**Purpose**

To compute one of the matrix products B = alpha*op( H ) * A, or B = alpha*A * op( H ), where alpha is a scalar, A and B are m-by-n matrices, H is an upper Hessenberg matrix, and op( H ) is one of op( H ) = H or op( H ) = H', the transpose of H.

SUBROUTINE MB01UD( SIDE, TRANS, M, N, ALPHA, H, LDH, A, LDA, B, $ LDB, INFO ) C .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, LDA, LDB, LDH, M, N DOUBLE PRECISION ALPHA C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), B(LDB,*), H(LDH,*)

**Mode Parameters**

SIDE CHARACTER*1 Specifies whether the Hessenberg matrix H appears on the left or right in the matrix product as follows: = 'L': B = alpha*op( H ) * A; = 'R': B = alpha*A * op( H ). TRANS CHARACTER*1 Specifies the form of op( H ) to be used in the matrix multiplication as follows: = 'N': op( H ) = H; = 'T': op( H ) = H'; = 'C': op( H ) = H'.

M (input) INTEGER The number of rows of the matrices A and B. M >= 0. N (input) INTEGER The number of columns of the matrices A and B. N >= 0. ALPHA (input) DOUBLE PRECISION The scalar alpha. When alpha is zero then H is not referenced and A need not be set before entry. H (input) DOUBLE PRECISION array, dimension (LDH,k) where k is M when SIDE = 'L' and is N when SIDE = 'R'. On entry with SIDE = 'L', the leading M-by-M upper Hessenberg part of this array must contain the upper Hessenberg matrix H. On entry with SIDE = 'R', the leading N-by-N upper Hessenberg part of this array must contain the upper Hessenberg matrix H. The elements below the subdiagonal are not referenced, except possibly for those in the first column, which could be overwritten, but are restored on exit. LDH INTEGER The leading dimension of the array H. LDH >= max(1,k), where k is M when SIDE = 'L' and is N when SIDE = 'R'. A (input) DOUBLE PRECISION array, dimension (LDA,N) The leading M-by-N part of this array must contain the matrix A. LDA INTEGER The leading dimension of the array A. LDA >= max(1,M). B (output) DOUBLE PRECISION array, dimension (LDB,N) The leading M-by-N part of this array contains the computed product. LDB INTEGER The leading dimension of the array B. LDB >= max(1,M).

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

The required matrix product is computed in two steps. In the first step, the upper triangle of H is used; in the second step, the contribution of the subdiagonal is added. A fast BLAS 3 DTRMM operation is used in the first step.

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**Program Text**

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