NAG Library Routine Document
F08YYF (ZTGSNA)
1 Purpose
F08YYF (ZTGSNA) estimates condition numbers for specified eigenvalues and/or eigenvectors of a complex matrix pair in generalized Schur form.
2 Specification
SUBROUTINE F08YYF ( |
JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) |
INTEGER |
N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, IWORK(*), INFO |
REAL (KIND=nag_wp) |
S(*), DIF(*) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), WORK(max(1,LWORK)) |
LOGICAL |
SELECT(*) |
CHARACTER(1) |
JOB, HOWMNY |
|
The routine may be called by its
LAPACK
name ztgsna.
3 Description
F08YYF (ZTGSNA) estimates condition numbers for specified eigenvalues and/or right eigenvectors of an n by n matrix pair S,T in generalized Schur form. The routine actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.
The pair
S,T are in generalized Schur form if
S and
T are upper triangular as returned, for example, by
F08XNF (ZGGES) or
F08XPF (ZGGESX), or
F08XSF (ZHGEQZ) with
JOB='S'. The diagonal elements define the generalized eigenvalues
αi,βi, for
i=1,2,…,n, of the pair
S,T and the eigenvalues are given by
so that
where
xi is the corresponding (right) eigenvector.
If
S and
T are the result of a generalized Schur factorization of a matrix pair
A,B
then the eigenvalues and condition numbers of the pair
S,T are the same as those of the pair
A,B.
Let
α,β≠0,0 be a simple generalized eigenvalue of
A,B. Then the reciprocal of the condition number of the eigenvalue
λ=α/β is defined as
where
x and
y are the right and left eigenvectors of
A,B corresponding to
λ. If both
α and
β are zero, then
A,B is singular and
sλ=-1 is returned.
If
U and
V are unitary transformations such that
where
S22 and
T22 are
n-1 by
n-1 matrices, then the reciprocal condition number is given by
where
σminZ denotes the smallest singular value of the
2n-1 by
2n-1 matrix
and
⊗ is the Kronecker product.
See Sections 2.4.8 and 4.11 of
Anderson et al. (1999) and
Kågström and Poromaa (1996) for further details and information.
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs
ACM Trans. Math. Software 22 78–103
5 Parameters
- 1: JOB – CHARACTER(1)Input
On entry: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.
- JOB='E'
- Condition numbers for eigenvalues only are computed.
- JOB='V'
- Condition numbers for eigenvectors only are computed.
- JOB='B'
- Condition numbers for both eigenvalues and eigenvectors are computed.
Constraint:
JOB='E', 'V' or 'B'.
- 2: HOWMNY – CHARACTER(1)Input
On entry: indicates how many condition numbers are to be computed.
- HOWMNY='A'
- Condition numbers for all eigenpairs are computed.
- HOWMNY='S'
- Condition numbers for selected eigenpairs (as specified by SELECT) are computed.
Constraint:
HOWMNY='A' or 'S'.
- 3: SELECT(*) – LOGICAL arrayInput
Note: the dimension of the array
SELECT
must be at least
max1,N if
HOWMNY='S', and at least
1 otherwise.
On entry: specifies the eigenpairs for which condition numbers are to be computed if
HOWMNY='S'. To select condition numbers for the eigenpair corresponding to the eigenvalue
λj,
SELECTj must be set to .TRUE..
If
HOWMNY='A',
SELECT is not referenced.
- 4: N – INTEGERInput
On entry: n, the order of the matrix pair S,T.
Constraint:
N≥0.
- 5: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the upper triangular matrix S.
- 6: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08YYF (ZTGSNA) is called.
Constraint:
LDA≥max1,N.
- 7: B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
B
must be at least
max1,N.
On entry: the upper triangular matrix T.
- 8: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F08YYF (ZTGSNA) is called.
Constraint:
LDB≥max1,N.
- 9: VL(LDVL,*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
VL
must be at least
max1,MM if
JOB='E' or
'B', and at least
1 otherwise.
On entry: if
JOB='E' or
'B',
VL must contain left eigenvectors of
S,T, corresponding to the eigenpairs specified by
HOWMNY and
SELECT. The eigenvectors must be stored in consecutive columns of
VL, as returned by
F08WNF (ZGGEV) or
F08YXF (ZTGEVC).
If
JOB='V',
VL is not referenced.
- 10: LDVL – INTEGERInput
On entry: the first dimension of the array
VL as declared in the (sub)program from which F08YYF (ZTGSNA) is called.
Constraints:
- if JOB='E' or 'B', LDVL≥ max1,N ;
- otherwise LDVL≥1.
- 11: VR(LDVR,*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
VR
must be at least
max1,MM if
JOB='E' or
'B', and at least
1 otherwise.
On entry: if
JOB='E' or
'B',
VR must contain right eigenvectors of
S,T, corresponding to the eigenpairs specified by
HOWMNY and
SELECT. The eigenvectors must be stored in consecutive columns of
VR, as returned by
F08WNF (ZGGEV) or
F08YXF (ZTGEVC).
If
JOB='V',
VR is not referenced.
- 12: LDVR – INTEGERInput
On entry: the first dimension of the array
VR as declared in the (sub)program from which F08YYF (ZTGSNA) is called.
Constraints:
- if JOB='E' or 'B', LDVR≥ max1,N ;
- otherwise LDVR≥1.
- 13: S(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
S
must be at least
max1,MM if
JOB='E' or
'B', and at least
1 otherwise.
On exit: if
JOB='E' or
'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array.
If
JOB='V',
S is not referenced.
- 14: DIF(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
DIF
must be at least
max1,MM if
JOB='V' or
'B', and at least
1 otherwise.
On exit: if
JOB='V' or
'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute
DIFj,
DIFj is set to
0; this can only occur when the true value would be very small anyway.
If
JOB='E',
DIF is not referenced.
- 15: MM – INTEGERInput
On entry: the number of elements in the arrays
S and
DIF.
Constraint:
MM≥N.
- 16: M – INTEGEROutput
On exit: the number of elements of the arrays
S and
DIF used to store the specified condition numbers; for each selected eigenvalue one element is used.
If
HOWMNY='A',
M is set to
N.
- 17: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 18: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08YYF (ZTGSNA) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the minimum size of the
WORK array, returns this value as the first entry of the
WORK array, and no error message related to
LWORK is issued.
Constraints:
if
LWORK≠-1,
- if JOB='V' or 'B', LWORK≥max1,2×N×N;
- otherwise LWORK≥max1,N.
- 19: IWORK(*) – INTEGER arrayWorkspace
-
Note: the dimension of the array
IWORK
must be at least
N+2.
If
JOB='E',
IWORK is not referenced.
- 20: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
7 Accuracy
None.
8 Further Comments
An approximate asymptotic error bound on the chordal distance between the computed eigenvalue
λ~ and the corresponding exact eigenvalue
λ is
where
ε is the
machine precision.
An approximate asymptotic error bound for the right or left computed eigenvectors
x~ or
y~ corresponding to the right and left eigenvectors
x and
y is given by
The real analogue of this routine is
F08YLF (DTGSNA).
9 Example
This example estimates condition numbers and approximate error estimates for all the eigenvalues and right eigenvectors of the pair
S,T given by
and
The eigenvalues and eigenvectors are computed by calling
F08YXF (ZTGEVC).
9.1 Program Text
Program Text (f08yyfe.f90)
9.2 Program Data
Program Data (f08yyfe.d)
9.3 Program Results
Program Results (f08yyfe.r)