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 ZUCPL (NDIM,NLEN,NLIM,MAXPQ,MAXVW,M,MVEC,NORMS,ZWK, $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 "zucpl.doc" in C the current directory. C C External routines used: C double precision dlamch(ch) C LAPACK routine, computes machine-related constants. C double precision dznrm2(n,x,incx) C BLAS-1 routine, computes the 2-norm of x. C subroutine zaxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C double precision zdotu(n,x,incx,y,incy) C BLAS-1 routine, computes y^H * x. C subroutine zqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine zqrsl(x,ldx,n,k,qraux,y,qy,qty,b,rsd,xb,job,info) C LINPACK routine, applies the QR factorization of x. C subroutine zrandn(n,x,seed) C Library routine, fill x with random numbers. C subroutine zrotg(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 zsvdc(x,ldx,n,p,s,e,u,ldu,v,ldv,work,job,info) C LINPACK routine, computes the SVD of x. C subroutine zucpl1(ndim,nlen,m,n,k,kstar,l,lstar,mk,mkstar,nl, C nlstar,vf,ierr,adjust,norm1,norm2,zwk,dwk,iwk,idx,vecs) C Low-level routine, rebuilds vectors PQ in CPL. C subroutine zucpl2(ndim,nlen,m,n,k,kstar,l,lstar,mk,mkstar,nl, C nlstar,vf,ierr,adjust,norm1,norm2,zwk,dwk,iwk,idx,vecs) C Low-level routine, rebuilds vectors VW in CPL. C double precision function zucpll(i,n) C User-supplied routine, computes inner recurrence coefficients. C double precision zucplom(n) C User-supplied routine, specifies the QMR scaling factors. C double precision function zucplu(i,n) C User-supplied routine, computes inner recurrence coefficients. C C Noel M. Nachtigal C March 1, 1992 C C********************************************************************** C INTRINSIC CDABS, DABS, DBLE, DCMPLX, DCONJG, DMAX1, DREAL, DSQRT INTRINSIC MAX0, MIN0, MOD EXTERNAL DLAMCH, DZNRM2, ZAXPBY, ZDOTU, ZQRDC, ZQRSL, ZRANDN EXTERNAL ZROTG, ZSVDC EXTERNAL ZUCPL1, ZUCPL2, ZUCPLL, ZUCPLO, ZUCPLU DOUBLE COMPLEX ZDOTU, ZUCPLL, ZUCPLU DOUBLE PRECISION DLAMCH, DZNRM2, ZUCPLO C INTEGER INFO(4), M, MAXPQ, MAXVW, MVEC, NDIM, NLEN, NLIM INTEGER IDX(6,NLIM+2), IWK(M,13) DOUBLE COMPLEX VECS(NDIM,5*MVEC+4-1), ZWK(M,8*M+7) DOUBLE PRECISION DWK(M,11), NORMS(2), TOL C C Common block variables. C C C Common block ZUCPLX. C DOUBLE PRECISION NORMA COMMON /ZUCPLX/NORMA C C Miscellaneous parameters. C DOUBLE COMPLEX ZONE, ZZERO PARAMETER (ZONE = (1.0D0,0.0D0),ZZERO = (0.0D0,0.0D0)) 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 COMPLEX ZTMP, ZTMP1, ZTMP2 DOUBLE PRECISION DTMP, DTMP1, DTMP2, DTMP3, DTMP4 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 ZAXPBY (NLEN,VECS(1,IDX(2,1)),ZONE,VECS(1,2),ZZERO,VECS(1,$IDX(2,1))) CALL ZAXPBY (NLEN,VECS(1,1),ZZERO,VECS(1,1),ZZERO,VECS(1,1)) R0 = DZNRM2(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 ZRANDN (NLEN,VECS(1,3),1) CALL ZAXPBY (NLEN,VECS(1,IDX(3,1)),ZONE,VECS(1,3),ZZERO,VECS(1, $IDX(3,1))) C C Scale the first pair of Lanczos vectors and check for invariant C subspaces. C DTMP1 = R0 DTMP2 = DZNRM2(NLEN,VECS(1,IDX(3,1)),1) IF (DTMP1.LT.TNRM) IERR = IERR + 16 IF (DTMP2.LT.TNRM) IERR = IERR + 32 IF (IERR.NE.0) GO TO 750 DWK(IDX(1,1),3) = DONE ZWK(IDX(1,1),IDX(1,1)) = ZDOTU(NLEN,VECS(1,IDX(2,1)),1,VECS(1,$IDX(3,1)),1) / ( DTMP1 * DTMP2 ) IF ((DTMP1.GE.TMAX).OR.(DTMP1.LE.TMIN)) THEN ZTMP = DCMPLX(DONE / DTMP1,DZERO) CALL ZAXPBY (NLEN,VECS(1,IDX(2,1)),ZTMP,VECS(1,IDX(2,1)),ZZERO, $VECS(1,IDX(2,1))) DTMP1 = DONE END IF IF ((DTMP2.GE.TMAX).OR.(DTMP2.LE.TMIN)) THEN ZTMP = DCMPLX(DONE / DTMP2,DZERO) CALL ZAXPBY (NLEN,VECS(1,IDX(3,1)),ZTMP,VECS(1,IDX(3,1)),ZZERO,$VECS(1,IDX(3,1))) DTMP2 = DONE END IF DWK(IDX(1,1),4) = DONE / DTMP1 DWK(IDX(1,1),5) = 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),1) = ZUCPLO(1) ZWK(IDX(1,1),8*M+4) = DWK(IDX(1,1),1) * R0 MAXOMG = DONE / DWK(IDX(1,1),1) 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),11) = DZERO KBLKSZ = N - MK IBUILT = .FALSE. IERR = 0 C C Clear current column of U. C DO 40 I = 1, M ZWK(IDX(1,I),6*M+IDX(1,N)) = ZZERO 40 CONTINUE ZWK(IDX(1,N),6*M+IDX(1,N)) = ZONE 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),9) = DONE DWK(IDX(1,1),10) = DONE DWK(IDX(1,1),7) = DWK(IDX(1,1),4) DWK(IDX(1,1),8) = DWK(IDX(1,1),5) INNER = .FALSE. CALL ZAXPBY (NLEN,VECS(1,IDX(4,1)),ZONE,VECS(1,IDX(2,1)),ZZERO, $VECS(1,IDX(4,1))) CALL ZAXPBY (NLEN,VECS(1,IDX(5,1)),ZONE,VECS(1,IDX(3,1)),ZZERO,$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 ZTMP = ZZERO DO 50 J = MAX0(NLSTAR,IWK(IDX(1,I),3)), N-1 ZTMP = ZTMP + ZWK(IDX(1,I),IDX(1,J)) * ZWK(IDX(1,J),2*M+ $IDX(1,N-1)) 50 CONTINUE ZTMP = ( ZWK(IDX(1,I),M+IDX(1,N-1)) - ZTMP ) / ZWK(IDX(1,N)$,2*M+IDX(1,N-1)) ZWK(IDX(1,I),IDX(1,N)) = ZTMP ZWK(IDX(1,N),IDX(1,I)) = ZTMP * DWK(IDX(1,N),3) / DWK(IDX(1,I), $3) 60 CONTINUE C C Step 2: C Compute k^\star. C I = KSTAR DO 70 J = I+1, K IF (IWK(IDX(1,J+1),1).LE.NL-1) KSTAR = J 70 CONTINUE MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Check memory allocation (MKSTAR.GE.PQBEG). Update the range of C valid indices: the new vectors p_n and q_n will overwrite the C vectors at location N-MVEC. C IF (MKSTAR.LT.PQBEG) THEN IERR = 64 GO TO 750 END IF PQBEG = MAX0(PQBEG,N-MVEC+1) C C Step 3: C Compute F_{n,1:n-1} = F_{n,m_{k^\star}:n-1}. C DO 90 I = MKSTAR, N-1 ZTMP = ZZERO DO 80 J = MAX0(NL,IWK(IDX(1,I),5)), I+1 ZTMP = ZTMP + ZWK(IDX(1,N),IDX(1,J)) * ZWK(IDX(1,J),2*M+$IDX(1,I)) 80 CONTINUE ZWK(IDX(1,N),M+IDX(1,I)) = ZTMP DTMP = CDABS(ZTMP) / DWK(IDX(1,I),9) 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 = CDABS(ZWK(IDX(1,MK),5*M+IDX(1,MK))).EQ.DZERO ELSE DO 110 I = MK, N-1 DO 100 J = MK, N-1 ZWK(I-MK+1,4*M+J-MK+1) = ZWK(IDX(1,I),5*M+IDX(1,J)) 100 CONTINUE 110 CONTINUE CALL ZSVDC (ZWK(1,4*M+1),M,KBLKSZ,KBLKSZ,ZWK(1,8*M+1),ZWK(1,8*M $+3),ZZERO,0,ZZERO,0,ZWK(1,8*M+2),0,I) IF (I.NE.0) THEN IERR = -I GO TO 750 END IF DTMP = DZERO IF (DREAL(ZWK(1,8*M+1)).NE.DZERO) THEN DTMP = DABS( DREAL(ZWK(KBLKSZ,8*M+1)) / DREAL(ZWK(1,8*M+1))) END IF INNER = DTMP.LT.(DBLE(NLEN) * DLAMCH('E')) END IF KEND = K IF (INNER) KEND = K - 1 C C Step 5: C Compute U_{m_i:m_{i+1}-1,n}, for i = k^\star, ..., kend, kend = k C or kend = k-1. C IF (KEND.EQ.K) IWK(IDX(1,K+1+1),1) = N DO 150 I = KSTAR, KEND MI = IWK(IDX(1,I+1),1) MIP1 = IWK(IDX(1,I+1+1),1) IBLKSZ = MIP1 - MI IF (IBLKSZ.EQ.1) THEN ZWK(IDX(1,MI),6*M+IDX(1,N)) = ZWK(IDX(1,N),M+IDX(1,MI)) /$ZWK(IDX(1,MI),5*M+IDX(1,MI)) * DWK(IDX(1,MI),3) / DWK(IDX(1,N),3) ELSE DO 130 J = MI, MIP1-1 ZWK(J-MI+1,8*M+1) = ZWK(IDX(1,N),M+IDX(1,J)) * DWK(IDX(1, $J),3) / DWK(IDX(1,N),3) DO 120 IJ = MI, MIP1-1 ZWK(J-MI+1,4*M+IJ-MI+1) = ZWK(IDX(1,J),5*M+IDX(1,IJ)) 120 CONTINUE 130 CONTINUE CALL ZQRDC (ZWK(1,4*M+1),M,IBLKSZ,IBLKSZ,ZWK(1,8*M+3),0,$ZZERO,0) CALL ZQRSL (ZWK(1,4*M+1),M,IBLKSZ,IBLKSZ,ZWK(1,8*M+3),ZWK(1, $8*M+1),$ ZZERO,ZWK(1,8*M+1),ZWK(1,8*M+2),ZWK(1,8*M+1), $ZZERO,100,J) DO 140 J = MI, MIP1-1 ZWK(IDX(1,J),6*M+IDX(1,N)) = ZWK(J-MI+1,8*M+2) 140 CONTINUE END IF 150 CONTINUE C C Step 6: C Build the part common to both inner and regular vectors. C DWK(IDX(1,N),7) = DWK(IDX(1,N),4) DWK(IDX(1,N),8) = DWK(IDX(1,N),5) CALL ZAXPBY (NLEN,VECS(1,3),ZONE,VECS(1,IDX(2,N)),ZZERO,VECS(1,3)) CALL ZAXPBY (NLEN,VECS(1,4),ZONE,VECS(1,IDX(3,N)),ZZERO,VECS(1,4)) DO 160 I = MKSTAR, MK-1 ZTMP = ZWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),7) / DWK(IDX(1$,N),7) CALL ZAXPBY (NLEN,VECS(1,3),ZONE,VECS(1,3),-ZTMP,VECS(1,IDX(4,I $))) ZTMP = ZWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8) / DWK(IDX(1$,N),8) * DWK(IDX(1,N),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,4),ZONE,VECS(1,4),-ZTMP,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, $' (PQ) as regular' END IF END IF C C If building an inner vector, set the coefficients U_{m_k:n-1,n}. C IF (INNER) THEN DO 170 I = MK, N-1 ZWK(IDX(1,I),6*M+IDX(1,N)) = ZUCPLU(I,N) 170 CONTINUE END IF ELSE C C PQ is not being rerun, determine whether it is being rebuilt. C IF ((VF.NE.0).AND.(N.LE.NMAX)) THEN WRITE (VF,'(A15,I8)') 'Rebuilding P&Q:', N END IF C C If E_k is singular, build an inner vector. C IF (INNER) THEN IF (VF.NE.0) WRITE (VF,'(A31)')$'... 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 ZAXPBY (NLEN,VECS(1,IDX(4,N)),ZONE,VECS(1,3),ZZERO,VECS(1, $IDX(4,N))) CALL ZAXPBY (NLEN,VECS(1,IDX(5,N)),ZONE,VECS(1,4),ZZERO,VECS(1,$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 ZTMP = ZWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),7) / DWK(IDX(1 $,N),7) CALL ZAXPBY (NLEN,VECS(1,IDX(4,N)),ZONE,VECS(1,IDX(4,N)),-ZTMP,$VECS(1,IDX(4,I))) ZTMP = ZWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8) / DWK(IDX(1 $,N),8) * DWK(IDX(1,N),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,IDX(5,N)),ZONE,VECS(1,IDX(5,N)),-ZTMP,$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 ZWK(IDX(1,N),5*M+IDX(1,N)) = ZDOTU(NLEN,VECS(1,IDX(5,N)),1,VECS(1, $IDX(2,NP1)),1) * DWK(IDX(1,N),7) * DWK(IDX(1,N),8) C C Step 9: C Compute the last column of E_k^{-1}. C IF (KBLKSZ.EQ.1) THEN ZWK(IDX(1,MK),7*M+IDX(1,K)) = ZONE / ZWK(IDX(1,MK),5*M+IDX(1,MK$)) ELSE DO 210 I = 1, KBLKSZ-1 ZWK(I,8*M+1) = ZZERO 210 CONTINUE ZWK(KBLKSZ,8*M+1) = ZONE CALL ZQRSL (ZWK(1,4*M+1),M,KBLKSZ,KBLKSZ,ZWK(1,8*M+3),ZWK(1,8*M $+1),$ ZZERO,ZWK(1,8*M+1),ZWK(1,8*M+2),ZWK(1,8*M+1),ZZERO, $100,J) DO 220 I = MK, N-1 ZWK(IDX(1,I),7*M+IDX(1,K)) = ZWK(I-MK+1,8*M+2) 220 CONTINUE END IF C C Check for zero norms. C DWK(IDX(1,N),9) = DZNRM2(NLEN,VECS(1,IDX(4,N)),1) * DWK(IDX(1,N),7$) DWK(IDX(1,N),10) = DZNRM2(NLEN,VECS(1,IDX(5,N)),1) * DWK(IDX(1,N), $8) IF (DWK(IDX(1,N),9).EQ.DZERO) GO TO 360 IF (DWK(IDX(1,N),10).EQ.DZERO) GO TO 360 C C If PQ is being rerun, then skip to the end. C IF (RERUN) GO TO 360 C C Step 7: C Build the full column G_{m_{k-1}:n-1,n-1} and compute its norm. C DTMP1 = DZERO DTMP2 = DZERO DO 240 I = IWK(IDX(1,K-1+1),1), N-1 ZTMP = DZERO DO 230 J = MAX0(I,NLSTAR), N ZTMP = ZTMP + ZWK(IDX(1,I),6*M+IDX(1,J)) * ZWK(IDX(1,J),2*M+$IDX(1,N-1)) 230 CONTINUE DTMP = CDABS(ZTMP) DTMP1 = DTMP1 + DTMP * DWK(IDX(1,I),9) DTMP2 = DTMP2 + DTMP * DWK(IDX(1,I),10) / DWK(IDX(1,I),3) 240 CONTINUE DTMP1 = DTMP1 / DWK(IDX(1,N-1),9) DTMP2 = DTMP2 * DWK(IDX(1,N-1),3) / DWK(IDX(1,N-1),10) 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 ZTMP = ZWK(IDX(1,N),5*M+IDX(1,N)) * ZWK(IDX(1,N),2*M+IDX(1,N-1)) $* DWK(IDX(1,N-1),3) / DWK(IDX(1,N),3) DO 250 I = MK, N-1 DTMP = CDABS(ZTMP * ZWK(IDX(1,I),7*M+IDX(1,K))) DTMP3 = DTMP3 + DTMP * DWK(IDX(1,I),9) DTMP4 = DTMP4 + DTMP * DWK(IDX(1,I),10) / DWK(IDX(1,I),3) 250 CONTINUE DTMP3 = DTMP3 / DWK(IDX(1,N),9) DTMP4 = DTMP4 * DWK(IDX(1,N),3) / DWK(IDX(1,N),10) C C Steps 7 and 9: C Check G_{m_{k-1}:n-1,n-1} and G_{m_k:n-1,n}. C DTMP = DMAX1(DTMP1,DTMP2,DTMP3,DTMP4) INNER = DTMP.GT.NORM1 IF (.NOT.INNER) GO TO 360 DWK(IDX(1,N),11) = DTMP C C If G_{m_{k-1}:n-1,n-1} is bad, build inner vectors. C IF (DMAX1(DTMP1,DTMP2).GT.NORM1) GO TO 300 C C If G_{m_k:n-1,n} is bad, check the inner vectors. C This only applies if m_k > 1. C IF (MK.LE.1) GO TO 300 C C Build the inner vectors to compute the 2nd term at the next step. C Get the coefficients U_{m_k:n-1,n} in a temporary location. C Build the inner vectors, their norms are needed. C IBUILT = .TRUE. DO 260 I = MK, N-1 ZWK(IDX(1,I),6*M+IDX(1,NP1)) = ZUCPLU(I,N) ZTMP = ZWK(IDX(1,I),6*M+IDX(1,NP1)) * DWK(IDX(1,I),7) /$DWK(IDX(1,N),7) CALL ZAXPBY (NLEN,VECS(1,3),ZONE,VECS(1,3),-ZTMP,VECS(1,IDX(4,I $))) ZTMP = ZWK(IDX(1,I),6*M+IDX(1,NP1)) * DWK(IDX(1,I),8) /$DWK(IDX(1,N),8) * DWK(IDX(1,N),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,4),ZONE,VECS(1,4),-ZTMP,VECS(1,IDX(5,I $))) 260 CONTINUE DWK(IDX(1,NP1),9) = DZNRM2(NLEN,VECS(1,3),1) * DWK(IDX(1,N),7) DWK(IDX(1,NP1),10) = DZNRM2(NLEN,VECS(1,4),1) * DWK(IDX(1,N),8) IF (DWK(IDX(1,NP1),9).EQ.DZERO) GO TO 310 IF (DWK(IDX(1,NP1),10).EQ.DZERO) GO TO 310 C C Build the 2nd term for the next step, G_{m_{k-1}:m_k-1,n}, inner. C Compute the norm of G_{m_{k-1}:m_k-1,n}. C ZTMP = ZZERO DO 270 J = MKSTAR, MK-1 ZTMP = ZTMP + ZWK(IDX(1,J),6*M+IDX(1,N)) * ZWK(IDX(1,J),5*M+$IDX(1,MK)) / DWK(IDX(1,J),3) 270 CONTINUE DO 280 J = MK, N-1 ZTMP = ZTMP + ZWK(IDX(1,J),6*M+IDX(1,NP1)) * ZWK(IDX(1,J),5*M+ $IDX(1,MK)) / DWK(IDX(1,J),3) 280 CONTINUE DTMP1 = DZERO DTMP2 = DZERO DTMP3 = DMAX1(DTMP3,DTMP4) ZTMP = ZWK(IDX(1,N),M+IDX(1,MK)) - ZTMP * DWK(IDX(1,N),3) ZTMP = ZTMP * ZWK(IDX(1,MK),2*M+IDX(1,MK-1)) * DWK(IDX(1,MK-1),3)$ / DWK(IDX(1,MK),3) DO 290 I = IWK(IDX(1,K-1+1),1), MK-1 DTMP = CDABS(ZTMP * ZWK(IDX(1,I),7*M+IDX(1,K-1))) DTMP1 = DTMP1 + DTMP * DWK(IDX(1,I),9) DTMP2 = DTMP2 + DTMP * DWK(IDX(1,I),10) / DWK(IDX(1,I),3) 290 CONTINUE DTMP1 = DTMP1 / DWK(IDX(1,NP1),9) DTMP2 = DTMP2 * DWK(IDX(1,N),3) / DWK(IDX(1,NP1),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 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 ZUCPL1 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL,NLSTAR, $VF,IERR,ADJUST,NORM1,NORM2,ZWK,DWK,IWK,IDX,VECS) IF (IERR.NE.0) GO TO 750 INNER = .FALSE. KBLKSZ = N - MK RERUN = .TRUE. NP1 = N + 1 GO TO 190 END IF C C The temporary vectors contain either just partial inner vectors, C or the completed ones, depending on whether IBUILT is TRUE or C FALSE. In either case, replace the regular vectors. C 310 CALL ZAXPBY (NLEN,VECS(1,IDX(4,N)),ZONE,VECS(1,3),ZZERO,VECS(1,$IDX(4,N))) CALL ZAXPBY (NLEN,VECS(1,IDX(5,N)),ZONE,VECS(1,4),ZZERO,VECS(1, $IDX(5,N))) C C Step 11: C Get the coefficients U_{m_k:n-1,n} and build inner vectors. C IF (IBUILT) THEN DO 320 I = MK, N-1 ZWK(IDX(1,I),6*M+IDX(1,N)) = ZWK(IDX(1,I),6*M+IDX(1,NP1)) 320 CONTINUE DWK(IDX(1,N),9) = DWK(IDX(1,NP1),9) DWK(IDX(1,N),10) = DWK(IDX(1,NP1),10) ELSE DO 330 I = MK, N-1 ZWK(IDX(1,I),6*M+IDX(1,N)) = ZUCPLU(I,N) ZTMP = ZWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),7) /$DWK(IDX(1,N),7) CALL ZAXPBY (NLEN,VECS(1,IDX(4,N)),ZONE,VECS(1,IDX(4,N)),- $ZTMP,VECS(1,IDX(4,I))) ZTMP = ZWK(IDX(1,I),6*M+IDX(1,N)) * DWK(IDX(1,I),8) /$DWK(IDX(1,N),8) * DWK(IDX(1,N),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,IDX(5,N)),ZONE,VECS(1,IDX(5,N)),- $ZTMP,VECS(1,IDX(5,I))) 330 CONTINUE DWK(IDX(1,N),9) = DZNRM2(NLEN,VECS(1,IDX(4,N)),1) * DWK(IDX(1,N$),7) DWK(IDX(1,N),10) = DZNRM2(NLEN,VECS(1,IDX(5,N)),1) * DWK(IDX(1, $N),8) END IF C C Step 11: C Compute A p_n and q_n^T A p_n. C Have the caller carry out AXB, then return here. C CALL AXB (VECS(1,IDX(4,N)),VECS(1,IDX(2,NP1))) C 340 INFO(2) = 1 INFO(3) = IDX(4,N) INFO(4) = IDX(2,NP1) RETLBL = 350 RETURN 350 ZWK(IDX(1,N),5*M+IDX(1,N)) = ZDOTU(NLEN,VECS(1,IDX(5,N)),1,VECS(1,$IDX(2,NP1)),1) * DWK(IDX(1,N),7) * DWK(IDX(1,N),8) 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),9).LT.TNRM) IERR = IERR + 16 IF (DWK(IDX(1,N),10).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 = DZNRM2(NLEN,VECS(1,IDX(2,NP1)),1) * DWK(IDX(1,N),7) / $DWK(IDX(1,N),9) DTMP2 = DZNRM2(NLEN,VECS(1,IDX(3,NP1)),1) * DWK(IDX(1,N),8) /$DWK(IDX(1,N),10) NORMA = DMAX1(DTMP1,DTMP2,NORMA) END IF DTMP = CDABS(ZWK(IDX(1,N),5*M+IDX(1,N))) / ( DWK(IDX(1,N),9) * $DWK(IDX(1,N),10) ) NORMA = DMAX1(DTMP,NORMA) NORM1 = DMAX1(ADJUST*NORMA,NORM1) NORM2 = DMAX1(ADJUST*NORMA,NORM2) C C Build v_{n+1} and w_{n+1}. C There are three cases: C N > NMAX : the VW sequence is being built new. C N >= NLMAX : the VW sequence is being rebuilt, build as if new. C N < NLMAX : the PQ sequence is being rebuilt, just rerun VW. C The first two are identical in terms of code. C IWK(IDX(1,N),12) = K IWK(IDX(1,N),13) = KSTAR DWK(IDX(1,N),6) = DZERO IBUILT = .FALSE. LBLKSZ = N - NL + 1 C C Clear current column of L. C DO 380 I = 1, M ZWK(IDX(1,I),2*M+IDX(1,N)) = ZZERO 380 CONTINUE C C Step 14: C Update E^{(n-1)} to E^{(n)}. C DO 400 I = MK, N-1 ZTMP = ZZERO DO 390 J = MAX0(MKSTAR,IWK(IDX(1,I),4)), N-1 ZTMP = ZTMP + ZWK(IDX(1,J),6*M+IDX(1,N)) * ZWK(IDX(1,J),5*M+$IDX(1,I)) / DWK(IDX(1,J),3) 390 CONTINUE ZTMP = ZWK(IDX(1,N),M+IDX(1,I)) - ZTMP * DWK(IDX(1,N),3) ZWK(IDX(1,N),5*M+IDX(1,I)) = ZTMP ZWK(IDX(1,I),5*M+IDX(1,N)) = ZTMP * DWK(IDX(1,I),3) / DWK(IDX(1 $,N),3) DTMP1 = CDABS(ZWK(IDX(1,N),5*M+IDX(1,I))) / (DWK(IDX(1,N),10$) * DWK(IDX(1,I),9)) DTMP2 = CDABS(ZWK(IDX(1,I),5*M+IDX(1,N))) / (DWK(IDX(1,I),10 $) * DWK(IDX(1,N),9)) NORMA = DMAX1(NORMA,DTMP1,DTMP2) NORM1 = DMAX1(ADJUST*NORMA,NORM1) NORM2 = DMAX1(ADJUST*NORMA,NORM2) 400 CONTINUE C C Step 15: C Compute l^\star. C I = LSTAR DO 410 J = I+1, L IF (IWK(IDX(1,J+1),2).LE.MK) LSTAR = J 410 CONTINUE NLSTAR = IWK(IDX(1,LSTAR+1),2) C C Check memory allocation (NLSTAR.GE.VWBEG). Update the range of C valid indices: the new vectors v_{n+1} and w_{n+1} will overwrite C the vectors at location N+1-MVEC. C IF (NLSTAR.LT.VWBEG) THEN IERR = 64 GO TO 750 END IF VWBEG = MAX0(VWBEG,NP1-MVEC+1) C C Step 16: C Compute F_{1:n,n} = F_{n_{l^\star}:n,n}. C DO 430 I = NLSTAR, N ZTMP = ZZERO DO 420 J = MAX0(MK,IWK(IDX(1,I),6)), I ZTMP = ZTMP + ZWK(IDX(1,J),6*M+IDX(1,I)) * ZWK(IDX(1,J),5*M+$IDX(1,N)) / DWK(IDX(1,J),3) 420 CONTINUE ZWK(IDX(1,I),M+IDX(1,N)) = ZTMP * DWK(IDX(1,I),3) DTMP = CDABS(ZTMP) / DWK(IDX(1,N),9) 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 = CDABS(ZWK(IDX(1,NL),IDX(1,NL))).EQ.DZERO ELSE DO 450 I = NL, N DO 440 J = NL, N ZWK(I-NL+1,4*M+J-NL+1) = ZWK(IDX(1,I),IDX(1,J)) 440 CONTINUE 450 CONTINUE CALL ZSVDC (ZWK(1,4*M+1),M,LBLKSZ,LBLKSZ,ZWK(1,8*M+1),ZWK(1,8*M $+3),ZZERO,0,ZZERO,0,ZWK(1,8*M+2),0,I) IF (I.NE.0) THEN IERR = -I GO TO 750 END IF DTMP = DZERO IF (DREAL(ZWK(1,8*M+1)).NE.DZERO) THEN DTMP = DABS( DREAL(ZWK(LBLKSZ,8*M+1)) / DREAL(ZWK(1,8*M+1))) END IF INNER = DTMP.LT.(DBLE(NLEN) * DLAMCH('E')) END IF LEND = L IF (INNER) LEND = L - 1 C C Step 18: C Compute L_{n_i:n_{i+1}-1,n}, for i = l^\star, ..., lend, lend = l C or lend = l-1. C IF (LEND.EQ.L) IWK(IDX(1,L+1+1),2) = NP1 DO 490 I = LSTAR, LEND NI = IWK(IDX(1,I+1),2) NIP1 = IWK(IDX(1,I+1+1),2) IBLKSZ = NIP1 - NI IF (IBLKSZ.EQ.1) THEN ZWK(IDX(1,NI),2*M+IDX(1,N)) = ZWK(IDX(1,NI),M+IDX(1,N)) /$ZWK(IDX(1,NI),IDX(1,NI)) ELSE DO 470 J = NI, NIP1-1 ZWK(J-NI+1,8*M+1) = ZWK(IDX(1,J),M+IDX(1,N)) DO 460 IJ = NI, NIP1-1 ZWK(J-NI+1,4*M+IJ-NI+1) = ZWK(IDX(1,J),IDX(1,IJ)) 460 CONTINUE 470 CONTINUE CALL ZQRDC (ZWK(1,4*M+1),M,IBLKSZ,IBLKSZ,ZWK(1,8*M+3),0, $ZZERO,0) CALL ZQRSL (ZWK(1,4*M+1),M,IBLKSZ,IBLKSZ,ZWK(1,8*M+3),ZWK(1,$8*M+1), $ZZERO,ZWK(1,8*M+1),ZWK(1,8*M+2),ZWK(1,8*M+1),$ZZERO,100,J) DO 480 J = NI, NIP1-1 ZWK(IDX(1,J),2*M+IDX(1,N)) = ZWK(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 ZWK(IDX(1,NP1),2*M+IDX(1,N)) = ZZERO DO 500 I = NLSTAR, NL-1 ZTMP = ZWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),4) / DWK(IDX(1 $,N),7) CALL ZAXPBY (NLEN,VECS(1,IDX(2,NP1)),ZONE,VECS(1,IDX(2,NP1)),-$ZTMP,VECS(1,IDX(2,I))) ZTMP = ZWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),5) / DWK(IDX(1 $,N),8) * DWK(IDX(1,N),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,IDX(3,NP1)),ZONE,VECS(1,IDX(3,NP1)),-$ZTMP,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 ZWK(IDX(1,I),2*M+IDX(1,N)) = ZUCPLL(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)') $'... moment matrix D is singular' GO TO 630 END IF END IF C C Either D_l is nonsingular, or VW is being rerun. For the latter, C just finish building the vectors. For the former, the check in C Step 20 (H_{n_{l-1}:n,n}) could be done. However, the look-ahead C strategy requires the smaller of the checks in Step 20 and Step C 22, in case the norm estimates need updating. Hence, Step 20 and C Step 22 are switched, and their checks are later performed C together. C C Save the common part of the vectors. C CALL ZAXPBY (NLEN,VECS(1,3),ZONE,VECS(1,IDX(2,NP1)),ZZERO,VECS(1,3$)) CALL ZAXPBY (NLEN,VECS(1,4),ZONE,VECS(1,IDX(3,NP1)),ZZERO,VECS(1,4 $)) C C Step 21. C Build regular vectors. C DO 520 I = NL, N ZTMP = ZWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),4) / DWK(IDX(1$,N),7) CALL ZAXPBY (NLEN,VECS(1,IDX(2,NP1)),ZONE,VECS(1,IDX(2,NP1)),- $ZTMP,VECS(1,IDX(2,I))) ZTMP = ZWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),5) / DWK(IDX(1$,N),8) * DWK(IDX(1,N),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,IDX(3,NP1)),ZONE,VECS(1,IDX(3,NP1)),- $ZTMP,VECS(1,IDX(3,I))) 520 CONTINUE C C Compute scale factors for the new vectors. C 530 DTMP3 = DZNRM2(NLEN,VECS(1,IDX(2,NP1)),1) DTMP4 = DZNRM2(NLEN,VECS(1,IDX(3,NP1)),1) DTMP1 = DWK(IDX(1,N),7) * DTMP3 DTMP2 = DWK(IDX(1,N),8) * DTMP4 ZWK(IDX(1,NP1),2*M+IDX(1,N)) = DCMPLX(DTMP1,DZERO) IF (DTMP1.LT.TNRM) IERR = IERR + 16 IF (DTMP2.LT.TNRM) IERR = IERR + 32 IF (IERR.NE.0) GO TO 670 DWK(IDX(1,NP1),3) = DWK(IDX(1,N),3) * DTMP1 / DTMP2 ZWK(IDX(1,NP1),IDX(1,NP1)) = ZDOTU(NLEN,VECS(1,IDX(3,NP1)),1,$VECS(1,IDX(2,NP1)),1) / ( DTMP3 * DTMP4 ) IF ((DTMP3.GE.TMAX).OR.(DTMP3.LE.TMIN)) THEN ZTMP = DCMPLX(DONE / DTMP3,DZERO) CALL ZAXPBY (NLEN,VECS(1,IDX(2,NP1)),ZTMP,VECS(1,IDX(2,NP1)), $ZZERO,VECS(1,IDX(2,NP1))) DTMP3 = DONE END IF IF ((DTMP4.GE.TMAX).OR.(DTMP4.LE.TMIN)) THEN ZTMP = DCMPLX(DONE / DTMP4,DZERO) CALL ZAXPBY (NLEN,VECS(1,IDX(3,NP1)),ZTMP,VECS(1,IDX(3,NP1)),$ZZERO,VECS(1,IDX(3,NP1))) DTMP4 = DONE END IF DWK(IDX(1,NP1),4) = DONE / DTMP3 DWK(IDX(1,NP1),5) = DONE / DTMP4 C C Step 22: C Compute the last column of D_l^{-1}. C IF (LBLKSZ.EQ.1) THEN ZWK(IDX(1,NL),3*M+IDX(1,L)) = ZONE / ZWK(IDX(1,NL),IDX(1,NL)) ELSE DO 540 I = 1, LBLKSZ-1 ZWK(I,8*M+1) = ZZERO 540 CONTINUE ZWK(LBLKSZ,8*M+1) = ZONE CALL ZQRSL (ZWK(1,4*M+1),M,LBLKSZ,LBLKSZ,ZWK(1,8*M+3),ZWK(1,8*M $+1),$ ZZERO,ZWK(1,8*M+1),ZWK(1,8*M+2),ZWK(1,8*M+1),ZZERO, $100,J) DO 550 I = NL, N ZWK(IDX(1,I),3*M+IDX(1,L)) = ZWK(I-NL+1,8*M+2) 550 CONTINUE END IF C C If VW is being rerun, then skip to the end. C IF (RERUN) GO TO 670 C C Step 20: C Build the full column H_{n_{l-1}:n,n} and compute its norm. C DTMP1 = DZERO DTMP2 = DZERO DO 570 I = IWK(IDX(1,L-1+1),2), N ZTMP = ZZERO DO 560 J = MAX0(1,I-1), N ZTMP = ZTMP + ZWK(IDX(1,I),2*M+IDX(1,J)) * ZWK(IDX(1,J),6*M+$IDX(1,N)) 560 CONTINUE DTMP = CDABS(ZTMP) DTMP1 = DTMP1 + DTMP DTMP2 = DTMP2 + DTMP / DWK(IDX(1,I),3) 570 CONTINUE DTMP2 = DTMP2 * DWK(IDX(1,N),3) 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 ZTMP = ZWK(IDX(1,NP1),IDX(1,NP1)) * ZWK(IDX(1,NP1),2*M+IDX(1,N)) $* DWK(IDX(1,N),3) / DWK(IDX(1,NP1),3) DO 580 I = NL, N DTMP = CDABS(ZTMP * ZWK(IDX(1,I),3*M+IDX(1,L))) DTMP3 = DTMP3 + DTMP DTMP4 = DTMP4 + DTMP / DWK(IDX(1,I),3) 580 CONTINUE DTMP4 = DTMP4 * DWK(IDX(1,NP1),3) C C Steps 20 and 22: C Check H_{n_{l-1}:n,n} and H_{n_l:n-1,n+1}. C DTMP = DMAX1(DTMP1,DTMP2,DTMP3,DTMP4) INNER = DTMP.GT.NORM2 IF (.NOT.INNER) GO TO 670 DWK(IDX(1,N),6) = DTMP C C If H_{n_{l-1}:n,n} is bad, build inner vectors. C IF (DMAX1(DTMP1,DTMP2).GT.NORM2) GO TO 630 C C If H_{n_l:n-1,n+1} is bad, check the inner vectors. C This only applies if n_l > 1. C IF (NL.LE.1) GO TO 630 C C Build the inner vectors to compute the 2nd term at the next step. C Get the coefficients L_{n_l:n+1,n} in a temporary location. C Build inner vectors in VECS(1,3) and VECS(1,4). C IBUILT = .TRUE. DO 590 I = NL, N ZWK(IDX(1,I),2*M+IDX(1,NP1)) = ZUCPLL(I,N) ZTMP = ZWK(IDX(1,I),2*M+IDX(1,NP1)) * DWK(IDX(1,I),4) /$DWK(IDX(1,N),7) CALL ZAXPBY (NLEN,VECS(1,3),ZONE,VECS(1,3),-ZTMP,VECS(1,IDX(2,I $))) ZTMP = ZWK(IDX(1,I),2*M+IDX(1,NP1)) * DWK(IDX(1,I),5) /$DWK(IDX(1,N),8) * DWK(IDX(1,N),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,4),ZONE,VECS(1,4),-ZTMP,VECS(1,IDX(3,I $))) 590 CONTINUE IDTMP3 = DZNRM2(NLEN,VECS(1,3),1) IDTMP4 = DZNRM2(NLEN,VECS(1,4),1) IDTMP1 = DWK(IDX(1,N),7) * IDTMP3 IDTMP2 = DWK(IDX(1,N),8) * IDTMP4 ZWK(IDX(1,NP1),2*M+IDX(1,NP1)) = DCMPLX(IDTMP1,DZERO) IF (IDTMP1.LT.TNRM) IERR = IERR + 16 IF (IDTMP2.LT.TNRM) IERR = IERR + 32 IF (IERR.NE.0) GO TO 640 C C Build the 2nd term for the next step, H_{n_{l-1}:n_l-1,n+1}, C inner. Compute the norm of H_{n_{l-1}:n_l-1,n+1}. C ZTMP = ZZERO DO 600 J = NLSTAR, NL-1 ZTMP = ZTMP + ZWK(IDX(1,NL),IDX(1,J)) * ZWK(IDX(1,I),2*M+IDX(1,$N)) 600 CONTINUE DO 610 J = NL, N ZTMP = ZTMP + ZWK(IDX(1,NL),IDX(1,J)) * ZWK(IDX(1,I),2*M+IDX(1, $NP1)) 610 CONTINUE DTMP1 = DZERO DTMP2 = DZERO DTMP3 = DMAX1(DTMP3,DTMP4) ZTMP = ( ZWK(IDX(1,NL),M+IDX(1,N)) - ZTMP ) / ZWK(IDX(1,NP1),2*M+$IDX(1,NP1)) ZTMP = ZTMP * ZWK(IDX(1,NL),2*M+IDX(1,NL-1)) * DWK(IDX(1,NL-1),3) $/ DWK(IDX(1,NL),3) DO 620 I = IWK(IDX(1,L-1+1),2), NL-1 DTMP = CDABS(ZWK(IDX(1,I),3*M+IDX(1,L-1)) * ZTMP) DTMP1 = DTMP1 + DTMP DTMP2 = DTMP2 + DTMP / DWK(IDX(1,I),3) 620 CONTINUE DTMP2 = DTMP2 * DWK(IDX(1,N),3) C C Compare the inner and regular versions of the 2nd term at the next C step. Build the vector corresponding to the smaller term. C INNER = DTMP3.GT.DMAX1(DTMP1,DTMP2) IF (.NOT.INNER) GO TO 670 C C Build inner vectors. C Check whether the V&W block has to be forced to close. C 630 IF (VF.NE.0) WRITE (VF,'(A7,I5,A14)') 'Vector ',NP1,$' (VW) is inner' IF (NP1-NL.EQ.MAXVW) THEN CALL ZUCPL2 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL,NLSTAR, $VF,IERR,ADJUST,NORM1,NORM2,ZWK,DWK,IWK,IDX,VECS) IF (IERR.NE.0) GO TO 750 LBLKSZ = N - NL + 1 INNER = .FALSE. RERUN = .TRUE. NP1 = N + 1 GO TO 530 END IF C C The temporary vectors contain either just partial inner vectors, C or the completed ones, depending on whether IBUILT is TRUE or C FALSE. In either case, replace the regular vectors. C 640 CALL ZAXPBY (NLEN,VECS(1,IDX(2,NP1)),ZONE,VECS(1,3),ZZERO,VECS(1,$IDX(2,NP1))) CALL ZAXPBY (NLEN,VECS(1,IDX(3,NP1)),ZONE,VECS(1,4),ZZERO,VECS(1, $IDX(3,NP1))) C C Step 24: C Get the coefficients L_{n_l:n+1,n} and build inner vectors. C IF (IBUILT) THEN DO 650 I = NL, NP1 ZWK(IDX(1,I),2*M+IDX(1,N)) = ZWK(IDX(1,I),2*M+IDX(1,NP1)) 650 CONTINUE DTMP1 = IDTMP1 DTMP2 = IDTMP2 DTMP3 = IDTMP3 DTMP4 = IDTMP4 ELSE DO 660 I = NL, N ZWK(IDX(1,I),2*M+IDX(1,N)) = ZUCPLL(I,N) ZTMP = ZWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),4) /$DWK(IDX(1,N),7) CALL ZAXPBY (NLEN,VECS(1,IDX(2,NP1)),ZONE,VECS(1,IDX(2,NP1)) $,-ZTMP,VECS(1,IDX(2,I))) ZTMP = ZWK(IDX(1,I),2*M+IDX(1,N)) * DWK(IDX(1,I),5) /$DWK(IDX(1,N),8) * DWK(IDX(1,N),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,IDX(3,NP1)),ZONE,VECS(1,IDX(3,NP1)) $,-ZTMP,VECS(1,IDX(3,I))) 660 CONTINUE DTMP3 = DZNRM2(NLEN,VECS(1,IDX(2,NP1)),1) DTMP4 = DZNRM2(NLEN,VECS(1,IDX(3,NP1)),1) DTMP1 = DWK(IDX(1,N),7) * DTMP3 DTMP2 = DWK(IDX(1,N),8) * DTMP4 ZWK(IDX(1,NP1),2*M+IDX(1,N)) = DCMPLX(DTMP1,DZERO) IF (DTMP1.LT.TNRM) IERR = IERR + 16 IF (DTMP2.LT.TNRM) IERR = IERR + 32 END IF IF (IERR.NE.0) GO TO 670 DWK(IDX(1,NP1),3) = DWK(IDX(1,N),3) * DTMP1 / DTMP2 ZWK(IDX(1,NP1),IDX(1,NP1)) = ZDOTU(NLEN,VECS(1,IDX(3,NP1)),1,$VECS(1,IDX(2,NP1)),1) / ( DTMP3 * DTMP4 ) IF ((DTMP3.GE.TMAX).OR.(DTMP3.LE.TMIN)) THEN ZTMP = DCMPLX(DONE / DTMP3,DZERO) CALL ZAXPBY (NLEN,VECS(1,IDX(2,NP1)),ZTMP,VECS(1,IDX(2,NP1)), $ZZERO,VECS(1,IDX(2,NP1))) DTMP3 = DONE END IF IF ((DTMP4.GE.TMAX).OR.(DTMP4.LE.TMIN)) THEN ZTMP = DCMPLX(DONE / DTMP4,DZERO) CALL ZAXPBY (NLEN,VECS(1,IDX(3,NP1)),ZTMP,VECS(1,IDX(3,NP1)),$ZZERO,VECS(1,IDX(3,NP1))) DTMP4 = DONE END IF DWK(IDX(1,NP1),4) = DONE / DTMP3 DWK(IDX(1,NP1),5) = 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),1) = ZUCPLO(NQMR+1) MAXOMG = DMAX1(MAXOMG,DONE/DWK(IDX(1,NQMR+1),1)) C C Compute the starting index IBASE for the column of \hat{R}. C IBASE = MAX0(1,IWK(IDX(1,NQMR),5)-1) ZWK(IDX(1,IBASE),8*M+5) = ZZERO C C Multiply the new column by the previous omegas. C DO 690 J = IWK(IDX(1,NQMR),5), NQMR+1 ZWK(IDX(1,J),8*M+5) = DWK(IDX(1,J),1) * ZWK(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 ZROT: C CALL ZROT (1,ZWK(IDX(1,J-1),8*M+5),1,ZWK(IDX(1,J),8*M+5),1,DWK(IDX(1,J),2),ZWK(IDX(1,J),8*M+6)) C DO 700 J = IBASE+1, NQMR ZTMP1 = ZWK(IDX(1,J),8*M+5) ZTMP2 = ZWK(IDX(1,J-1),8*M+5) ZWK(IDX(1,J-1),8*M+5) = DWK(IDX(1,J),2) * ZTMP2 + ZWK(IDX(1,J),$8*M+6) * ZTMP1 ZWK(IDX(1,J),8*M+5) = DWK(IDX(1,J),2) * ZTMP1 - $DCONJG(ZWK(IDX(1,J),8*M+6)) * ZTMP2 700 CONTINUE C C Compute the rotation for the last element (this also applies it). C CALL ZROTG (ZWK(IDX(1,NQMR),8*M+5),ZWK(IDX(1,NQMR+1),8*M+5),$DWK(IDX(1,NQMR+1),2),ZWK(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 ZWK(IDX(1,NQMR+1),8*M+4) = DZERO C CALL ZROT (1,ZWK(IDX(1,NQMR),8*M+4),1,ZWK(IDX(1,NQMR+1),8*M+4),1,DWK(IDX(1,NQMR+1),2),ZWK(IDX(1,NQMR+1),8*M+6)) C ZWK(IDX(1,NQMR+1),8*M+4) = -DCONJG(ZWK(IDX(1,NQMR+1),8*M+6)) * $ZWK(IDX(1,NQMR),8*M+4) ZWK(IDX(1,NQMR),8*M+4) = DWK(IDX(1,NQMR+1),2) * ZWK(IDX(1,NQMR)$,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 ZTMP2 = ZZERO ZTMP = DWK(IDX(1,NQMR),7) DO 710 J = IBASE, NQMR-1 ZTMP1 = ZWK(IDX(1,J),8*M+5) * ZWK(IDX(1,J),8*M+7) / ZTMP IF (CDABS(ZTMP1).EQ.DZERO) GO TO 710 CALL ZAXPBY (NLEN,VECS(1,IDX(6,NQMR)),ZTMP2,VECS(1,IDX(6,NQMR)) $,-ZTMP1,VECS(1,IDX(6,J))) ZTMP2 = ZONE 710 CONTINUE CALL ZAXPBY (NLEN,VECS(1,IDX(6,NQMR)),ZTMP2,VECS(1,IDX(6,NQMR)),$ZONE,VECS(1,IDX(4,NQMR))) ZTMP = ZTMP / ZWK(IDX(1,NQMR),8*M+5) C C Compute the new QMR iterate, then scale the search direction. C ZTMP1 = ZTMP * ZWK(IDX(1,NQMR),8*M+4) CALL ZAXPBY (NLEN,VECS(1,1),ZONE,VECS(1,1),ZTMP1,VECS(1,IDX(6,NQMR $))) ZWK(IDX(1,NQMR),8*M+7) = ZTMP DTMP = CDABS(ZTMP) IF ((DTMP.GE.TMAX).OR.(DTMP.LE.TMIN)) THEN ZWK(IDX(1,NQMR),8*M+7) = ZONE CALL ZAXPBY (NLEN,VECS(1,IDX(6,NQMR)),ZTMP,VECS(1,IDX(6,NQMR)),$ZZERO,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 * CDABS(ZWK(IDX(1,NQMR+1),8*M+ $4)) / R0 UCHK = UNRM IF ((TRES.EQ.0).AND.(UNRM/TOL.GT.DTEN).AND.(N.LT.NLIM)) GO TO 730 C C Have the caller carry out AXB, then return here. C CALL AXB (VECS(1,1),VECS(1,3)) C INFO(2) = 1 INFO(3) = 1 INFO(4) = 3 RETLBL = 720 RETURN 720 CALL ZAXPBY (NLEN,VECS(1,3),ZONE,VECS(1,2),-ZONE,VECS(1,3)) RESN = DZNRM2(NLEN,VECS(1,3),1) / R0 UCHK = RESN C C Output the convergence history. C 730 IF (VF.NE.0) WRITE (VF,'(I8,2E11.4)') NQMR, UNRM, RESN IF (TF.NE.0) WRITE (TF,'(I8,2E11.4)') NQMR, UNRM, RESN C C Check for convergence or termination. Stop if: C 1. algorithm converged; C 2. there is an error condition; C 3. the residual norm upper bound is smaller than the computed C residual norm by a factor of at least 100; C 4. algorithm exceeded the iterations limit. C IF (RESN.LE.TOL) THEN IERR = 0 GO TO 750 ELSE IF (IERR.NE.0) THEN GO TO 750 ELSE IF (UNRM.LT.UCHK/DHUN) THEN IERR = 4 GO TO 750 ELSE IF (NQMR.GE.NLIM) THEN IERR = 4 GO TO 750 END IF GO TO 680 C C Update the running counter. C 740 IF (IERR.NE.0) GO TO 750 IERR = 4 IF (N.GE.NLIM) GO TO 750 N = N + 1 GO TO 30 C C Done. C 750 RETLBL = 0 NLIM = NQMR INFO(1) = IERR MAXPQ = MPQBLT MAXVW = MVWBLT NORMS(1) = NORM1 NORMS(2) = NORM2 C RETURN END C C********************************************************************** C SUBROUTINE ZUCPL1 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL,$ NLSTAR,VF,IERR,ADJUST,NORM1,NORM2,ZWK,DWK,IWK, $IDX,VECS) C C Purpose: C This subroutine rebuilds the data for vectors p_n and q_n at the C point where a block was forced to close. The routine is called C internally by the coupled Lanczos code, and is not meant to be C called directly by the user code. C C Parameters: C See descriptions in the main routine ZUCPL. C C External routines used: C double precision dznrm2(n,x,incx) C BLAS-1 routine, computes the 2-norm of x. C subroutine zqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine zqrsl(x,ldx,n,k,qraux,y,qy,qty,b,rsd,xb,job,info) C LINPACK routine, applies the QR factorization of x. C subroutine zaxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C double precision zdotu(n,x,incx,y,incy) C BLAS-1 routine, computes y^H * x. C C Noel M. Nachtigal C May 31, 1993 C C********************************************************************** C INTRINSIC DMAX1, MAX0 EXTERNAL ZAXPBY, ZQRDC, ZQRSL C INTEGER IERR, K, KSTAR, L, LSTAR, M, MK, MKSTAR, NL, NLSTAR, N INTEGER IDX(6,*), IWK(M,13), NDIM, NLEN, VF DOUBLE COMPLEX VECS(NDIM,*), ZWK(M,8*M+7) DOUBLE PRECISION ADJUST, DWK(M,11), NORM1, NORM2 C C Miscellaneous parameters. C DOUBLE COMPLEX ZONE, ZZERO PARAMETER (ZONE = (1.0D0,0.0D0),ZZERO = (0.0D0,0.0D0)) DOUBLE PRECISION DZERO PARAMETER (DZERO = 0.0D0) C C Local variables. C INTEGER I, IJ, J, KBLKSZ, NP1 DOUBLE COMPLEX ZTMP1, ZTMP2 DOUBLE PRECISION DTMP1, DTMP2 C C Find the index of the vector pair with the smallest pass value. C IERR = 0 IF (VF.NE.0) WRITE (VF,'(A23)') 'PQ block did not close:' J = MK + 1 DTMP1 = DWK(IDX(1,J),11) DO 100 I = MK+2, N DTMP2 = DWK(IDX(1,I),11) IF (DTMP2.GT.DZERO) THEN IF ((DTMP1.EQ.DZERO).OR.(DTMP2.LT.DTMP1)) THEN J = I DTMP1 = DTMP2 END IF END IF 100 CONTINUE IF (DTMP1.EQ.DZERO) THEN IF (VF.NE.0) WRITE (VF,'(A47)')$'... 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 ZWK(IDX(1,N),6*M+IDX(1,N)) = ZONE NP1 = N + 1 DWK(IDX(1,N),11) = 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 ZWK(IDX(1,MK),6*M+IDX(1,NP1)) = ZWK(IDX(1,MK),6*M+IDX(1,N)) ZWK(IDX(1,MK),6*M+IDX(1,N)) = ZWK(IDX(1,N),M+IDX(1,MK)) / $ZWK(IDX(1,MK),5*M+IDX(1,MK)) * DWK(IDX(1,MK),3) / DWK(IDX(1,N),3) ELSE DO 130 J = MK, N-1 ZWK(J-MK+1,8*M+1) = ZWK(IDX(1,N),M+IDX(1,J)) * DWK(IDX(1,J),$3) / DWK(IDX(1,N),3) DO 120 IJ = MK, N-1 ZWK(J-MK+1,4*M+IJ-MK+1) = ZWK(IDX(1,J),5*M+IDX(1,IJ)) 120 CONTINUE 130 CONTINUE CALL ZQRDC (ZWK(1,4*M+1),M,KBLKSZ,KBLKSZ,ZWK(1,8*M+3),0,ZZERO,0 $) CALL ZQRSL (ZWK(1,4*M+1),M,KBLKSZ,KBLKSZ,ZWK(1,8*M+3),ZWK(1,8*M$+1), $ZZERO,ZWK(1,8*M+1),ZWK(1,8*M+2),ZWK(1,8*M+1),ZZERO,$100,J) DO 140 J = MK, N-1 ZWK(IDX(1,J),6*M+IDX(1,NP1)) = ZWK(IDX(1,J),6*M+IDX(1,N)) ZWK(IDX(1,J),6*M+IDX(1,N)) = ZWK(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 ZTMP1 = ZWK(IDX(1,I),6*M+IDX(1,N)) - ZWK(IDX(1,I),6*M+IDX(1,NP1 $)) ZTMP2 = ZTMP1 * DWK(IDX(1,I),7) / DWK(IDX(1,N),7) CALL ZAXPBY (NLEN,VECS(1,IDX(4,N)),ZONE,VECS(1,IDX(4,N)),-ZTMP2$,VECS(1,IDX(4,I))) ZTMP2 = ZTMP1 * DWK(IDX(1,I),8) / DWK(IDX(1,N),8) * DWK(IDX(1,N $),3) / DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,IDX(5,N)),ZONE,VECS(1,IDX(5,N)),-ZTMP2$,VECS(1,IDX(5,I))) 150 CONTINUE C RETURN END C C********************************************************************** C SUBROUTINE ZUCPL2 (NDIM,NLEN,M,N,K,KSTAR,L,LSTAR,MK,MKSTAR,NL, $NLSTAR,VF,IERR,ADJUST,NORM1,NORM2,ZWK,DWK,IWK,$ 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 ZUCPL. C C External routines used: C subroutine zaxpby(n,z,a,x,b,y) C Library routine, computes z = a * x + b * y. C subroutine zqrdc(x,ldx,n,p,qraux,jpvt,work,job) C LINPACK routine, computes the QR factorization of x. C subroutine zqrsl(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 ZAXPBY, ZQRDC, ZQRSL C INTEGER IERR, K, KSTAR, L, LSTAR, M, MK, MKSTAR, NL, NLSTAR, N INTEGER IDX(6,*), IWK(M,13), NDIM, NLEN, VF DOUBLE COMPLEX VECS(NDIM,*), ZWK(M,8*M+7) DOUBLE PRECISION ADJUST, DWK(M,11), NORM1, NORM2 C C Miscellaneous parameters. C DOUBLE COMPLEX ZONE, ZZERO PARAMETER (ZONE = (1.0D0,0.0D0),ZZERO = (0.0D0,0.0D0)) DOUBLE PRECISION DZERO PARAMETER (DZERO = 0.0D0) C C Local variables. C INTEGER I, IJ, J, LBLKSZ, NP1 DOUBLE COMPLEX SCALV, SCALW, ZTMP1, ZTMP2 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)') 'VW block did not close:' J = NL DTMP1 = DWK(IDX(1,J),6) DO 100 I = NL+1, N DTMP2 = DWK(IDX(1,I),6) 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), $NORM1, NORM2 IF (IWK(IDX(1,J),7).EQ.N) RETURN N = IWK(IDX(1,J),7) L = IWK(IDX(1,N),9) NL = IWK(IDX(1,L+1),2) K = IWK(IDX(1,N),12) MK = IWK(IDX(1,K+1),1) LSTAR = IWK(IDX(1,N),11) NLSTAR = IWK(IDX(1,LSTAR+1),2) KSTAR = IWK(IDX(1,N),13) MKSTAR = IWK(IDX(1,KSTAR+1),1) C C Initialize local variables. C NP1 = N + 1 DWK(IDX(1,N),6) = DZERO LBLKSZ = N - NL + 1 C C Step 15: C Compute l^\star. C I = LSTAR DO 110 J = I+1, L IF (IWK(IDX(1,J+1),2).LE.MK) LSTAR = J 110 CONTINUE NLSTAR = IWK(IDX(1,LSTAR+1),2) C C Compute L_{n_l:n,n}. Save the old coefficients. C IWK(IDX(1,L+1+1),2) = NP1 IF (LBLKSZ.EQ.1) THEN ZWK(IDX(1,NL),2*M+IDX(1,NP1)) = ZWK(IDX(1,NL),2*M+IDX(1,N)) ZWK(IDX(1,NL),2*M+IDX(1,N)) = ZWK(IDX(1,NL),M+IDX(1,N)) /$ZWK(IDX(1,NL),IDX(1,NL)) ELSE DO 130 J = NL, N ZWK(J-NL+1,8*M+1) = ZWK(IDX(1,J),M+IDX(1,N)) DO 120 IJ = NL, N ZWK(J-NL+1,4*M+IJ-NL+1) = ZWK(IDX(1,J),IDX(1,IJ)) 120 CONTINUE 130 CONTINUE CALL ZQRDC (ZWK(1,4*M+1),M,LBLKSZ,LBLKSZ,ZWK(1,8*M+3),0,ZZERO,0 $) CALL ZQRSL (ZWK(1,4*M+1),M,LBLKSZ,LBLKSZ,ZWK(1,8*M+3),ZWK(1,8*M$+1), $ZZERO,ZWK(1,8*M+1),ZWK(1,8*M+2),ZWK(1,8*M+1),ZZERO,$100,J) DO 140 J = NL, N ZWK(IDX(1,J),2*M+IDX(1,NP1)) = ZWK(IDX(1,J),2*M+IDX(1,N)) ZWK(IDX(1,J),2*M+IDX(1,N)) = ZWK(J-NL+1,8*M+2) 140 CONTINUE END IF C C Convert inner vectors to regular vectors. C SCALV = ZWK(IDX(1,NP1),2*M+IDX(1,N)) * DWK(IDX(1,NP1),4) SCALW = ZWK(IDX(1,NP1),2*M+IDX(1,N)) * DWK(IDX(1,NP1),5) * $DWK(IDX(1,N),3) / DWK(IDX(1,NP1),3) ZWK(IDX(1,NP1),2*M+IDX(1,N)) = ZZERO DO 150 I = NL, N ZTMP1 = ZWK(IDX(1,I),2*M+IDX(1,N)) - ZWK(IDX(1,I),2*M+IDX(1,NP1$)) ZTMP2 = ZTMP1 * DWK(IDX(1,I),4) / SCALV CALL ZAXPBY (NLEN,VECS(1,IDX(2,NP1)),ZONE,VECS(1,IDX(2,NP1)),- $ZTMP2,VECS(1,IDX(2,I))) ZTMP2 = ZTMP1 * DWK(IDX(1,I),5) / SCALW * DWK(IDX(1,N),3) /$DWK(IDX(1,I),3) CALL ZAXPBY (NLEN,VECS(1,IDX(3,NP1)),ZONE,VECS(1,IDX(3,NP1)),- \$ZTMP2,VECS(1,IDX(3,I))) 150 CONTINUE C RETURN END C C********************************************************************** C DOUBLE COMPLEX FUNCTION ZUCPLL(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 ZUCPLX. C DOUBLE PRECISION NORMA COMMON /ZUCPLX/NORMA C INTRINSIC DCMPLX C INTEGER I, J C IF ((I.LT.1).OR.(J.LT.1)) THEN ZUCPLL = (0.0D0,0.0D0) ELSE IF (I.EQ.J) THEN ZUCPLL = DCMPLX(NORMA,0.0D0) ELSE ZUCPLL = (0.0D0,0.0D0) END IF C RETURN END C C********************************************************************** C DOUBLE PRECISION FUNCTION ZUCPLO (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 ZUCPLO = 1.0D0 C RETURN END C C********************************************************************** C DOUBLE COMPLEX FUNCTION ZUCPLU(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 ZUCPLU = (0.0D0,0.0D0) ELSE ZUCPLU = (0.0D0,0.0D0) END IF C RETURN END C C**********************************************************************