ZTRSNA(3F) ZTRSNA(3F)
ZTRSNA - estimate reciprocal condition numbers for specified eigenvalues
and/or right eigenvectors of a complex upper triangular matrix T (or of
any matrix Q*T*Q**H with Q unitary)
SUBROUTINE ZTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S,
SEP, MM, M, WORK, LDWORK, RWORK, INFO )
CHARACTER HOWMNY, JOB
INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
LOGICAL SELECT( * )
DOUBLE PRECISION RWORK( * ), S( * ), SEP( * )
COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK(
LDWORK, * )
ZTRSNA estimates reciprocal condition numbers for specified eigenvalues
and/or right eigenvectors of a complex upper triangular matrix T (or of
any matrix Q*T*Q**H with Q unitary).
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 j-th eigenpair, SELECT(j) 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) COMPLEX*16 array, dimension (LDT,N)
The upper triangular matrix T.
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ZTRSNA(3F) ZTRSNA(3F)
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL (input) COMPLEX*16 array, dimension (LDVL,M)
If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or
of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs
specified by HOWMNY and SELECT. The eigenvectors must be stored
in consecutive columns of VL, as returned by ZHSEIN or ZTREVC.
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) COMPLEX*16 array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or
of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs
specified by HOWMNY and SELECT. The eigenvectors must be stored
in consecutive columns of VR, as returned by ZHSEIN or ZTREVC.
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. 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. 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) COMPLEX*16 array, dimension (LDWORK,N+1)
If JOB = 'E', WORK is not referenced.
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ZTRSNA(3F) ZTRSNA(3F)
LDWORK (input) INTEGER
The leading dimension of the array WORK. LDWORK >= 1; and if JOB
= 'V' or 'B', LDWORK >= N.
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
If JOB = 'E', RWORK 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'*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)
ZTRSNA(3F) ZTRSNA(3F)
ZTRSNA - estimate reciprocal condition numbers for specified eigenvalues
and/or right eigenvectors of a complex upper triangular matrix T (or of
any matrix Q*T*Q**H with Q unitary)
SUBROUTINE ZTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S,
SEP, MM, M, WORK, LDWORK, RWORK, INFO )
CHARACTER HOWMNY, JOB
INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
LOGICAL SELECT( * )
DOUBLE PRECISION RWORK( * ), S( * ), SEP( * )
COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK(
LDWORK, * )
ZTRSNA estimates reciprocal condition numbers for specified eigenvalues
and/or right eigenvectors of a complex upper triangular matrix T (or of
any matrix Q*T*Q**H with Q unitary).
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 j-th eigenpair, SELECT(j) 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) COMPLEX*16 array, dimension (LDT,N)
The upper triangular matrix T.
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ZTRSNA(3F) ZTRSNA(3F)
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL (input) COMPLEX*16 array, dimension (LDVL,M)
If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or
of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs
specified by HOWMNY and SELECT. The eigenvectors must be stored
in consecutive columns of VL, as returned by ZHSEIN or ZTREVC.
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) COMPLEX*16 array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or
of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs
specified by HOWMNY and SELECT. The eigenvectors must be stored
in consecutive columns of VR, as returned by ZHSEIN or ZTREVC.
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. 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. 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) COMPLEX*16 array, dimension (LDWORK,N+1)
If JOB = 'E', WORK is not referenced.
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ZTRSNA(3F) ZTRSNA(3F)
LDWORK (input) INTEGER
The leading dimension of the array WORK. LDWORK >= 1; and if JOB
= 'V' or 'B', LDWORK >= N.
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
If JOB = 'E', RWORK 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'*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)
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