**Purpose**

To overwrite general real m-by-n matrices C and D, or their transposes, with [ op(C) ] Q * [ ] if TRANQ = 'N', or [ op(D) ] T [ op(C) ] Q * [ ] if TRANQ = 'T', [ op(D) ] where Q is defined as the 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) ). Blocked version.

SUBROUTINE MB04QB( TRANC, TRAND, TRANQ, STOREV, STOREW, M, N, K, $ V, LDV, W, LDW, C, LDC, D, LDD, CS, TAU, DWORK, $ LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER STOREV, STOREW, TRANC, TRAND, TRANQ INTEGER INFO, K, LDC, LDD, LDV, LDW, LDWORK, M, N C .. Array Arguments .. DOUBLE PRECISION C(LDC,*), CS(*), D(LDD,*), DWORK(*), TAU(*), $ V(LDV,*), W(LDW,*)

**Mode Parameters**

TRANC CHARACTER*1 Specifies the form of op( C ) as follows: = 'N': op( C ) = C; = 'T': op( C ) = C'; = 'C': op( C ) = C'. TRAND CHARACTER*1 Specifies the form of op( D ) as follows: = 'N': op( D ) = D; = 'T': op( D ) = D'; = 'C': op( D ) = D'. TRANQ CHARACTER*1 = 'N': apply Q; = 'T': apply Q'. STOREV CHARACTER*1 Specifies how the vectors which define the concatenated Householder reflectors contained in V are stored: = 'C': columnwise; = 'R': rowwise. STOREW CHARACTER*1 Specifies how the vectors which define the concatenated Householder reflectors contained in W are stored: = 'C': columnwise; = 'R': rowwise.

M (input) INTEGER The number of rows of the matrices op(C) and op(D). M >= 0. N (input) INTEGER The number of columns of the matrices op(C) and op(D). N >= 0. K (input) INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. V (input) DOUBLE PRECISION array, dimension (LDV,K) if STOREV = 'C', (LDV,M) if STOREV = 'R' On entry with STOREV = 'C', the leading M-by-K part of this array must contain in its columns the vectors which define the elementary reflectors F(i). On entry with STOREV = 'R', the leading K-by-M part of this array must contain in its rows the vectors which define the elementary reflectors F(i). LDV INTEGER The leading dimension of the array V. LDV >= MAX(1,M), if STOREV = 'C'; LDV >= MAX(1,K), if STOREV = 'R'. W (input) DOUBLE PRECISION array, dimension (LDW,K) if STOREW = 'C', (LDW,M) if STOREW = 'R' On entry with STOREW = 'C', the leading M-by-K part of this array must contain in its columns the vectors which define the elementary reflectors H(i). On entry with STOREW = 'R', the leading K-by-M part of this array must contain in its rows the vectors which define the elementary reflectors H(i). LDW INTEGER The leading dimension of the array W. LDW >= MAX(1,M), if STOREW = 'C'; LDW >= MAX(1,K), if STOREW = 'R'. C (input/output) DOUBLE PRECISION array, dimension (LDC,N) if TRANC = 'N', (LDC,M) if TRANC = 'T' or TRANC = 'C' On entry with TRANC = 'N', the leading M-by-N part of this array must contain the matrix C. On entry with TRANC = 'C' or TRANC = 'T', the leading N-by-M part of this array must contain the transpose of the matrix C. On exit with TRANC = 'N', the leading M-by-N part of this array contains the updated matrix C. On exit with TRANC = 'C' or TRANC = 'T', the leading N-by-M part of this array contains the transpose of the updated matrix C. LDC INTEGER The leading dimension of the array C. LDC >= MAX(1,M), if TRANC = 'N'; LDC >= MAX(1,N), if TRANC = 'T' or TRANC = 'C'. D (input/output) DOUBLE PRECISION array, dimension (LDD,N) if TRAND = 'N', (LDD,M) if TRAND = 'T' or TRAND = 'C' On entry with TRAND = 'N', the leading M-by-N part of this array must contain the matrix D. On entry with TRAND = 'C' or TRAND = 'T', the leading N-by-M part of this array must contain the transpose of the matrix D. On exit with TRAND = 'N', the leading M-by-N part of this array contains the updated matrix D. On exit with TRAND = 'C' or TRAND = 'T', the leading N-by-M part of this array contains the transpose of the updated matrix D. LDD INTEGER The leading dimension of the array D. LDD >= MAX(1,M), if TRAND = 'N'; LDD >= MAX(1,N), if TRAND = 'T' or TRAND = '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. On exit, if INFO = -20, DWORK(1) returns the minimum value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= MAX(1,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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