Purpose
To move the eigenvalues with strictly negative real parts of an
N-by-N complex skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form to the leading principal subpencil, while
keeping the triangular form. On entry, we have
( A D ) ( B F )
S = ( ), H = ( ),
( 0 A' ) ( 0 -B' )
where A and B are upper triangular.
S and H are transformed by a unitary matrix Q such that
( Aout Dout )
Sout = J Q' J' S Q = ( ), and
( 0 Aout' )
(1)
( Bout Fout ) ( 0 I )
Hout = J Q' J' H Q = ( ), with J = ( ),
( 0 -Bout' ) ( -I 0 )
where Aout and Bout remain in upper triangular form. The notation
M' denotes the conjugate transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the unitary matrix Q
that fulfills (1) is computed.
Specification
SUBROUTINE MB3JZP( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
$ LDQ, NEIG, TOL, DWORK, ZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER INFO, LDA, LDB, LDD, LDF, LDQ, N, NEIG
DOUBLE PRECISION TOL
C .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ F( LDF, * ), Q( LDQ, * ), ZWORK( * )
DOUBLE PRECISION DWORK( * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
Specifies whether or not the unitary transformations
should be accumulated in the array Q, as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the unitary matrix Q is returned;
= 'U': the array Q contains a unitary matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the unitary transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix A.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Aout.
The strictly lower triangular part of this array is not
referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
D (input/output) COMPLEX*16 array, dimension (LDD, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the skew-Hermitian
matrix D.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Dout.
The strictly lower triangular part of this array is not
referenced.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N/2).
B (input/output) COMPLEX*16 array, dimension (LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix B.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Bout.
The strictly lower triangular part of this array is not
referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (input/output) COMPLEX*16 array, dimension (LDF, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the Hermitian matrix
F.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Fout.
The strictly lower triangular part of this array is not
referenced.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and H.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the unitary transformation matrix Q.
If COMPQ = 'N' this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
NEIG (output) INTEGER
The number of eigenvalues in aS - bH with strictly
negative real part.
Tolerances
TOL DOUBLE PRECISION
The tolerance used to decide the sign of the eigenvalues.
If the user sets TOL > 0, then the given value of TOL is
used. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by MIN(N,10)*EPS, is
used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH). A larger value might be
needed for pencils with multiple eigenvalues.
Workspace
DWORK DOUBLE PRECISION array, dimension (N/2) ZWORK COMPLEX*16 array, dimension (N/2)Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value.
Method
The algorithm reorders the eigenvalues like the following scheme:
Step 1: Reorder the eigenvalues in the subpencil aA - bB.
I. Reorder the eigenvalues with negative real parts to the
top.
II. Reorder the eigenvalues with positive real parts to the
bottom.
Step 2: Reorder the remaining eigenvalues with negative real parts.
I. Exchange the eigenvalues between the last diagonal block
in aA - bB and the last diagonal block in aS - bH.
II. Move the eigenvalues in the N/2-th place to the (MM+1)-th
place, where MM denotes the current number of eigenvalues
with negative real parts in aA - bB.
The algorithm uses a sequence of unitary transformations as
described on page 43 in [1]. To achieve those transformations the
elementary SLICOT Library subroutines MB03DZ and MB03HZ are called
for the corresponding matrix structures.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
For large values of N, the routine applies the transformations on panels of columns. The user may specify in INFO the desired number of columns. If on entry INFO <= 0, then the routine estimates a suitable value of this number.Example
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