255 lines
7.1 KiB
Fortran
255 lines
7.1 KiB
Fortran
*> \brief \b DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download DGEQRT3 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrt3.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrt3.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrt3.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LDA, M, N, LDT
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION A( LDA, * ), T( LDT, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> DGEQRT3 recursively computes a QR factorization of a real M-by-N
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*> matrix A, using the compact WY representation of Q.
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*>
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*> Based on the algorithm of Elmroth and Gustavson,
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*> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] M
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*> \verbatim
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*> M is INTEGER
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*> The number of rows of the matrix A. M >= N.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of columns of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is DOUBLE PRECISION array, dimension (LDA,N)
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*> On entry, the real M-by-N matrix A. On exit, the elements on and
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*> above the diagonal contain the N-by-N upper triangular matrix R; the
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*> elements below the diagonal are the columns of V. See below for
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*> further details.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,M).
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*> \endverbatim
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*>
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*> \param[out] T
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*> \verbatim
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*> T is DOUBLE PRECISION array, dimension (LDT,N)
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*> The N-by-N upper triangular factor of the block reflector.
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*> The elements on and above the diagonal contain the block
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*> reflector T; the elements below the diagonal are not used.
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*> See below for further details.
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*> \endverbatim
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*>
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*> \param[in] LDT
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*> \verbatim
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*> LDT is INTEGER
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*> The leading dimension of the array T. LDT >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup doubleGEcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> The matrix V stores the elementary reflectors H(i) in the i-th column
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*> below the diagonal. For example, if M=5 and N=3, the matrix V is
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*>
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*> V = ( 1 )
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*> ( v1 1 )
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*> ( v1 v2 1 )
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*> ( v1 v2 v3 )
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*> ( v1 v2 v3 )
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*>
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*> where the vi's represent the vectors which define H(i), which are returned
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*> in the matrix A. The 1's along the diagonal of V are not stored in A. The
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*> block reflector H is then given by
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*>
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*> H = I - V * T * V**T
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*>
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*> where V**T is the transpose of V.
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*>
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*> For details of the algorithm, see Elmroth and Gustavson (cited above).
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*> \endverbatim
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*>
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* =====================================================================
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RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDA, M, N, LDT
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION A( LDA, * ), T( LDT, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ONE
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PARAMETER ( ONE = 1.0D+00 )
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* ..
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* .. Local Scalars ..
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INTEGER I, I1, J, J1, N1, N2, IINFO
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* ..
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* .. External Subroutines ..
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EXTERNAL DLARFG, DTRMM, DGEMM, XERBLA
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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IF( N .LT. 0 ) THEN
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INFO = -2
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ELSE IF( M .LT. N ) THEN
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INFO = -1
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ELSE IF( LDA .LT. MAX( 1, M ) ) THEN
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INFO = -4
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ELSE IF( LDT .LT. MAX( 1, N ) ) THEN
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INFO = -6
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DGEQRT3', -INFO )
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RETURN
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END IF
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*
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IF( N.EQ.1 ) THEN
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*
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* Compute Householder transform when N=1
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*
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CALL DLARFG( M, A(1,1), A( MIN( 2, M ), 1 ), 1, T(1,1) )
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*
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ELSE
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*
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* Otherwise, split A into blocks...
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*
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N1 = N/2
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N2 = N-N1
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J1 = MIN( N1+1, N )
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I1 = MIN( N+1, M )
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*
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* Compute A(1:M,1:N1) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
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*
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CALL DGEQRT3( M, N1, A, LDA, T, LDT, IINFO )
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*
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* Compute A(1:M,J1:N) = Q1^H A(1:M,J1:N) [workspace: T(1:N1,J1:N)]
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*
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DO J=1,N2
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DO I=1,N1
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T( I, J+N1 ) = A( I, J+N1 )
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END DO
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END DO
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CALL DTRMM( 'L', 'L', 'T', 'U', N1, N2, ONE,
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& A, LDA, T( 1, J1 ), LDT )
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*
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CALL DGEMM( 'T', 'N', N1, N2, M-N1, ONE, A( J1, 1 ), LDA,
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& A( J1, J1 ), LDA, ONE, T( 1, J1 ), LDT)
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*
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CALL DTRMM( 'L', 'U', 'T', 'N', N1, N2, ONE,
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& T, LDT, T( 1, J1 ), LDT )
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*
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CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( J1, 1 ), LDA,
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& T( 1, J1 ), LDT, ONE, A( J1, J1 ), LDA )
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*
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CALL DTRMM( 'L', 'L', 'N', 'U', N1, N2, ONE,
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& A, LDA, T( 1, J1 ), LDT )
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*
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DO J=1,N2
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DO I=1,N1
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A( I, J+N1 ) = A( I, J+N1 ) - T( I, J+N1 )
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END DO
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END DO
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*
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* Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
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*
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CALL DGEQRT3( M-N1, N2, A( J1, J1 ), LDA,
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& T( J1, J1 ), LDT, IINFO )
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*
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* Compute T3 = T(1:N1,J1:N) = -T1 Y1^H Y2 T2
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*
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DO I=1,N1
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DO J=1,N2
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T( I, J+N1 ) = (A( J+N1, I ))
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END DO
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END DO
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*
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CALL DTRMM( 'R', 'L', 'N', 'U', N1, N2, ONE,
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& A( J1, J1 ), LDA, T( 1, J1 ), LDT )
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*
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CALL DGEMM( 'T', 'N', N1, N2, M-N, ONE, A( I1, 1 ), LDA,
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& A( I1, J1 ), LDA, ONE, T( 1, J1 ), LDT )
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*
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CALL DTRMM( 'L', 'U', 'N', 'N', N1, N2, -ONE, T, LDT,
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& T( 1, J1 ), LDT )
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*
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CALL DTRMM( 'R', 'U', 'N', 'N', N1, N2, ONE,
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& T( J1, J1 ), LDT, T( 1, J1 ), LDT )
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*
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* Y = (Y1,Y2); R = [ R1 A(1:N1,J1:N) ]; T = [T1 T3]
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* [ 0 R2 ] [ 0 T2]
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*
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END IF
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*
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RETURN
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*
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* End of DGEQRT3
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*
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END
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