8*M+16) DTMP2 = DTMP2 * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,NP1),8*M+17) 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 = DTMP3.GT.DMAX1(DTMP1,DTMP2) 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 DUCPL1 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL,NLSTAR, $VF,IERR,ADJUST,NORM1,NORM2,DWK,IWK,IDX,VECS) IF (IERR.NE.0) GO TO 750 INNER = .FALSE. KBLKSZ = N - MK RERUN = .TRUE. NP1 = N + 1 GO TO 190 END IF C C The temporary vectors contain either just partial inner vectors, C or the completed ones, depending on whether IBUILT is TRUE or C FALSE. In either case, replace the regular vectors. C 310 CALL DAXPBY (NLEN,VECS(1,IDX(4,N)),DONE,VECS(1,3),DZERO,VECS(1,$IDX(4,N))) CALL DAXPBY (NLEN,VECS(1,IDX(5,N)),DONE,VECS(1,4),DZERO,VECS(1, $IDX(5,N))) C C Step 11: C Get the coefficients U_{m_k:n-1,n} and build inner vectors. C IF (IBUILT) THEN DO 320 I = MK, N-1 DWK(IDX(1,I),6*M+IDX(1,N)) = DWK(IDX(1,I),6*M+IDX(1,NP1)) 320 CONTINUE DWK(IDX(1,N),8*M+16) = DWK(IDX(1,NP1),8*M+16) DWK(IDX(1,N),8*M+17) = DWK(IDX(1,NP1),8*M+17) ELSE DO 330 I = MK, N-1 DWK(IDX(1,I),6*M+IDX(1,N)) = DUCPLU(I,N) DTMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+14)$ / DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,IDX(4,N)),DONE,VECS(1,IDX(4,N)),- $DTMP,VECS(1,IDX(4,I))) DTMP = DWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8*M+15)$ / DWK(IDX(1,N),8*M+15) * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M+ $10) CALL DAXPBY (NLEN,VECS(1,IDX(5,N)),DONE,VECS(1,IDX(5,N)),-$DTMP,VECS(1,IDX(5,I))) 330 CONTINUE DWK(IDX(1,N),8*M+16) = DNRM2(NLEN,VECS(1,IDX(4,N)),1) * $DWK(IDX(1,N),8*M+14) DWK(IDX(1,N),8*M+17) = DNRM2(NLEN,VECS(1,IDX(5,N)),1) *$DWK(IDX(1,N),8*M+15) END IF C C Step 11: C Compute A p_n and q_n^T A p_n. C Have the caller carry out AXB, then return here. C CALL AXB (VECS(1,IDX(4,N)),VECS(1,IDX(2,NP1))) C 340 INFO(2) = 1 INFO(3) = IDX(4,N) INFO(4) = IDX(2,NP1) RETLBL = 350 RETURN 350 DWK(IDX(1,N),5*M+IDX(1,N)) = DDOT(NLEN,VECS(1,IDX(5,N)),1,VECS(1, $IDX(2,NP1)),1) * DWK(IDX(1,N),8*M+14) * DWK(IDX(1,N),8*M+15) C C Step 10: C If regular vectors were built, update the counters. C 360 IF (.NOT.INNER) THEN K = K + 1 MK = IWK(IDX(1,K+1),1) END IF C C Update counters. C MPQBLT = MAX0(MPQBLT,MK-IWK(IDX(1,K-1+1),1)) MKMAX = MAX0(MK,MKMAX) IWK(IDX(1,N),6) = MKSTAR IWK(IDX(1,N),4) = MK C C Check for invariant subspaces. C IF (DWK(IDX(1,N),8*M+16).LT.TNRM) IERR = IERR + 16 IF (DWK(IDX(1,N),8*M+17).LT.TNRM) IERR = IERR + 32 IF (IERR.NE.0) GO TO 750 C C Step 13: C Compute A^T q_n. C Have the caller carry out ATXB, then return here. C CALL ATXB (VECS(1,IDX(5,N)),VECS(1,IDX(3,NP1))) C INFO(2) = 2 INFO(3) = IDX(5,N) INFO(4) = IDX(3,NP1) RETLBL = 370 RETURN C C Update the norm estimates. C 370 IF (N.LE.NUMCHK) THEN DTMP1 = DNRM2(NLEN,VECS(1,IDX(2,NP1)),1) * DWK(IDX(1,N),8*M+14)$ / DWK(IDX(1,N),8*M+16) DTMP2 = DNRM2(NLEN,VECS(1,IDX(3,NP1)),1) * DWK(IDX(1,N),8*M+15) $/ DWK(IDX(1,N),8*M+17) NORMA = DMAX1(DTMP1,DTMP2,NORMA) END IF DTMP = DABS(DWK(IDX(1,N),5*M+IDX(1,N))) / ( DWK(IDX(1,N),8*M+16)$* DWK(IDX(1,N),8*M+17) ) NORMA = DMAX1(DTMP,NORMA) NORM1 = DMAX1(ADJUST*NORMA,NORM1) NORM2 = DMAX1(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+13) = DZERO 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)) = DZERO 380 CONTINUE C C Step 14: C Update E^{(n-1)} to E^{(n)}. C DO 400 I = MK, N-1 DTMP = DZERO DO 390 J = MAX0(MKSTAR,IWK(IDX(1,I),4)), N-1 DTMP = DTMP + DWK(IDX(1,J),6*M+IDX(1,N)) * DWK(IDX(1,J),5*M+ $IDX(1,I)) / DWK(IDX(1,J),8*M+10) 390 CONTINUE DTMP = DWK(IDX(1,N),M+IDX(1,I)) - DTMP * DWK(IDX(1,N),8*M+$10) DWK(IDX(1,N),5*M+IDX(1,I)) = DTMP DWK(IDX(1,I),5*M+IDX(1,N)) = DTMP * DWK(IDX(1,I),8*M+10) / $DWK(IDX(1,N),8*M+10) DTMP1 = DABS(DWK(IDX(1,N),5*M+IDX(1,I))) / (DWK(IDX(1,N),8*M$+17) * DWK(IDX(1,I),8*M+16)) DTMP2 = DABS(DWK(IDX(1,I),5*M+IDX(1,N))) / (DWK(IDX(1,I),8*M $+17) * DWK(IDX(1,N),8*M+16)) NORMA = DMAX1(NORMA,DTMP1,DTMP2) NORM1 = DMAX1(ADJUST*NORMA,NORM1) NORM2 = DMAX1(ADJUST*NORMA,NORM2) 400 CONTINUE C C Step 15: C Compute l^\star. C I = LSTAR DO 410 J = I+1, L IF (IWK(IDX(1,J+1),2).LE.MK) LSTAR = J 410 CONTINUE NLSTAR = IWK(IDX(1,LSTAR+1),2) C C Check memory allocation (NLSTAR.GE.VWBEG). Update the range of C valid indices: the new vectors v_{n+1} and w_{n+1} will overwrite C the vectors at location N+1-MVEC. C IF (NLSTAR.LT.VWBEG) THEN IERR = 64 GO TO 750 END IF VWBEG = MAX0(VWBEG,NP1-MVEC+1) C C Step 16: C Compute F_{1:n,n} = F_{n_{l^\star}:n,n}. C DO 430 I = NLSTAR, N DTMP = DZERO DO 420 J = MAX0(MK,IWK(IDX(1,I),6)), I DTMP = DTMP + DWK(IDX(1,J),6*M+IDX(1,I)) * DWK(IDX(1,J),5*M+$IDX(1,N)) / DWK(IDX(1,J),8*M+10) 420 CONTINUE DWK(IDX(1,I),M+IDX(1,N)) = DTMP * DWK(IDX(1,I),8*M+10) DTMP = DABS(DTMP) / DWK(IDX(1,N),8*M+16) NORMA = DMAX1(NORMA,DTMP) NORM1 = DMAX1(ADJUST*NORMA,NORM1) NORM2 = DMAX1(ADJUST*NORMA,NORM2) 430 CONTINUE C C Step 17: C Check whether D_l is nonsingular. C IF (LBLKSZ.EQ.1) THEN INNER = DABS(DWK(IDX(1,NL),IDX(1,NL))).EQ.DZERO 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 DSVDC (DWK(1,4*M+1),M,LBLKSZ,LBLKSZ,DWK(1,8*M+1),DWK(1,8*M $+3),DZERO,0,DZERO,0,DWK(1,8*M+2),0,I) IF (I.NE.0) THEN IERR = -I GO TO 750 END IF DTMP = DZERO IF (DWK(1,8*M+1).NE.DZERO) DTMP = DWK(LBLKSZ,8*M+1) / DWK(1,8*M$+1) INNER = DTMP.LT.(DBLE(NLEN) * DLAMCH('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)) / $DWK(IDX(1,NI),IDX(1,NI)) ELSE DO 470 J = NI, NIP1-1 DWK(J-NI+1,8*M+1) = DWK(IDX(1,J),M+IDX(1,N)) DO 460 IJ = NI, NIP1-1 DWK(J-NI+1,4*M+IJ-NI+1) = DWK(IDX(1,J),IDX(1,IJ)) 460 CONTINUE 470 CONTINUE CALL DQRDC (DWK(1,4*M+1),M,IBLKSZ,IBLKSZ,DWK(1,8*M+3),0,$DZERO,0) CALL DQRSL (DWK(1,4*M+1),M,IBLKSZ,IBLKSZ,DWK(1,8*M+3),DWK(1, $8*M+1),$ DZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1), $DZERO,100,J) DO 480 J = NI, NIP1-1 DWK(IDX(1,J),2*M+IDX(1,N)) = DWK(J-NI+1,8*M+2) 480 CONTINUE END IF 490 CONTINUE C C Step 19: C Build the part common to both inner and regular vectors. C Assume that V(NP1) = A p_n and W(NP1) = A^T q_n. C DWK(IDX(1,NP1),2*M+IDX(1,N)) = DZERO DO 500 I = NLSTAR, NL-1 DTMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+11) /$DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,IDX(2,NP1)),- $DTMP,VECS(1,IDX(2,I))) DTMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+12) /$DWK(IDX(1,N),8*M+15) * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M+10) CALL DAXPBY (NLEN,VECS(1,IDX(3,NP1)),DONE,VECS(1,IDX(3,NP1)),- $DTMP,VECS(1,IDX(3,I))) 500 CONTINUE C C Check whether VW is being rerun. C RERUN = (N.LE.NMAX).AND.(N.LT.NLMAX) IF (RERUN) THEN C C Check whether D_l has become singular (this should not happen). C IF ((IWK(IDX(1,L+1+1),2).EQ.NP1).AND.INNER) THEN IERR = 128 GO TO 750 END IF C C Determine whether this was an inner vector or not. C INNER = IWK(IDX(1,L+1+1),2).NE.NP1 IF (VF.NE.0) THEN IF (INNER) THEN WRITE (VF,'(A17,I5,A14)') 'Rerunning vector ',NP1,$' (VW) as inner' ELSE WRITE (VF,'(A17,I5,A16)') 'Rerunning vector ',NP1, $' (VW) as regular' END IF END IF C C If building an inner vector, set the coefficients L_{n_l:n,n}. C IF (INNER) THEN DO 510 I = NL, N DWK(IDX(1,I),2*M+IDX(1,N)) = DUCPLL(I,N) 510 CONTINUE END IF ELSE C C VW is not being rerun, determine whether it is being rebuilt. C IF ((VF.NE.0).AND.(N.LE.NMAX)) THEN WRITE (VF,'(A15,I8)') 'Rebuilding V&W:', NP1 END IF C C If E_k is singular, build an inner vector. C IF (INNER) THEN IF (VF.NE.0) WRITE (VF,'(A31)')$'... 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 DAXPBY (NLEN,VECS(1,3),DONE,VECS(1,IDX(2,NP1)),DZERO,VECS(1,3 $)) CALL DAXPBY (NLEN,VECS(1,4),DONE,VECS(1,IDX(3,NP1)),DZERO,VECS(1,4$)) C C Step 21. C Build regular vectors. C DO 520 I = NL, N DTMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+11) / $DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,IDX(2,NP1)),-$DTMP,VECS(1,IDX(2,I))) DTMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+12) / $DWK(IDX(1,N),8*M+15) * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M+10) CALL DAXPBY (NLEN,VECS(1,IDX(3,NP1)),DONE,VECS(1,IDX(3,NP1)),-$DTMP,VECS(1,IDX(3,I))) 520 CONTINUE C C Compute scale factors for the new vectors. C 530 DTMP3 = DNRM2(NLEN,VECS(1,IDX(2,NP1)),1) DTMP4 = DNRM2(NLEN,VECS(1,IDX(3,NP1)),1) DTMP1 = DWK(IDX(1,N),8*M+14) * DTMP3 DTMP2 = DWK(IDX(1,N),8*M+15) * DTMP4 DWK(IDX(1,NP1),2*M+IDX(1,N)) = DTMP1 IF (DTMP1.LT.TNRM) IERR = IERR + 16 IF (DTMP2.LT.TNRM) IERR = IERR + 32 IF (IERR.NE.0) GO TO 670 DWK(IDX(1,NP1),8*M+10) = DWK(IDX(1,N),8*M+10) * DTMP1 / DTMP2 DWK(IDX(1,NP1),IDX(1,NP1)) = DDOT(NLEN,VECS(1,IDX(3,NP1)),1,VECS(1 $,IDX(2,NP1)),1) / ( DTMP3 * DTMP4 ) IF ((DTMP3.GE.TMAX).OR.(DTMP3.LE.TMIN)) THEN DTMP = DONE / DTMP3 CALL DAXPBY (NLEN,VECS(1,IDX(2,NP1)),DTMP,VECS(1,IDX(2,NP1)),$DZERO,VECS(1,IDX(2,NP1))) DTMP3 = DONE END IF IF ((DTMP4.GE.TMAX).OR.(DTMP4.LE.TMIN)) THEN DTMP = DONE / DTMP4 CALL DAXPBY (NLEN,VECS(1,IDX(3,NP1)),DTMP,VECS(1,IDX(3,NP1)), $DZERO,VECS(1,IDX(3,NP1))) DTMP4 = DONE END IF DWK(IDX(1,NP1),8*M+11) = DONE / DTMP3 DWK(IDX(1,NP1),8*M+12) = DONE / DTMP4 C C Step 22: C Compute the last column of D_l^{-1}. C IF (LBLKSZ.EQ.1) THEN DWK(IDX(1,NL),3*M+IDX(1,L)) = DONE / DWK(IDX(1,NL),IDX(1,NL)) ELSE DO 540 I = 1, LBLKSZ-1 DWK(I,8*M+1) = DZERO 540 CONTINUE DWK(LBLKSZ,8*M+1) = DONE CALL DQRSL (DWK(1,4*M+1),M,LBLKSZ,LBLKSZ,DWK(1,8*M+3),DWK(1,8*M$+1), $DZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1),DZERO,$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 DTMP1 = DZERO DTMP2 = DZERO DO 570 I = IWK(IDX(1,L-1+1),2), N DTMP = DZERO DO 560 J = MAX0(1,I-1), N DTMP = DTMP + DWK(IDX(1,I),2*M+IDX(1,J)) * DWK(IDX(1,J),6*M+ $IDX(1,N)) 560 CONTINUE DTMP = DABS(DTMP) DTMP1 = DTMP1 + DTMP DTMP2 = DTMP2 + DTMP / DWK(IDX(1,I),8*M+10) 570 CONTINUE DTMP2 = DTMP2 * DWK(IDX(1,N),8*M+10) C C Step 22: C Build the 2nd term for the next step, H_{n_l:n,n+1}, regular. C Compute the norm of H_{n_l:n,n+1}. C DTMP3 = DZERO DTMP4 = DZERO DTMP5 = DWK(IDX(1,NP1),IDX(1,NP1)) * DWK(IDX(1,NP1),2*M+IDX(1,N))$* DWK(IDX(1,N),8*M+10) / DWK(IDX(1,NP1),8*M+10) DO 580 I = NL, N DTMP = DABS(DTMP5 * DWK(IDX(1,I),3*M+IDX(1,L))) DTMP3 = DTMP3 + DTMP DTMP4 = DTMP4 + DTMP / DWK(IDX(1,I),8*M+10) 580 CONTINUE DTMP4 = DTMP4 * DWK(IDX(1,NP1),8*M+10) C C Steps 20 and 22: C Check H_{n_{l-1}:n,n} and H_{n_l:n-1,n+1}. C DTMP = DMAX1(DTMP1,DTMP2,DTMP3,DTMP4) INNER = DTMP.GT.NORM2 IF (.NOT.INNER) GO TO 670 DWK(IDX(1,N),8*M+13) = DTMP C C If H_{n_{l-1}:n,n} is bad, build inner vectors. C IF (DMAX1(DTMP1,DTMP2).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 vectors in VECS(1,3) and VECS(1,4). C IBUILT = .TRUE. DO 590 I = NL, N DWK(IDX(1,I),2*M+IDX(1,NP1)) = DUCPLL(I,N) DTMP = DWK(IDX(1,I),2*M+IDX(1,NP1)) * DWK(IDX(1,I),8*M+11 $) / DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,3),DONE,VECS(1,3),-DTMP,VECS(1,IDX(2,I$))) DTMP = DWK(IDX(1,I),2*M+IDX(1,NP1)) * DWK(IDX(1,I),8*M+12 $) / DWK(IDX(1,N),8*M+15) * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M$+10) CALL DAXPBY (NLEN,VECS(1,4),DONE,VECS(1,4),-DTMP,VECS(1,IDX(3,I $))) 590 CONTINUE IDTMP3 = DNRM2(NLEN,VECS(1,3),1) IDTMP4 = DNRM2(NLEN,VECS(1,4),1) IDTMP1 = DWK(IDX(1,N),8*M+14) * IDTMP3 IDTMP2 = DWK(IDX(1,N),8*M+15) * IDTMP4 DWK(IDX(1,NP1),2*M+IDX(1,NP1)) = IDTMP1 IF (IDTMP1.LT.TNRM) IERR = IERR + 16 IF (IDTMP2.LT.TNRM) IERR = IERR + 32 IF (IERR.NE.0) GO TO 640 C C Build the 2nd term for the next step, H_{n_{l-1}:n_l-1,n+1}, C inner. Compute the norm of H_{n_{l-1}:n_l-1,n+1}. C DTMP = DZERO DO 600 J = NLSTAR, NL-1 DTMP = DTMP + DWK(IDX(1,NL),IDX(1,J)) * DWK(IDX(1,I),2*M+IDX(1,$N)) 600 CONTINUE DO 610 J = NL, N DTMP = DTMP + DWK(IDX(1,NL),IDX(1,J)) * DWK(IDX(1,I),2*M+IDX(1, $NP1)) 610 CONTINUE DTMP1 = DZERO DTMP2 = DZERO DTMP3 = DMAX1(DTMP3,DTMP4) DTMP = ( DWK(IDX(1,NL),M+IDX(1,N)) - DTMP ) / DWK(IDX(1,NP1),2*M+$IDX(1,NP1)) DTMP5 = DTMP * DWK(IDX(1,NL),2*M+IDX(1,NL-1)) * DWK(IDX(1,NL-1), $8*M+10) / DWK(IDX(1,NL),8*M+10) DO 620 I = IWK(IDX(1,L-1+1),2), NL-1 DTMP = DABS(DWK(IDX(1,I),3*M+IDX(1,L-1)) * DTMP5) DTMP1 = DTMP1 + DTMP DTMP2 = DTMP2 + DTMP / DWK(IDX(1,I),8*M+10) 620 CONTINUE DTMP2 = DTMP2 * DWK(IDX(1,N),8*M+10) C C Compare the inner and regular versions of the 2nd term at the next C step. Build the vector corresponding to the smaller term. C INNER = DTMP3.GT.DMAX1(DTMP1,DTMP2) IF (.NOT.INNER) GO TO 670 C C Build inner vectors. C Check whether the V&W block has to be forced to close. C 630 IF (VF.NE.0) WRITE (VF,'(A7,I5,A14)') 'Vector ',NP1,$' (VW) is inner' IF (NP1-NL.EQ.MAXVW) THEN CALL DUCPL2 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL,NLSTAR, $VF,IERR,ADJUST,NORM1,NORM2,DWK,IWK,IDX,VECS) IF (IERR.NE.0) GO TO 750 LBLKSZ = N - NL + 1 INNER = .FALSE. RERUN = .TRUE. NP1 = N + 1 GO TO 530 END IF C C The temporary vectors contain either just partial inner vectors, C or the completed ones, depending on whether IBUILT is TRUE or C FALSE. In either case, replace the regular vectors. C 640 CALL DAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,3),DZERO,VECS(1,$IDX(2,NP1))) CALL DAXPBY (NLEN,VECS(1,IDX(3,NP1)),DONE,VECS(1,4),DZERO,VECS(1, $IDX(3,NP1))) C C Step 24: C Get the coefficients L_{n_l:n+1,n} and build inner vectors. C IF (IBUILT) THEN DO 650 I = NL, NP1 DWK(IDX(1,I),2*M+IDX(1,N)) = DWK(IDX(1,I),2*M+IDX(1,NP1)) 650 CONTINUE DTMP1 = IDTMP1 DTMP2 = IDTMP2 DTMP3 = IDTMP3 DTMP4 = IDTMP4 ELSE DO 660 I = NL, N DWK(IDX(1,I),2*M+IDX(1,N)) = DUCPLL(I,N) DTMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+11$) / DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,IDX(2,NP1)) $,-DTMP,VECS(1,IDX(2,I))) DTMP = DWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),8*M+12$) / DWK(IDX(1,N),8*M+15) * DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M $+10) CALL DAXPBY (NLEN,VECS(1,IDX(3,NP1)),DONE,VECS(1,IDX(3,NP1))$,-DTMP,VECS(1,IDX(3,I))) 660 CONTINUE DTMP3 = DNRM2(NLEN,VECS(1,IDX(2,NP1)),1) DTMP4 = DNRM2(NLEN,VECS(1,IDX(3,NP1)),1) DTMP1 = DWK(IDX(1,N),8*M+14) * DTMP3 DTMP2 = DWK(IDX(1,N),8*M+15) * DTMP4 DWK(IDX(1,NP1),2*M+IDX(1,N)) = DTMP1 IF (DTMP1.LT.TNRM) IERR = IERR + 16 IF (DTMP2.LT.TNRM) IERR = IERR + 32 END IF IF (IERR.NE.0) GO TO 670 DWK(IDX(1,NP1),8*M+10) = DWK(IDX(1,N),8*M+10) * DTMP1 / DTMP2 DWK(IDX(1,NP1),IDX(1,NP1)) = DDOT(NLEN,VECS(1,IDX(3,NP1)),1,VECS(1 $,IDX(2,NP1)),1) / ( DTMP3 * DTMP4 ) IF ((DTMP3.GE.TMAX).OR.(DTMP3.LE.TMIN)) THEN DTMP = DONE / DTMP3 CALL DAXPBY (NLEN,VECS(1,IDX(2,NP1)),DTMP,VECS(1,IDX(2,NP1)),$DZERO,VECS(1,IDX(2,NP1))) DTMP3 = DONE END IF IF ((DTMP4.GE.TMAX).OR.(DTMP4.LE.TMIN)) THEN DTMP = DONE / DTMP4 CALL DAXPBY (NLEN,VECS(1,IDX(3,NP1)),DTMP,VECS(1,IDX(3,NP1)), $DZERO,VECS(1,IDX(3,NP1))) DTMP4 = DONE END IF DWK(IDX(1,NP1),8*M+11) = DONE / DTMP3 DWK(IDX(1,NP1),8*M+12) = DONE / DTMP4 C C Step 23: C If regular vectors were built, update the counters. C 670 IF (.NOT.INNER) THEN L = L + 1 NL = IWK(IDX(1,L+1),2) END IF C C Update counters. C MVWBLT = MAX0(MVWBLT,NL-IWK(IDX(1,L-1+1),2)) NLMAX = MAX0(NL,NLMAX) IWK(IDX(1,N),5) = NLSTAR IWK(IDX(1,N),3) = NL C C Update the counter for steps taken. C NMAX = MAX0(NMAX,N) IF ((N.LT.MKMAX).OR.(N.LT.NLMAX-1)) GO TO 740 C C The QMR code starts here. C At this point, (N.GE.MKMAX).AND.(N.GE.NLMAX-1), so that steps up C to MIN(MKMAX,NLMAX-1) are guaranteed not to be rebuilt. Also, no C errors are allowed in IERR, other than possibly having found one C or both invariant subspaces, in which case all remaining iterates C are computed. C IEND = MIN0(MKMAX,NLMAX-1) IF (IERR.NE.0) IEND = N 680 IF (NQMR.GT.IEND-1) GO TO 740 C C Update the QMR iteration counter. C NQMR = NQMR + 1 C C Get the next scaling factor omega(i) and update MAXOMG. C DWK(IDX(1,NQMR+1),8*M+8) = DUCPLO(NQMR+1) MAXOMG = DMAX1(MAXOMG,DONE/DWK(IDX(1,NQMR+1),8*M+8)) C C Compute the starting index IBASE for the column of \hat{R}. C IBASE = MAX0(1,IWK(IDX(1,NQMR),5)-1) DWK(IDX(1,IBASE),8*M+5) = DZERO C C Multiply the new column by the previous omegas. C DO 690 J = IWK(IDX(1,NQMR),5), NQMR+1 DWK(IDX(1,J),8*M+5) = DWK(IDX(1,J),8*M+8) * DWK(IDX(1,J),2*M+$IDX(1,NQMR)) 690 CONTINUE C C Apply the previous rotations. C The loop below explicitly implements a call to DROT: C CALL DROT (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 DTMP1 = DWK(IDX(1,J),8*M+5) DTMP2 = DWK(IDX(1,J-1),8*M+5) DWK(IDX(1,J-1),8*M+5) = DWK(IDX(1,J),8*M+9) * DTMP2 + DWK(IDX(1 $,J),8*M+6) * DTMP1 DWK(IDX(1,J),8*M+5) = DWK(IDX(1,J),8*M+9) * DTMP1 - DWK(IDX(1$,J),8*M+6) * DTMP2 700 CONTINUE C C Compute the rotation for the last element (this also applies it). C CALL DROTG (DWK(IDX(1,NQMR),8*M+5),DWK(IDX(1,NQMR+1),8*M+5), $DWK(IDX(1,NQMR+1),8*M+9),DWK(IDX(1,NQMR+1),8*M+6)) C C Apply the new rotation to the right-hand side vector. C Could be replaced with: C DWK(IDX(1,NQMR+1),8*M+4) = DZERO C CALL DROT (1,DWK(IDX(1,NQMR),8*M+4),1,DWK(IDX(1,NQMR+1),8*M+4),1,DWK(IDX(1,NQMR+1),8*M+9),DWK(IDX(1,NQMR+1),8*M+6)) C DWK(IDX(1,NQMR+1),8*M+4) = -DWK(IDX(1,NQMR+1),8*M+6) * DWK(IDX(1,$NQMR),8*M+4) DWK(IDX(1,NQMR),8*M+4) = DWK(IDX(1,NQMR+1),8*M+9) * DWK(IDX(1, $NQMR),8*M+4) C C Compute the next search direction s_i. C This is more complicated than it might have to be because storage C for the vectors VECS(1,IDX(6,NQMR)) is minimized. C DTMP2 = DZERO DTMP = DWK(IDX(1,NQMR),8*M+14) DO 710 J = IBASE, NQMR-1 DTMP1 = DWK(IDX(1,J),8*M+5) * DWK(IDX(1,J),8*M+7) / DTMP IF (DTMP1.EQ.DZERO) GO TO 710 CALL DAXPBY (NLEN,VECS(1,IDX(6,NQMR)),DTMP2,VECS(1,IDX(6,NQMR))$,-DTMP1,VECS(1,IDX(6,J))) DTMP2 = DONE 710 CONTINUE CALL DAXPBY (NLEN,VECS(1,IDX(6,NQMR)),DTMP2,VECS(1,IDX(6,NQMR)), $DONE,VECS(1,IDX(4,NQMR))) DTMP = DTMP / DWK(IDX(1,NQMR),8*M+5) C C Compute the new QMR iterate, then scale the search direction. C DTMP1 = DTMP * DWK(IDX(1,NQMR),8*M+4) CALL DAXPBY (NLEN,VECS(1,1),DONE,VECS(1,1),DTMP1,VECS(1,IDX(6,NQMR$))) DWK(IDX(1,NQMR),8*M+7) = DTMP DTMP = DABS(DTMP) IF ((DTMP.GE.TMAX).OR.(DTMP.LE.TMIN)) THEN DWK(IDX(1,NQMR),8*M+7) = DONE CALL DAXPBY (NLEN,VECS(1,IDX(6,NQMR)),DTMP,VECS(1,IDX(6,NQMR)), $DZERO,VECS(1,IDX(6,NQMR))) END IF C C Compute the residual norm upper bound. C If the scaled upper bound is within one order of magnitude of the C target convergence norm, compute the true residual norm. C UNRM = DSQRT(DBLE(NQMR+1)) * MAXOMG * DABS(DWK(IDX(1,NQMR+1),8*M+4$)) / 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 DAXPBY (NLEN,VECS(1,3),DONE,VECS(1,2),-DONE,VECS(1,3)) RESN = DNRM2(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 DUCPL1 (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 vectors p_n and q_n 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 DUCPL. C C External routines used: C subroutine daxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C double precision ddot(n,x,incx,y,incy) C BLAS-1 routine, computes y^H * x. C double precision dnrm2(n,x,incx) C BLAS-1 routine, computes the 2-norm of x. C subroutine dqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine dqrsl(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 DMAX1, MAX0 EXTERNAL DAXPBY, DQRDC, DQRSL C INTEGER IERR, K, KSTAR, L, LSTAR, M, MK, MKSTAR, NL, NLSTAR, N INTEGER IDX(6,*), IWK(M,13), NDIM, NLEN, VF DOUBLE PRECISION ADJUST, DWK(M,8*M+18), NORM1, NORM2 DOUBLE PRECISION VECS(NDIM,*) C C Miscellaneous parameters. C DOUBLE PRECISION DONE, DZERO PARAMETER (DONE = 1.0D0,DZERO = 0.0D0) C C Local variables. C INTEGER I, IJ, J, KBLKSZ, NP1 DOUBLE PRECISION DTMP1, DTMP2 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 DTMP1 = DWK(IDX(1,J),8*M+18) DO 100 I = MK+2, N DTMP2 = DWK(IDX(1,I),8*M+18) IF (DTMP2.GT.DZERO) THEN IF ((DTMP1.EQ.DZERO).OR.(DTMP2.LT.DTMP1)) THEN J = I DTMP1 = DTMP2 END IF END IF 100 CONTINUE IF (DTMP1.EQ.DZERO) THEN IF (VF.NE.0) WRITE (VF,'(A47)') $'... no new norm estimates available (aborting).' IERR = 8 RETURN END IF NORM1 = ADJUST * DTMP1 NORM2 = DMAX1(NORM1,NORM2) IF (VF.NE.0) WRITE (VF,'(A40,I5,2E11.4)')$ '... updated norms, restarting from step:', IWK(IDX(1,J),7), $NORM1, NORM2 IF (IWK(IDX(1,J),7).EQ.N) RETURN N = IWK(IDX(1,J),7) L = IWK(IDX(1,N),9) NL = IWK(IDX(1,L+1),2) K = IWK(IDX(1,N),8) MK = IWK(IDX(1,K+1),1) LSTAR = IWK(IDX(1,N),11) NLSTAR = IWK(IDX(1,LSTAR+1),2) KSTAR = IWK(IDX(1,N),10) MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Initialize local variables. C DWK(IDX(1,N),6*M+IDX(1,N)) = DONE NP1 = N + 1 DWK(IDX(1,N),8*M+18) = DZERO KBLKSZ = N - MK C C Step 2: C Compute k^\star. C I = KSTAR DO 110 J = I+1, K IF (IWK(IDX(1,J+1),1).LE.NL-1) KSTAR = J 110 CONTINUE MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Compute U_{m_k:n-1,n}. Save the old coefficients. C IWK(IDX(1,K+1+1),1) = N IF (KBLKSZ.EQ.1) THEN DWK(IDX(1,MK),6*M+IDX(1,NP1)) = DWK(IDX(1,MK),6*M+IDX(1,N)) DWK(IDX(1,MK),6*M+IDX(1,N)) = DWK(IDX(1,N),M+IDX(1,MK)) /$DWK(IDX(1,MK),5*M+IDX(1,MK)) * DWK(IDX(1,MK),8*M+10) / DWK(IDX(1,N $),8*M+10) ELSE DO 130 J = MK, N-1 DWK(J-MK+1,8*M+1) = DWK(IDX(1,N),M+IDX(1,J)) * DWK(IDX(1,J),$8*M+10) / DWK(IDX(1,N),8*M+10) 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 DQRDC (DWK(1,4*M+1),M,KBLKSZ,KBLKSZ,DWK(1,8*M+3),0,DZERO,0 $) CALL DQRSL (DWK(1,4*M+1),M,KBLKSZ,KBLKSZ,DWK(1,8*M+3),DWK(1,8*M$+1), $DZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1),DZERO,$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 DTMP1 = DWK(IDX(1,I),6*M+IDX(1,N)) - DWK(IDX(1,I),6*M+IDX(1,NP1 $)) DTMP2 = DTMP1 * DWK(IDX(1,I),8*M+14) / DWK(IDX(1,N),8*M+14) CALL DAXPBY (NLEN,VECS(1,IDX(4,N)),DONE,VECS(1,IDX(4,N)),-DTMP2$,VECS(1,IDX(4,I))) DTMP2 = DTMP1 * DWK(IDX(1,I),8*M+15) / DWK(IDX(1,N),8*M+15) * $DWK(IDX(1,N),8*M+10) / DWK(IDX(1,I),8*M+10) CALL DAXPBY (NLEN,VECS(1,IDX(5,N)),DONE,VECS(1,IDX(5,N)),-DTMP2$,VECS(1,IDX(5,I))) 150 CONTINUE C RETURN END C C********************************************************************** C SUBROUTINE DUCPL2 (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 vectors v_{n+1} and w_{n+1} C at the point where a block was forced to close. The routine is C called internally by the coupled Lanczos code, and is not meant C to be called directly by the user code. C C Parameters: C See descriptions in the main routine DUCPL. C C External routines used: C subroutine daxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C subroutine dqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine dqrsl(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 DABS, DMAX1, MAX0 EXTERNAL DAXPBY, DQRDC, DQRSL C INTEGER IERR, K, KSTAR, L, LSTAR, M, MK, MKSTAR, NL, NLSTAR, N INTEGER IDX(6,*), IWK(M,13), NDIM, NLEN, VF DOUBLE PRECISION ADJUST, DWK(M,8*M+18), NORM1, NORM2 DOUBLE PRECISION VECS(NDIM,*) C C Miscellaneous parameters. C DOUBLE PRECISION DONE, DZERO PARAMETER (DONE = 1.0D0,DZERO = 0.0D0) C C Local variables. C INTEGER I, IJ, J, LBLKSZ, NP1 DOUBLE PRECISION DTMP1, DTMP2, SCALV, SCALW 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 DTMP1 = DWK(IDX(1,J),8*M+13) DO 100 I = NL+1, N DTMP2 = DWK(IDX(1,I),8*M+13) IF (DTMP2.GT.DZERO) THEN IF ((DTMP1.EQ.DZERO).OR.(DTMP2.LT.DTMP1)) THEN J = I DTMP1 = DTMP2 END IF END IF 100 CONTINUE IF (DTMP1.EQ.DZERO) THEN IF (VF.NE.0) WRITE (VF,'(A47)') $'... no new norm estimates available (aborting).' IERR = 8 RETURN END IF NORM2 = ADJUST * DTMP1 NORM1 = DMAX1(NORM1,NORM2) IF (VF.NE.0) WRITE (VF,'(A40,I5,2E11.4)')$ '... updated norms, restarting from step:', IWK(IDX(1,J),7), $NORM1, NORM2 IF (IWK(IDX(1,J),7).EQ.N) RETURN N = IWK(IDX(1,J),7) L = IWK(IDX(1,N),9) NL = IWK(IDX(1,L+1),2) K = IWK(IDX(1,N),12) MK = IWK(IDX(1,K+1),1) LSTAR = IWK(IDX(1,N),11) NLSTAR = IWK(IDX(1,LSTAR+1),2) KSTAR = IWK(IDX(1,N),13) MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Initialize local variables. C NP1 = N + 1 DWK(IDX(1,N),8*M+13) = DZERO LBLKSZ = N - NL + 1 C C Step 15: C Compute l^\star. C I = LSTAR DO 110 J = I+1, L IF (IWK(IDX(1,J+1),2).LE.MK) LSTAR = J 110 CONTINUE NLSTAR = IWK(IDX(1,LSTAR+1),2) C C Compute L_{n_l:n,n}. Save the old coefficients. C IWK(IDX(1,L+1+1),2) = NP1 IF (LBLKSZ.EQ.1) THEN DWK(IDX(1,NL),2*M+IDX(1,NP1)) = DWK(IDX(1,NL),2*M+IDX(1,N)) DWK(IDX(1,NL),2*M+IDX(1,N)) = DWK(IDX(1,NL),M+IDX(1,N)) /$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 DQRDC (DWK(1,4*M+1),M,LBLKSZ,LBLKSZ,DWK(1,8*M+3),0,DZERO,0 $) CALL DQRSL (DWK(1,4*M+1),M,LBLKSZ,LBLKSZ,DWK(1,8*M+3),DWK(1,8*M$+1), $DZERO,DWK(1,8*M+1),DWK(1,8*M+2),DWK(1,8*M+1),DZERO,$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+11) SCALW = DWK(IDX(1,NP1),2*M+IDX(1,N)) * DWK(IDX(1,NP1),8*M+12) * $DWK(IDX(1,N),8*M+10) / DWK(IDX(1,NP1),8*M+10) DWK(IDX(1,NP1),2*M+IDX(1,N)) = DZERO DO 150 I = NL, N DTMP1 = DWK(IDX(1,I),2*M+IDX(1,N)) - DWK(IDX(1,I),2*M+IDX(1,NP1$)) DTMP2 = DTMP1 * DWK(IDX(1,I),8*M+11) / SCALV CALL DAXPBY (NLEN,VECS(1,IDX(2,NP1)),DONE,VECS(1,IDX(2,NP1)),- $DTMP2,VECS(1,IDX(2,I))) DTMP2 = DTMP1 * DWK(IDX(1,I),8*M+12) / SCALW * DWK(IDX(1,N),8*M$+10) / DWK(IDX(1,I),8*M+10) CALL DAXPBY (NLEN,VECS(1,IDX(3,NP1)),DONE,VECS(1,IDX(3,NP1)),- \$DTMP2,VECS(1,IDX(3,I))) 150 CONTINUE C RETURN END C C********************************************************************** C DOUBLE PRECISION FUNCTION DUCPLL(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 DUCPLX. C DOUBLE PRECISION NORMA COMMON /DUCPLX/NORMA C C INTEGER I, J C IF ((I.LT.1).OR.(J.LT.1)) THEN DUCPLL = 0.0D0 ELSE IF (I.EQ.J) THEN DUCPLL = NORMA ELSE DUCPLL = 0.0D0 END IF C RETURN END C C********************************************************************** C DOUBLE PRECISION FUNCTION DUCPLO (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 DUCPLO = 1.0D0 C RETURN END C C********************************************************************** C DOUBLE PRECISION FUNCTION DUCPLU(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 DUCPLU = 0.0D0 ELSE DUCPLU = 0.0D0 END IF C RETURN END C C**********************************************************************