Purpose
To compute an orthogonal matrix Q and an orthogonal symplectic matrix U for a real regular 2-by-2 or 4-by-4 skew-Hamiltonian/ Hamiltonian pencil a J B' J' B - b D with ( B11 B12 ) ( D11 D12 ) B = ( ), D = ( ), ( 0 B22 ) ( 0 -D11' ) such that J Q' J' D Q and U' B Q keep block triangular form, but the eigenvalues are reordered.Specification
SUBROUTINE MB03GD( N, B, LDB, D, LDD, MACPAR, Q, LDQ, U, LDU, $ DWORK, LDWORK, INFO ) C .. Scalar Arguments .. INTEGER INFO, LDB, LDD, LDQ, LDU, LDWORK, N C .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), D( LDD, * ), DWORK( * ), $ MACPAR( * ), Q( LDQ, * ), U( LDU, * )Arguments
Input/Output Parameters
N (input) INTEGER The order of the pencil a J B' J' B - b D. N = 2 or N = 4. B (input) DOUBLE PRECISION array, dimension (LDB, N) The leading N-by-N part of this array must contain the non-trivial factor of the decomposition of the skew-Hamiltonian input matrix J B' J' B. The (2,1) block is not referenced. LDB INTEGER The leading dimension of the array B. LDB >= N. D (input) DOUBLE PRECISION array, dimension (LDD, N) The leading N/2-by-N part of this array must contain the first block row of the second matrix of a J B' J' B - b D. The matrix D has to be Hamiltonian. The strict lower triangle of the (1,2) block is not referenced. LDD INTEGER The leading dimension of the array D. LDD >= 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. U (output) DOUBLE PRECISION array, dimension (LDU, N) The leading N-by-N part of this array contains the orthogonal symplectic transformation matrix U. LDU INTEGER The leading dimension of the array U. LDU >= N.Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK) If N = 2 then DWORK is not referenced. LDWORK INTEGER The length of the array DWORK. If N = 2 then LDWORK >= 0; if N = 4 then LDWORK >= 12.Error Indicator
INFO INTEGER = 0: succesful exit; = 1: B11 or B22 is a (numerically) singular matrix.Method
The algorithm uses orthogonal transformations as described on page 22 in [1], but with an improved implementation.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
The algorithm is numerically backward stable.Further Comments
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