**Purpose**

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

SUBROUTINE MB01RB( SIDE, UPLO, TRANS, M, N, ALPHA, BETA, R, LDR, $ A, LDA, B, LDB, INFO ) C .. Scalar Arguments .. CHARACTER SIDE, TRANS, UPLO INTEGER INFO, LDA, LDB, LDR, M, N DOUBLE PRECISION ALPHA, BETA C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), B(LDB,*), R(LDR,*)

**Mode Parameters**

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

M (input) INTEGER _ The order of the matrices R and R, the number of rows of the matrix op( A ) and the number of columns of the matrix B, for SIDE = 'L', or the number of rows of the matrix B and the number of columns of the matrix op( A ), for SIDE = 'R'. M >= 0. N (input) INTEGER The number of rows of the matrix B and the number of columns of the matrix op( A ), for SIDE = 'L', or the number of rows of the matrix op( A ) and the number of columns of the matrix B, for SIDE = 'R'. N >= 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 A 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). A (input) DOUBLE PRECISION array, dimension (LDA,k), where k = N when SIDE = 'L', and TRANS = 'N', or SIDE = 'R', and TRANS <> 'T'; k = M when SIDE = 'R', and TRANS = 'N', or SIDE = 'L', and TRANS <> 'T'. On entry, if SIDE = 'L', and TRANS = 'N', or SIDE = 'R', and TRANS <> 'T', the leading M-by-N part of this array must contain the matrix A. On entry, if SIDE = 'R', and TRANS = 'N', or SIDE = 'L', and TRANS <> 'T', the leading N-by-M part of this array must contain the matrix A. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,l), where l = M when SIDE = 'L', and TRANS = 'N', or SIDE = 'R', and TRANS <> 'T'; l = N when SIDE = 'R', and TRANS = 'N', or SIDE = 'L', and TRANS <> 'T'. B (input) DOUBLE PRECISION array, dimension (LDB,p), where p = M when SIDE = 'L'; p = N when SIDE = 'R'. On entry, the leading N-by-M part, if SIDE = 'L', or M-by-N part, if SIDE = 'R', of this array must contain the matrix B. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,N), if SIDE = 'L'; LDB >= MAX(1,M), if SIDE = 'R'.

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

The matrix expression is evaluated taking the triangular structure into account. A block algorithm is used.

The main application of this routine is when the result should be a symmetric matrix, e.g., when B = X*op( A )', for (1), or B = op( A )'*X, for (2), where B is already available and X = X'. The required triangle only is computed and overwritten, contrary to a general matrix multiplication operation. This is a BLAS 3 version of the SLICOT Library routine MB01RX.

**Program Text**

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