#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int cgegv_(char *jobvl, char *jobvr, integer *n, complex *a, integer *lda, complex *b, integer *ldb, complex *alpha, complex *beta, complex *vl, integer *ldvl, complex *vr, integer *ldvr, complex * work, integer *lwork, real *rwork, integer *info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= This routine is deprecated and has been replaced by routine CGGEV. CGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B). Given two square matrices A and B, the generalized nonsymmetric eigenvalue problem (GNEP) is to find the eigenvalues lambda and corresponding (non-zero) eigenvectors x such that A*x = lambda*B*x. An alternate form is to find the eigenvalues mu and corresponding eigenvectors y such that mu*A*y = B*y. These two forms are equivalent with mu = 1/lambda and x = y if neither lambda nor mu is zero. In order to deal with the case that lambda or mu is zero or small, two values alpha and beta are returned for each eigenvalue, such that lambda = alpha/beta and mu = beta/alpha. The vectors x and y in the above equations are right eigenvectors of the matrix pair (A,B). Vectors u and v satisfying u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B are left eigenvectors of (A,B). Note: this routine performs "full balancing" on A and B -- see "Further Details", below. Arguments ========= JOBVL (input) CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors (returned in VL). JOBVR (input) CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors (returned in VR). N (input) INTEGER The order of the matrices A, B, VL, and VR. N >= 0. A (input/output) COMPLEX array, dimension (LDA, N) On entry, the matrix A. If JOBVL = 'V' or JOBVR = 'V', then on exit A contains the Schur form of A from the generalized Schur factorization of the pair (A,B) after balancing. If no eigenvectors were computed, then only the diagonal elements of the Schur form will be correct. See CGGHRD and CHGEQZ for details. LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/output) COMPLEX array, dimension (LDB, N) On entry, the matrix B. If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the upper triangular matrix obtained from B in the generalized Schur factorization of the pair (A,B) after balancing. If no eigenvectors were computed, then only the diagonal elements of B will be correct. See CGGHRD and CHGEQZ for details. LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). ALPHA (output) COMPLEX array, dimension (N) The complex scalars alpha that define the eigenvalues of GNEP. BETA (output) COMPLEX array, dimension (N) The complex scalars beta that define the eigenvalues of GNEP. Together, the quantities alpha = ALPHA(j) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed. VL (output) COMPLEX array, dimension (LDVL,N) If JOBVL = 'V', the left eigenvectors u(j) are stored in the columns of VL, in the same order as their eigenvalues. Each eigenvector is scaled so that its largest component has abs(real part) + abs(imag. part) = 1, except for eigenvectors corresponding to an eigenvalue with alpha = beta = 0, which are set to zero. 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) COMPLEX array, dimension (LDVR,N) If JOBVR = 'V', the right eigenvectors x(j) are stored in the columns of VR, in the same order as their eigenvalues. Each eigenvector is scaled so that its largest component has abs(real part) + abs(imag. part) = 1, except for eigenvectors corresponding to an eigenvalue with alpha = beta = 0, which are set to zero. 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) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. 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 MAX( 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. RWORK (workspace/output) REAL array, dimension (8*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. No eigenvectors have been calculated, 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 CTGEVC =N+8: error return from CGGBAK (computing VL) =N+9: error return from CGGBAK (computing VR) =N+10: error return from CLASCL (various calls) Further Details =============== Balancing --------- This driver calls CGGBAL to both permute and scale rows and columns of A and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R will be upper triangular except for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible. The diagonal scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the elements that start out zero.) After the eigenvalues and eigenvectors of the balanced matrices have been computed, CGGBAK transforms the eigenvectors back to what they would have been (in perfect arithmetic) if they had not been balanced. Contents of A and B on Exit -------- -- - --- - -- ---- If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both), then on exit the arrays A and B will contain the complex Schur form[*] of the "balanced" versions of A and B. If no eigenvectors are computed, then only the diagonal blocks will be correct. [*] In other words, upper triangular form. ===================================================================== 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; static real c_b29 = 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, i__3, i__4; real r__1, r__2, r__3, r__4; complex q__1, q__2; /* Builtin functions */ double r_imag(complex *); /* Local variables */ static integer jc, nb, in, jr, nb1, nb2, nb3, ihi, ilo; static real eps; static logical ilv; static real absb, anrm, bnrm; static integer itau; static real temp; static logical ilvl, ilvr; static integer lopt; static real anrm1, anrm2, bnrm1, bnrm2, absai, scale, absar, sbeta; extern logical lsame_(char *, char *); static integer ileft, iinfo, icols, iwork, 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 *); 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 *); static real salfai; extern /* Subroutine */ int clascl_(char *, integer *, integer *, real *, real *, integer *, integer *, complex *, integer *, integer *), cgeqrf_(integer *, integer *, complex *, integer *, complex *, complex *, integer *, integer *); static real salfar; 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 ctgevc_(char *, char *, logical *, integer *, complex *, integer *, complex *, integer *, complex *, integer *, complex *, integer *, integer *, integer *, complex *, real *, integer *); static real safmax; static char chtemp[1]; static logical ldumma[1]; extern /* Subroutine */ int chgeqz_(char *, char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, complex *, complex *, complex *, integer *, complex *, integer *, complex *, integer *, real *, integer *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); static integer ijobvl, iright; static logical ilimit; static integer ijobvr; extern /* Subroutine */ int cungqr_(integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, integer *); static integer lwkmin; extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, complex *, integer *, integer *); static integer irwork, lwkopt; static logical lquery; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --alpha; --beta; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1; vr -= vr_offset; --work; --rwork; /* 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 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 (*ldvl < 1 || ilvl && *ldvl < *n) { *info = -11; } else if (*ldvr < 1 || ilvr && *ldvr < *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); /* Computing MAX */ i__1 = *n << 1, i__2 = *n * (nb + 1); lopt = max(i__1,i__2); work[1].r = (real) lopt, work[1].i = 0.f; } if (*info != 0) { i__1 = -(*info); xerbla_("CGEGV ", &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"); safmin += safmin; safmax = 1.f / safmin; /* Scale A */ anrm = clange_("M", n, n, &a[a_offset], lda, &rwork[1]); anrm1 = anrm; anrm2 = 1.f; if (anrm < 1.f) { if (safmax * anrm < 1.f) { anrm1 = safmin; anrm2 = safmax * anrm; } } if (anrm > 0.f) { clascl_("G", &c_n1, &c_n1, &anrm, &c_b29, n, n, &a[a_offset], lda, & iinfo); if (iinfo != 0) { *info = *n + 10; return 0; } } /* Scale B */ bnrm = clange_("M", n, n, &b[b_offset], ldb, &rwork[1]); bnrm1 = bnrm; bnrm2 = 1.f; if (bnrm < 1.f) { if (safmax * bnrm < 1.f) { bnrm1 = safmin; bnrm2 = safmax * bnrm; } } if (bnrm > 0.f) { clascl_("G", &c_n1, &c_n1, &bnrm, &c_b29, n, n, &b[b_offset], ldb, & iinfo); if (iinfo != 0) { *info = *n + 10; return 0; } } /* Permute the matrix to make it more nearly triangular Also "balance" the matrix. */ ileft = 1; iright = *n + 1; irwork = iright + *n; 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 L80; } /* Reduce B to triangular form, and initialize VL and/or VR */ irows = ihi + 1 - ilo; if (ilv) { icols = *n + 1 - ilo; } else { icols = irows; } itau = 1; iwork = itau + irows; i__1 = *lwork + 1 - iwork; cgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], 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 L80; } i__1 = *lwork + 1 - iwork; cunmqr_("L", "C", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, & work[itau], &a[ilo + ilo * a_dim1], 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 L80; } if (ilvl) { claset_("Full", n, n, &c_b1, &c_b2, &vl[vl_offset], ldvl); i__1 = irows - 1; i__2 = irows - 1; clacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vl[ilo + 1 + ilo * vl_dim1], ldvl); i__1 = *lwork + 1 - iwork; cungqr_(&irows, &irows, &irows, &vl[ilo + ilo * vl_dim1], ldvl, &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 L80; } } if (ilvr) { claset_("Full", n, n, &c_b1, &c_b2, &vr[vr_offset], ldvr); } /* Reduce to generalized Hessenberg form */ if (ilv) { /* Eigenvectors requested -- work on whole matrix. */ cgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset], ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &iinfo); } else { cgghrd_("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, &iinfo); } if (iinfo != 0) { *info = *n + 5; goto L80; } /* Perform QZ algorithm */ iwork = itau; if (ilv) { *(unsigned char *)chtemp = 'S'; } else { *(unsigned char *)chtemp = 'E'; } i__1 = *lwork + 1 - iwork; chgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[ b_offset], ldb, &alpha[1], &beta[1], &vl[vl_offset], ldvl, &vr[ vr_offset], ldvr, &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 L80; } if (ilv) { /* Compute Eigenvectors */ if (ilvl) { if (ilvr) { *(unsigned char *)chtemp = 'B'; } else { *(unsigned char *)chtemp = 'L'; } } else { *(unsigned char *)chtemp = 'R'; } ctgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[ iwork], &rwork[irwork], &iinfo); if (iinfo != 0) { *info = *n + 7; goto L80; } /* Undo balancing on VL and VR, rescale */ if (ilvl) { cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &vl[vl_offset], ldvl, &iinfo); if (iinfo != 0) { *info = *n + 8; goto L80; } i__1 = *n; for (jc = 1; jc <= i__1; ++jc) { temp = 0.f; i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ i__3 = jr + jc * vl_dim1; r__3 = temp, r__4 = (r__1 = vl[i__3].r, dabs(r__1)) + ( r__2 = r_imag(&vl[jr + jc * vl_dim1]), dabs(r__2)) ; temp = dmax(r__3,r__4); /* L10: */ } if (temp < safmin) { goto L30; } temp = 1.f / temp; i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { i__3 = jr + jc * vl_dim1; i__4 = jr + jc * vl_dim1; q__1.r = temp * vl[i__4].r, q__1.i = temp * vl[i__4].i; vl[i__3].r = q__1.r, vl[i__3].i = q__1.i; /* L20: */ } L30: ; } } if (ilvr) { cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &vr[vr_offset], ldvr, &iinfo); if (iinfo != 0) { *info = *n + 9; goto L80; } i__1 = *n; for (jc = 1; jc <= i__1; ++jc) { temp = 0.f; i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ i__3 = jr + jc * vr_dim1; r__3 = temp, r__4 = (r__1 = vr[i__3].r, dabs(r__1)) + ( r__2 = r_imag(&vr[jr + jc * vr_dim1]), dabs(r__2)) ; temp = dmax(r__3,r__4); /* L40: */ } if (temp < safmin) { goto L60; } temp = 1.f / temp; i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { i__3 = jr + jc * vr_dim1; i__4 = jr + jc * vr_dim1; q__1.r = temp * vr[i__4].r, q__1.i = temp * vr[i__4].i; vr[i__3].r = q__1.r, vr[i__3].i = q__1.i; /* L50: */ } L60: ; } } /* End of eigenvector calculation */ } /* Undo scaling in alpha, beta Note: this does not give the alpha and beta for the unscaled problem. Un-scaling is limited to avoid underflow in alpha and beta if they are significant. */ i__1 = *n; for (jc = 1; jc <= i__1; ++jc) { i__2 = jc; absar = (r__1 = alpha[i__2].r, dabs(r__1)); absai = (r__1 = r_imag(&alpha[jc]), dabs(r__1)); i__2 = jc; absb = (r__1 = beta[i__2].r, dabs(r__1)); i__2 = jc; salfar = anrm * alpha[i__2].r; salfai = anrm * r_imag(&alpha[jc]); i__2 = jc; sbeta = bnrm * beta[i__2].r; ilimit = FALSE_; scale = 1.f; /* Check for significant underflow in imaginary part of ALPHA Computing MAX */ r__1 = safmin, r__2 = eps * absar, r__1 = max(r__1,r__2), r__2 = eps * absb; if (dabs(salfai) < safmin && absai >= dmax(r__1,r__2)) { ilimit = TRUE_; /* Computing MAX */ r__1 = safmin, r__2 = anrm2 * absai; scale = safmin / anrm1 / dmax(r__1,r__2); } /* Check for significant underflow in real part of ALPHA Computing MAX */ r__1 = safmin, r__2 = eps * absai, r__1 = max(r__1,r__2), r__2 = eps * absb; if (dabs(salfar) < safmin && absar >= dmax(r__1,r__2)) { ilimit = TRUE_; /* Computing MAX Computing MAX */ r__3 = safmin, r__4 = anrm2 * absar; r__1 = scale, r__2 = safmin / anrm1 / dmax(r__3,r__4); scale = dmax(r__1,r__2); } /* Check for significant underflow in BETA Computing MAX */ r__1 = safmin, r__2 = eps * absar, r__1 = max(r__1,r__2), r__2 = eps * absai; if (dabs(sbeta) < safmin && absb >= dmax(r__1,r__2)) { ilimit = TRUE_; /* Computing MAX Computing MAX */ r__3 = safmin, r__4 = bnrm2 * absb; r__1 = scale, r__2 = safmin / bnrm1 / dmax(r__3,r__4); scale = dmax(r__1,r__2); } /* Check for possible overflow when limiting scaling */ if (ilimit) { /* Computing MAX */ r__1 = dabs(salfar), r__2 = dabs(salfai), r__1 = max(r__1,r__2), r__2 = dabs(sbeta); temp = scale * safmin * dmax(r__1,r__2); if (temp > 1.f) { scale /= temp; } if (scale < 1.f) { ilimit = FALSE_; } } /* Recompute un-scaled ALPHA, BETA if necessary. */ if (ilimit) { i__2 = jc; salfar = scale * alpha[i__2].r * anrm; salfai = scale * r_imag(&alpha[jc]) * anrm; i__2 = jc; q__2.r = scale * beta[i__2].r, q__2.i = scale * beta[i__2].i; q__1.r = bnrm * q__2.r, q__1.i = bnrm * q__2.i; sbeta = q__1.r; } i__2 = jc; q__1.r = salfar, q__1.i = salfai; alpha[i__2].r = q__1.r, alpha[i__2].i = q__1.i; i__2 = jc; beta[i__2].r = sbeta, beta[i__2].i = 0.f; /* L70: */ } L80: work[1].r = (real) lwkopt, work[1].i = 0.f; return 0; /* End of CGEGV */ } /* cgegv_ */