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*> \brief \b DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRT3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N, LDT
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQRT3 recursively computes a QR factorization of a real M-by-N
*> matrix A, using the compact WY representation of Q.
*>
*> Based on the algorithm of Elmroth and Gustavson,
*> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the real M-by-N matrix A. On exit, the elements on and
*> above the diagonal contain the N-by-N upper triangular matrix R; the
*> elements below the diagonal are the columns of V. See below for
*> further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The N-by-N upper triangular factor of the block reflector.
*> The elements on and above the diagonal contain the block
*> reflector T; the elements below the diagonal are not used.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix V stores the elementary reflectors H(i) in the i-th column
*> below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*> V = ( 1 )
*> ( v1 1 )
*> ( v1 v2 1 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> where the vi's represent the vectors which define H(i), which are returned
*> in the matrix A. The 1's along the diagonal of V are not stored in A. The
*> block reflector H is then given by
*>
*> H = I - V * T * V**T
*>
*> where V**T is the transpose of V.
*>
*> For details of the algorithm, see Elmroth and Gustavson (cited above).
*> \endverbatim
*>
* =====================================================================
RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, LDT
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+00 )
* ..
* .. Local Scalars ..
INTEGER I, I1, J, J1, N1, N2, IINFO
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DTRMM, DGEMM, XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N .LT. 0 ) THEN
INFO = -2
ELSE IF( M .LT. N ) THEN
INFO = -1
ELSE IF( LDA .LT. MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LDT .LT. MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRT3', -INFO )
RETURN
END IF
*
IF( N.EQ.1 ) THEN
*
* Compute Householder transform when N=1
*
CALL DLARFG( M, A(1,1), A( MIN( 2, M ), 1 ), 1, T(1,1) )
*
ELSE
*
* Otherwise, split A into blocks...
*
N1 = N/2
N2 = N-N1
J1 = MIN( N1+1, N )
I1 = MIN( N+1, M )
*
* Compute A(1:M,1:N1) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
*
CALL DGEQRT3( M, N1, A, LDA, T, LDT, IINFO )
*
* Compute A(1:M,J1:N) = Q1^H A(1:M,J1:N) [workspace: T(1:N1,J1:N)]
*
DO J=1,N2
DO I=1,N1
T( I, J+N1 ) = A( I, J+N1 )
END DO
END DO
CALL DTRMM( 'L', 'L', 'T', 'U', N1, N2, ONE,
& A, LDA, T( 1, J1 ), LDT )
*
CALL DGEMM( 'T', 'N', N1, N2, M-N1, ONE, A( J1, 1 ), LDA,
& A( J1, J1 ), LDA, ONE, T( 1, J1 ), LDT)
*
CALL DTRMM( 'L', 'U', 'T', 'N', N1, N2, ONE,
& T, LDT, T( 1, J1 ), LDT )
*
CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( J1, 1 ), LDA,
& T( 1, J1 ), LDT, ONE, A( J1, J1 ), LDA )
*
CALL DTRMM( 'L', 'L', 'N', 'U', N1, N2, ONE,
& A, LDA, T( 1, J1 ), LDT )
*
DO J=1,N2
DO I=1,N1
A( I, J+N1 ) = A( I, J+N1 ) - T( I, J+N1 )
END DO
END DO
*
* Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
*
CALL DGEQRT3( M-N1, N2, A( J1, J1 ), LDA,
& T( J1, J1 ), LDT, IINFO )
*
* Compute T3 = T(1:N1,J1:N) = -T1 Y1^H Y2 T2
*
DO I=1,N1
DO J=1,N2
T( I, J+N1 ) = (A( J+N1, I ))
END DO
END DO
*
CALL DTRMM( 'R', 'L', 'N', 'U', N1, N2, ONE,
& A( J1, J1 ), LDA, T( 1, J1 ), LDT )
*
CALL DGEMM( 'T', 'N', N1, N2, M-N, ONE, A( I1, 1 ), LDA,
& A( I1, J1 ), LDA, ONE, T( 1, J1 ), LDT )
*
CALL DTRMM( 'L', 'U', 'N', 'N', N1, N2, -ONE, T, LDT,
& T( 1, J1 ), LDT )
*
CALL DTRMM( 'R', 'U', 'N', 'N', N1, N2, ONE,
& T( J1, J1 ), LDT, T( 1, J1 ), LDT )
*
* Y = (Y1,Y2); R = [ R1 A(1:N1,J1:N) ]; T = [T1 T3]
* [ 0 R2 ] [ 0 T2]
*
END IF
*
RETURN
*
* End of DGEQRT3
*
END
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