table of contents
| trsna(3) | LAPACK | trsna(3) |
NAME¶
trsna - trsna: eig condition numbers
SYNOPSIS¶
Functions¶
subroutine ctrsna (job, howmny, select, n, t, ldt, vl,
ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)
CTRSNA subroutine dtrsna (job, howmny, select, n, t, ldt, vl,
ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)
DTRSNA subroutine strsna (job, howmny, select, n, t, ldt, vl,
ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)
STRSNA subroutine ztrsna (job, howmny, select, n, t, ldt, vl,
ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)
ZTRSNA
Detailed Description¶
Function Documentation¶
subroutine ctrsna (character job, character howmny, logical, dimension( * ) select, integer n, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) sep, integer mm, integer m, complex, dimension( ldwork, * ) work, integer ldwork, real, dimension( * ) rwork, integer info)¶
CTRSNA
Purpose:
CTRSNA 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).
Parameters
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).
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
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 j-th eigenpair, SELECT(j) must be set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is COMPLEX array, dimension (LDT,N)
The upper triangular matrix T.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL
VL is COMPLEX 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
CHSEIN or CTREVC.
If JOB = 'V', VL is not referenced.
LDVL
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
VR
VR is COMPLEX 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
CHSEIN or CTREVC.
If JOB = 'V', VR is not referenced.
LDVR
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
S
S is REAL 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
SEP is 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.
If JOB = 'E', SEP is not referenced.
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.
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.
WORK
WORK is COMPLEX array, dimension (LDWORK,N+6)
If JOB = 'E', WORK is not referenced.
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
RWORK
RWORK is REAL array, dimension (N)
If JOB = 'E', RWORK is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The reciprocal of the condition number of an eigenvalue lambda is
defined as
S(lambda) = |v**H*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding
to lambda; v**H 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)
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
Purpose:
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.
Parameters
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).
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
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.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is DOUBLE PRECISION array, dimension (LDT,N)
The upper quasi-triangular matrix T, in Schur canonical form.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
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.
LDVL
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
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.
LDVR
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
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.
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.
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.
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.
WORK
WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
If JOB = 'E', WORK is not referenced.
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
IWORK
IWORK is INTEGER array, dimension (2*(N-1))
If JOB = 'E', IWORK is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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)
subroutine strsna (character job, character howmny, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) sep, integer mm, integer m, real, dimension( ldwork, * ) work, integer ldwork, integer, dimension( * ) iwork, integer info)¶
STRSNA
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.
Parameters
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).
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
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.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T, in Schur canonical form.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL
VL is 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
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
VR
VR is 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
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
S
S is 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
SEP is 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
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.
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.
WORK
WORK is REAL array, dimension (LDWORK,N+6)
If JOB = 'E', WORK is not referenced.
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
IWORK
IWORK is INTEGER array, dimension (2*(N-1))
If JOB = 'E', IWORK is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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)
subroutine ztrsna (character job, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, complex*16, dimension( ldwork, * ) work, integer ldwork, double precision, dimension( * ) rwork, integer info)¶
ZTRSNA
Purpose:
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).
Parameters
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).
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
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 j-th eigenpair, SELECT(j) must be set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is COMPLEX*16 array, dimension (LDT,N)
The upper triangular matrix T.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL
VL is 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
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
VR
VR is 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
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
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. 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
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.
If JOB = 'E', SEP is not referenced.
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.
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.
WORK
WORK is COMPLEX*16 array, dimension (LDWORK,N+6)
If JOB = 'E', WORK is not referenced.
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
RWORK
RWORK is DOUBLE PRECISION array, dimension (N)
If JOB = 'E', RWORK is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The reciprocal of the condition number of an eigenvalue lambda is
defined as
S(lambda) = |v**H*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding
to lambda; v**H 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)
Author¶
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