#include "blaswrap.h" #include "f2c.h" /* 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 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 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 (external procedure) 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 (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 LWORK >= max(8*N,6*N+16). 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. 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_b38 = 0.f; static real c_b39 = 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 integer i__, ip; static real dif[2]; static integer ihi, ilo; static real eps, anrm, bnrm; static integer idum[1], ierr, itau, iwrk; static real pvsl, pvsr; 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 *), 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 *); a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --alphar; --alphai; --beta; vsl_dim1 = *ldvsl; vsl_offset = 1 + vsl_dim1; vsl -= vsl_offset; vsr_dim1 = *ldvsr; vsr_offset = 1 + vsr_dim1; 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.) */ if (*info == 0) { if (*n > 0) { /* Computing MAX */ i__1 = *n << 3, i__2 = *n * 6 + 16; minwrk = max(i__1,i__2); maxwrk = minwrk - *n + *n * ilaenv_(&c__1, "SGEQRF", " ", n, & c__1, n, &c__0, (ftnlen)6, (ftnlen)1); /* Computing MAX */ i__1 = maxwrk, i__2 = minwrk - *n + *n * ilaenv_(&c__1, "SORMQR", " ", n, &c__1, n, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); if (ilvsl) { /* Computing MAX */ i__1 = maxwrk, i__2 = minwrk - *n + *n * ilaenv_(&c__1, "SOR" "GQR", " ", n, &c__1, n, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); } } else { minwrk = 1; maxwrk = 1; } 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[ilo + ilo * b_dim1], 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[ilo + ilo * b_dim1], ldb, & work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwrk], &i__1, & ierr); /* Initialize VSL (Workspace: need N, prefer N*NB) */ if (ilvsl) { slaset_("Full", n, n, &c_b38, &c_b39, &vsl[vsl_offset], ldvsl); if (irows > 1) { i__1 = irows - 1; i__2 = irows - 1; slacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vsl[ ilo + 1 + ilo * vsl_dim1], ldvsl); } i__1 = *lwork + 1 - iwrk; sorgqr_(&irows, &irows, &irows, &vsl[ilo + ilo * vsl_dim1], ldvsl, & work[itau], &work[iwrk], &i__1, &ierr); } /* Initialize VSR */ if (ilvsr) { slaset_("Full", n, n, &c_b38, &c_b39, &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[i__ + i__ * a_dim1] / 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[i__ + (i__ + 1) * a_dim1] / 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[i__ + i__ * b_dim1] / 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_ */