NAG Library Routine Document
F08YLF (DTGSNA)
1 Purpose
F08YLF (DTGSNA) estimates condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair in generalized real Schur form.
2 Specification
SUBROUTINE F08YLF ( |
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) |
A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), S(*), DIF(*), WORK(max(1,LWORK)) |
LOGICAL |
SELECT(*) |
CHARACTER(1) |
JOB, HOWMNY |
|
The routine may be called by its
LAPACK
name dtgsna.
3 Description
F08YLF (DTGSNA) estimates condition numbers for specified eigenvalues and/or right eigenvectors of an n by n matrix pair S,T in real 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 real generalized Schur form if
S is block upper triangular with
1 by
1 and
2 by
2 diagonal blocks and
T is upper triangular as returned, for example, by
F08XAF (DGGES) or
F08XBF (DGGESX), or
F08XEF (DHGEQZ) with
JOB='S'. The diagonal elements, or blocks, 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.
The definition of the reciprocal of the estimated condition number of the right eigenvector x and the left eigenvector y corresponding to the simple eigenvalue λ depends upon whether λ is a real eigenvalue, or one of a complex conjugate pair.
If the eigenvalue
λ is real and
U and
V are orthogonal 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.
If
λ is part of a complex conjugate pair and
U and
V are orthogonal transformations such that
where
S11 and
T11 are two by two matrices,
S22 and
T22 are
n-2 by
n-2 matrices, and
S11,T11 corresponds to the complex conjugate eigenvalue pair
λ,
λ-, then there exist unitary matrices
U1 and
V1 such that
The eigenvalues are given by
λ=s11/t11 and
λ-=s22/t22. Then the Frobenius norm-based, estimated reciprocal condition number is bounded by
where
Rez denotes the real part of
z,
d1=Difs11,t11,s22,t22=σminZ1,
Z1 is the complex two by two matrix
and
d2 is an upper bound on
DifS11,T11,S22,T22; i.e., an upper bound on
σminZ2, where
Z2 is the
2n-2 by
2n-2 matrix
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 real eigenvalue
λj,
SELECTj must be set .TRUE.. To select condition numbers corresponding to a complex conjugate pair of eigenvalues
λj and
λj+1,
SELECTj and/or
SELECTj+1 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,*) – REAL (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the upper quasi-triangular matrix S.
- 6: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08YLF (DTGSNA) is called.
Constraint:
LDA≥max1,N.
- 7: B(LDB,*) – REAL (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 F08YLF (DTGSNA) is called.
Constraint:
LDB≥max1,N.
- 9: VL(LDVL,*) – REAL (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
F08WAF (DGGEV) or
F08YKF (DTGEVC).
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 F08YLF (DTGSNA) is called.
Constraints:
- if JOB='E' or 'B', LDVL≥ max1,N ;
- otherwise LDVL≥1.
- 11: VR(LDVR,*) – REAL (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
F08WAF (DGGEV) or
F08YKF (DTGEVC).
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 F08YLF (DTGSNA) 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.
On exit: 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
Sj,
DIFj, and the
jth columns of
VL and
VR all correspond to the same eigenpair (but not in general the
jth eigenpair, unless all eigenpairs are selected).
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.
On exit: 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
DIF are set to the same value. 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 real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If
HOWMNY='A',
M is set to
N.
- 17: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
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 F08YLF (DTGSNA) 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≥2×N×N+2+16;
- otherwise LWORK≥max1,N.
- 19: IWORK(*) – INTEGER arrayWorkspace
-
Note: the dimension of the array
IWORK
must be at least
N+6.
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 complex analogue of this routine is
F08YYF (ZTGSNA).
9 Example
This example estimates condition numbers and approximate error estimates for all the eigenvalues and eigenvalues and right eigenvectors of the pair
S,T given by
The eigenvalues and eigenvectors are computed by calling
F08YKF (DTGEVC).
9.1 Program Text
Program Text (f08ylfe.f90)
9.2 Program Data
Program Data (f08ylfe.d)
9.3 Program Results
Program Results (f08ylfe.r)