**Purpose**

To compute either the upper or lower triangular part of one of the matrix formulas _ R = alpha*R + beta*op( H )*B, (1) _ R = alpha*R + beta*B*op( H ), (2) _ where alpha and beta are scalars, H, B, R, and R are m-by-m matrices, H is an upper Hessenberg matrix, and op( H ) is one of op( H ) = H or op( H ) = H', the transpose of H. The result is overwritten on R.

SUBROUTINE MB01RY( SIDE, UPLO, TRANS, M, ALPHA, BETA, R, LDR, H, $ LDH, B, LDB, DWORK, INFO ) C .. Scalar Arguments .. CHARACTER SIDE, TRANS, UPLO INTEGER INFO, LDB, LDH, LDR, M DOUBLE PRECISION ALPHA, BETA C .. Array Arguments .. DOUBLE PRECISION B(LDB,*), DWORK(*), H(LDH,*), R(LDR,*)

**Mode Parameters**

SIDE CHARACTER*1 Specifies whether the Hessenberg matrix H appears on the left or right in the matrix product as follows: _ = 'L': R = alpha*R + beta*op( H )*B; _ = 'R': R = alpha*R + beta*B*op( H ). UPLO CHARACTER*1 _ Specifies which triangles of the matrices R and R are computed and given, respectively, as follows: = 'U': the upper triangular part; = 'L': the lower triangular part. 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 order of the matrices R, R, H and B. M >= 0. ALPHA (input) DOUBLE PRECISION The scalar alpha. When alpha is zero then R need not be set before entry. BETA (input) DOUBLE PRECISION The scalar beta. When beta is zero then H and B are not referenced. R (input/output) DOUBLE PRECISION array, dimension (LDR,M) On entry with UPLO = 'U', the leading M-by-M upper triangular part of this array must contain the upper triangular part of the matrix R; the strictly lower triangular part of the array is not referenced. On entry with UPLO = 'L', the leading M-by-M lower triangular part of this array must contain the lower triangular part of the matrix R; the strictly upper triangular part of the array is not referenced. On exit, the leading M-by-M upper triangular part (if UPLO = 'U'), or lower triangular part (if UPLO = 'L') of this array contains the corresponding triangular part of _ the computed matrix R. LDR INTEGER The leading dimension of array R. LDR >= MAX(1,M). H (input) DOUBLE PRECISION array, dimension (LDH,M) On entry, the leading M-by-M upper Hessenberg part of this array must contain the upper Hessenberg part of the 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 array H. LDH >= MAX(1,M). B (input) DOUBLE PRECISION array, dimension (LDB,M) On entry, the leading M-by-M part of this array must contain the matrix B. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,M).

DWORK DOUBLE PRECISION array, dimension (LDWORK) LDWORK >= M, if beta <> 0 and SIDE = 'L'; LDWORK >= 0, if beta = 0 or SIDE = 'R'. This array is not referenced when beta = 0 or SIDE = 'R'.

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

The matrix expression is efficiently evaluated taking the Hessenberg/triangular structure into account. BLAS 2 operations are used. A block algorithm can be constructed; it can use BLAS 3 GEMM operations for most computations, and calls of this BLAS 2 algorithm for computing the triangles.

The main application of this routine is when the result should be a symmetric matrix, e.g., when B = X*op( H )', for (1), or B = op( H )'*X, for (2), where B is already available and X = X'.

**Program Text**

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