
stgsna.f(3) LAPACK stgsna.f(3)
NAME
stgsna.f 
SYNOPSIS
Functions/Subroutines
subroutine stgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM,
M, WORK, LWORK, IWORK, INFO)
STGSNA
Function/Subroutine Documentation
subroutine stgsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN,
real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real,
dimension( ldvl, * )VL, integerLDVL, real, dimension( ldvr, * )VR, integerLDVR, real,
dimension( * )S, real, dimension( * )DIF, integerMM, integerM, real, dimension( * )WORK,
integerLWORK, integer, dimension( * )IWORK, integerINFO)
STGSNA
Purpose:
STGSNA estimates reciprocal condition numbers for specified
eigenvalues and/or eigenvectors of a matrix pair (A, B) in
generalized real Schur canonical form (or of any matrix pair
(Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
Z**T denotes the transpose of Z.
(A, B) must be in generalized real Schur form (as returned by SGGES),
i.e. A is block upper triangular with 1by1 and 2by2 diagonal
blocks. B is upper triangular.
Parameters:
JOB
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
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 square matrix pair (A, B). N >= 0.
A
A is REAL array, dimension (LDA,N)
The upper quasitriangular matrix A in the pair (A,B).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is REAL array, dimension (LDB,N)
The upper triangular matrix B in the pair (A,B).
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
VL
VL is REAL 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 STGEVC.
If JOB = 'V', VL is not referenced.
LDVL
LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1.
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
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns ov VR, as returned by STGEVC.
If JOB = 'V', VR is not referenced.
LDVR
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1.
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), DIF(j), 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.
DIF
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. For a complex eigenvector two
consecutive elements of DIF are set to the same value. 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.
If JOB = 'E', DIF is not referenced.
MM
MM is INTEGER
The number of elements in the arrays S and DIF. MM >= M.
M
M is INTEGER
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.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
If LWORK = 1, then a workspace query is assumed; the routine
only calculates the optimal 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 by XERBLA.
IWORK
IWORK is INTEGER array, dimension (N + 6)
If JOB = 'E', IWORK is not referenced.
INFO
INFO is INTEGER
=0: Successful exit
<0: If INFO = i, the ith argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details:
The reciprocal of the condition number of a generalized eigenvalue
w = (a, b) is defined as
S(w) = (u**TAv**2 + u**TBv**2)**(1/2) / (norm(u)*norm(v))
where u and v are the left and right eigenvectors of (A, B)
corresponding to w; z denotes the absolute value of the complex
number, and norm(u) denotes the 2norm of the vector u.
The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
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 ith
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 DIF(i) of right eigenvector u
and left eigenvector v corresponding to the generalized eigenvalue w
is defined as follows:
a) If the ith eigenvalue w = (a,b) is real
Suppose U and V are orthogonal transformations such that
U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
( 0 S22 ),( 0 T22 ) n1
1 n1 1 n1
Then the reciprocal condition number DIF(i) is
Difl((a, b), (S22, T22)) = sigmamin( Zl ),
where sigmamin(Zl) denotes the smallest singular value of the
2(n1)by2(n1) matrix
Zl = [ kron(a, In1) kron(1, S22) ]
[ kron(b, In1) kron(1, T22) ] .
Here In1 is the identity matrix of size n1. kron(X, Y) is the
Kronecker product between the matrices X and Y.
Note that if the default method for computing DIF(i) is wanted
(see SLATDF), then the parameter DIFDRI (see below) should be
changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
See STGSYL for more details.
b) If the ith and (i+1)th eigenvalues are complex conjugate pair,
Suppose U and V are orthogonal transformations such that
U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
( 0 S22 ),( 0 T22) n2
2 n2 2 n2
and (S11, T11) corresponds to the complex conjugate eigenvalue
pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
that
U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
( 0 s22 ) ( 0 t22 )
where the generalized eigenvalues w = s11/t11 and
conjg(w) = s22/t22.
Then the reciprocal condition number DIF(i) is bounded by
min( d1, max( 1, real(s11)/real(s22) )*d2 )
where, d1 = Difl((s11, t11), (s22, t22)) = sigmamin(Z1), where
Z1 is the complex 2by2 matrix
Z1 = [ s11 s22 ]
[ t11 t22 ],
This is done by computing (using real arithmetic) the
roots of the characteristical polynomial det(Z1**T * Z1  lambda I),
where Z1**T denotes the transpose of Z1 and det(X) denotes
the determinant of X.
and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
upper bound on sigmamin(Z2), where Z2 is (2n2)by(2n2)
Z2 = [ kron(S11**T, In2) kron(I2, S22) ]
[ kron(T11**T, In2) kron(I2, T22) ]
Note that if the default method for computing DIF is wanted (see
SLATDF), then the parameter DIFDRI (see below) should be changed
from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
for more details.
For each eigenvalue/vector specified by SELECT, DIF stores a
Frobenius normbased estimate of Difl.
An approximate error bound for the ith computed eigenvector VL(i) or
VR(i) is given by
EPS * norm(A, B) / DIF(i).
See ref. [23] for more details and further references.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901
87 Umea, Sweden.
References:
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
RealTime Applications, Kluwer Academic Publ. 1993, pp 195218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software,
Report UMINF  94.04, Department of Computing Science, Umea
University, S901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACKStyle Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF  93.23,
Department of Computing Science, Umea University, S901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
No 1, 1996.
Definition at line 380 of file stgsna.f.
Author
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