CTGSNA(3) LAPACK routine of NEC Numeric Library Collection CTGSNA(3)
NAME
CTGSNA
SYNOPSIS
SUBROUTINE CTGSNA (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL,
VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
PURPOSE
CTGSNA estimates reciprocal condition numbers for specified
eigenvalues and/or eigenvectors of a matrix pair (A, B).
(A, B) must be in generalized Schur canonical form, that is, A and
B are both upper triangular.
ARGUMENTS
JOB (input)
JOB is CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (DIF):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (DIF);
= 'B': for both eigenvalues and eigenvectors (S and DIF).
HOWMNY (input)
HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT (input)
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 corresponding j-th eigenvalue and/or eigenvector,
SELECT(j) must be set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
N (input)
N is INTEGER
The order of the square matrix pair (A, B). N >= 0.
A (input)
A is COMPLEX array, dimension (LDA,N)
The upper triangular matrix A in the pair (A,B).
LDA (input)
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input)
B is COMPLEX array, dimension (LDB,N)
The upper triangular matrix B in the pair (A, B).
LDB (input)
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
VL (input)
VL is COMPLEX array, dimension (LDVL,M)
IF JOB = 'E' or 'B', VL must contain left eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns of VL, as returned by CTGEVC.
If JOB = 'V', VL is not referenced.
LDVL (input)
LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1; and
If JOB = 'E' or 'B', LDVL >= N.
VR (input)
VR is COMPLEX array, dimension (LDVR,M)
IF JOB = 'E' or 'B', VR must contain right eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns of VR, as returned by CTGEVC.
If JOB = 'V', VR is not referenced.
LDVR (input)
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1;
If JOB = 'E' or 'B', LDVR >= N.
S (output)
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.
If JOB = 'V', S is not referenced.
DIF (output)
DIF 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 the eigenvalues cannot be reordered to compute DIF(j),
DIF(j) is set to 0; this can only occur when the true value
would be very small anyway.
For each eigenvalue/vector specified by SELECT, DIF stores
a Frobenius norm-based estimate of Difl.
If JOB = 'E', DIF is not referenced.
MM (input)
MM is INTEGER
The number of elements in the arrays S and DIF. MM >= M.
M (output)
M is INTEGER
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.
WORK (output)
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
IWORK (output)
IWORK is INTEGER array, dimension (N+2)
If JOB = 'E', IWORK is not referenced.
INFO (output)
INFO is 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 the i-th generalized
eigenvalue w = (a, b) is defined as
S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of (A, B)
corresponding to w; |z| denotes the absolute value of the complex
number, and norm(u) denotes the 2-norm of the vector u. The pair
(a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the
matrix pair (A, B). If both a and b equal zero, then (A,B) is
singular and S(I) = -1 is returned.
An approximate error bound on the chordal distance between the i-th
computed generalized eigenvalue w and the corresponding exact
eigenvalue lambda is
chord(w, lambda) <= EPS * norm(A, B) / S(I),
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u
and left eigenvector v corresponding to the generalized eigenvalue w
is defined as follows. Suppose
(A, B) = ( a * ) ( b * ) 1
( 0 A22 ),( 0 B22 ) n-1
1 n-1 1 n-1
Then the reciprocal condition number DIF(I) is
Difl[(a, b), (A22, B22)] = sigma-min( Zl )
where sigma-min(Zl) denotes the smallest singular value of
Zl = [ kron(a, In-1) -kron(1, A22) ]
[ kron(b, In-1) -kron(1, B22) ].
Here In-1 is the identity matrix of size n-1 and X**H is the conjugate
transpose of X. kron(X, Y) is the Kronecker product between the
matrices X and Y.
We approximate the smallest singular value of Zl with an upper
bound. This is done by CLATDF.
An approximate error bound for a computed eigenvector VL(i) or
VR(i) is given by
EPS * norm(A, B) / DIF(i).
LAPACK routine 31 October 2017 CTGSNA(3)