/* 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 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 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 (input) 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 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 (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= 2*N. If SENSE = 'E', 'V', or 'B', LWORK >= MAX(2*N, 2*SDIM*(N-SDIM)). RWORK (workspace) REAL array, dimension ( 8*N ) Real workspace. IWORK (workspace/output) INTEGER array, dimension (LIWORK) Not referenced if SENSE = 'N'. On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. LIWORK (input) INTEGER The dimension of the array WORK. LIWORK >= N+2. 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 ijob; static real anrm, bnrm; static integer ierr, itau, iwrk, i__; 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 *); static real pl; 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 real pr; 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, wantsv; static real dif[2]; static integer ihi, ilo; static real eps; #define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1 #define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)] #define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1 #define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)] #define vsl_subscr(a_1,a_2) (a_2)*vsl_dim1 + a_1 #define vsl_ref(a_1,a_2) vsl[vsl_subscr(a_1,a_2)] a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; --alpha; --beta; vsl_dim1 = *ldvsl; vsl_offset = 1 + vsl_dim1 * 1; vsl -= vsl_offset; vsr_dim1 = *ldvsr; vsr_offset = 1 + vsr_dim1 * 1; 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"); if (wantsn) { ijob = 0; iwork[1] = 1; } 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.) */ minwrk = 1; if (*info == 0 && *lwork >= 1) { /* Computing MAX */ i__1 = 1, i__2 = *n << 1; minwrk = max(i__1,i__2); maxwrk = *n + *n * ilaenv_(&c__1, "CGEQRF", " ", n, &c__1, n, &c__0, ( ftnlen)6, (ftnlen)1); if (ilvsl) { /* Computing MAX */ i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "CUNGQR", " ", n, & c__1, n, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); } work[1].r = (real) maxwrk, work[1].i = 0.f; } if (! wantsn) { liwmin = *n + 2; } else { liwmin = 1; } iwork[1] = liwmin; if (*info == 0 && *lwork < minwrk) { *info = -21; } else if (*info == 0 && ijob >= 1) { if (*liwork < liwmin) { *info = -24; } } if (*info != 0) { i__1 = -(*info); xerbla_("CGGESX", &i__1); 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_ref(ilo, ilo), 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_ref(ilo, ilo), ldb, &work[ itau], &a_ref(ilo, ilo), 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); i__1 = irows - 1; i__2 = irows - 1; clacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vsl_ref(ilo + 1, ilo), ldvsl); i__1 = *lwork + 1 - iwrk; cungqr_(&irows, &irows, &irows, &vsl_ref(ilo, ilo), 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 { rconde[1] = pl; rconde[2] = pl; 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); } /* L20: */ 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_ */
/* Subroutine */ int sgges_(char *jobvsl, char *jobvsr, char *sort, L_fp selctg, integer *n, real *a, integer *lda, real *b, integer *ldb, integer *sdim, real *alphar, real *alphai, real *beta, real *vsl, integer *ldvsl, real *vsr, integer *ldvsr, real *work, integer *lwork, logical *bwork, integer *info) { /* -- LAPACK driver routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Purpose ======= SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized real Schur form (S,T), optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T ) Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T.The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). (If only the generalized eigenvalues are needed, use the driver SGGEV instead, which is faster.) 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 both being zero. A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be "standardized" by making the corresponding elements of T have the form: [ a 0 ] [ 0 b ] and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues. 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 (input) LOGICAL FUNCTION of three REAL 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. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected. Note that in the ill-conditioned case, a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in this case. N (input) INTEGER The order of the matrices A, B, VSL, and VSR. N >= 0. A (input/output) REAL 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) REAL 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. (Complex conjugate pairs for which SELCTG is true for either eigenvalue count as 2.) ALPHAR (output) REAL array, dimension (N) ALPHAI (output) REAL array, dimension (N) BETA (output) REAL array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative. Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio. However, ALPHAR and ALPHAI 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) REAL 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) REAL 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. WORK (workspace/output) REAL array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= 8*N+16. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK 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 ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in SHGEQZ. =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 STGSEN. ===================================================================== Decode the input arguments Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static integer c__0 = 0; static integer c_n1 = -1; static real c_b33 = 0.f; static real c_b34 = 1.f; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, vsl_dim1, vsl_offset, vsr_dim1, vsr_offset, i__1, i__2; real r__1; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static real anrm, bnrm; static integer idum[1], ierr, itau, iwrk; static real pvsl, pvsr; static integer i__; extern logical lsame_(char *, char *); static integer ileft, icols; static logical cursl, ilvsl, ilvsr; static integer irows; static logical lst2sl; extern /* Subroutine */ int slabad_(real *, real *); static integer ip; extern /* Subroutine */ int sggbak_(char *, char *, integer *, integer *, integer *, real *, real *, integer *, real *, integer *, integer * ), sggbal_(char *, integer *, real *, integer *, real *, integer *, integer *, integer *, real *, real *, real *, integer *); static logical ilascl, ilbscl; extern doublereal slamch_(char *), slange_(char *, integer *, integer *, real *, integer *, real *); static real safmin; extern /* Subroutine */ int sgghrd_(char *, char *, integer *, integer *, integer *, real *, integer *, real *, integer *, real *, integer * , real *, integer *, integer *); static real safmax; extern /* Subroutine */ int xerbla_(char *, integer *); static real bignum; extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, real *, integer *, integer *, real *, integer *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); static integer ijobvl, iright; extern /* Subroutine */ int sgeqrf_(integer *, integer *, real *, integer *, real *, real *, integer *, integer *); static integer ijobvr; extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *), slaset_(char *, integer *, integer *, real *, real *, real *, integer *); static real anrmto, bnrmto; static logical lastsl; extern /* Subroutine */ int shgeqz_(char *, char *, char *, integer *, integer *, integer *, real *, integer *, real *, integer *, real * , real *, real *, real *, integer *, real *, integer *, real *, integer *, integer *), stgsen_(integer *, logical *, logical *, logical *, integer *, real *, integer *, real *, integer *, real *, real *, real *, real *, integer *, real *, integer *, integer *, real *, real *, real *, real *, integer *, integer *, integer *, integer *); static integer minwrk, maxwrk; static real smlnum; extern /* Subroutine */ int sorgqr_(integer *, integer *, integer *, real *, integer *, real *, real *, integer *, integer *); static logical wantst, lquery; extern /* Subroutine */ int sormqr_(char *, char *, integer *, integer *, integer *, real *, integer *, real *, real *, integer *, real *, integer *, integer *); static real dif[2]; static integer ihi, ilo; static real eps; #define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1] #define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1] #define vsl_ref(a_1,a_2) vsl[(a_2)*vsl_dim1 + a_1] a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; --alphar; --alphai; --beta; vsl_dim1 = *ldvsl; vsl_offset = 1 + vsl_dim1 * 1; vsl -= vsl_offset; vsr_dim1 = *ldvsr; vsr_offset = 1 + vsr_dim1 * 1; vsr -= vsr_offset; --work; --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"); /* Test the input arguments */ *info = 0; lquery = *lwork == -1; if (ijobvl <= 0) { *info = -1; } else if (ijobvr <= 0) { *info = -2; } else if (! wantst && ! lsame_(sort, "N")) { *info = -3; } else if (*n < 0) { *info = -5; } else if (*lda < max(1,*n)) { *info = -7; } else if (*ldb < max(1,*n)) { *info = -9; } 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.) */ minwrk = 1; if (*info == 0 && (*lwork >= 1 || lquery)) { minwrk = (*n + 1) * 7 + 16; maxwrk = (*n + 1) * 7 + *n * ilaenv_(&c__1, "SGEQRF", " ", n, &c__1, n, &c__0, (ftnlen)6, (ftnlen)1) + 16; if (ilvsl) { /* Computing MAX */ i__1 = maxwrk, i__2 = (*n + 1) * 7 + *n * ilaenv_(&c__1, "SORGQR", " ", n, &c__1, n, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); } work[1] = (real) maxwrk; } if (*lwork < minwrk && ! lquery) { *info = -19; } if (*info != 0) { i__1 = -(*info); xerbla_("SGGES ", &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"); safmin = slamch_("S"); safmax = 1.f / safmin; slabad_(&safmin, &safmax); smlnum = sqrt(safmin) / eps; bignum = 1.f / smlnum; /* Scale A if max element outside range [SMLNUM,BIGNUM] */ anrm = slange_("M", n, n, &a[a_offset], lda, &work[1]); ilascl = FALSE_; if (anrm > 0.f && anrm < smlnum) { anrmto = smlnum; ilascl = TRUE_; } else if (anrm > bignum) { anrmto = bignum; ilascl = TRUE_; } if (ilascl) { slascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, & ierr); } /* Scale B if max element outside range [SMLNUM,BIGNUM] */ bnrm = slange_("M", n, n, &b[b_offset], ldb, &work[1]); ilbscl = FALSE_; if (bnrm > 0.f && bnrm < smlnum) { bnrmto = smlnum; ilbscl = TRUE_; } else if (bnrm > bignum) { bnrmto = bignum; ilbscl = TRUE_; } if (ilbscl) { slascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, & ierr); } /* Permute the matrix to make it more nearly triangular (Workspace: need 6*N + 2*N space for storing balancing factors) */ ileft = 1; iright = *n + 1; iwrk = iright + *n; sggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[ ileft], &work[iright], &work[iwrk], &ierr); /* Reduce B to triangular form (QR decomposition of B) (Workspace: need N, prefer N*NB) */ irows = ihi + 1 - ilo; icols = *n + 1 - ilo; itau = iwrk; iwrk = itau + irows; i__1 = *lwork + 1 - iwrk; sgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwrk], & i__1, &ierr); /* Apply the orthogonal transformation to matrix A (Workspace: need N, prefer N*NB) */ i__1 = *lwork + 1 - iwrk; sormqr_("L", "T", &irows, &icols, &irows, &b_ref(ilo, ilo), ldb, &work[ itau], &a_ref(ilo, ilo), lda, &work[iwrk], &i__1, &ierr); /* Initialize VSL (Workspace: need N, prefer N*NB) */ if (ilvsl) { slaset_("Full", n, n, &c_b33, &c_b34, &vsl[vsl_offset], ldvsl); i__1 = irows - 1; i__2 = irows - 1; slacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vsl_ref(ilo + 1, ilo), ldvsl); i__1 = *lwork + 1 - iwrk; sorgqr_(&irows, &irows, &irows, &vsl_ref(ilo, ilo), ldvsl, &work[itau] , &work[iwrk], &i__1, &ierr); } /* Initialize VSR */ if (ilvsr) { slaset_("Full", n, n, &c_b33, &c_b34, &vsr[vsr_offset], ldvsr); } /* Reduce to generalized Hessenberg form (Workspace: none needed) */ sgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset], ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &ierr); /* Perform QZ algorithm, computing Schur vectors if desired (Workspace: need N) */ iwrk = itau; i__1 = *lwork + 1 - iwrk; shgeqz_("S", jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[ b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vsl[vsl_offset] , ldvsl, &vsr[vsr_offset], ldvsr, &work[iwrk], &i__1, &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 if desired (Workspace: need 4*N+16 ) */ *sdim = 0; if (wantst) { /* Undo scaling on eigenvalues before SELCTGing */ if (ilascl) { slascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphar[1], n, &ierr); slascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphai[1], n, &ierr); } if (ilbscl) { slascl_("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)(&alphar[i__], &alphai[i__], &beta[i__]); /* L10: */ } i__1 = *lwork - iwrk + 1; stgsen_(&c__0, &ilvsl, &ilvsr, &bwork[1], n, &a[a_offset], lda, &b[ b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vsl[ vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, sdim, &pvsl, & pvsr, dif, &work[iwrk], &i__1, idum, &c__1, &ierr); if (ierr == 1) { *info = *n + 3; } } /* Apply back-permutation to VSL and VSR (Workspace: none needed) */ if (ilvsl) { sggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsl[ vsl_offset], ldvsl, &ierr); } if (ilvsr) { sggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsr[ vsr_offset], ldvsr, &ierr); } /* Check if unscaling would cause over/underflow, if so, rescale (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I) */ if (ilascl) { i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if (alphai[i__] != 0.f) { if (alphar[i__] / safmax > anrmto / anrm || safmin / alphar[ i__] > anrm / anrmto) { work[1] = (r__1 = a_ref(i__, i__) / alphar[i__], dabs( r__1)); beta[i__] *= work[1]; alphar[i__] *= work[1]; alphai[i__] *= work[1]; } else if (alphai[i__] / safmax > anrmto / anrm || safmin / alphai[i__] > anrm / anrmto) { work[1] = (r__1 = a_ref(i__, i__ + 1) / alphai[i__], dabs( r__1)); beta[i__] *= work[1]; alphar[i__] *= work[1]; alphai[i__] *= work[1]; } } /* L50: */ } } if (ilbscl) { i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if (alphai[i__] != 0.f) { if (beta[i__] / safmax > bnrmto / bnrm || safmin / beta[i__] > bnrm / bnrmto) { work[1] = (r__1 = b_ref(i__, i__) / beta[i__], dabs(r__1)) ; beta[i__] *= work[1]; alphar[i__] *= work[1]; alphai[i__] *= work[1]; } } /* L60: */ } } /* Undo scaling */ if (ilascl) { slascl_("H", &c__0, &c__0, &anrmto, &anrm, n, n, &a[a_offset], lda, & ierr); slascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphar[1], n, & ierr); slascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphai[1], n, & ierr); } if (ilbscl) { slascl_("U", &c__0, &c__0, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, & ierr); slascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, & ierr); } if (wantst) { /* Check if reordering is correct */ lastsl = TRUE_; lst2sl = TRUE_; *sdim = 0; ip = 0; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { cursl = (*selctg)(&alphar[i__], &alphai[i__], &beta[i__]); if (alphai[i__] == 0.f) { if (cursl) { ++(*sdim); } ip = 0; if (cursl && ! lastsl) { *info = *n + 2; } } else { if (ip == 1) { /* Last eigenvalue of conjugate pair */ cursl = cursl || lastsl; lastsl = cursl; if (cursl) { *sdim += 2; } ip = -1; if (cursl && ! lst2sl) { *info = *n + 2; } } else { /* First eigenvalue of conjugate pair */ ip = 1; } } lst2sl = lastsl; lastsl = cursl; /* L30: */ } } L40: work[1] = (real) maxwrk; return 0; /* End of SGGES */ } /* sgges_ */
/* Subroutine */ int zgges_(char *jobvsl, char *jobvsr, char *sort, L_fp delctg, integer *n, doublecomplex *a, integer *lda, doublecomplex *b, integer *ldb, integer *sdim, doublecomplex *alpha, doublecomplex * beta, doublecomplex *vsl, integer *ldvsl, doublecomplex *vsr, integer *ldvsr, doublecomplex *work, integer *lwork, doublereal *rwork, logical *bwork, integer *info) { /* -- LAPACK driver routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Purpose ======= ZGGES computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized complex Schur form (S, T), and optionally left and/or right 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. The leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces). (If only the generalized eigenvalues are needed, use the driver ZGGEV instead, which is faster.) 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, and even for both being zero. A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers. 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 DELZTG). DELZTG (input) LOGICAL FUNCTION of two COMPLEX*16 arguments DELZTG must be declared EXTERNAL in the calling subroutine. If SORT = 'N', DELZTG is not referenced. If SORT = 'S', DELZTG is used to select eigenvalues to sort to the top left of the Schur form. An eigenvalue ALPHA(j)/BETA(j) is selected if DELZTG(ALPHA(j),BETA(j)) is true. Note that a selected complex eigenvalue may no longer satisfy DELZTG(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+2 (See INFO below). N (input) INTEGER The order of the matrices A, B, VSL, and VSR. N >= 0. A (input/output) COMPLEX*16 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*16 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 DELZTG is true. ALPHA (output) COMPLEX*16 array, dimension (N) BETA (output) COMPLEX*16 array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by ZGGES. The 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*16 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*16 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. WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,2*N). For good performance, LWORK must generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK (workspace) DOUBLE PRECISION array, dimension (8*N) 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 ZHGEQZ =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy DELZTG=.TRUE. This could also be caused due to scaling. =N+3: reordering falied in ZTGSEN. ===================================================================== Decode the input arguments Parameter adjustments */ /* Table of constant values */ static doublecomplex c_b1 = {0.,0.}; static doublecomplex c_b2 = {1.,0.}; 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 doublereal anrm, bnrm; static integer idum[1], ierr, itau, iwrk; static doublereal pvsl, pvsr; static integer i__; extern logical lsame_(char *, char *); static integer ileft, icols; static logical cursl, ilvsl, ilvsr; static integer irwrk, irows; extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); extern doublereal dlamch_(char *); extern /* Subroutine */ int zggbak_(char *, char *, integer *, integer *, integer *, doublereal *, doublereal *, integer *, doublecomplex *, integer *, integer *), zggbal_(char *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer * , integer *, doublereal *, doublereal *, doublereal *, integer *); static logical ilascl, ilbscl; extern /* Subroutine */ int xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern doublereal zlange_(char *, integer *, integer *, doublecomplex *, integer *, doublereal *); static doublereal bignum; static integer ijobvl, iright; extern /* Subroutine */ int zgghrd_(char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer * ), zlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublecomplex *, integer *, integer *); static integer ijobvr; extern /* Subroutine */ int zgeqrf_(integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, integer * ); static doublereal anrmto; static integer lwkmin; static logical lastsl; static doublereal bnrmto; extern /* Subroutine */ int zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *), zlaset_(char *, integer *, integer *, doublecomplex *, doublecomplex *, doublecomplex *, integer *), zhgeqz_( char *, char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, integer *), ztgsen_(integer *, logical *, logical *, logical *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublecomplex *, integer *, integer *, integer *, integer *); static doublereal smlnum; static logical wantst, lquery; static integer lwkopt; extern /* Subroutine */ int zungqr_(integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, integer *), zunmqr_(char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, integer *); static doublereal dif[2]; static integer ihi, ilo; static doublereal eps; #define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1 #define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)] #define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1 #define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)] #define vsl_subscr(a_1,a_2) (a_2)*vsl_dim1 + a_1 #define vsl_ref(a_1,a_2) vsl[vsl_subscr(a_1,a_2)] a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; --alpha; --beta; vsl_dim1 = *ldvsl; vsl_offset = 1 + vsl_dim1 * 1; vsl -= vsl_offset; vsr_dim1 = *ldvsr; vsr_offset = 1 + vsr_dim1 * 1; vsr -= vsr_offset; --work; --rwork; --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"); /* Test the input arguments */ *info = 0; lquery = *lwork == -1; if (ijobvl <= 0) { *info = -1; } else if (ijobvr <= 0) { *info = -2; } else if (! wantst && ! lsame_(sort, "N")) { *info = -3; } else if (*n < 0) { *info = -5; } else if (*lda < max(1,*n)) { *info = -7; } else if (*ldb < max(1,*n)) { *info = -9; } else if (*ldvsl < 1 || ilvsl && *ldvsl < *n) { *info = -14; } else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) { *info = -16; } /* 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.) */ lwkmin = 1; if (*info == 0 && (*lwork >= 1 || lquery)) { /* Computing MAX */ i__1 = 1, i__2 = *n << 1; lwkmin = max(i__1,i__2); lwkopt = *n + *n * ilaenv_(&c__1, "ZGEQRF", " ", n, &c__1, n, &c__0, ( ftnlen)6, (ftnlen)1); if (ilvsl) { /* Computing MAX */ i__1 = lwkopt, i__2 = *n + *n * ilaenv_(&c__1, "ZUNGQR", " ", n, & c__1, n, &c_n1, (ftnlen)6, (ftnlen)1); lwkopt = max(i__1,i__2); } work[1].r = (doublereal) lwkopt, work[1].i = 0.; } if (*lwork < lwkmin && ! lquery) { *info = -18; } if (*info != 0) { i__1 = -(*info); xerbla_("ZGGES ", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ work[1].r = (doublereal) lwkopt, work[1].i = 0.; if (*n == 0) { *sdim = 0; return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S"); bignum = 1. / smlnum; dlabad_(&smlnum, &bignum); smlnum = sqrt(smlnum) / eps; bignum = 1. / smlnum; /* Scale A if max element outside range [SMLNUM,BIGNUM] */ anrm = zlange_("M", n, n, &a[a_offset], lda, &rwork[1]); ilascl = FALSE_; if (anrm > 0. && anrm < smlnum) { anrmto = smlnum; ilascl = TRUE_; } else if (anrm > bignum) { anrmto = bignum; ilascl = TRUE_; } if (ilascl) { zlascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, & ierr); } /* Scale B if max element outside range [SMLNUM,BIGNUM] */ bnrm = zlange_("M", n, n, &b[b_offset], ldb, &rwork[1]); ilbscl = FALSE_; if (bnrm > 0. && bnrm < smlnum) { bnrmto = smlnum; ilbscl = TRUE_; } else if (bnrm > bignum) { bnrmto = bignum; ilbscl = TRUE_; } if (ilbscl) { zlascl_("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; zggbal_("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; zgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwrk], & i__1, &ierr); /* Apply the orthogonal transformation to matrix A (Complex Workspace: need N, prefer N*NB) */ i__1 = *lwork + 1 - iwrk; zunmqr_("L", "C", &irows, &icols, &irows, &b_ref(ilo, ilo), ldb, &work[ itau], &a_ref(ilo, ilo), lda, &work[iwrk], &i__1, &ierr); /* Initialize VSL (Complex Workspace: need N, prefer N*NB) */ if (ilvsl) { zlaset_("Full", n, n, &c_b1, &c_b2, &vsl[vsl_offset], ldvsl); i__1 = irows - 1; i__2 = irows - 1; zlacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vsl_ref(ilo + 1, ilo), ldvsl); i__1 = *lwork + 1 - iwrk; zungqr_(&irows, &irows, &irows, &vsl_ref(ilo, ilo), ldvsl, &work[itau] , &work[iwrk], &i__1, &ierr); } /* Initialize VSR */ if (ilvsr) { zlaset_("Full", n, n, &c_b1, &c_b2, &vsr[vsr_offset], ldvsr); } /* Reduce to generalized Hessenberg form (Workspace: none needed) */ zgghrd_(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; zhgeqz_("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 L30; } /* Sort eigenvalues ALPHA/BETA if desired (Workspace: none needed) */ if (wantst) { /* Undo scaling on eigenvalues before selecting */ if (ilascl) { zlascl_("G", &c__0, &c__0, &anrm, &anrmto, n, &c__1, &alpha[1], n, &ierr); } if (ilbscl) { zlascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, &c__1, &beta[1], n, &ierr); } /* Select eigenvalues */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { bwork[i__] = (*delctg)(&alpha[i__], &beta[i__]); /* L10: */ } i__1 = *lwork - iwrk + 1; ztgsen_(&c__0, &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, &pvsl, &pvsr, dif, &work[iwrk], &i__1, idum, &c__1, &ierr); if (ierr == 1) { *info = *n + 3; } } /* Apply back-permutation to VSL and VSR (Workspace: none needed) */ if (ilvsl) { zggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, & vsl[vsl_offset], ldvsl, &ierr); } if (ilvsr) { zggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, & vsr[vsr_offset], ldvsr, &ierr); } /* Undo scaling */ if (ilascl) { zlascl_("U", &c__0, &c__0, &anrmto, &anrm, n, n, &a[a_offset], lda, & ierr); zlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alpha[1], n, & ierr); } if (ilbscl) { zlascl_("U", &c__0, &c__0, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, & ierr); zlascl_("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 = (*delctg)(&alpha[i__], &beta[i__]); if (cursl) { ++(*sdim); } if (cursl && ! lastsl) { *info = *n + 2; } lastsl = cursl; /* L20: */ } } L30: work[1].r = (doublereal) lwkopt, work[1].i = 0.; return 0; /* End of ZGGES */ } /* zgges_ */
/* Subroutine */ int dgegs_(char *jobvsl, char *jobvsr, integer *n, doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal * alphar, doublereal *alphai, doublereal *beta, doublereal *vsl, integer *ldvsl, doublereal *vsr, integer *ldvsr, doublereal *work, integer *lwork, integer *info) { /* -- LAPACK driver routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Purpose ======= This routine is deprecated and has been replaced by routine DGGES. DGEGS computes for a pair of N-by-N real nonsymmetric matrices A, B: the generalized eigenvalues (alphar +/- alphai*i, beta), the real Schur form (A, B), and optionally left and/or right Schur vectors (VSL and VSR). (If only the generalized eigenvalues are needed, use the driver DGEGV instead.) A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, 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, and even for both being zero. A good beginning reference is the book, "Matrix Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press) The (generalized) Schur form of a pair of matrices is the result of multiplying both matrices on the left by one orthogonal matrix and both on the right by another orthogonal matrix, these two orthogonal matrices being chosen so as to bring the pair of matrices into (real) Schur form. A pair of matrices A, B is in generalized real Schur form if B is upper triangular with non-negative diagonal and A is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of A will be "standardized" by making the corresponding elements of B have the form: [ a 0 ] [ 0 b ] and the pair of corresponding 2-by-2 blocks in A and B will have a complex conjugate pair of generalized eigenvalues. The left and right Schur vectors are the columns of VSL and VSR, respectively, where VSL and VSR are the orthogonal matrices which reduce A and B to Schur form: Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T B (VSR) ) 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. N (input) INTEGER The order of the matrices A, B, VSL, and VSR. N >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA, N) On entry, the first of the pair of matrices whose generalized eigenvalues and (optionally) Schur vectors are to be computed. On exit, the generalized Schur form of A. Note: to avoid overflow, the Frobenius norm of the matrix A should be less than the overflow threshold. LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB, N) On entry, the second of the pair of matrices whose generalized eigenvalues and (optionally) Schur vectors are to be computed. On exit, the generalized Schur form of B. Note: to avoid overflow, the Frobenius norm of the matrix B should be less than the overflow threshold. LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). ALPHAR (output) DOUBLE PRECISION array, dimension (N) ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA (output) DOUBLE PRECISION array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, j=1,...,N and BETA(j),j=1,...,N are the diagonals of the complex Schur form (A,B) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative. Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION array, dimension (LDVSL,N) If JOBVSL = 'V', VSL will contain the left Schur vectors. (See "Purpose", above.) 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) DOUBLE PRECISION array, dimension (LDVSR,N) If JOBVSR = 'V', VSR will contain the right Schur vectors. (See "Purpose", above.) Not referenced if JOBVSR = 'N'. LDVSR (input) INTEGER The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,4*N). For good performance, LWORK must generally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR The optimal LWORK is 2*N + N*(NB+1). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 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 ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N. > N: errors that usually indicate LAPACK problems: =N+1: error return from DGGBAL =N+2: error return from DGEQRF =N+3: error return from DORMQR =N+4: error return from DORGQR =N+5: error return from DGGHRD =N+6: error return from DHGEQZ (other than failed iteration) =N+7: error return from DGGBAK (computing VSL) =N+8: error return from DGGBAK (computing VSR) =N+9: error return from DLASCL (various places) ===================================================================== Decode the input arguments Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static integer c_n1 = -1; static doublereal c_b36 = 0.; static doublereal c_b37 = 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; /* Local variables */ static doublereal anrm, bnrm; static integer itau, lopt; extern logical lsame_(char *, char *); static integer ileft, iinfo, icols; static logical ilvsl; static integer iwork; static logical ilvsr; static integer irows; extern /* Subroutine */ int dggbak_(char *, char *, integer *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); static integer nb; extern /* Subroutine */ int dggbal_(char *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *); extern doublereal dlamch_(char *), dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dgghrd_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *); static logical ilascl, ilbscl; extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *); static doublereal safmin; extern /* Subroutine */ int dlaset_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); static doublereal bignum; extern /* Subroutine */ int dhgeqz_(char *, char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *); static integer ijobvl, iright, ijobvr; extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *); static doublereal anrmto; static integer lwkmin, nb1, nb2, nb3; static doublereal bnrmto; extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); static doublereal smlnum; static integer lwkopt; static logical lquery; static integer ihi, ilo; static doublereal eps; #define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1] #define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1] #define vsl_ref(a_1,a_2) vsl[(a_2)*vsl_dim1 + a_1] a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; --alphar; --alphai; --beta; vsl_dim1 = *ldvsl; vsl_offset = 1 + vsl_dim1 * 1; vsl -= vsl_offset; vsr_dim1 = *ldvsr; vsr_offset = 1 + vsr_dim1 * 1; vsr -= vsr_offset; --work; /* 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_; } /* Test the input arguments Computing MAX */ i__1 = *n << 2; lwkmin = max(i__1,1); lwkopt = lwkmin; work[1] = (doublereal) lwkopt; lquery = *lwork == -1; *info = 0; if (ijobvl <= 0) { *info = -1; } else if (ijobvr <= 0) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*lda < max(1,*n)) { *info = -5; } else if (*ldb < max(1,*n)) { *info = -7; } else if (*ldvsl < 1 || ilvsl && *ldvsl < *n) { *info = -12; } else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) { *info = -14; } else if (*lwork < lwkmin && ! lquery) { *info = -16; } if (*info == 0) { nb1 = ilaenv_(&c__1, "DGEQRF", " ", n, n, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); nb2 = ilaenv_(&c__1, "DORMQR", " ", n, n, n, &c_n1, (ftnlen)6, ( ftnlen)1); nb3 = ilaenv_(&c__1, "DORGQR", " ", n, n, n, &c_n1, (ftnlen)6, ( ftnlen)1); /* Computing MAX */ i__1 = max(nb1,nb2); nb = max(i__1,nb3); lopt = (*n << 1) + *n * (nb + 1); work[1] = (doublereal) lopt; } if (*info != 0) { i__1 = -(*info); xerbla_("DGEGS ", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Get machine constants */ eps = dlamch_("E") * dlamch_("B"); safmin = dlamch_("S"); smlnum = *n * safmin / eps; bignum = 1. / smlnum; /* Scale A if max element outside range [SMLNUM,BIGNUM] */ anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]); ilascl = FALSE_; if (anrm > 0. && anrm < smlnum) { anrmto = smlnum; ilascl = TRUE_; } else if (anrm > bignum) { anrmto = bignum; ilascl = TRUE_; } if (ilascl) { dlascl_("G", &c_n1, &c_n1, &anrm, &anrmto, n, n, &a[a_offset], lda, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } } /* Scale B if max element outside range [SMLNUM,BIGNUM] */ bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1]); ilbscl = FALSE_; if (bnrm > 0. && bnrm < smlnum) { bnrmto = smlnum; ilbscl = TRUE_; } else if (bnrm > bignum) { bnrmto = bignum; ilbscl = TRUE_; } if (ilbscl) { dlascl_("G", &c_n1, &c_n1, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } } /* Permute the matrix to make it more nearly triangular Workspace layout: (2*N words -- "work..." not actually used) left_permutation, right_permutation, work... */ ileft = 1; iright = *n + 1; iwork = iright + *n; dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[ ileft], &work[iright], &work[iwork], &iinfo); if (iinfo != 0) { *info = *n + 1; goto L10; } /* Reduce B to triangular form, and initialize VSL and/or VSR Workspace layout: ("work..." must have at least N words) left_permutation, right_permutation, tau, work... */ irows = ihi + 1 - ilo; icols = *n + 1 - ilo; itau = iwork; iwork = itau + irows; i__1 = *lwork + 1 - iwork; dgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1; lwkopt = max(i__1,i__2); } if (iinfo != 0) { *info = *n + 2; goto L10; } i__1 = *lwork + 1 - iwork; dormqr_("L", "T", &irows, &icols, &irows, &b_ref(ilo, ilo), ldb, &work[ itau], &a_ref(ilo, ilo), lda, &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1; lwkopt = max(i__1,i__2); } if (iinfo != 0) { *info = *n + 3; goto L10; } if (ilvsl) { dlaset_("Full", n, n, &c_b36, &c_b37, &vsl[vsl_offset], ldvsl); i__1 = irows - 1; i__2 = irows - 1; dlacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vsl_ref(ilo + 1, ilo), ldvsl); i__1 = *lwork + 1 - iwork; dorgqr_(&irows, &irows, &irows, &vsl_ref(ilo, ilo), ldvsl, &work[itau] , &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1; lwkopt = max(i__1,i__2); } if (iinfo != 0) { *info = *n + 4; goto L10; } } if (ilvsr) { dlaset_("Full", n, n, &c_b36, &c_b37, &vsr[vsr_offset], ldvsr); } /* Reduce to generalized Hessenberg form */ dgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset], ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &iinfo); if (iinfo != 0) { *info = *n + 5; goto L10; } /* Perform QZ algorithm, computing Schur vectors if desired Workspace layout: ("work..." must have at least 1 word) left_permutation, right_permutation, work... */ iwork = itau; i__1 = *lwork + 1 - iwork; dhgeqz_("S", jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[ b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vsl[vsl_offset] , ldvsl, &vsr[vsr_offset], ldvsr, &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1; lwkopt = max(i__1,i__2); } if (iinfo != 0) { if (iinfo > 0 && iinfo <= *n) { *info = iinfo; } else if (iinfo > *n && iinfo <= *n << 1) { *info = iinfo - *n; } else { *info = *n + 6; } goto L10; } /* Apply permutation to VSL and VSR */ if (ilvsl) { dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsl[ vsl_offset], ldvsl, &iinfo); if (iinfo != 0) { *info = *n + 7; goto L10; } } if (ilvsr) { dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsr[ vsr_offset], ldvsr, &iinfo); if (iinfo != 0) { *info = *n + 8; goto L10; } } /* Undo scaling */ if (ilascl) { dlascl_("H", &c_n1, &c_n1, &anrmto, &anrm, n, n, &a[a_offset], lda, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } dlascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alphar[1], n, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } dlascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alphai[1], n, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } } if (ilbscl) { dlascl_("U", &c_n1, &c_n1, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } dlascl_("G", &c_n1, &c_n1, &bnrmto, &bnrm, n, &c__1, &beta[1], n, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } } L10: work[1] = (doublereal) lwkopt; return 0; /* End of DGEGS */ } /* dgegs_ */
/* Subroutine */ int cgegs_(char *jobvsl, char *jobvsr, integer *n, complex * a, integer *lda, complex *b, integer *ldb, complex *alpha, complex * beta, complex *vsl, integer *ldvsl, complex *vsr, integer *ldvsr, complex *work, integer *lwork, real *rwork, integer *info) { /* -- LAPACK driver routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Purpose ======= This routine is deprecated and has been replaced by routine CGGES. CGEGS computes for a pair of N-by-N complex nonsymmetric matrices A, B: the generalized eigenvalues (alpha, beta), the complex Schur form (A, B), and optionally left and/or right Schur vectors (VSL and VSR). (If only the generalized eigenvalues are needed, use the driver CGEGV instead.) A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, 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, and even for both being zero. A good beginning reference is the book, "Matrix Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press) The (generalized) Schur form of a pair of matrices is the result of multiplying both matrices on the left by one unitary matrix and both on the right by another unitary matrix, these two unitary matrices being chosen so as to bring the pair of matrices into upper triangular form with the diagonal elements of B being non-negative real numbers (this is also called complex Schur form.) The left and right Schur vectors are the columns of VSL and VSR, respectively, where VSL and VSR are the unitary matrices which reduce A and B to Schur form: Schur form of (A,B) = ( (VSL)**H A (VSR), (VSL)**H B (VSR) ) 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. 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 whose generalized eigenvalues and (optionally) Schur vectors are to be computed. On exit, the generalized Schur form of A. 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 whose generalized eigenvalues and (optionally) Schur vectors are to be computed. On exit, the generalized Schur form of B. LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). 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), j=1,...,N and BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by CGEGS. The 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. (See "Purpose", above.) 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. (See "Purpose", above.) Not referenced if JOBVSR = 'N'. LDVSR (input) INTEGER The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. WORK (workspace/output) COMPLEX array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,2*N). For good performance, LWORK must generally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute: NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR; the optimal LWORK is N*(NB+1). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK (workspace) REAL array, dimension (3*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: errors that usually indicate LAPACK problems: =N+1: error return from CGGBAL =N+2: error return from CGEQRF =N+3: error return from CUNMQR =N+4: error return from CUNGQR =N+5: error return from CGGHRD =N+6: error return from CHGEQZ (other than failed iteration) =N+7: error return from CGGBAK (computing VSL) =N+8: error return from CGGBAK (computing VSR) =N+9: error return from CLASCL (various places) ===================================================================== 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_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, i__3; /* Local variables */ static real anrm, bnrm; static integer itau, lopt; extern logical lsame_(char *, char *); static integer ileft, iinfo, icols; static logical ilvsl; static integer iwork; static logical ilvsr; static integer 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 *); static integer nb; 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 *); extern doublereal slamch_(char *); extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex *, integer *, complex *, integer *), claset_(char *, integer *, integer *, complex *, complex *, complex *, integer *); static real safmin; extern /* Subroutine */ int xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); 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 *); static integer ijobvl, iright, ijobvr; static real anrmto; static integer lwkmin, nb1, nb2, nb3; static real bnrmto; extern /* Subroutine */ int cungqr_(integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, integer *), cunmqr_(char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, complex *, integer *, integer *); static real smlnum; static integer irwork, lwkopt; static logical lquery; static integer ihi, ilo; static real eps; #define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1 #define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)] #define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1 #define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)] #define vsl_subscr(a_1,a_2) (a_2)*vsl_dim1 + a_1 #define vsl_ref(a_1,a_2) vsl[vsl_subscr(a_1,a_2)] a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; --alpha; --beta; vsl_dim1 = *ldvsl; vsl_offset = 1 + vsl_dim1 * 1; vsl -= vsl_offset; vsr_dim1 = *ldvsr; vsr_offset = 1 + vsr_dim1 * 1; vsr -= vsr_offset; --work; --rwork; /* 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_; } /* Test the input arguments Computing MAX */ i__1 = *n << 1; lwkmin = max(i__1,1); lwkopt = lwkmin; work[1].r = (real) lwkopt, work[1].i = 0.f; lquery = *lwork == -1; *info = 0; if (ijobvl <= 0) { *info = -1; } else if (ijobvr <= 0) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*lda < max(1,*n)) { *info = -5; } else if (*ldb < max(1,*n)) { *info = -7; } else if (*ldvsl < 1 || ilvsl && *ldvsl < *n) { *info = -11; } else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) { *info = -13; } else if (*lwork < lwkmin && ! lquery) { *info = -15; } if (*info == 0) { nb1 = ilaenv_(&c__1, "CGEQRF", " ", n, n, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); nb2 = ilaenv_(&c__1, "CUNMQR", " ", n, n, n, &c_n1, (ftnlen)6, ( ftnlen)1); nb3 = ilaenv_(&c__1, "CUNGQR", " ", n, n, n, &c_n1, (ftnlen)6, ( ftnlen)1); /* Computing MAX */ i__1 = max(nb1,nb2); nb = max(i__1,nb3); lopt = *n * (nb + 1); work[1].r = (real) lopt, work[1].i = 0.f; } if (*info != 0) { i__1 = -(*info); xerbla_("CGEGS ", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Get machine constants */ eps = slamch_("E") * slamch_("B"); safmin = slamch_("S"); smlnum = *n * safmin / 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_n1, &c_n1, &anrm, &anrmto, n, n, &a[a_offset], lda, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } } /* 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_n1, &c_n1, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } } /* Permute the matrix to make it more nearly triangular */ ileft = 1; iright = *n + 1; irwork = iright + *n; iwork = 1; cggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &rwork[ ileft], &rwork[iright], &rwork[irwork], &iinfo); if (iinfo != 0) { *info = *n + 1; goto L10; } /* Reduce B to triangular form, and initialize VSL and/or VSR */ irows = ihi + 1 - ilo; icols = *n + 1 - ilo; itau = iwork; iwork = itau + irows; i__1 = *lwork + 1 - iwork; cgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__3 = iwork; i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1; lwkopt = max(i__1,i__2); } if (iinfo != 0) { *info = *n + 2; goto L10; } i__1 = *lwork + 1 - iwork; cunmqr_("L", "C", &irows, &icols, &irows, &b_ref(ilo, ilo), ldb, &work[ itau], &a_ref(ilo, ilo), lda, &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__3 = iwork; i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1; lwkopt = max(i__1,i__2); } if (iinfo != 0) { *info = *n + 3; goto L10; } if (ilvsl) { claset_("Full", n, n, &c_b1, &c_b2, &vsl[vsl_offset], ldvsl); i__1 = irows - 1; i__2 = irows - 1; clacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vsl_ref(ilo + 1, ilo), ldvsl); i__1 = *lwork + 1 - iwork; cungqr_(&irows, &irows, &irows, &vsl_ref(ilo, ilo), ldvsl, &work[itau] , &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__3 = iwork; i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1; lwkopt = max(i__1,i__2); } if (iinfo != 0) { *info = *n + 4; goto L10; } } if (ilvsr) { claset_("Full", n, n, &c_b1, &c_b2, &vsr[vsr_offset], ldvsr); } /* Reduce to generalized Hessenberg form */ cgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset], ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &iinfo); if (iinfo != 0) { *info = *n + 5; goto L10; } /* Perform QZ algorithm, computing Schur vectors if desired */ iwork = itau; i__1 = *lwork + 1 - iwork; 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[iwork], &i__1, &rwork[irwork], & iinfo); if (iinfo >= 0) { /* Computing MAX */ i__3 = iwork; i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1; lwkopt = max(i__1,i__2); } if (iinfo != 0) { if (iinfo > 0 && iinfo <= *n) { *info = iinfo; } else if (iinfo > *n && iinfo <= *n << 1) { *info = iinfo - *n; } else { *info = *n + 6; } goto L10; } /* Apply permutation to VSL and VSR */ if (ilvsl) { cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, & vsl[vsl_offset], ldvsl, &iinfo); if (iinfo != 0) { *info = *n + 7; goto L10; } } if (ilvsr) { cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, & vsr[vsr_offset], ldvsr, &iinfo); if (iinfo != 0) { *info = *n + 8; goto L10; } } /* Undo scaling */ if (ilascl) { clascl_("U", &c_n1, &c_n1, &anrmto, &anrm, n, n, &a[a_offset], lda, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } clascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alpha[1], n, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } } if (ilbscl) { clascl_("U", &c_n1, &c_n1, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } clascl_("G", &c_n1, &c_n1, &bnrmto, &bnrm, n, &c__1, &beta[1], n, & iinfo); if (iinfo != 0) { *info = *n + 9; return 0; } } L10: work[1].r = (real) lwkopt, work[1].i = 0.f; return 0; /* End of CGEGS */ } /* cgegs_ */