/* ztgsna.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__1 = 1; static doublecomplex c_b19 = {1.,0.}; static doublecomplex c_b20 = {0.,0.}; static logical c_false = FALSE_; static integer c__3 = 3; /* Subroutine */ int ztgsna_(char *job, char *howmny, logical *select, integer *n, doublecomplex *a, integer *lda, doublecomplex *b, integer *ldb, doublecomplex *vl, integer *ldvl, doublecomplex *vr, integer * ldvr, doublereal *s, doublereal *dif, integer *mm, integer *m, doublecomplex *work, integer *lwork, integer *iwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1; doublereal d__1, d__2; doublecomplex z__1; /* Builtin functions */ double z_abs(doublecomplex *); /* Local variables */ integer i__, k, n1, n2, ks; doublereal eps, cond; integer ierr, ifst; doublereal lnrm; doublecomplex yhax, yhbx; integer ilst; doublereal rnrm, scale; extern logical lsame_(char *, char *); extern /* Double Complex */ VOID zdotc_(doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); integer lwmin; extern /* Subroutine */ int zgemv_(char *, integer *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *); logical wants; doublecomplex dummy[1]; extern doublereal dlapy2_(doublereal *, doublereal *); extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); doublecomplex dummy1[1]; extern doublereal dznrm2_(integer *, doublecomplex *, integer *), dlamch_( char *); extern /* Subroutine */ int xerbla_(char *, integer *); doublereal bignum; logical wantbh, wantdf, somcon; extern /* Subroutine */ int zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *), ztgexc_(logical *, logical *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer *, integer *, integer *); doublereal smlnum; logical lquery; extern /* Subroutine */ int ztgsyl_(char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, doublecomplex *, integer *, integer *, integer *); /* -- LAPACK routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* ZTGSNA estimates reciprocal condition numbers for specified */ /* eigenvalues and/or eigenvectors of a matrix pair (A, B). */ /* (A, B) must be in generalized Schur canonical form, that is, A and */ /* B are both upper triangular. */ /* Arguments */ /* ========= */ /* JOB (input) CHARACTER*1 */ /* Specifies whether condition numbers are required for */ /* eigenvalues (S) or eigenvectors (DIF): */ /* = 'E': for eigenvalues only (S); */ /* = 'V': for eigenvectors only (DIF); */ /* = 'B': for both eigenvalues and eigenvectors (S and DIF). */ /* HOWMNY (input) CHARACTER*1 */ /* = 'A': compute condition numbers for all eigenpairs; */ /* = 'S': compute condition numbers for selected eigenpairs */ /* specified by the array SELECT. */ /* SELECT (input) LOGICAL array, dimension (N) */ /* If HOWMNY = 'S', SELECT specifies the eigenpairs for which */ /* condition numbers are required. To select condition numbers */ /* for the corresponding j-th eigenvalue and/or eigenvector, */ /* SELECT(j) must be set to .TRUE.. */ /* If HOWMNY = 'A', SELECT is not referenced. */ /* N (input) INTEGER */ /* The order of the square matrix pair (A, B). N >= 0. */ /* A (input) COMPLEX*16 array, dimension (LDA,N) */ /* The upper triangular matrix A in the pair (A,B). */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,N). */ /* B (input) COMPLEX*16 array, dimension (LDB,N) */ /* The upper triangular matrix B in the pair (A, B). */ /* LDB (input) INTEGER */ /* The leading dimension of the array B. LDB >= max(1,N). */ /* VL (input) COMPLEX*16 array, dimension (LDVL,M) */ /* IF JOB = 'E' or 'B', VL must contain left eigenvectors of */ /* (A, B), corresponding to the eigenpairs specified by HOWMNY */ /* and SELECT. The eigenvectors must be stored in consecutive */ /* columns of VL, as returned by ZTGEVC. */ /* If JOB = 'V', VL is not referenced. */ /* LDVL (input) INTEGER */ /* The leading dimension of the array VL. LDVL >= 1; and */ /* If JOB = 'E' or 'B', LDVL >= N. */ /* VR (input) COMPLEX*16 array, dimension (LDVR,M) */ /* IF JOB = 'E' or 'B', VR must contain right eigenvectors of */ /* (A, B), corresponding to the eigenpairs specified by HOWMNY */ /* and SELECT. The eigenvectors must be stored in consecutive */ /* columns of VR, as returned by ZTGEVC. */ /* If JOB = 'V', VR is not referenced. */ /* LDVR (input) INTEGER */ /* The leading dimension of the array VR. LDVR >= 1; */ /* If JOB = 'E' or 'B', LDVR >= N. */ /* S (output) DOUBLE PRECISION array, dimension (MM) */ /* If JOB = 'E' or 'B', the reciprocal condition numbers of the */ /* selected eigenvalues, stored in consecutive elements of the */ /* array. */ /* If JOB = 'V', S is not referenced. */ /* DIF (output) DOUBLE PRECISION array, dimension (MM) */ /* If JOB = 'V' or 'B', the estimated reciprocal condition */ /* numbers of the selected eigenvectors, stored in consecutive */ /* elements of the array. */ /* If the eigenvalues cannot be reordered to compute DIF(j), */ /* DIF(j) is set to 0; this can only occur when the true value */ /* would be very small anyway. */ /* For each eigenvalue/vector specified by SELECT, DIF stores */ /* a Frobenius norm-based estimate of Difl. */ /* If JOB = 'E', DIF is not referenced. */ /* MM (input) INTEGER */ /* The number of elements in the arrays S and DIF. MM >= M. */ /* M (output) INTEGER */ /* The number of elements of the arrays S and DIF used to store */ /* the specified condition numbers; for each selected eigenvalue */ /* one element is used. If HOWMNY = 'A', M is set to N. */ /* WORK (workspace/output) COMPLEX*16 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,N). */ /* If JOB = 'V' or 'B', LWORK >= max(1,2*N*N). */ /* IWORK (workspace) INTEGER array, dimension (N+2) */ /* If JOB = 'E', IWORK is not referenced. */ /* INFO (output) INTEGER */ /* = 0: Successful exit */ /* < 0: If INFO = -i, the i-th argument had an illegal value */ /* Further Details */ /* =============== */ /* The reciprocal of the condition number of the i-th generalized */ /* eigenvalue w = (a, b) is defined as */ /* S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v)) */ /* where u and v are the right and left eigenvectors of (A, B) */ /* corresponding to w; |z| denotes the absolute value of the complex */ /* number, and norm(u) denotes the 2-norm of the vector u. The pair */ /* (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the */ /* matrix pair (A, B). If both a and b equal zero, then (A,B) is */ /* singular and S(I) = -1 is returned. */ /* An approximate error bound on the chordal distance between the i-th */ /* computed generalized eigenvalue w and the corresponding exact */ /* eigenvalue lambda is */ /* chord(w, lambda) <= EPS * norm(A, B) / S(I), */ /* where EPS is the machine precision. */ /* The reciprocal of the condition number of the right eigenvector u */ /* and left eigenvector v corresponding to the generalized eigenvalue w */ /* is defined as follows. Suppose */ /* (A, B) = ( a * ) ( b * ) 1 */ /* ( 0 A22 ),( 0 B22 ) n-1 */ /* 1 n-1 1 n-1 */ /* Then the reciprocal condition number DIF(I) is */ /* Difl[(a, b), (A22, B22)] = sigma-min( Zl ) */ /* where sigma-min(Zl) denotes the smallest singular value of */ /* Zl = [ kron(a, In-1) -kron(1, A22) ] */ /* [ kron(b, In-1) -kron(1, B22) ]. */ /* Here In-1 is the identity matrix of size n-1 and X' is the conjugate */ /* transpose of X. kron(X, Y) is the Kronecker product between the */ /* matrices X and Y. */ /* We approximate the smallest singular value of Zl with an upper */ /* bound. This is done by ZLATDF. */ /* An approximate error bound for a computed eigenvector VL(i) or */ /* VR(i) is given by */ /* EPS * norm(A, B) / DIF(i). */ /* See ref. [2-3] for more details and further references. */ /* Based on contributions by */ /* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */ /* Umea University, S-901 87 Umea, Sweden. */ /* References */ /* ========== */ /* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */ /* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */ /* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */ /* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */ /* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */ /* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */ /* Estimation: Theory, Algorithms and Software, Report */ /* UMINF - 94.04, Department of Computing Science, Umea University, */ /* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. */ /* To appear in Numerical Algorithms, 1996. */ /* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */ /* for Solving the Generalized Sylvester Equation and Estimating the */ /* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */ /* Department of Computing Science, Umea University, S-901 87 Umea, */ /* Sweden, December 1993, Revised April 1994, Also as LAPACK Working */ /* Note 75. */ /* To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. Local Arrays .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Decode and test the input parameters */ /* Parameter adjustments */ --select; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1; vr -= vr_offset; --s; --dif; --work; --iwork; /* Function Body */ wantbh = lsame_(job, "B"); wants = lsame_(job, "E") || wantbh; wantdf = lsame_(job, "V") || wantbh; somcon = lsame_(howmny, "S"); *info = 0; lquery = *lwork == -1; if (! wants && ! wantdf) { *info = -1; } else if (! lsame_(howmny, "A") && ! somcon) { *info = -2; } else if (*n < 0) { *info = -4; } else if (*lda < max(1,*n)) { *info = -6; } else if (*ldb < max(1,*n)) { *info = -8; } else if (wants && *ldvl < *n) { *info = -10; } else if (wants && *ldvr < *n) { *info = -12; } else { /* Set M to the number of eigenpairs for which condition numbers */ /* are required, and test MM. */ if (somcon) { *m = 0; i__1 = *n; for (k = 1; k <= i__1; ++k) { if (select[k]) { ++(*m); } /* L10: */ } } else { *m = *n; } if (*n == 0) { lwmin = 1; } else if (lsame_(job, "V") || lsame_(job, "B")) { lwmin = (*n << 1) * *n; } else { lwmin = *n; } work[1].r = (doublereal) lwmin, work[1].i = 0.; if (*mm < *m) { *info = -15; } else if (*lwork < lwmin && ! lquery) { *info = -18; } } if (*info != 0) { i__1 = -(*info); xerbla_("ZTGSNA", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S") / eps; bignum = 1. / smlnum; dlabad_(&smlnum, &bignum); ks = 0; i__1 = *n; for (k = 1; k <= i__1; ++k) { /* Determine whether condition numbers are required for the k-th */ /* eigenpair. */ if (somcon) { if (! select[k]) { goto L20; } } ++ks; if (wants) { /* Compute the reciprocal condition number of the k-th */ /* eigenvalue. */ rnrm = dznrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); lnrm = dznrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); zgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 + 1] , &c__1, &c_b20, &work[1], &c__1); zdotc_(&z__1, n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1); yhax.r = z__1.r, yhax.i = z__1.i; zgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 + 1] , &c__1, &c_b20, &work[1], &c__1); zdotc_(&z__1, n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1); yhbx.r = z__1.r, yhbx.i = z__1.i; d__1 = z_abs(&yhax); d__2 = z_abs(&yhbx); cond = dlapy2_(&d__1, &d__2); if (cond == 0.) { s[ks] = -1.; } else { s[ks] = cond / (rnrm * lnrm); } } if (wantdf) { if (*n == 1) { d__1 = z_abs(&a[a_dim1 + 1]); d__2 = z_abs(&b[b_dim1 + 1]); dif[ks] = dlapy2_(&d__1, &d__2); } else { /* Estimate the reciprocal condition number of the k-th */ /* eigenvectors. */ /* Copy the matrix (A, B) to the array WORK and move the */ /* (k,k)th pair to the (1,1) position. */ zlacpy_("Full", n, n, &a[a_offset], lda, &work[1], n); zlacpy_("Full", n, n, &b[b_offset], ldb, &work[*n * *n + 1], n); ifst = k; ilst = 1; ztgexc_(&c_false, &c_false, n, &work[1], n, &work[*n * *n + 1] , n, dummy, &c__1, dummy1, &c__1, &ifst, &ilst, &ierr) ; if (ierr > 0) { /* Ill-conditioned problem - swap rejected. */ dif[ks] = 0.; } else { /* Reordering successful, solve generalized Sylvester */ /* equation for R and L, */ /* A22 * R - L * A11 = A12 */ /* B22 * R - L * B11 = B12, */ /* and compute estimate of Difl[(A11,B11), (A22, B22)]. */ n1 = 1; n2 = *n - n1; i__ = *n * *n + 1; ztgsyl_("N", &c__3, &n2, &n1, &work[*n * n1 + n1 + 1], n, &work[1], n, &work[n1 + 1], n, &work[*n * n1 + n1 + i__], n, &work[i__], n, &work[n1 + i__], n, & scale, &dif[ks], dummy, &c__1, &iwork[1], &ierr); } } } L20: ; } work[1].r = (doublereal) lwmin, work[1].i = 0.; return 0; /* End of ZTGSNA */ } /* ztgsna_ */