#include "blaswrap.h" #include "f2c.h" /* 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) { /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 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. ===================================================================== Decode and test the input parameters Parameter adjustments */ /* 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; /* 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 */ static integer i__, k, n1, n2, ks; static doublereal eps, cond; static integer ierr, ifst; static doublereal lnrm; static doublecomplex yhax, yhbx; static integer ilst; static doublereal rnrm, scale; extern logical lsame_(char *, char *); extern /* Double Complex */ VOID zdotc_(doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); static integer lwmin; extern /* Subroutine */ int zgemv_(char *, integer *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *); static logical wants; static doublecomplex dummy[1]; extern doublereal dlapy2_(doublereal *, doublereal *); extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); static doublecomplex dummy1[1]; extern doublereal dznrm2_(integer *, doublecomplex *, integer *), dlamch_( char *); extern /* Subroutine */ int xerbla_(char *, integer *); static doublereal bignum; static 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 *); static doublereal smlnum; static 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 *); --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_ */