DTRSNA - estimate reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q orthogonal)

SUBROUTINE DTRSNA(

JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, INFO )

CHARACTER HOWMNY, JOB

INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N

LOGICAL SELECT( * )

INTEGER IWORK( * )

DOUBLE PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * )

DTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q orthogonal). T must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign.

JOB (input) CHARACTER*1

Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP):

= 'E': for eigenvalues only (S);

= 'V': for eigenvectors only (SEP);

= 'B': for both eigenvalues and eigenvectors (S and SEP).

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 matrix T. N >= 0.

T (input) DOUBLE PRECISION array, dimension (LDT,N)

The upper quasi-triangular matrix T, in Schur canonical form.

LDT (input) INTEGER

The leading dimension of the array T. LDT >= max(1,N).

VL (input) DOUBLE PRECISION array, dimension (LDVL,M)

If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by DHSEIN or DTREVC. 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) DOUBLE PRECISION array, dimension (LDVR,M)

If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by DHSEIN or DTREVC. If JOB = 'V', VR is not referenced.

LDVR (input) INTEGER

The leading dimension of the array VR. LDVR >= 1; and 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), SEP(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.

SEP (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 SEP are set to the same value. If the eigenvalues cannot be reordered to compute SEP(j), SEP(j) is set to 0; this can only occur when the true value would be very small anyway. If JOB = 'E', SEP is not referenced.

MM (input) INTEGER

The number of elements in the arrays S (if JOB = 'E' or 'B') and/or SEP (if JOB = 'V' or 'B'). MM >= M.

M (output) INTEGER

The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = 'A', M is set to N.

WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+1)

If JOB = 'E', WORK is not referenced.

LDWORK (input) INTEGER

The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.

IWORK (workspace) INTEGER array, dimension (N)

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 an eigenvalue lambda is defined as

S(lambda) = |v'*u| / (norm(u)*norm(v))

where u and v are the right and left eigenvectors of T corresponding to lambda; v' denotes the conjugate-transpose of v, and norm(u) denotes the Euclidean norm. These reciprocal condition numbers always lie between zero (very badly conditioned) and one (very well conditioned). If n = 1, S(lambda) is defined to be 1.

An approximate error bound for a computed eigenvalue W(i) is given by

EPS * norm(T) / S(i)

where EPS is the machine precision.

The reciprocal of the condition number of the right eigenvector u corresponding to lambda is defined as follows. Suppose

T = ( lambda c )

( 0 T22 )

Then the reciprocal condition number is

SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )

where sigma-min denotes the smallest singular value. We approximate the smallest singular value by the reciprocal of an estimate of the one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to be abs(T(1,1)).

An approximate error bound for a computed right eigenvector VR(i) is given by

EPS * norm(T) / SEP(i)