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SUBROUTINE MB01RY( SIDE, UPLO, TRANS, M, ALPHA, BETA, R, LDR, H,
$ LDH, B, LDB, DWORK, INFO )
C
C RELEASE 4.0, WGS COPYRIGHT 1999.
C
C PURPOSE
C
C To compute either the upper or lower triangular part of one of the
C matrix formulas
C _
C R = alpha*R + beta*op( H )*B, (1)
C _
C R = alpha*R + beta*B*op( H ), (2)
C _
C where alpha and beta are scalars, H, B, R, and R are m-by-m
C matrices, H is an upper Hessenberg matrix, and op( H ) is one of
C
C op( H ) = H or op( H ) = H', the transpose of H.
C
C The result is overwritten on R.
C
C ARGUMENTS
C
C Mode Parameters
C
C SIDE CHARACTER*1
C Specifies whether the Hessenberg matrix H appears on the
C left or right in the matrix product as follows:
C _
C = 'L': R = alpha*R + beta*op( H )*B;
C _
C = 'R': R = alpha*R + beta*B*op( H ).
C
C UPLO CHARACTER*1 _
C Specifies which triangles of the matrices R and R are
C computed and given, respectively, as follows:
C = 'U': the upper triangular part;
C = 'L': the lower triangular part.
C
C TRANS CHARACTER*1
C Specifies the form of op( H ) to be used in the matrix
C multiplication as follows:
C = 'N': op( H ) = H;
C = 'T': op( H ) = H';
C = 'C': op( H ) = H'.
C
C Input/Output Parameters
C
C M (input) INTEGER _
C The order of the matrices R, R, H and B. M >= 0.
C
C ALPHA (input) DOUBLE PRECISION
C The scalar alpha. When alpha is zero then R need not be
C set before entry.
C
C BETA (input) DOUBLE PRECISION
C The scalar beta. When beta is zero then H and B are not
C referenced.
C
C R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
C On entry with UPLO = 'U', the leading M-by-M upper
C triangular part of this array must contain the upper
C triangular part of the matrix R; the strictly lower
C triangular part of the array is not referenced.
C On entry with UPLO = 'L', the leading M-by-M lower
C triangular part of this array must contain the lower
C triangular part of the matrix R; the strictly upper
C triangular part of the array is not referenced.
C On exit, the leading M-by-M upper triangular part (if
C UPLO = 'U'), or lower triangular part (if UPLO = 'L') of
C this array contains the corresponding triangular part of
C _
C the computed matrix R.
C
C LDR INTEGER
C The leading dimension of array R. LDR >= MAX(1,M).
C
C H (input) DOUBLE PRECISION array, dimension (LDH,M)
C On entry, the leading M-by-M upper Hessenberg part of
C this array must contain the upper Hessenberg part of the
C matrix H.
C The elements below the subdiagonal are not referenced,
C except possibly for those in the first column, which
C could be overwritten, but are restored on exit.
C
C LDH INTEGER
C The leading dimension of array H. LDH >= MAX(1,M).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,M)
C On entry, the leading M-by-M part of this array must
C contain the matrix B.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,M).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C LDWORK >= M, if beta <> 0 and SIDE = 'L';
C LDWORK >= 0, if beta = 0 or SIDE = 'R'.
C This array is not referenced when beta = 0 or SIDE = 'R'.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C The matrix expression is efficiently evaluated taking the
C Hessenberg/triangular structure into account. BLAS 2 operations
C are used. A block algorithm can be constructed; it can use BLAS 3
C GEMM operations for most computations, and calls of this BLAS 2
C algorithm for computing the triangles.
C
C FURTHER COMMENTS
C
C The main application of this routine is when the result should
C be a symmetric matrix, e.g., when B = X*op( H )', for (1), or
C B = op( H )'*X, for (2), where B is already available and X = X'.
C
C CONTRIBUTORS
C
C V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1999.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Elementary matrix operations, matrix algebra, matrix operations.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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,*)
C .. Local Scalars ..
LOGICAL LSIDE, LTRANS, LUPLO
INTEGER I, J
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL DDOT, LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DLASCL, DLASET, DSCAL, DSWAP,
$ DTRMV, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN
C .. Executable Statements ..
C
C Test the input scalar arguments.
C
INFO = 0
LSIDE = LSAME( SIDE, 'L' )
LUPLO = LSAME( UPLO, 'U' )
LTRANS = LSAME( TRANS, 'T' ) .OR. LSAME( TRANS, 'C' )
C
IF( ( .NOT.LSIDE ).AND.( .NOT.LSAME( SIDE, 'R' ) ) )THEN
INFO = -1
ELSE IF( ( .NOT.LUPLO ).AND.( .NOT.LSAME( UPLO, 'L' ) ) )THEN
INFO = -2
ELSE IF( ( .NOT.LTRANS ).AND.( .NOT.LSAME( TRANS, 'N' ) ) )THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( LDR.LT.MAX( 1, M ) ) THEN
INFO = -8
ELSE IF( LDH.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
INFO = -12
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'MB01RY', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( M.EQ.0 )
$ RETURN
C
IF ( BETA.EQ.ZERO ) THEN
IF ( ALPHA.EQ.ZERO ) THEN
C
C Special case when both alpha = 0 and beta = 0.
C
CALL DLASET( UPLO, M, M, ZERO, ZERO, R, LDR )
ELSE
C
C Special case beta = 0.
C
IF ( ALPHA.NE.ONE )
$ CALL DLASCL( UPLO, 0, 0, ONE, ALPHA, M, M, R, LDR, INFO )
END IF
RETURN
END IF
C
C General case: beta <> 0.
C Compute the required triangle of (1) or (2) using BLAS 2
C operations.
C
IF( LSIDE ) THEN
C
C To avoid repeated references to the subdiagonal elements of H,
C these are swapped with the corresponding elements of H in the
C first column, and are finally restored.
C
IF( M.GT.2 )
$ CALL DSWAP( M-2, H( 3, 2 ), LDH+1, H( 3, 1 ), 1 )
C
IF( LUPLO ) THEN
IF ( LTRANS ) THEN
C
DO 20 J = 1, M
C
C Multiply the transposed upper triangle of the leading
C j-by-j submatrix of H by the leading part of the j-th
C column of B.
C
CALL DCOPY( J, B( 1, J ), 1, DWORK, 1 )
CALL DTRMV( 'Upper', TRANS, 'Non-unit', J, H, LDH,
$ DWORK, 1 )
C
C Add the contribution of the subdiagonal of H to
C the j-th column of the product.
C
DO 10 I = 1, MIN( J, M - 1 )
R( I, J ) = ALPHA*R( I, J ) + BETA*( DWORK( I ) +
$ H( I+1, 1 )*B( I+1, J ) )
10 CONTINUE
C
20 CONTINUE
C
R( M, M ) = ALPHA*R( M, M ) + BETA*DWORK( M )
C
ELSE
C
DO 40 J = 1, M
C
C Multiply the upper triangle of the leading j-by-j
C submatrix of H by the leading part of the j-th column
C of B.
C
CALL DCOPY( J, B( 1, J ), 1, DWORK, 1 )
CALL DTRMV( 'Upper', TRANS, 'Non-unit', J, H, LDH,
$ DWORK, 1 )
IF( J.LT.M ) THEN
C
C Multiply the remaining right part of the leading
C j-by-M submatrix of H by the trailing part of the
C j-th column of B.
C
CALL DGEMV( TRANS, J, M-J, BETA, H( 1, J+1 ), LDH,
$ B( J+1, J ), 1, ALPHA, R( 1, J ), 1 )
ELSE
CALL DSCAL( M, ALPHA, R( 1, M ), 1 )
END IF
C
C Add the contribution of the subdiagonal of H to
C the j-th column of the product.
C
R( 1, J ) = R( 1, J ) + BETA*DWORK( 1 )
C
DO 30 I = 2, J
R( I, J ) = R( I, J ) + BETA*( DWORK( I ) +
$ H( I, 1 )*B( I-1, J ) )
30 CONTINUE
C
40 CONTINUE
C
END IF
C
ELSE
C
IF ( LTRANS ) THEN
C
DO 60 J = M, 1, -1
C
C Multiply the transposed upper triangle of the trailing
C (M-j+1)-by-(M-j+1) submatrix of H by the trailing part
C of the j-th column of B.
C
CALL DCOPY( M-J+1, B( J, J ), 1, DWORK( J ), 1 )
CALL DTRMV( 'Upper', TRANS, 'Non-unit', M-J+1,
$ H( J, J ), LDH, DWORK( J ), 1 )
IF( J.GT.1 ) THEN
C
C Multiply the remaining left part of the trailing
C (M-j+1)-by-(j-1) submatrix of H' by the leading
C part of the j-th column of B.
C
CALL DGEMV( TRANS, J-1, M-J+1, BETA, H( 1, J ),
$ LDH, B( 1, J ), 1, ALPHA, R( J, J ),
$ 1 )
ELSE
CALL DSCAL( M, ALPHA, R( 1, 1 ), 1 )
END IF
C
C Add the contribution of the subdiagonal of H to
C the j-th column of the product.
C
DO 50 I = J, M - 1
R( I, J ) = R( I, J ) + BETA*( DWORK( I ) +
$ H( I+1, 1 )*B( I+1, J ) )
50 CONTINUE
C
R( M, J ) = R( M, J ) + BETA*DWORK( M )
60 CONTINUE
C
ELSE
C
DO 80 J = M, 1, -1
C
C Multiply the upper triangle of the trailing
C (M-j+1)-by-(M-j+1) submatrix of H by the trailing
C part of the j-th column of B.
C
CALL DCOPY( M-J+1, B( J, J ), 1, DWORK( J ), 1 )
CALL DTRMV( 'Upper', TRANS, 'Non-unit', M-J+1,
$ H( J, J ), LDH, DWORK( J ), 1 )
C
C Add the contribution of the subdiagonal of H to
C the j-th column of the product.
C
DO 70 I = MAX( J, 2 ), M
R( I, J ) = ALPHA*R( I, J ) + BETA*( DWORK( I )
$ + H( I, 1 )*B( I-1, J ) )
70 CONTINUE
C
80 CONTINUE
C
R( 1, 1 ) = ALPHA*R( 1, 1 ) + BETA*DWORK( 1 )
C
END IF
END IF
C
IF( M.GT.2 )
$ CALL DSWAP( M-2, H( 3, 2 ), LDH+1, H( 3, 1 ), 1 )
C
ELSE
C
C Row-wise calculations are used for H, if SIDE = 'R' and
C TRANS = 'T'.
C
IF( LUPLO ) THEN
IF( LTRANS ) THEN
R( 1, 1 ) = ALPHA*R( 1, 1 ) +
$ BETA*DDOT( M, B, LDB, H, LDH )
C
DO 90 J = 2, M
CALL DGEMV( 'NoTranspose', J, M-J+2, BETA,
$ B( 1, J-1 ), LDB, H( J, J-1 ), LDH,
$ ALPHA, R( 1, J ), 1 )
90 CONTINUE
C
ELSE
C
DO 100 J = 1, M - 1
CALL DGEMV( 'NoTranspose', J, J+1, BETA, B, LDB,
$ H( 1, J ), 1, ALPHA, R( 1, J ), 1 )
100 CONTINUE
C
CALL DGEMV( 'NoTranspose', M, M, BETA, B, LDB,
$ H( 1, M ), 1, ALPHA, R( 1, M ), 1 )
C
END IF
C
ELSE
C
IF( LTRANS ) THEN
C
CALL DGEMV( 'NoTranspose', M, M, BETA, B, LDB, H, LDH,
$ ALPHA, R( 1, 1 ), 1 )
C
DO 110 J = 2, M
CALL DGEMV( 'NoTranspose', M-J+1, M-J+2, BETA,
$ B( J, J-1 ), LDB, H( J, J-1 ), LDH, ALPHA,
$ R( J, J ), 1 )
110 CONTINUE
C
ELSE
C
DO 120 J = 1, M - 1
CALL DGEMV( 'NoTranspose', M-J+1, J+1, BETA,
$ B( J, 1 ), LDB, H( 1, J ), 1, ALPHA,
$ R( J, J ), 1 )
120 CONTINUE
C
R( M, M ) = ALPHA*R( M, M ) +
$ BETA*DDOT( M, B( M, 1 ), LDB, H( 1, M ), 1 )
C
END IF
END IF
END IF
C
RETURN
C *** Last line of MB01RY ***
END
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