#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int cggesx_(char *jobvsl, char *jobvsr, char *sort, L_fp selctg, char *sense, integer *n, complex *a, integer *lda, complex *b, integer *ldb, integer *sdim, complex *alpha, complex *beta, complex * vsl, integer *ldvsl, complex *vsr, integer *ldvsr, real *rconde, real *rcondv, complex *work, integer *lwork, real *rwork, integer *iwork, integer *liwork, logical *bwork, integer *info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= CGGESX computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the complex Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H ) where (VSR)**H is the conjugate-transpose of VSR. Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or for both being zero. A pair of matrices (S,T) is in generalized complex Schur form if T is upper triangular with non-negative diagonal and S is upper triangular. Arguments ========= JOBVSL (input) CHARACTER*1 = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors. JOBVSR (input) CHARACTER*1 = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors. SORT (input) CHARACTER*1 Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form. = 'N': Eigenvalues are not ordered; = 'S': Eigenvalues are ordered (see SELCTG). SELCTG (external procedure) LOGICAL FUNCTION of two COMPLEX arguments SELCTG must be declared EXTERNAL in the calling subroutine. If SORT = 'N', SELCTG is not referenced. If SORT = 'S', SELCTG is used to select eigenvalues to sort to the top left of the Schur form. Note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case INFO is set to N+3 see INFO below). SENSE (input) CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N' : None are computed; = 'E' : Computed for average of selected eigenvalues only; = 'V' : Computed for selected deflating subspaces only; = 'B' : Computed for both. If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. N (input) INTEGER The order of the matrices A, B, VSL, and VSR. N >= 0. A (input/output) COMPLEX array, dimension (LDA, N) On entry, the first of the pair of matrices. On exit, A has been overwritten by its generalized Schur form S. LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/output) COMPLEX array, dimension (LDB, N) On entry, the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T. LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). SDIM (output) INTEGER If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which SELCTG is true. ALPHA (output) COMPLEX array, dimension (N) BETA (output) COMPLEX array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T). BETA(j) will be non-negative real. Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). VSL (output) COMPLEX array, dimension (LDVSL,N) If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'. LDVSL (input) INTEGER The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N. VSR (output) COMPLEX array, dimension (LDVSR,N) If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'. LDVSR (input) INTEGER The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. RCONDE (output) REAL array, dimension ( 2 ) If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'. RCONDV (output) REAL array, dimension ( 2 ) If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition number for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'. WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may not be large enough. If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. RWORK (workspace) REAL array, dimension ( 8*N ) Real workspace. IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. LIWORK (input) INTEGER The dimension of the array WORK. If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise LIWORK >= N+2. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. BWORK (workspace) LOGICAL array, dimension (N) Not referenced if SORT = 'N'. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. = 1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in CHGEQZ =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling. =N+3: reordering failed in CTGSEN. ===================================================================== Decode the input arguments 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__0 = 0; static integer c_n1 = -1; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, vsl_dim1, vsl_offset, vsr_dim1, vsr_offset, i__1, i__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__; static real pl, pr, dif[2]; static integer ihi, ilo; static real eps; static integer ijob; static real anrm, bnrm; static integer ierr, itau, iwrk, lwrk; extern logical lsame_(char *, char *); static integer ileft, icols; static logical cursl, ilvsl, ilvsr; static integer irwrk, irows; extern /* Subroutine */ int cggbak_(char *, char *, integer *, integer *, integer *, real *, real *, integer *, complex *, integer *, integer *), cggbal_(char *, integer *, complex *, integer *, complex *, integer *, integer *, integer *, real *, real *, real *, integer *), slabad_(real *, real *); extern doublereal clange_(char *, integer *, integer *, complex *, integer *, real *); extern /* Subroutine */ int cgghrd_(char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, complex *, integer *, complex *, integer *, integer *), clascl_(char *, integer *, integer *, real *, real *, integer *, integer *, complex *, integer *, integer *); static logical ilascl, ilbscl; extern /* Subroutine */ int cgeqrf_(integer *, integer *, complex *, integer *, complex *, 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); extern doublereal slamch_(char *); static real bignum; extern /* Subroutine */ int chgeqz_(char *, char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, complex *, complex *, complex *, integer *, complex *, integer *, complex *, integer *, real *, integer *), ctgsen_(integer *, logical *, logical *, logical *, integer *, complex *, integer *, complex *, integer *, complex *, complex *, complex *, integer *, complex *, integer *, integer *, real *, real *, real *, complex *, integer *, integer *, integer *, integer *); static integer ijobvl, iright, ijobvr; static logical wantsb; static integer liwmin; static logical wantse, lastsl; static real anrmto, bnrmto; extern /* Subroutine */ int cungqr_(integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, integer *); static integer minwrk, maxwrk; static logical wantsn; static real smlnum; extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, complex *, integer *, integer *); static logical wantst, lquery, wantsv; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --alpha; --beta; vsl_dim1 = *ldvsl; vsl_offset = 1 + vsl_dim1; vsl -= vsl_offset; vsr_dim1 = *ldvsr; vsr_offset = 1 + vsr_dim1; vsr -= vsr_offset; --rconde; --rcondv; --work; --rwork; --iwork; --bwork; /* Function Body */ if (lsame_(jobvsl, "N")) { ijobvl = 1; ilvsl = FALSE_; } else if (lsame_(jobvsl, "V")) { ijobvl = 2; ilvsl = TRUE_; } else { ijobvl = -1; ilvsl = FALSE_; } if (lsame_(jobvsr, "N")) { ijobvr = 1; ilvsr = FALSE_; } else if (lsame_(jobvsr, "V")) { ijobvr = 2; ilvsr = TRUE_; } else { ijobvr = -1; ilvsr = FALSE_; } wantst = lsame_(sort, "S"); wantsn = lsame_(sense, "N"); wantse = lsame_(sense, "E"); wantsv = lsame_(sense, "V"); wantsb = lsame_(sense, "B"); lquery = *lwork == -1 || *liwork == -1; if (wantsn) { ijob = 0; } else if (wantse) { ijob = 1; } else if (wantsv) { ijob = 2; } else if (wantsb) { ijob = 4; } /* Test the input arguments */ *info = 0; if (ijobvl <= 0) { *info = -1; } else if (ijobvr <= 0) { *info = -2; } else if (! wantst && ! lsame_(sort, "N")) { *info = -3; } else if (! (wantsn || wantse || wantsv || wantsb) || ! wantst && ! wantsn) { *info = -5; } else if (*n < 0) { *info = -6; } else if (*lda < max(1,*n)) { *info = -8; } else if (*ldb < max(1,*n)) { *info = -10; } else if (*ldvsl < 1 || ilvsl && *ldvsl < *n) { *info = -15; } else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) { *info = -17; } /* Compute workspace (Note: Comments in the code beginning "Workspace:" describe the minimal amount of workspace needed at that point in the code, as well as the preferred amount for good performance. NB refers to the optimal block size for the immediately following subroutine, as returned by ILAENV.) */ if (*info == 0) { if (*n > 0) { minwrk = *n << 1; maxwrk = *n * (ilaenv_(&c__1, "CGEQRF", " ", n, &c__1, n, &c__0, ( ftnlen)6, (ftnlen)1) + 1); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * (ilaenv_(&c__1, "CUNMQR", " ", n, & c__1, n, &c_n1, (ftnlen)6, (ftnlen)1) + 1); maxwrk = max(i__1,i__2); if (ilvsl) { /* Computing MAX */ i__1 = maxwrk, i__2 = *n * (ilaenv_(&c__1, "CUNGQR", " ", n, & c__1, n, &c_n1, (ftnlen)6, (ftnlen)1) + 1); maxwrk = max(i__1,i__2); } lwrk = maxwrk; if (ijob >= 1) { /* Computing MAX */ i__1 = lwrk, i__2 = *n * *n / 2; lwrk = max(i__1,i__2); } } else { minwrk = 1; maxwrk = 1; lwrk = 1; } work[1].r = (real) lwrk, work[1].i = 0.f; if (wantsn || *n == 0) { liwmin = 1; } else { liwmin = *n + 2; } iwork[1] = liwmin; if (*lwork < minwrk && ! lquery) { *info = -21; } else if (*liwork < liwmin && ! lquery) { *info = -24; } } if (*info != 0) { i__1 = -(*info); xerbla_("CGGESX", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { *sdim = 0; return 0; } /* Get machine constants */ eps = slamch_("P"); smlnum = slamch_("S"); bignum = 1.f / smlnum; slabad_(&smlnum, &bignum); smlnum = sqrt(smlnum) / eps; bignum = 1.f / smlnum; /* Scale A if max element outside range [SMLNUM,BIGNUM] */ anrm = clange_("M", n, n, &a[a_offset], lda, &rwork[1]); ilascl = FALSE_; if (anrm > 0.f && anrm < smlnum) { anrmto = smlnum; ilascl = TRUE_; } else if (anrm > bignum) { anrmto = bignum; ilascl = TRUE_; } if (ilascl) { clascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, & ierr); } /* Scale B if max element outside range [SMLNUM,BIGNUM] */ bnrm = clange_("M", n, n, &b[b_offset], ldb, &rwork[1]); ilbscl = FALSE_; if (bnrm > 0.f && bnrm < smlnum) { bnrmto = smlnum; ilbscl = TRUE_; } else if (bnrm > bignum) { bnrmto = bignum; ilbscl = TRUE_; } if (ilbscl) { clascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, & ierr); } /* Permute the matrix to make it more nearly triangular (Real Workspace: need 6*N) */ ileft = 1; iright = *n + 1; irwrk = iright + *n; cggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &rwork[ ileft], &rwork[iright], &rwork[irwrk], &ierr); /* Reduce B to triangular form (QR decomposition of B) (Complex Workspace: need N, prefer N*NB) */ irows = ihi + 1 - ilo; icols = *n + 1 - ilo; itau = 1; iwrk = itau + irows; i__1 = *lwork + 1 - iwrk; cgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[ iwrk], &i__1, &ierr); /* Apply the unitary transformation to matrix A (Complex Workspace: need N, prefer N*NB) */ i__1 = *lwork + 1 - iwrk; cunmqr_("L", "C", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, & work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwrk], &i__1, & ierr); /* Initialize VSL (Complex Workspace: need N, prefer N*NB) */ if (ilvsl) { claset_("Full", n, n, &c_b1, &c_b2, &vsl[vsl_offset], ldvsl); if (irows > 1) { i__1 = irows - 1; i__2 = irows - 1; clacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vsl[ ilo + 1 + ilo * vsl_dim1], ldvsl); } i__1 = *lwork + 1 - iwrk; cungqr_(&irows, &irows, &irows, &vsl[ilo + ilo * vsl_dim1], ldvsl, & work[itau], &work[iwrk], &i__1, &ierr); } /* Initialize VSR */ if (ilvsr) { claset_("Full", n, n, &c_b1, &c_b2, &vsr[vsr_offset], ldvsr); } /* Reduce to generalized Hessenberg form (Workspace: none needed) */ cgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset], ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &ierr); *sdim = 0; /* Perform QZ algorithm, computing Schur vectors if desired (Complex Workspace: need N) (Real Workspace: need N) */ iwrk = itau; i__1 = *lwork + 1 - iwrk; chgeqz_("S", jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[ b_offset], ldb, &alpha[1], &beta[1], &vsl[vsl_offset], ldvsl, & vsr[vsr_offset], ldvsr, &work[iwrk], &i__1, &rwork[irwrk], &ierr); if (ierr != 0) { if (ierr > 0 && ierr <= *n) { *info = ierr; } else if (ierr > *n && ierr <= *n << 1) { *info = ierr - *n; } else { *info = *n + 1; } goto L40; } /* Sort eigenvalues ALPHA/BETA and compute the reciprocal of condition number(s) */ if (wantst) { /* Undo scaling on eigenvalues before SELCTGing */ if (ilascl) { clascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alpha[1], n, &ierr); } if (ilbscl) { clascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &ierr); } /* Select eigenvalues */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { bwork[i__] = (*selctg)(&alpha[i__], &beta[i__]); /* L10: */ } /* Reorder eigenvalues, transform Generalized Schur vectors, and compute reciprocal condition numbers (Complex Workspace: If IJOB >= 1, need MAX(1, 2*SDIM*(N-SDIM)) otherwise, need 1 ) */ i__1 = *lwork - iwrk + 1; ctgsen_(&ijob, &ilvsl, &ilvsr, &bwork[1], n, &a[a_offset], lda, &b[ b_offset], ldb, &alpha[1], &beta[1], &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, sdim, &pl, &pr, dif, &work[iwrk], & i__1, &iwork[1], liwork, &ierr); if (ijob >= 1) { /* Computing MAX */ i__1 = maxwrk, i__2 = (*sdim << 1) * (*n - *sdim); maxwrk = max(i__1,i__2); } if (ierr == -21) { /* not enough complex workspace */ *info = -21; } else { if (ijob == 1 || ijob == 4) { rconde[1] = pl; rconde[2] = pr; } if (ijob == 2 || ijob == 4) { rcondv[1] = dif[0]; rcondv[2] = dif[1]; } if (ierr == 1) { *info = *n + 3; } } } /* Apply permutation to VSL and VSR (Workspace: none needed) */ if (ilvsl) { cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, & vsl[vsl_offset], ldvsl, &ierr); } if (ilvsr) { cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, & vsr[vsr_offset], ldvsr, &ierr); } /* Undo scaling */ if (ilascl) { clascl_("U", &c__0, &c__0, &anrmto, &anrm, n, n, &a[a_offset], lda, & ierr); clascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alpha[1], n, & ierr); } if (ilbscl) { clascl_("U", &c__0, &c__0, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, & ierr); clascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, & ierr); } if (wantst) { /* Check if reordering is correct */ lastsl = TRUE_; *sdim = 0; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { cursl = (*selctg)(&alpha[i__], &beta[i__]); if (cursl) { ++(*sdim); } if (cursl && ! lastsl) { *info = *n + 2; } lastsl = cursl; /* L30: */ } } L40: work[1].r = (real) maxwrk, work[1].i = 0.f; iwork[1] = liwmin; return 0; /* End of CGGESX */ } /* cggesx_ */