**Purpose**

To generate an orthogonal symplectic matrix U, which is defined 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) ). as returned by MB04PU. The matrix U is returned in terms of its first N rows [ U1 U2 ] U = [ ]. [ -U2 U1 ]

SUBROUTINE MB04WP( N, ILO, U1, LDU1, U2, LDU2, CS, TAU, DWORK, $ LDWORK, INFO ) C .. Scalar Arguments .. INTEGER ILO, INFO, LDU1, LDU2, LDWORK, N C .. Array Arguments .. DOUBLE PRECISION CS(*), DWORK(*), U1(LDU1,*), U2(LDU2,*), TAU(*)

**Input/Output Parameters**

N (input) INTEGER The order of the matrices U1 and U2. N >= 0. ILO (input) INTEGER ILO must have the same value as in the previous call of MB04PU. U is equal to the unit matrix except in the submatrix U([ilo+1:n n+ilo+1:2*n], [ilo+1:n n+ilo+1:2*n]). 1 <= ILO <= N, if N > 0; ILO = 1, if N = 0. U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,N) On entry, the leading N-by-N part of this array must contain in its i-th column the vector which defines the elementary reflector F(i). On exit, the leading N-by-N part of this array contains the matrix U1. LDU1 INTEGER The leading dimension of the array U1. LDU1 >= MAX(1,N). U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,N) On entry, the leading N-by-N part of this array must contain in its i-th column the vector which defines the elementary reflector H(i) and, on the subdiagonal, the scalar factor of H(i). On exit, the leading N-by-N part of this array contains the matrix U2. LDU2 INTEGER The leading dimension of the array U2. LDU2 >= MAX(1,N). CS (input) DOUBLE PRECISION array, dimension (2N-2) On entry, the first 2N-2 elements of this array must contain the cosines and sines of the symplectic Givens rotations G(i). TAU (input) DOUBLE PRECISION array, dimension (N-1) On entry, the first N-1 elements of this array must contain the scalar factors of the elementary reflectors F(i).

DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK. On exit, if INFO = -10, DWORK(1) returns the minimum value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= MAX(1,2*(N-ILO)). For optimum performance LDWORK should be larger. (See SLICOT Library routine MB04WD). 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.

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

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

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**Program Text**

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