Purpose
To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH with
( A D ) ( C V )
S = ( ) and H = ( ). (1)
( E A' ) ( W -C' )
Optionally, if JOB = 'T', decompositions of S and H will be
computed via orthogonal transformations Q1 and Q2 as follows:
( Aout Dout )
Q1' S J Q1 J' = ( ),
( 0 Aout' )
( Bout Fout )
J' Q2' J S Q2 = ( ) =: T, (2)
( 0 Bout' )
( C1out Vout ) ( 0 I )
Q1' H Q2 = ( ), where J = ( )
( 0 C2out' ) ( -I 0 )
and Aout, Bout, C1out are upper triangular, C2out is upper quasi-
triangular and Dout and Fout are skew-symmetric. The notation M'
denotes the transpose of the matrix M.
Optionally, if COMPQ1 = 'I' or COMPQ1 = 'U', then the orthogonal
transformation matrix Q1 will be computed.
Optionally, if COMPQ2 = 'I' or COMPQ2 = 'U', then the orthogonal
transformation matrix Q2 will be computed.
Specification
SUBROUTINE MB04BP( JOB, COMPQ1, COMPQ2, N, A, LDA, DE, LDDE, C1,
$ LDC1, VW, LDVW, Q1, LDQ1, Q2, LDQ2, B, LDB, F,
$ LDF, C2, LDC2, ALPHAR, ALPHAI, BETA, IWORK,
$ LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ1, COMPQ2, JOB
INTEGER INFO, LDA, LDB, LDC1, LDC2, LDDE, LDF, LDQ1,
$ LDQ2, LDVW, LDWORK, LIWORK, N
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), C1( LDC1, * ),
$ C2( LDC2, * ), DE( LDDE, * ), DWORK( * ),
$ F( LDF, * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
$ VW( LDVW, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (2).
= 'T': put S and H into the forms in (2) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ1 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q1, as follows:
= 'N': Q1 is not computed;
= 'I': the array Q1 is initialized internally to the unit
matrix, and the orthogonal matrix Q1 is returned;
= 'U': the array Q1 contains an orthogonal matrix Q on
entry, and the product Q*Q1 is returned, where Q1
is the product of the orthogonal transformations
that are applied to the pencil aS - bH to reduce
S and H to the forms in (2), for COMPQ1 = 'I'.
COMPQ2 CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q2, as follows:
= 'N': Q2 is not computed;
= 'I': on exit, the array Q2 contains the orthogonal
matrix Q2;
= 'U': on exit, the array Q2 contains the matrix product
J*Q*J'*Q2, where Q2 is the product of the
orthogonal transformations that are applied to
the pencil aS - bH to reduce S and H to the forms
in (2), for COMPQ2 = 'I'.
Setting COMPQ2 <> 'N' assumes COMPQ2 = COMPQ1.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Aout; otherwise, it contains the
upper triangular matrix A obtained just before the
application of the periodic QZ algorithm.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
DE (input/output) DOUBLE PRECISION array, dimension
(LDDE, N/2+1)
On entry, the leading N/2-by-N/2 strictly lower triangular
part of this array must contain the strictly lower
triangular part of the skew-symmetric matrix E, and the
N/2-by-N/2 strictly upper triangular part of the submatrix
in the columns 2 to N/2+1 of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix D.
The entries on the diagonal and the first superdiagonal of
this array need not be set, but are assumed to be zero.
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of the submatrix in the columns
2 to N/2+1 of this array contains the strictly upper
triangular part of the skew-symmetric matrix Dout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array contains the strictly upper triangular part
of the skew-symmetric matrix D just before the application
of the periodic QZ algorithm. The remaining entries are
meaningless.
LDDE INTEGER
The leading dimension of the array DE.
LDDE >= MAX(1, N/2).
C1 (input/output) DOUBLE PRECISION array, dimension
(LDC1, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix C1 = C.
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix C1out; otherwise, it contains the
upper triangular matrix C1 obtained just before the
application of the periodic QZ algorithm.
LDC1 INTEGER
The leading dimension of the array C1.
LDC1 >= MAX(1, N/2).
VW (input/output) DOUBLE PRECISION array, dimension
(LDVW, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
symmetric matrix W, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
symmetric matrix V.
On exit, if JOB = 'T', the N/2-by-N/2 part in the columns
2 to N/2+1 of this array contains the matrix Vout.
If JOB = 'E', the N/2-by-N/2 part in the columns 2 to
N/2+1 of this array contains the matrix V just before the
application of the periodic QZ algorithm.
LDVW INTEGER
The leading dimension of the array VW.
LDVW >= MAX(1, N/2).
Q1 (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
On entry, if COMPQ1 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q and the transformation matrix Q1
used to transform the matrices S and H.
On exit, if COMPQ1 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q1.
If COMPQ1 = 'N', this array is not referenced.
LDQ1 INTEGER
The leading dimension of the array Q1.
LDQ1 >= 1, if COMPQ1 = 'N';
LDQ1 >= MAX(1, N), if COMPQ1 = 'I' or COMPQ1 = 'U'.
Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N)
On exit, if COMPQ2 = 'U', then the leading N-by-N part of
this array contains the product of the matrix J*Q*J' and
the transformation matrix Q2 used to transform the
matrices S and H.
On exit, if COMPQ2 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q2.
If COMPQ2 = 'N', this array is not referenced.
LDQ2 INTEGER
The leading dimension of the array Q2.
LDQ2 >= 1, if COMPQ2 = 'N';
LDQ2 >= MAX(1, N), if COMPQ2 = 'I' or COMPQ2 = 'U'.
B (output) DOUBLE PRECISION array, dimension (LDB, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix Bout; otherwise, it contains the
upper triangular matrix B obtained just before the
application of the periodic QZ algorithm.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (output) DOUBLE PRECISION array, dimension (LDF, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
upper triangular part of this array contains the strictly
upper triangular part of the skew-symmetric matrix Fout.
If JOB = 'E', the leading N/2-by-N/2 strictly upper
triangular part of this array contains the strictly upper
triangular part of the skew-symmetric matrix F just before
the application of the periodic QZ algorithm.
The entries on the leading N/2-by-N/2 lower triangular
part of this array are not referenced.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
C2 (output) DOUBLE PRECISION array, dimension (LDC2, N/2)
On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
array contains the matrix C2out; otherwise, it contains
the upper Hessenberg matrix C2 obtained just before the
application of the periodic QZ algorithm.
LDC2 INTEGER
The leading dimension of the array C2.
LDC2 >= MAX(1, N/2).
ALPHAR (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/Hamiltonian structure of the
pencil, for every eigenvalue lambda, -lambda is also an
eigenvalue, and thus it has only to be saved once in
ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.
As a consequence, pairs of complex eigenvalues, stored in
consecutive locations, are not complex conjugate.
Workspace
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = 3, IWORK(1) contains the number of
possibly inaccurate eigenvalues, q <= N/2, and IWORK(2),
..., IWORK(q+1) indicate their indices. Specifically, a
positive value is an index of a real or purely imaginary
eigenvalue, corresponding to a 1-by-1 block, while the
absolute value of a negative entry in IWORK is an index to
the first eigenvalue in a pair of consecutively stored
eigenvalues, corresponding to a 2-by-2 block. A 2-by-2
block may have two complex, two real, two purely
imaginary, or one real and one purely imaginary
eigenvalues.
For i = q+2, ..., 2*q+1, IWORK(i) contains a pointer to
the starting location in DWORK of the i-th quadruple of
1-by-1 blocks, if IWORK(i-q) > 0, or 2-by-2 blocks,
if IWORK(i-q) < 0.
IWORK(2*q+2) contains the number s of finite eigenvalues
corresponding to the 1-by-1 blocks, and IWORK(2*q+3)
contains the number t of the 2-by-2 blocks.
If INFO = 0, then q = 0.
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= MAX(N+12, 2*N+3).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2), ..., DWORK(5) contain the
Frobenius norms of the factors of the formal matrix
product used by the algorithm; in addition, DWORK(6), ...,
DWORK(5+4*s) contain the s quadruple values corresponding
to the used 1-by-1 blocks. Their eigenvalues are finite
real and purely imaginary. (Such an eigenvalue is obtained
from -i*sqrt(a1*a3/a2/a4), but always taking a positive
sign, where a1, ..., a4 are the corresponding quadruple
values.)
Moreover, DWORK(6+4*s), ..., DWORK(5+4*s+16*t) contain the
t groups of quadruple 2-by-2 matrices corresponding to the
used 2-by-2 blocks. Their eigenvalue pairs are finite,
either complex, or placed on the real and imaginary axes.
(Such an eigenvalue pair is obtained from the eigenvalues
of the matrix product A1*inv(A2)*A3*inv(A4), where A1,
..., A4 define the corresponding 2-by-2 matrix quadruple;
such matrix products are not evaluated.)
On exit, if INFO = -27, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
If JOB = 'E' and COMPQ1 = 'N' and COMPQ2 = 'N',
LDWORK >= N**2 + MAX(L, 36);
if JOB = 'T' or COMPQ1 <> 'N' or COMPQ2 <> 'N',
LDWORK >= 2*N**2 + MAX(L, 36);
where
L = 4*N + 4, if N/2 is even, and
L = 4*N , if N/2 is odd.
For good performance LDWORK should generally be larger.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: problem during computation of the eigenvalues;
= 2: periodic QZ algorithm did not converge in the SLICOT
Library subroutine MB03BD;
= 3: some eigenvalues might be inaccurate, and details can
be found in IWORK and DWORK. This is a warning.
Method
The algorithm uses Givens rotations and Householder reflections to annihilate elements in S, T, and H such that A, B, and C1 are upper triangular and C2 is upper Hessenberg. Finally, the periodic QZ algorithm is applied to transform C2 to upper quasi-triangular form while A, B, and C1 stay in upper triangular form. See also page 27 in [1] for more details.References
[1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
For large values of N, the routine applies the transformations for reducing T 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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