#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int sggev_(char *jobvl, char *jobvr, integer *n, real *a, integer *lda, real *b, integer *ldb, real *alphar, real *alphai, real *beta, real *vl, integer *ldvl, real *vr, integer *ldvr, real *work, integer *lwork, integer *info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= SGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*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. The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j). The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies u(j)**H * A = lambda(j) * u(j)**H * B . where u(j)**H is the conjugate-transpose of u(j). Arguments ========= JOBVL (input) CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors. JOBVR (input) CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors. N (input) INTEGER The order of the matrices A, B, VL, and VR. N >= 0. A (input/output) REAL array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/output) REAL array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). 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. 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). VL (output) REAL array, dimension (LDVL,N) If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). Each eigenvector is scaled so the largest component has abs(real part)+abs(imag. part)=1. Not referenced if JOBVL = 'N'. LDVL (input) INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N. VR (output) REAL array, dimension (LDVR,N) If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each eigenvector is scaled so the largest component has abs(real part)+abs(imag. part)=1. Not referenced if JOBVR = 'N'. LDVR (input) INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= 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. LWORK >= max(1,8*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. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. = 1,...,N: The QZ iteration failed. No eigenvectors have been calculated, 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: error return from STGEVC. ===================================================================== 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_b36 = 0.f; static real c_b37 = 1.f; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1, i__2; real r__1, r__2, r__3, r__4; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer jc, in, jr, ihi, ilo; static real eps; static logical ilv; static real anrm, bnrm; static integer ierr, itau; static real temp; static logical ilvl, ilvr; static integer iwrk; extern logical lsame_(char *, char *); static integer ileft, icols, irows; 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 *); extern /* Subroutine */ int xerbla_(char *, integer *), sgghrd_( char *, char *, integer *, integer *, integer *, real *, integer * , real *, integer *, real *, integer *, real *, integer *, integer *); static logical ldumma[1]; static char chtemp[1]; 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 *), stgevc_( char *, char *, logical *, integer *, real *, integer *, real *, integer *, real *, integer *, real *, integer *, integer *, integer *, real *, integer *); static real anrmto, bnrmto; extern /* Subroutine */ int shgeqz_(char *, char *, char *, integer *, integer *, integer *, real *, integer *, real *, integer *, real * , real *, real *, real *, integer *, real *, integer *, real *, integer *, integer *); static integer minwrk, maxwrk; static real smlnum; extern /* Subroutine */ int sorgqr_(integer *, integer *, integer *, real *, integer *, real *, real *, integer *, integer *); static logical 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; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1; vr -= vr_offset; --work; /* Function Body */ if (lsame_(jobvl, "N")) { ijobvl = 1; ilvl = FALSE_; } else if (lsame_(jobvl, "V")) { ijobvl = 2; ilvl = TRUE_; } else { ijobvl = -1; ilvl = FALSE_; } if (lsame_(jobvr, "N")) { ijobvr = 1; ilvr = FALSE_; } else if (lsame_(jobvr, "V")) { ijobvr = 2; ilvr = TRUE_; } else { ijobvr = -1; ilvr = FALSE_; } ilv = ilvl || ilvr; /* Test the input arguments */ *info = 0; lquery = *lwork == -1; 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 (*ldvl < 1 || ilvl && *ldvl < *n) { *info = -12; } else if (*ldvr < 1 || ilvr && *ldvr < *n) { *info = -14; } /* 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. The workspace is computed assuming ILO = 1 and IHI = N, the worst case.) */ if (*info == 0) { /* Computing MAX */ i__1 = 1, i__2 = *n << 3; minwrk = max(i__1,i__2); /* Computing MAX */ i__1 = 1, i__2 = *n * (ilaenv_(&c__1, "SGEQRF", " ", n, &c__1, n, & c__0, (ftnlen)6, (ftnlen)1) + 7); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * (ilaenv_(&c__1, "SORMQR", " ", n, &c__1, n, &c__0, (ftnlen)6, (ftnlen)1) + 7); maxwrk = max(i__1,i__2); if (ilvl) { /* Computing MAX */ i__1 = maxwrk, i__2 = *n * (ilaenv_(&c__1, "SORGQR", " ", n, & c__1, n, &c_n1, (ftnlen)6, (ftnlen)1) + 7); maxwrk = max(i__1,i__2); } work[1] = (real) maxwrk; if (*lwork < minwrk && ! lquery) { *info = -16; } } if (*info != 0) { i__1 = -(*info); xerbla_("SGGEV ", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 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 = 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 matrices A, B to isolate eigenvalues if possible (Workspace: need 6*N) */ 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; if (ilv) { icols = *n + 1 - ilo; } else { icols = irows; } 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 VL (Workspace: need N, prefer N*NB) */ if (ilvl) { slaset_("Full", n, n, &c_b36, &c_b37, &vl[vl_offset], ldvl) ; if (irows > 1) { i__1 = irows - 1; i__2 = irows - 1; slacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vl[ ilo + 1 + ilo * vl_dim1], ldvl); } i__1 = *lwork + 1 - iwrk; sorgqr_(&irows, &irows, &irows, &vl[ilo + ilo * vl_dim1], ldvl, &work[ itau], &work[iwrk], &i__1, &ierr); } /* Initialize VR */ if (ilvr) { slaset_("Full", n, n, &c_b36, &c_b37, &vr[vr_offset], ldvr) ; } /* Reduce to generalized Hessenberg form (Workspace: none needed) */ if (ilv) { /* Eigenvectors requested -- work on whole matrix. */ sgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset], ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &ierr); } else { sgghrd_("N", "N", &irows, &c__1, &irows, &a[ilo + ilo * a_dim1], lda, &b[ilo + ilo * b_dim1], ldb, &vl[vl_offset], ldvl, &vr[ vr_offset], ldvr, &ierr); } /* Perform QZ algorithm (Compute eigenvalues, and optionally, the Schur forms and Schur vectors) (Workspace: need N) */ iwrk = itau; if (ilv) { *(unsigned char *)chtemp = 'S'; } else { *(unsigned char *)chtemp = 'E'; } i__1 = *lwork + 1 - iwrk; shgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[ b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &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 L110; } /* Compute Eigenvectors (Workspace: need 6*N) */ if (ilv) { if (ilvl) { if (ilvr) { *(unsigned char *)chtemp = 'B'; } else { *(unsigned char *)chtemp = 'L'; } } else { *(unsigned char *)chtemp = 'R'; } stgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[ iwrk], &ierr); if (ierr != 0) { *info = *n + 2; goto L110; } /* Undo balancing on VL and VR and normalization (Workspace: none needed) */ if (ilvl) { sggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, & vl[vl_offset], ldvl, &ierr); i__1 = *n; for (jc = 1; jc <= i__1; ++jc) { if (alphai[jc] < 0.f) { goto L50; } temp = 0.f; if (alphai[jc] == 0.f) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ r__2 = temp, r__3 = (r__1 = vl[jr + jc * vl_dim1], dabs(r__1)); temp = dmax(r__2,r__3); /* L10: */ } } else { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ r__3 = temp, r__4 = (r__1 = vl[jr + jc * vl_dim1], dabs(r__1)) + (r__2 = vl[jr + (jc + 1) * vl_dim1], dabs(r__2)); temp = dmax(r__3,r__4); /* L20: */ } } if (temp < smlnum) { goto L50; } temp = 1.f / temp; if (alphai[jc] == 0.f) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vl[jr + jc * vl_dim1] *= temp; /* L30: */ } } else { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vl[jr + jc * vl_dim1] *= temp; vl[jr + (jc + 1) * vl_dim1] *= temp; /* L40: */ } } L50: ; } } if (ilvr) { sggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, & vr[vr_offset], ldvr, &ierr); i__1 = *n; for (jc = 1; jc <= i__1; ++jc) { if (alphai[jc] < 0.f) { goto L100; } temp = 0.f; if (alphai[jc] == 0.f) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ r__2 = temp, r__3 = (r__1 = vr[jr + jc * vr_dim1], dabs(r__1)); temp = dmax(r__2,r__3); /* L60: */ } } else { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ r__3 = temp, r__4 = (r__1 = vr[jr + jc * vr_dim1], dabs(r__1)) + (r__2 = vr[jr + (jc + 1) * vr_dim1], dabs(r__2)); temp = dmax(r__3,r__4); /* L70: */ } } if (temp < smlnum) { goto L100; } temp = 1.f / temp; if (alphai[jc] == 0.f) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vr[jr + jc * vr_dim1] *= temp; /* L80: */ } } else { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vr[jr + jc * vr_dim1] *= temp; vr[jr + (jc + 1) * vr_dim1] *= temp; /* L90: */ } } L100: ; } } /* End of eigenvector calculation */ } /* Undo scaling if necessary */ 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); } L110: work[1] = (real) maxwrk; return 0; /* End of SGGEV */ } /* sggev_ */