SUBROUTINE MB01RD( UPLO, TRANS, M, N, ALPHA, BETA, R, LDR, A, LDA,
$ X, LDX, DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To compute the matrix formula
C _
C R = alpha*R + beta*op( A )*X*op( A )',
C _
C where alpha and beta are scalars, R, X, and R are symmetric
C matrices, A is a general matrix, and op( A ) is one of
C
C op( A ) = A or op( A ) = A'.
C
C The result is overwritten on R.
C
C ARGUMENTS
C
C Mode Parameters
C
C UPLO CHARACTER*1 _
C Specifies which triangles of the symmetric matrices R, R,
C and X are given as follows:
C = 'U': the upper triangular part is given;
C = 'L': the lower triangular part is given.
C
C TRANS CHARACTER*1
C Specifies the form of op( A ) to be used in the matrix
C multiplication as follows:
C = 'N': op( A ) = A;
C = 'T': op( A ) = A';
C = 'C': op( A ) = A'.
C
C Input/Output Parameters
C
C M (input) INTEGER _
C The order of the matrices R and R and the number of rows
C of the matrix op( A ). M >= 0.
C
C N (input) INTEGER
C The order of the matrix X and the number of columns of the
C the matrix op( A ). N >= 0.
C
C ALPHA (input) DOUBLE PRECISION
C The scalar alpha. When alpha is zero then R need not be
C set before entry, except when R is identified with X in
C the call (which is possible only in this case).
C
C BETA (input) DOUBLE PRECISION
C The scalar beta. When beta is zero then A and X 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 symmetric matrix R; the strictly
C lower triangular part of the array is used as workspace.
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 symmetric matrix R; the strictly
C upper triangular part of the array is used as workspace.
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. If beta <> 0, the remaining
C strictly triangular part of this array contains the
C corresponding part of the matrix expression
C beta*op( A )*T*op( A )', where T is the triangular matrix
C defined in the Method section.
C
C LDR INTEGER
C The leading dimension of array R. LDR >= MAX(1,M).
C
C A (input) DOUBLE PRECISION array, dimension (LDA,k)
C where k is N when TRANS = 'N' and is M when TRANS = 'T' or
C TRANS = 'C'.
C On entry with TRANS = 'N', the leading M-by-N part of this
C array must contain the matrix A.
C On entry with TRANS = 'T' or TRANS = 'C', the leading
C N-by-M part of this array must contain the matrix A.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,l),
C where l is M when TRANS = 'N' and is N when TRANS = 'T' or
C TRANS = 'C'.
C
C X (input/output) DOUBLE PRECISION array, dimension (LDX,N)
C On entry, if UPLO = 'U', the leading N-by-N upper
C triangular part of this array must contain the upper
C triangular part of the symmetric matrix X and the strictly
C lower triangular part of the array is not referenced.
C On entry, if UPLO = 'L', the leading N-by-N lower
C triangular part of this array must contain the lower
C triangular part of the symmetric matrix X and the strictly
C upper triangular part of the array is not referenced.
C On exit, each diagonal element of this array has half its
C input value, but the other elements are not modified.
C
C LDX INTEGER
C The leading dimension of array X. LDX >= MAX(1,N).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, the leading M-by-N part of this
C array (with the leading dimension MAX(1,M)) returns the
C matrix product beta*op( A )*T, where T is the triangular
C matrix defined in the Method section.
C This array is not referenced when beta = 0.
C
C LDWORK The length of the array DWORK.
C LDWORK >= MAX(1,M*N), if beta <> 0;
C LDWORK >= 1, if beta = 0.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -k, the k-th argument had an illegal
C value.
C
C METHOD
C
C The matrix expression is efficiently evaluated taking the symmetry
C into account. Specifically, let X = T + T', with T an upper or
C lower triangular matrix, defined by
C
C T = triu( X ) - (1/2)*diag( X ), if UPLO = 'U',
C T = tril( X ) - (1/2)*diag( X ), if UPLO = 'L',
C
C where triu, tril, and diag denote the upper triangular part, lower
C triangular part, and diagonal part of X, respectively. Then,
C
C op( A )*X*op( A )' = B + B',
C
C where B := op( A )*T*op( A )'. Matrix B is not symmetric, but it
C can be written as tri( B ) + stri( B ), where tri denotes the
C triangular part specified by UPLO, and stri denotes the remaining
C strictly triangular part. Let R = V + V', with V defined as T
C above. Then, the required triangular part of the result can be
C written as
C
C alpha*V + beta*tri( B ) + beta*(stri( B ))' +
C alpha*diag( V ) + beta*diag( tri( B ) ).
C
C REFERENCES
C
C None.
C
C NUMERICAL ASPECTS
C
C The algorithm requires approximately
C
C 2 2
C M x N + 1/2 x N x M
C
C operations.
C
C CONTRIBUTORS
C
C V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1997.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Mar. 2004,
C Apr. 2004, Sep. 2013.
C
C KEYWORDS
C
C Elementary matrix operations, matrix algebra, matrix operations.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE, HALF
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, HALF = 0.5D0 )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDA, LDR, LDWORK, LDX, M, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), R(LDR,*), X(LDX,*)
C .. Local Scalars ..
CHARACTER*12 NTRAN
LOGICAL LTRANS, LUPLO
INTEGER J, JWORK, LDW, NROWA
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMM, DLACPY, DLASCL, DLASET,
$ DSCAL, DTRMM, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX
C .. Executable Statements ..
C
C Test the input scalar arguments.
C
INFO = 0
LUPLO = LSAME( UPLO, 'U' )
LTRANS = LSAME( TRANS, 'T' ) .OR. LSAME( TRANS, 'C' )
C
IF ( LTRANS ) THEN
NROWA = N
NTRAN = 'No transpose'
ELSE
NROWA = M
NTRAN = 'Transpose'
END IF
C
LDW = MAX( 1, M )
C
IF( ( .NOT.LUPLO ).AND.( .NOT.LSAME( UPLO, 'L' ) ) )THEN
INFO = -1
ELSE IF( ( .NOT.LTRANS ).AND.( .NOT.LSAME( TRANS, 'N' ) ) )THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDR.LT.LDW ) THEN
INFO = -8
ELSE IF( LDA.LT.MAX( 1, NROWA ) ) THEN
INFO = -10
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( ( BETA.NE.ZERO .AND. LDWORK.LT.MAX( 1, M*N ) )
$ .OR.( BETA.EQ.ZERO .AND. LDWORK.LT.1 ) ) THEN
INFO = -14
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'MB01RD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
CALL DSCAL( N, HALF, X, LDX+1 )
IF ( M.EQ.0 )
$ RETURN
C
IF ( BETA.EQ.ZERO .OR. N.EQ.0 ) THEN
IF ( ALPHA.EQ.ZERO ) THEN
C
C Special case alpha = 0.
C
CALL DLASET( UPLO, M, M, ZERO, ZERO, R, LDR )
ELSE
C
C Special case beta = 0 or N = 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. Efficiently compute
C _
C R = alpha*R + beta*op( A )*X*op( A )',
C
C as described in the Method section.
C
C Compute W = beta*op( A )*T in DWORK.
C Workspace: need M*N.
C
C (Note: Comments in the code beginning "Workspace:" describe the
C minimal amount of real workspace needed at that point in the
C code.)
C
IF( LTRANS ) THEN
JWORK = 1
C
DO 10 J = 1, N
CALL DCOPY( M, A(J,1), LDA, DWORK(JWORK), 1 )
JWORK = JWORK + LDW
10 CONTINUE
C
ELSE
CALL DLACPY( 'Full', M, N, A, LDA, DWORK, LDW )
END IF
C
CALL DTRMM( 'Right', UPLO, 'No transpose', 'Non-unit', M, N, BETA,
$ X, LDX, DWORK, LDW )
C
C Compute Y = alpha*V + W*op( A )' in R. First, set to zero the
C strictly triangular part of R not specified by UPLO. That part
C will then contain beta*stri( B ).
C
IF ( ALPHA.NE.ZERO ) THEN
IF ( M.GT.1 ) THEN
IF ( LUPLO ) THEN
CALL DLASET( 'Lower', M-1, M-1, ZERO, ZERO, R(2,1), LDR )
ELSE
CALL DLASET( 'Upper', M-1, M-1, ZERO, ZERO, R(1,2), LDR )
END IF
END IF
CALL DSCAL( M, HALF, R, LDR+1 )
END IF
C
CALL DGEMM( 'No transpose', NTRAN, M, M, N, ONE, DWORK, LDW, A,
$ LDA, ALPHA, R, LDR )
C
C Add the term corresponding to B', with B = op( A )*T*op( A )'.
C
IF( LUPLO ) THEN
C
DO 20 J = 1, M
CALL DAXPY( J, ONE, R(J,1), LDR, R(1,J), 1 )
20 CONTINUE
C
ELSE
C
DO 30 J = 1, M
CALL DAXPY( J, ONE, R(1,J), 1, R(J,1), LDR )
30 CONTINUE
C
END IF
C
RETURN
C *** Last line of MB01RD ***
END