DTGSNA - estimate 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', Q*B*Z') with orthogonal matrices Q and Z, where Z' denotes the transpose of Z

SUBROUTINE DTGSNA(

JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO )

CHARACTER HOWMNY, JOB

INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N

LOGICAL SELECT( * )

INTEGER IWORK( * )

DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

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', Q*B*Z') with orthogonal matrices Q and Z, where Z' 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 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.

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 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.

N (input) INTEGER

The order of the square matrix pair (A, B). N >= 0.

A (input) DOUBLE PRECISION array, dimension (LDA,N)

The upper quasi-triangular matrix A in the pair (A,B).

LDA (input) INTEGER

The leading dimension of the array A. LDA >= max(1,N).

B (input) DOUBLE PRECISION 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) 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.

LDVL (input) INTEGER

The leading dimension of the array VL. LDVL >= 1. If JOB = 'E' or 'B', LDVL >= N.

VR (input) 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.

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. 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 j-th columns of VL and VR all correspond to the same eigenpair (but not in general the j-th eigenpair, unless all eigenpairs are selected). 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. 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.

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 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.

WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)

If JOB = 'E', WORK is not referenced. Otherwise, on exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= 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.

IWORK (workspace) INTEGER array, dimension (N + 6)

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

The reciprocal of the condition number of a generalized eigenvalue w = (a, b) is defined as

S(w) = (|u'Av|**2 + |u'Bv|**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 2-norm of the vector u.

The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv) 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 DIF(i) of right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows:

a) If the i-th eigenvalue w = (a,b) is real

Suppose U and V are orthogonal transformations such that

U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 ( 0 S22 ),( 0 T22 ) n-1 1 n-1 1 n-1

Then the reciprocal condition number DIF(i) is

Difl((a, b), (S22, T22)) = sigma-min( Zl ),

where sigma-min(Zl) denotes the smallest singular value of the 2(n-1)-by-2(n-1) matrix

Zl = [ kron(a, In-1) -kron(1, S22) ]

[ kron(b, In-1) -kron(1, T22) ] .

Here In-1 is the identity matrix of size n-1. 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 i-th and (i+1)-th eigenvalues are complex conjugate pair,

Suppose U and V are orthogonal transformations such that

U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 ( 0 S22 ),( 0 T22) n-2 2 n-2 2 n-2

and (S11, T11) corresponds to the complex conjugate eigenvalue pair (w, conjg(w)). There exist unitary matrices U1 and V1 such that

U1'*S11*V1 = ( s11 s12 ) and U1'*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)) = sigma-min(Z1), where Z1 is the complex 2-by-2 matrix

Z1 = [ s11 -s22 ]

[ t11 -t22 ],

This is done by computing (using real arithmetic) the

roots of the characteristical polynomial det(Z1' * Z1 - lambda I), where Z1' denotes the conjugate 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 sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)

Z2 = [ kron(S11', In-2) -kron(I2, S22) ]

[ kron(T11', In-2) -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 norm-based estimate of Difl.

An approximate error bound for the i-th 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

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[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.