SUBROUTINE ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, $ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, $ IWORK, INFO ) * * -- LAPACK routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER HOWMNY, JOB INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N * .. * .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) DOUBLE PRECISION DIF( * ), S( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ), $ VR( LDVR, * ), WORK( * ) * .. * * 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 .. DOUBLE PRECISION ZERO, ONE INTEGER IDIFJB PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, IDIFJB = 3 ) * .. * .. Local Scalars .. LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2 DOUBLE PRECISION BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM COMPLEX*16 YHAX, YHBX * .. * .. Local Arrays .. COMPLEX*16 DUMMY( 1 ), DUMMY1( 1 ) * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLAPY2, DZNRM2 COMPLEX*16 ZDOTC EXTERNAL LSAME, DLAMCH, DLAPY2, DZNRM2, ZDOTC * .. * .. External Subroutines .. EXTERNAL DLABAD, XERBLA, ZGEMV, ZLACPY, ZTGEXC, ZTGSYL * .. * .. Intrinsic Functions .. INTRINSIC ABS, DCMPLX, MAX * .. * .. Executable Statements .. * * Decode and test the input parameters * WANTBH = LSAME( JOB, 'B' ) WANTS = LSAME( JOB, 'E' ) .OR. WANTBH WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH * SOMCON = LSAME( HOWMNY, 'S' ) * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) * IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN INFO = -1 ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( WANTS .AND. LDVL.LT.N ) THEN INFO = -10 ELSE IF( WANTS .AND. LDVR.LT.N ) THEN INFO = -12 ELSE * * Set M to the number of eigenpairs for which condition numbers * are required, and test MM. * IF( SOMCON ) THEN M = 0 DO 10 K = 1, N IF( SELECT( K ) ) $ M = M + 1 10 CONTINUE ELSE M = N END IF * IF( N.EQ.0 ) THEN LWMIN = 1 ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN LWMIN = 2*N*N ELSE LWMIN = N END IF WORK( 1 ) = LWMIN * IF( MM.LT.M ) THEN INFO = -15 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -18 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZTGSNA', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Get machine constants * EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) KS = 0 DO 20 K = 1, N * * Determine whether condition numbers are required for the k-th * eigenpair. * IF( SOMCON ) THEN IF( .NOT.SELECT( K ) ) $ GO TO 20 END IF * KS = KS + 1 * IF( WANTS ) THEN * * Compute the reciprocal condition number of the k-th * eigenvalue. * RNRM = DZNRM2( N, VR( 1, KS ), 1 ) LNRM = DZNRM2( N, VL( 1, KS ), 1 ) CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), A, LDA, $ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 ) YHAX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 ) CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), B, LDB, $ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 ) YHBX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 ) COND = DLAPY2( ABS( YHAX ), ABS( YHBX ) ) IF( COND.EQ.ZERO ) THEN S( KS ) = -ONE ELSE S( KS ) = COND / ( RNRM*LNRM ) END IF END IF * IF( WANTDF ) THEN IF( N.EQ.1 ) THEN DIF( KS ) = DLAPY2( ABS( A( 1, 1 ) ), ABS( B( 1, 1 ) ) ) 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. * CALL ZLACPY( 'Full', N, N, A, LDA, WORK, N ) CALL ZLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N ) IFST = K ILST = 1 * CALL ZTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), $ N, DUMMY, 1, DUMMY1, 1, IFST, ILST, IERR ) * IF( IERR.GT.0 ) THEN * * Ill-conditioned problem - swap rejected. * DIF( KS ) = ZERO 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 CALL ZTGSYL( 'N', IDIFJB, N2, N1, WORK( N*N1+N1+1 ), $ N, WORK, N, WORK( N1+1 ), N, $ WORK( N*N1+N1+I ), N, WORK( I ), N, $ WORK( N1+I ), N, SCALE, DIF( KS ), DUMMY, $ 1, IWORK, IERR ) END IF END IF END IF * 20 CONTINUE WORK( 1 ) = LWMIN RETURN * * End of ZTGSNA * END