**Purpose**

To compute one of the matrix products A : = alpha*op( T ) * A, or A : = alpha*A * op( T ), where alpha is a scalar, A is an m-by-n matrix, T is a quasi- triangular matrix, and op( T ) is one of op( T ) = T or op( T ) = T', the transpose of T.

SUBROUTINE MB01UX( SIDE, UPLO, TRANS, M, N, ALPHA, T, LDT, A, LDA, $ DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER SIDE, TRANS, UPLO INTEGER INFO, LDA, LDT, LDWORK, M, N DOUBLE PRECISION ALPHA C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), DWORK(*), T(LDT,*)

**Mode Parameters**

SIDE CHARACTER*1 Specifies whether the upper quasi-triangular matrix H appears on the left or right in the matrix product as follows: = 'L': A := alpha*op( T ) * A; = 'R': A := alpha*A * op( T ). UPLO CHARACTER*1. Specifies whether the matrix T is an upper or lower quasi-triangular matrix as follows: = 'U': T is an upper quasi-triangular matrix; = 'L': T is a lower quasi-triangular matrix. TRANS CHARACTER*1 Specifies the form of op( T ) to be used in the matrix multiplication as follows: = 'N': op( T ) = T; = 'T': op( T ) = T'; = 'C': op( T ) = T'.

M (input) INTEGER The number of rows of the matrix A. M >= 0. N (input) INTEGER The number of columns of the matrix A. N >= 0. ALPHA (input) DOUBLE PRECISION The scalar alpha. When alpha is zero then T is not referenced and A need not be set before entry. T (input) DOUBLE PRECISION array, dimension (LDT,k) where k is M when SIDE = 'L' and is N when SIDE = 'R'. On entry with UPLO = 'U', the leading k-by-k upper Hessenberg part of this array must contain the upper quasi-triangular matrix T. The elements below the subdiagonal are not referenced. On entry with UPLO = 'L', the leading k-by-k lower Hessenberg part of this array must contain the lower quasi-triangular matrix T. The elements above the supdiagonal are not referenced. LDT INTEGER The leading dimension of the array T. LDT >= max(1,k), where k is M when SIDE = 'L' and is N when SIDE = 'R'. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading M-by-N part of this array must contain the matrix A. On exit, the leading M-by-N part of this array contains the computed product. LDA INTEGER The leading dimension of the array A. LDA >= max(1,M).

DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0 and ALPHA<>0, DWORK(1) returns the optimal value of LDWORK. On exit, if INFO = -12, DWORK(1) returns the minimum value of LDWORK. This array is not referenced when alpha = 0. LDWORK The length of the array DWORK. LDWORK >= 1, if alpha = 0 or MIN(M,N) = 0; LDWORK >= 2*(M-1), if SIDE = 'L'; LDWORK >= 2*(N-1), if SIDE = 'R'. For maximal efficiency LDWORK should be at least NOFF*N + M - 1, if SIDE = 'L'; NOFF*M + N - 1, if SIDE = 'R'; where NOFF is the number of nonzero elements on the subdiagonal (if UPLO = 'U') or supdiagonal (if UPLO = 'L') of T. 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 technique used in this routine is similiar to the technique used in the SLICOT [1] subroutine MB01UW developed by Vasile Sima. The required matrix product is computed in two steps. In the first step, the triangle of T specified by UPLO is used; in the second step, the contribution of the sub-/supdiagonal is added. If the workspace can accommodate parts of A, a fast BLAS 3 DTRMM operation is used in the first step.

[1] Benner, P., Mehrmann, V., Sima, V., Van Huffel, S., and Varga, A. SLICOT - A subroutine library in systems and control theory. In: Applied and computational control, signals, and circuits, Vol. 1, pp. 499-539, Birkhauser, Boston, 1999.

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**Program Text**

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