Purpose
To determine an orthogonal matrix Q, for a real regular 2-by-2 or 4-by-4 skew-Hamiltonian/Hamiltonian pencil ( A11 A12 ) ( B11 B12 ) aA - bB = a ( T ) - b ( T ) ( 0 A11 ) ( 0 -B11 ) T T in structured Schur form, such that J Q J (aA - bB) Q is still in structured Schur form but the eigenvalues are exchanged.Specification
SUBROUTINE MB03HD( N, A, LDA, B, LDB, MACPAR, Q, LDQ, DWORK, $ INFO ) C .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LDQ, N C .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ), $ MACPAR( * ), Q( LDQ, * )Arguments
Input/Output Parameters
N (input) INTEGER The order of the pencil aA - bB. N = 2 or N = 4. A (input) DOUBLE PRECISION array, dimension (LDA, N) If N = 4, the leading N/2-by-N upper trapezoidal part of this array must contain the first block row of the skew- Hamiltonian matrix A of the pencil aA - bB in structured Schur form. Only the entries (1,1), (1,2), (1,4), and (2,2) are referenced. If N = 2, this array is not referenced. LDA INTEGER The leading dimension of the array A. LDA >= N/2. B (input) DOUBLE PRECISION array, dimension (LDB, N) The leading N/2-by-N part of this array must contain the first block row of the Hamiltonian matrix B of the pencil aA - bB in structured Schur form. The entry (2,3) is not referenced. LDB INTEGER The leading dimension of the array B. LDB >= N/2. MACPAR (input) DOUBLE PRECISION array, dimension (2) Machine parameters: MACPAR(1) (machine precision)*base, DLAMCH( 'P' ); MACPAR(2) safe minimum, DLAMCH( 'S' ). This argument is not used for N = 2. Q (output) DOUBLE PRECISION array, dimension (LDQ, N) The leading N-by-N part of this array contains the orthogonal transformation matrix Q. LDQ INTEGER The leading dimension of the array Q. LDQ >= N.Workspace
DWORK DOUBLE PRECISION array, dimension (24) If N = 2, then DWORK is not referenced.Error Indicator
INFO INTEGER = 0: succesful exit; = 1: the leading N/2-by-N/2 block of the matrix B is numerically singular.Method
The algorithm uses orthogonal transformations as described on page 31 in [2]. The structure is exploited.References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H. Numerical computation of deflating subspaces of skew- Hamiltonian/Hamiltonian pencils. SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002. [2] 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
The algorithm is numerically backward stable.Further Comments
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