**Purpose**

To reduce a skew-Hamiltonian matrix, [ A G ] W = [ T ] , [ Q A ] where A is an N-by-N matrix and G, Q are N-by-N skew-symmetric matrices, to Paige/Van Loan (PVL) form. That is, an orthogonal symplectic matrix U is computed so that T [ Aout Gout ] U W U = [ T ] , [ 0 Aout ] where Aout is in upper Hessenberg form. Blocked version.

SUBROUTINE MB04RB( N, ILO, A, LDA, QG, LDQG, CS, TAU, DWORK, $ LDWORK, INFO ) C .. Scalar Arguments .. INTEGER ILO, INFO, LDA, LDQG, LDWORK, N C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), CS(*), DWORK(*), QG(LDQG,*), TAU(*)

**Input/Output Parameters**

N (input) INTEGER The order of the matrix A. N >= 0. ILO (input) INTEGER It is assumed that A is already upper triangular and Q is zero in rows and columns 1:ILO-1. ILO is normally set by a previous call to the SLICOT Library routine MB04DS; otherwise it should be set to 1. 1 <= ILO <= N+1, if N > 0; ILO = 1, if N = 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the matrix A. On exit, the leading N-by-N part of this array contains the matrix Aout and, in the zero part of Aout, information about the elementary reflectors used to compute the PVL factorization. LDA INTEGER The leading dimension of the array A. LDA >= MAX(1,N). QG (input/output) DOUBLE PRECISION array, dimension (LDQG,N+1) On entry, the leading N-by-N+1 part of this array must contain in columns 1:N the strictly lower triangular part of the matrix Q and in columns 2:N+1 the strictly upper triangular part of the matrix G. The parts containing the diagonal and the first superdiagonal of this array are not referenced. On exit, the leading N-by-N+1 part of this array contains in its first N-1 columns information about the elementary reflectors used to compute the PVL factorization and in its last N columns the strictly upper triangular part of the matrix Gout. LDQG INTEGER The leading dimension of the array QG. LDQG >= MAX(1,N). CS (output) DOUBLE PRECISION array, dimension (2N-2) On exit, the first 2N-2 elements of this array contain the cosines and sines of the symplectic Givens rotations used to compute the PVL factorization. TAU (output) DOUBLE PRECISION array, dimension (N-1) On exit, the first N-1 elements of this array contain the scalar factors of some of the elementary reflectors.

DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK, 8*N*NB + 3*NB, where NB is the optimal block size. On exit, if INFO = -10, DWORK(1) returns the minimum value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= MAX(1,N-1). If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the DWORK array, returns this value as the first entry of the DWORK array, and no error message related to LDWORK is issued by XERBLA.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

An algorithm similar to the block algorithm for the symplectic URV factorization described in [2] is used. The matrix U is represented as a product of symplectic reflectors and Givens rotations U = diag( H(1),H(1) ) G(1) diag( F(1),F(1) ) diag( H(2),H(2) ) G(2) diag( F(2),F(2) ) .... diag( H(n-1),H(n-1) ) G(n-1) diag( F(n-1),F(n-1) ). Each H(i) has the form H(i) = I - tau * v * v' where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in QG(i+2:n,i), and tau in QG(i+1,i). Each F(i) has the form F(i) = I - nu * w * w' where nu is a real scalar, and w is a real vector with w(1:i) = 0 and w(i+1) = 1; w(i+2:n) is stored on exit in A(i+2:n,i), and nu in TAU(i). Each G(i) is a Givens rotation acting on rows i+1 and n+i+1, where the cosine is stored in CS(2*i-1) and the sine in CS(2*i).

The algorithm requires O(N**3) floating point operations and is strongly backward stable.

[1] Van Loan, C.F. A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Linear Algebra and its Applications, 61, pp. 233-251, 1984. [2] Kressner, D. Block algorithms for orthogonal symplectic factorizations. BIT, 43(4), pp. 775-790, 2003.

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**Program Text**

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