
DTGSNA(l) ) DTGSNA(l)
NAME
DTGSNA  estimate reciprocal condition numbers for specified eigenvalues and/or eigenvec
tors of a matrix pair (A, B) in generalized real Schur canonical form (or of any matrix
pair (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where Z' denotes the transpose of
Z
SYNOPSIS
SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM,
M, WORK, LWORK, IWORK, INFO )
CHARACTER HOWMNY, JOB
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ), VL( LDVL, * ),
VR( LDVR, * ), WORK( * )
PURPOSE
DTGSNA estimates reciprocal condition numbers for specified eigenvalues and/or eigenvec
tors of a matrix pair (A, B) in generalized real Schur canonical form (or of any matrix
pair (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where Z' denotes the transpose of
Z. (A, B) must be in generalized real Schur form (as returned by DGGES), i.e. A is block
upper triangular with 1by1 and 2by2 diagonal blocks. B is upper triangular.
ARGUMENTS
JOB (input) CHARACTER*1
Specifies whether condition numbers are required for eigenvalues (S) or eigenvec
tors (DIF):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (DIF);
= 'B': for both eigenvalues and eigenvectors (S and DIF).
HOWMNY (input) CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs specified by the array
SELECT.
SELECT (input) 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 cor
responding 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 (input) INTEGER
The order of the square matrix pair (A, B). N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The upper quasitriangular matrix A in the pair (A,B).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The upper triangular matrix B in the pair (A,B).
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
VL (input) DOUBLE PRECISION 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 DTGEVC. If JOB = 'V', VL is not refer
enced.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1. If JOB = 'E' or 'B', LDVL >= N.
VR (input) DOUBLE PRECISION 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 DTGEVC. If JOB = 'V', VR is not ref
erenced.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1. If JOB = 'E' or 'B', LDVR >= N.
S (output) 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 ei
genvalues 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 (output) 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 eigenvec
tor 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 refer
enced.
MM (input) INTEGER
The number of elements in the arrays S and DIF. MM >= M.
M (output) INTEGER
The number of elements of the arrays S and DIF used to store the specified condi
tion 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 (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
If JOB = 'E', WORK is not referenced. Otherwise, on exit, if INFO = 0, WORK(1)
returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 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 (workspace) INTEGER array, dimension (N + 6)
If JOB = 'E', IWORK is not referenced.
INFO (output) INTEGER
=0: Successful exit
<0: If INFO = i, the ith argument had an illegal value
FURTHER DETAILS
The reciprocal of the condition number of a generalized eigenvalue w = (a, b) is defined
as
S(w) = (u'Av**2 + u'Bv**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'Av/u'Bv) 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'*(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 DLATDF), then the parameter DIFDRI (see below) should be
changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
See DTGSYL 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'*(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'*S11*V1 = ( s11 s12 ) and U1'*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' * Z1  lambda I),
where Z1' denotes the conjugate 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', In2) kron(I2, S22) ]
[ kron(T11', In2) kron(I2, T22) ]
Note that if the default method for computing DIF is wanted (see
DLATDF), then the parameter DIFDRI (see below) should be changed
from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
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.
Based on contributions by
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.
LAPACK version 3.0 15 June 2000 DTGSNA(l) 
