SUBROUTINE MB02XD( FORM, STOR, UPLO, F, M, N, NRHS, IPAR, LIPAR,
$ DPAR, LDPAR, A, LDA, B, LDB, ATA, LDATA, DWORK,
$ LDWORK, INFO )
C
C PURPOSE
C
C To solve a set of systems of linear equations, A'*A*X = B, or,
C in the implicit form, f(A)*X = B, with A'*A or f(A) positive
C definite, using symmetric Gaussian elimination.
C
C ARGUMENTS
C
C Mode Parameters
C
C FORM CHARACTER*1
C Specifies the form in which the matrix A is provided, as
C follows:
C = 'S' : standard form, the matrix A is given;
C = 'F' : the implicit, function form f(A) is provided.
C If FORM = 'F', then the routine F is called to compute the
C matrix A'*A.
C
C STOR CHARACTER*1
C Specifies the storage scheme for the symmetric
C matrix A'*A, as follows:
C = 'F' : full storage is used;
C = 'P' : packed storage is used.
C
C UPLO CHARACTER*1
C Specifies which part of the matrix A'*A is stored, as
C follows:
C = 'U' : the upper triagular part is stored;
C = 'L' : the lower triagular part is stored.
C
C Function Parameters
C
C F EXTERNAL
C If FORM = 'F', then F is a subroutine which calculates the
C value of f(A) = A'*A, for given A.
C If FORM = 'S', then F is not called.
C
C F must have the following interface:
C
C SUBROUTINE F( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR, A,
C $ LDA, ATA, LDATA, DWORK, LDWORK, INFO )
C
C where
C
C STOR (input) CHARACTER*1
C Specifies the storage scheme for the symmetric
C matrix A'*A, as follows:
C = 'F' : full storage is used;
C = 'P' : packed storage is used.
C
C UPLO (input) CHARACTER*1
C Specifies which part of the matrix A'*A is stored,
C as follows:
C = 'U' : the upper triagular part is stored;
C = 'L' : the lower triagular part is stored.
C
C N (input) INTEGER
C The order of the matrix A'*A. N >= 0.
C
C IPAR (input) INTEGER array, dimension (LIPAR)
C The integer parameters describing the structure of
C the matrix A.
C
C LIPAR (input) INTEGER
C The length of the array IPAR. LIPAR >= 0.
C
C DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
C The real parameters needed for solving the
C problem.
C
C LDPAR (input) INTEGER
C The length of the array DPAR. LDPAR >= 0.
C
C A (input) DOUBLE PRECISION array, dimension
C (LDA, NC), where NC is the number of columns.
C The leading NR-by-NC part of this array must
C contain the (compressed) representation of the
C matrix A, where NR is the number of rows of A
C (function of IPAR entries).
C
C LDA (input) INTEGER
C The leading dimension of the array A.
C LDA >= MAX(1,NR).
C
C ATA (output) DOUBLE PRECISION array,
C dimension (LDATA,N), if STOR = 'F',
C dimension (N*(N+1)/2), if STOR = 'P'.
C The leading N-by-N (if STOR = 'F'), or N*(N+1)/2
C (if STOR = 'P') part of this array contains the
C upper or lower triangle of the matrix A'*A,
C depending on UPLO = 'U', or UPLO = 'L',
C respectively, stored either as a two-dimensional,
C or one-dimensional array, depending on STOR.
C
C LDATA (input) INTEGER
C The leading dimension of the array ATA.
C LDATA >= MAX(1,N), if STOR = 'F'.
C LDATA >= 1, if STOR = 'P'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C The workspace array for subroutine F.
C
C LDWORK (input) INTEGER
C The size of the array DWORK (as large as needed
C in the subroutine F).
C
C INFO INTEGER
C Error indicator, set to a negative value if an
C input scalar argument is erroneous, and to
C positive values for other possible errors in the
C subroutine F. The LAPACK Library routine XERBLA
C should be used in conjunction with negative INFO.
C INFO must be zero if the subroutine finished
C successfully.
C
C Parameters marked with "(input)" must not be changed.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The number of rows of the matrix A. M >= 0.
C
C N (input) INTEGER
C The order of the matrix A'*A, the number of columns of the
C matrix A, and the number of rows of the matrix X. N >= 0.
C
C NRHS (input) INTEGER
C The number of columns of the matrices B and X. NRHS >= 0.
C
C IPAR (input) INTEGER array, dimension (LIPAR)
C If FORM = 'F', the integer parameters describing the
C structure of the matrix A.
C This parameter is ignored if FORM = 'S'.
C
C LIPAR (input) INTEGER
C The length of the array IPAR. LIPAR >= 0.
C
C DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
C If FORM = 'F', the real parameters needed for solving
C the problem.
C This parameter is ignored if FORM = 'S'.
C
C LDPAR (input) INTEGER
C The length of the array DPAR. LDPAR >= 0.
C
C A (input) DOUBLE PRECISION array,
C dimension (LDA, N), if FORM = 'S',
C dimension (LDA, NC), if FORM = 'F', where NC is
C the number of columns.
C If FORM = 'S', the leading M-by-N part of this array
C must contain the matrix A.
C If FORM = 'F', the leading NR-by-NC part of this array
C must contain an appropriate representation of matrix A,
C where NR is the number of rows.
C If FORM = 'F', this array is not referenced by this
C routine itself, except in the call to the routine F.
C
C LDA INTEGER
C The leading dimension of array A.
C LDA >= MAX(1,M), if FORM = 'S';
C LDA >= MAX(1,NR), if FORM = 'F'.
C
C B (input/output) DOUBLE PRECISION array, dimension
C (LDB, NRHS)
C On entry, the leading N-by-NRHS part of this array must
C contain the right hand side matrix B.
C On exit, if INFO = 0 and M (or NR) is nonzero, the leading
C N-by-NRHS part of this array contains the solution X of
C the set of systems of linear equations A'*A*X = B or
C f(A)*X = B. If M (or NR) is zero, then B is unchanged.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C ATA (output) DOUBLE PRECISION array,
C dimension (LDATA,N), if STOR = 'F',
C dimension (N*(N+1)/2), if STOR = 'P'.
C The leading N-by-N (if STOR = 'F'), or N*(N+1)/2 (if
C STOR = 'P') part of this array contains the upper or lower
C triangular Cholesky factor of the matrix A'*A, depending
C on UPLO = 'U', or UPLO = 'L', respectively, stored either
C as a two-dimensional, or one-dimensional array, depending
C on STOR.
C
C LDATA INTEGER
C The leading dimension of the array ATA.
C LDATA >= MAX(1,N), if STOR = 'F'.
C LDATA >= 1, if STOR = 'P'.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C
C LDWORK INTEGER
C The length of the array DWORK.
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 > 0: if INFO = i, then the (i,i) element of the
C triangular factor of the matrix A'*A is exactly
C zero (the matrix A'*A is exactly singular);
C if INFO = j > n, then F returned with INFO = j-n.
C
C METHOD
C
C The matrix A'*A is built either directly (if FORM = 'S'), or
C implicitly, by calling the routine F. Then, A'*A is Cholesky
C factored and its factor is used to solve the set of systems of
C linear equations, A'*A*X = B.
C
C REFERENCES
C
C [1] Golub, G.H. and van Loan, C.F.
C Matrix Computations. Third Edition.
C M. D. Johns Hopkins University Press, Baltimore, 1996.
C
C [2] Anderson, E., Bai, Z., Bischof, C., Blackford, Demmel, J.,
C Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S.,
C McKenney, A., Sorensen, D.
C LAPACK Users' Guide: Third Edition, SIAM, Philadelphia, 1999.
C
C NUMERICAL ASPECTS
C
C For speed, this routine does not check for near singularity of the
C matrix A'*A. If the matrix A is nearly rank deficient, then the
C computed X could be inaccurate. Estimates of the reciprocal
C condition numbers of the matrices A and A'*A can be obtained
C using LAPACK routines DGECON and DPOCON (DPPCON), respectively.
C
C The approximate number of floating point operations is
C (M+3)*N**2/2 + N**3/6 + NRHS*N**2, if FORM = 'S',
C f + N**3/6 + NRHS*N**2, if FORM = 'F',
C where M is the number of rows of A, and f is the number of
C floating point operations required by the subroutine F.
C
C CONTRIBUTORS
C
C V. Sima, Research Institute for Informatics, Bucharest, Mar. 2001.
C
C REVISIONS
C
C V. Sima, Mar. 2002.
C
C KEYWORDS
C
C Linear system of equations, matrix operations.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER FORM, STOR, UPLO
INTEGER INFO, LDA, LDATA, LDB, LDPAR, LDWORK, LIPAR, M,
$ N, NRHS
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ATA(*), B(LDB,*), DPAR(*), DWORK(*)
INTEGER IPAR(*)
C .. Local Scalars ..
INTEGER IERR, J, J1
LOGICAL FULL, MAT, UPPER
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DGEMV, DPOTRF, DPOTRS, DPPTRF, DPPTRS, DSYRK, F,
$ XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX
C ..
C .. Executable Statements ..
C
C Decode the scalar input parameters.
C
MAT = LSAME( FORM, 'S' )
FULL = LSAME( STOR, 'F' )
UPPER = LSAME( UPLO, 'U' )
C
C Check the scalar input parameters.
C
INFO = 0
IF( .NOT.( MAT .OR. LSAME( FORM, 'F' ) ) ) THEN
INFO = -1
ELSEIF ( .NOT.( FULL .OR. LSAME( STOR, 'P' ) ) ) THEN
INFO = -2
ELSEIF ( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -3
ELSEIF ( M.LT.0 ) THEN
INFO = -5
ELSEIF ( N.LT.0 ) THEN
INFO = -6
ELSEIF ( NRHS.LT.0 ) THEN
INFO = -7
ELSEIF ( .NOT. MAT .AND. LIPAR.LT.0 ) THEN
INFO = -9
ELSEIF ( .NOT. MAT .AND. LDPAR.LT.0 ) THEN
INFO = -11
ELSEIF ( LDA.LT.1 .OR. ( MAT .AND. LDA.LT.M ) ) THEN
INFO = -13
ELSEIF ( LDB.LT.MAX( 1, N ) ) THEN
INFO = -15
ELSEIF ( LDATA.LT.1 .OR. ( FULL .AND. LDATA.LT.N ) ) THEN
INFO = -17
ENDIF
C
C Return if there are illegal arguments.
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MB02XD', -INFO )
RETURN
ENDIF
C
C Quick return if possible.
C
IF ( N.EQ.0 .OR. ( MAT .AND. M.EQ.0 ) )
$ RETURN
C
C Build a triangle of the matrix A'*A.
C
IF ( MAT ) THEN
C
C Matrix A is given in the usual form.
C
IF ( FULL ) THEN
CALL DSYRK( UPLO, 'Transpose', N, M, ONE, A, LDA, ZERO,
$ ATA, LDATA )
ELSEIF ( UPPER ) THEN
J1 = 1
C
DO 10 J = 1, N
CALL DGEMV( 'Transpose', M, J, ONE, A, LDA, A(1,J), 1,
$ ZERO, ATA(J1), 1 )
J1 = J1 + J
10 CONTINUE
C
ELSE
J1 = 1
C
DO 20 J = 1, N
CALL DGEMV( 'Transpose', M, N-J+1, ONE, A(1,J), LDA,
$ A(1,J), 1, ZERO, ATA(J1), 1 )
J1 = J1 + N - J + 1
20 CONTINUE
C
ENDIF
C
ELSE
C
C Implicit form, A'*A = f(A).
C
CALL F( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR, A, LDA, ATA,
$ LDATA, DWORK, LDWORK, IERR )
IF ( IERR.NE.0 ) THEN
INFO = N + IERR
RETURN
ENDIF
C
ENDIF
C
C Factor the matrix A'*A.
C
IF ( FULL ) THEN
CALL DPOTRF( UPLO, N, ATA, LDATA, IERR )
ELSE
CALL DPPTRF( UPLO, N, ATA, IERR )
ENDIF
C
IF ( IERR.NE.0 ) THEN
INFO = IERR
RETURN
ENDIF
C
C Solve the set of linear systems.
C
IF ( FULL ) THEN
CALL DPOTRS( UPLO, N, NRHS, ATA, LDATA, B, LDB, IERR )
ELSE
CALL DPPTRS( UPLO, N, NRHS, ATA, B, LDB, IERR )
ENDIF
C
C *** Last line of MB02XD ***
END