STRSNA
Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH.
Purpose
STRSNA 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 SHSEQR), 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.
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 SHSEQR), 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.
Arguments
| 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) REAL 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) REAL 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 SHSEIN or STREVC. 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) REAL 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 SHSEIN or STREVC. 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) REAL 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) REAL 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) REAL array, dimension (LDWORK,N+6)
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 (2*(N-1))
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 an eigenvalue lambda is
defined as
S(lambda) = |v**T*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding
to lambda; v**T denotes the 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)
defined as
S(lambda) = |v**T*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding
to lambda; v**T denotes the 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)