**Purpose**

To generate a matrix Q with orthogonal columns (spanning an isotropic subspace), which is defined as the first n columns of a product of symplectic reflectors and Givens rotations, Q = 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(k),H(k) ) G(k) diag( F(k),F(k) ). The matrix Q is returned in terms of its first 2*M rows [ op( Q1 ) op( Q2 ) ] Q = [ ]. [ -op( Q2 ) op( Q1 ) ] Blocked version of the SLICOT Library routine MB04WU.

SUBROUTINE MB04WD( TRANQ1, TRANQ2, M, N, K, Q1, LDQ1, Q2, LDQ2, $ CS, TAU, DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER TRANQ1, TRANQ2 INTEGER INFO, K, LDQ1, LDQ2, LDWORK, M, N C .. Array Arguments .. DOUBLE PRECISION CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)

**Mode Parameters**

TRANQ1 CHARACTER*1 Specifies the form of op( Q1 ) as follows: = 'N': op( Q1 ) = Q1; = 'T': op( Q1 ) = Q1'; = 'C': op( Q1 ) = Q1'. TRANQ2 CHARACTER*1 Specifies the form of op( Q2 ) as follows: = 'N': op( Q2 ) = Q2; = 'T': op( Q2 ) = Q2'; = 'C': op( Q2 ) = Q2'.

M (input) INTEGER The number of rows of the matrices Q1 and Q2. M >= 0. N (input) INTEGER The number of columns of the matrices Q1 and Q2. M >= N >= 0. K (input) INTEGER The number of symplectic Givens rotations whose product partly defines the matrix Q. N >= K >= 0. Q1 (input/output) DOUBLE PRECISION array, dimension (LDQ1,N) if TRANQ1 = 'N', (LDQ1,M) if TRANQ1 = 'T' or TRANQ1 = 'C' On entry with TRANQ1 = 'N', the leading M-by-K part of this array must contain in its i-th column the vector which defines the elementary reflector F(i). On entry with TRANQ1 = 'T' or TRANQ1 = 'C', the leading K-by-M part of this array must contain in its i-th row the vector which defines the elementary reflector F(i). On exit with TRANQ1 = 'N', the leading M-by-N part of this array contains the matrix Q1. On exit with TRANQ1 = 'T' or TRANQ1 = 'C', the leading N-by-M part of this array contains the matrix Q1'. LDQ1 INTEGER The leading dimension of the array Q1. LDQ1 >= MAX(1,M), if TRANQ1 = 'N'; LDQ1 >= MAX(1,N), if TRANQ1 = 'T' or TRANQ1 = 'C'. Q2 (input/output) DOUBLE PRECISION array, dimension (LDQ2,N) if TRANQ2 = 'N', (LDQ2,M) if TRANQ2 = 'T' or TRANQ2 = 'C' On entry with TRANQ2 = 'N', the leading M-by-K part of this array must contain in its i-th column the vector which defines the elementary reflector H(i) and, on the diagonal, the scalar factor of H(i). On entry with TRANQ2 = 'T' or TRANQ2 = 'C', the leading K-by-M part of this array must contain in its i-th row the vector which defines the elementary reflector H(i) and, on the diagonal, the scalar factor of H(i). On exit with TRANQ2 = 'N', the leading M-by-N part of this array contains the matrix Q2. On exit with TRANQ2 = 'T' or TRANQ2 = 'C', the leading N-by-M part of this array contains the matrix Q2'. LDQ2 INTEGER The leading dimension of the array Q2. LDQ2 >= MAX(1,M), if TRANQ2 = 'N'; LDQ2 >= MAX(1,N), if TRANQ2 = 'T' or TRANQ2 = 'C'. CS (input) DOUBLE PRECISION array, dimension (2*K) On entry, the first 2*K elements of this array must contain the cosines and sines of the symplectic Givens rotations G(i). TAU (input) DOUBLE PRECISION array, dimension (K) On entry, the first K 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, MAX(M+N,8*N*NB + 15*NB*NB), where NB is the optimal block size determined by the function UE01MD. On exit, if INFO = -13, DWORK(1) returns the minimum value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= MAX(1,M+N). 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.

[1] 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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