dtgsna.f (3)  Linux Manuals
NAME
dtgsna.f 
SYNOPSIS
Functions/Subroutines
subroutine dtgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
DTGSNA
Function/Subroutine Documentation
subroutine dtgsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, * )B, integerLDB, double precision, dimension( ldvl, * )VL, integerLDVL, double precision, dimension( ldvr, * )VR, integerLDVR, double precision, dimension( * )S, double precision, dimension( * )DIF, integerMM, integerM, double precision, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)
DTGSNA
Purpose:

DTGSNA estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B) in generalized real Schur canonical form (or of any matrix pair (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where Z**T denotes the transpose of Z. (A, B) must be in generalized real Schur form (as returned by DGGES), i.e. A is block upper triangular with 1by1 and 2by2 diagonal blocks. B is upper triangular.
Parameters:

JOB
JOB is 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).
HOWMNYHOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT.
SELECTSELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the eigenpair corresponding to a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select condition numbers corresponding to a complex conjugate pair of eigenvalues w(j) and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced.
NN is INTEGER The order of the square matrix pair (A, B). N >= 0.
AA is DOUBLE PRECISION array, dimension (LDA,N) The upper quasitriangular matrix A in the pair (A,B).
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
BB is DOUBLE PRECISION array, dimension (LDB,N) The upper triangular matrix B in the pair (A,B).
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
VLVL is DOUBLE PRECISION 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 DTGEVC. If JOB = 'V', VL is not referenced.
LDVLLDVL is INTEGER The leading dimension of the array VL. LDVL >= 1. If JOB = 'E' or 'B', LDVL >= N.
VRVR is DOUBLE PRECISION 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 ov VR, as returned by DTGEVC. If JOB = 'V', VR is not referenced.
LDVRLDVR is INTEGER The leading dimension of the array VR. LDVR >= 1. If JOB = 'E' or 'B', LDVR >= N.
SS is 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. For a complex conjugate pair of eigenvalues two consecutive elements of S are set to the same value. Thus S(j), DIF(j), and the jth columns of VL and VR all correspond to the same eigenpair (but not in general the jth eigenpair, unless all eigenpairs are selected). If JOB = 'V', S is not referenced.
DIFDIF is 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. For a complex eigenvector two consecutive elements of DIF are set to the same value. 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. If JOB = 'E', DIF is not referenced.
MMMM is INTEGER The number of elements in the arrays S and DIF. MM >= M.
MM is INTEGER The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If HOWMNY = 'A', M is set to N.
WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N). If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16. 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.
IWORKIWORK is INTEGER array, dimension (N + 6) If JOB = 'E', IWORK is not referenced.
INFOINFO is INTEGER =0: Successful exit <0: If INFO = i, the ith argument had an illegal value
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
Further Details:

The reciprocal of the condition number of a generalized eigenvalue w = (a, b) is defined as S(w) = (u**TAv**2 + u**TBv**2)**(1/2) / (norm(u)*norm(v)) where u and v are the left and right eigenvectors of (A, B) corresponding to w; z denotes the absolute value of the complex number, and norm(u) denotes the 2norm of the vector u. The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv) 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 ith 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 DIF(i) of right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows: a) If the ith eigenvalue w = (a,b) is real Suppose U and V are orthogonal transformations such that U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 ( 0 S22 ),( 0 T22 ) n1 1 n1 1 n1 Then the reciprocal condition number DIF(i) is Difl((a, b), (S22, T22)) = sigmamin( Zl ), where sigmamin(Zl) denotes the smallest singular value of the 2(n1)by2(n1) matrix Zl = [ kron(a, In1) kron(1, S22) ] [ kron(b, In1) kron(1, T22) ] . Here In1 is the identity matrix of size n1. kron(X, Y) is the Kronecker product between the matrices X and Y. Note that if the default method for computing DIF(i) is wanted (see DLATDF), then the parameter DIFDRI (see below) should be changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL for more details. b) If the ith and (i+1)th eigenvalues are complex conjugate pair, Suppose U and V are orthogonal transformations such that U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 ( 0 S22 ),( 0 T22) n2 2 n2 2 n2 and (S11, T11) corresponds to the complex conjugate eigenvalue pair (w, conjg(w)). There exist unitary matrices U1 and V1 such that U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 ) ( 0 s22 ) ( 0 t22 ) where the generalized eigenvalues w = s11/t11 and conjg(w) = s22/t22. Then the reciprocal condition number DIF(i) is bounded by min( d1, max( 1, real(s11)/real(s22) )*d2 ) where, d1 = Difl((s11, t11), (s22, t22)) = sigmamin(Z1), where Z1 is the complex 2by2 matrix Z1 = [ s11 s22 ] [ t11 t22 ], This is done by computing (using real arithmetic) the roots of the characteristical polynomial det(Z1**T * Z1  lambda I), where Z1**T denotes the transpose of Z1 and det(X) denotes the determinant of X. and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an upper bound on sigmamin(Z2), where Z2 is (2n2)by(2n2) Z2 = [ kron(S11**T, In2) kron(I2, S22) ] [ kron(T11**T, In2) kron(I2, T22) ] Note that if the default method for computing DIF is wanted (see DLATDF), then the parameter DIFDRI (see below) should be changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL for more details. For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius normbased estimate of Difl. An approximate error bound for the ith computed eigenvector VL(i) or VR(i) is given by EPS * norm(A, B) / DIF(i). See ref. [23] for more details and further references.
Contributors:
 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901 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 RealTime Applications, Kluwer Academic Publ. 1993, pp 195218. [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, S901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996. [3] B. Kagstrom and P. Poromaa, LAPACKStyle 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, S901 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.
Definition at line 380 of file dtgsna.f.
Author
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