#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int chseqr_(char *job, char *compz, integer *n, integer *ilo, integer *ihi, complex *h__, integer *ldh, complex *w, complex *z__, integer *ldz, complex *work, integer *lwork, integer *info ) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= CHSEQR computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. Optionally Z may be postmultiplied into an input unitary matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H. Arguments ========= JOB (input) CHARACTER*1 = 'E': compute eigenvalues only; = 'S': compute eigenvalues and the Schur form T. COMPZ (input) CHARACTER*1 = 'N': no Schur vectors are computed; = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned; = 'V': Z must contain an unitary matrix Q on entry, and the product Q*Z is returned. N (input) INTEGER The order of the matrix H. N .GE. 0. ILO (input) INTEGER IHI (input) INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to CGEBAL, and then passed to CGEHRD when the matrix output by CGEBAL is reduced to Hessenberg form. Otherwise ILO and IHI should be set to 1 and N respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0. H (input/output) COMPLEX array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and JOB = 'S', H contains the upper triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and JOB = 'E', the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.) Unlike earlier versions of CHSEQR, this subroutine may explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. LDH (input) INTEGER The leading dimension of the array H. LDH .GE. max(1,N). W (output) COMPLEX array, dimension (N) The computed eigenvalues. If JOB = 'S', the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i). Z (input/output) COMPLEX array, dimension (LDZ,N) If COMPZ = 'N', Z is not referenced. If COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, Z contains the unitary matrix Z of the Schur vectors of H. If COMPZ = 'V', on entry Z must contain an N-by-N matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, if INFO = 0, Z contains Q*Z. Normally Q is the unitary matrix generated by CUNGHR after the call to CGEHRD which formed the Hessenberg matrix H. (The output value of Z when INFO.GT.0 is given under the description of INFO below.) LDZ (input) INTEGER The leading dimension of the array Z. if COMPZ = 'I' or COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1. WORK (workspace/output) COMPLEX array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns an estimate of the optimal value for LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then CHSEQR does a workspace query. In this case, CHSEQR checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed. INFO (output) INTEGER = 0: successful exit .LT. 0: if INFO = -i, the i-th argument had an illegal value .GT. 0: if INFO = i, CHSEQR failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 0 and JOB = 'E', then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO .GT. 0 and JOB = 'S', then on exit (*) (initial value of H)*U = U*(final value of H) where U is a unitary matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and COMPZ = 'V', then on exit (final value of Z) = (initial value of Z)*U where U is the unitary matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ = 'I', then on exit (final value of Z) = U where U is the unitary matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ = 'N', then Z is not accessed. ================================================================ Default values supplied by ILAENV(ISPEC,'CHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). It is suggested that these defaults be adjusted in order to attain best performance in each particular computational environment. ISPEC=1: The CLAHQR vs CLAQR0 crossover point. Default: 75. (Must be at least 11.) ISPEC=2: Recommended deflation window size. This depends on ILO, IHI and NS. NS is the number of simultaneous shifts returned by ILAENV(ISPEC=4). (See ISPEC=4 below.) The default for (IHI-ILO+1).LE.500 is NS. The default for (IHI-ILO+1).GT.500 is 3*NS/2. ISPEC=3: Nibble crossover point. (See ILAENV for details.) Default: 14% of deflation window size. ISPEC=4: Number of simultaneous shifts, NS, in a multi-shift QR iteration. If IHI-ILO+1 is ... greater than ...but less ... the or equal to ... than default is 1 30 NS - 2(+) 30 60 NS - 4(+) 60 150 NS = 10(+) 150 590 NS = ** 590 3000 NS = 64 3000 6000 NS = 128 6000 infinity NS = 256 (+) By default some or all matrices of this order are passed to the implicit double shift routine CLAHQR and NS is ignored. See ISPEC=1 above and comments in IPARM for details. The asterisks (**) indicate an ad-hoc function of N increasing from 10 to 64. ISPEC=5: Select structured matrix multiply. (See ILAENV for details.) Default: 3. ================================================================ Based on contributions by Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA ================================================================ References: K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002. ================================================================ ==== Matrices of order NTINY or smaller must be processed by . CLAHQR because of insufficient subdiagonal scratch space. . (This is a hard limit.) ==== ==== NL allocates some local workspace to help small matrices . through a rare CLAHQR failure. NL .GT. NTINY = 11 is . required and NL .LE. NMIN = ILAENV(ISPEC=1,...) is recom- . mended. (The default value of NMIN is 75.) Using NL = 49 . allows up to six simultaneous shifts and a 16-by-16 . deflation window. ==== ==== Decode and check the input parameters. ==== Parameter adjustments */ /* Table of constant values */ static complex c_b1 = {0.f,0.f}; static complex c_b2 = {1.f,0.f}; static integer c__1 = 1; static integer c__2 = 2; static integer c__49 = 49; /* System generated locals */ address a__1[2]; integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2, i__3[2]; real r__1, r__2, r__3; complex q__1; char ch__1[2]; /* Builtin functions Subroutine */ int s_cat(char *, char **, integer *, integer *, ftnlen); /* Local variables */ static complex hl[2401] /* was [49][49] */; static integer kbot, nmin; extern logical lsame_(char *, char *); extern /* Subroutine */ int ccopy_(integer *, complex *, integer *, complex *, integer *); static logical initz; static complex workl[49]; static logical wantt, wantz; extern /* Subroutine */ int claqr0_(logical *, logical *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, integer *, complex *, integer *, complex *, integer *, integer *), clahqr_(logical *, logical *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, integer *, complex *, integer *, integer *), clacpy_(char *, integer *, integer *, complex *, integer *, complex *, integer *), claset_(char *, integer *, integer *, complex *, complex *, complex *, integer *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); static logical lquery; h_dim1 = *ldh; h_offset = 1 + h_dim1; h__ -= h_offset; --w; z_dim1 = *ldz; z_offset = 1 + z_dim1; z__ -= z_offset; --work; /* Function Body */ wantt = lsame_(job, "S"); initz = lsame_(compz, "I"); wantz = initz || lsame_(compz, "V"); r__1 = (real) max(1,*n); q__1.r = r__1, q__1.i = 0.f; work[1].r = q__1.r, work[1].i = q__1.i; lquery = *lwork == -1; *info = 0; if (! lsame_(job, "E") && ! wantt) { *info = -1; } else if (! lsame_(compz, "N") && ! wantz) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*ilo < 1 || *ilo > max(1,*n)) { *info = -4; } else if (*ihi < min(*ilo,*n) || *ihi > *n) { *info = -5; } else if (*ldh < max(1,*n)) { *info = -7; } else if (*ldz < 1 || wantz && *ldz < max(1,*n)) { *info = -10; } else if (*lwork < max(1,*n) && ! lquery) { *info = -12; } if (*info != 0) { /* ==== Quick return in case of invalid argument. ==== */ i__1 = -(*info); xerbla_("CHSEQR", &i__1); return 0; } else if (*n == 0) { /* ==== Quick return in case N = 0; nothing to do. ==== */ return 0; } else if (lquery) { /* ==== Quick return in case of a workspace query ==== */ claqr0_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &w[1], ilo, ihi, &z__[z_offset], ldz, &work[1], lwork, info); /* ==== Ensure reported workspace size is backward-compatible with . previous LAPACK versions. ==== Computing MAX */ r__2 = work[1].r, r__3 = (real) max(1,*n); r__1 = dmax(r__2,r__3); q__1.r = r__1, q__1.i = 0.f; work[1].r = q__1.r, work[1].i = q__1.i; return 0; } else { /* ==== copy eigenvalues isolated by CGEBAL ==== */ if (*ilo > 1) { i__1 = *ilo - 1; i__2 = *ldh + 1; ccopy_(&i__1, &h__[h_offset], &i__2, &w[1], &c__1); } if (*ihi < *n) { i__1 = *n - *ihi; i__2 = *ldh + 1; ccopy_(&i__1, &h__[*ihi + 1 + (*ihi + 1) * h_dim1], &i__2, &w[* ihi + 1], &c__1); } /* ==== Initialize Z, if requested ==== */ if (initz) { claset_("A", n, n, &c_b1, &c_b2, &z__[z_offset], ldz); } /* ==== Quick return if possible ==== */ if (*ilo == *ihi) { i__1 = *ilo; i__2 = *ilo + *ilo * h_dim1; w[i__1].r = h__[i__2].r, w[i__1].i = h__[i__2].i; return 0; } /* ==== CLAHQR/CLAQR0 crossover point ==== Writing concatenation */ i__3[0] = 1, a__1[0] = job; i__3[1] = 1, a__1[1] = compz; s_cat(ch__1, a__1, i__3, &c__2, (ftnlen)2); nmin = ilaenv_(&c__1, "CHSEQR", ch__1, n, ilo, ihi, lwork, (ftnlen)6, (ftnlen)2); nmin = max(11,nmin); /* ==== CLAQR0 for big matrices; CLAHQR for small ones ==== */ if (*n > nmin) { claqr0_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &w[1], ilo, ihi, &z__[z_offset], ldz, &work[1], lwork, info); } else { /* ==== Small matrix ==== */ clahqr_(&wantt, &wantz, n, ilo, ihi, &h__[h_offset], ldh, &w[1], ilo, ihi, &z__[z_offset], ldz, info); if (*info > 0) { /* ==== A rare CLAHQR failure! CLAQR0 sometimes succeeds . when CLAHQR fails. ==== */ kbot = *info; if (*n >= 49) { /* ==== Larger matrices have enough subdiagonal scratch . space to call CLAQR0 directly. ==== */ claqr0_(&wantt, &wantz, n, ilo, &kbot, &h__[h_offset], ldh, &w[1], ilo, ihi, &z__[z_offset], ldz, &work[ 1], lwork, info); } else { /* ==== Tiny matrices don't have enough subdiagonal . scratch space to benefit from CLAQR0. Hence, . tiny matrices must be copied into a larger . array before calling CLAQR0. ==== */ clacpy_("A", n, n, &h__[h_offset], ldh, hl, &c__49); i__1 = *n + 1 + *n * 49 - 50; hl[i__1].r = 0.f, hl[i__1].i = 0.f; i__1 = 49 - *n; claset_("A", &c__49, &i__1, &c_b1, &c_b1, &hl[(*n + 1) * 49 - 49], &c__49); claqr0_(&wantt, &wantz, &c__49, ilo, &kbot, hl, &c__49, & w[1], ilo, ihi, &z__[z_offset], ldz, workl, & c__49, info); if (wantt || *info != 0) { clacpy_("A", n, n, hl, &c__49, &h__[h_offset], ldh); } } } } /* ==== Clear out the trash, if necessary. ==== */ if ((wantt || *info != 0) && *n > 2) { i__1 = *n - 2; i__2 = *n - 2; claset_("L", &i__1, &i__2, &c_b1, &c_b1, &h__[h_dim1 + 3], ldh); } /* ==== Ensure reported workspace size is backward-compatible with . previous LAPACK versions. ==== Computing MAX */ r__2 = (real) max(1,*n), r__3 = work[1].r; r__1 = dmax(r__2,r__3); q__1.r = r__1, q__1.i = 0.f; work[1].r = q__1.r, work[1].i = q__1.i; } /* ==== End of CHSEQR ==== */ return 0; } /* chseqr_ */