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LAPACK
3.9.0
LAPACK: Linear Algebra PACKage
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LAPACK
3.9.0
LAPACK: Linear Algebra PACKage
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| subroutine dtrsna | ( | character | JOB, |
| character | HOWMNY, | ||
| logical, dimension( * ) | SELECT, | ||
| integer | N, | ||
| double precision, dimension( ldt, * ) | T, | ||
| integer | LDT, | ||
| double precision, dimension( ldvl, * ) | VL, | ||
| integer | LDVL, | ||
| double precision, dimension( ldvr, * ) | VR, | ||
| integer | LDVR, | ||
| double precision, dimension( * ) | S, | ||
| double precision, dimension( * ) | SEP, | ||
| integer | MM, | ||
| integer | M, | ||
| double precision, dimension( ldwork, * ) | WORK, | ||
| integer | LDWORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | INFO | ||
| ) |
DTRSNA
Download DTRSNA + dependencies [TGZ] [ZIP] [TXT]
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.
| [in] | JOB | JOB is 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). |
| [in] | HOWMNY | HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT. |
| [in] | SELECT | SELECT 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. |
| [in] | N | N is INTEGER
The order of the matrix T. N >= 0. |
| [in] | T | T is DOUBLE PRECISION array, dimension (LDT,N)
The upper quasi-triangular matrix T, in Schur canonical form. |
| [in] | LDT | LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N). |
| [in] | VL | VL is 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. |
| [in] | LDVL | LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. |
| [in] | VR | VR is 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. |
| [in] | LDVR | LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. |
| [out] | S | S 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), 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. |
| [out] | SEP | SEP 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 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. |
| [in] | MM | MM is INTEGER
The number of elements in the arrays S (if JOB = 'E' or 'B')
and/or SEP (if JOB = 'V' or 'B'). MM >= M. |
| [out] | M | M is 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. |
| [out] | WORK | WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
If JOB = 'E', WORK is not referenced. |
| [in] | LDWORK | LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. |
| [out] | IWORK | IWORK is INTEGER array, dimension (2*(N-1))
If JOB = 'E', IWORK is not referenced. |
| [out] | INFO | INFO is 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**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) Definition at line 267 of file dtrsna.f.
1.8.16