You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
299 lines
8.4 KiB
299 lines
8.4 KiB
*> \brief \b DTPQRT2 computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download DTPQRT2 + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpqrt2.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpqrt2.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpqrt2.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE DTPQRT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* INTEGER INFO, LDA, LDB, LDT, N, M, L
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> DTPQRT2 computes a QR factorization of a real "triangular-pentagonal"
|
|
*> matrix C, which is composed of a triangular block A and pentagonal block B,
|
|
*> using the compact WY representation for Q.
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \param[in] M
|
|
*> \verbatim
|
|
*> M is INTEGER
|
|
*> The total number of rows of the matrix B.
|
|
*> M >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] N
|
|
*> \verbatim
|
|
*> N is INTEGER
|
|
*> The number of columns of the matrix B, and the order of
|
|
*> the triangular matrix A.
|
|
*> N >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] L
|
|
*> \verbatim
|
|
*> L is INTEGER
|
|
*> The number of rows of the upper trapezoidal part of B.
|
|
*> MIN(M,N) >= L >= 0. See Further Details.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] A
|
|
*> \verbatim
|
|
*> A is DOUBLE PRECISION array, dimension (LDA,N)
|
|
*> On entry, the upper triangular N-by-N matrix A.
|
|
*> On exit, the elements on and above the diagonal of the array
|
|
*> contain the upper triangular matrix R.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDA
|
|
*> \verbatim
|
|
*> LDA is INTEGER
|
|
*> The leading dimension of the array A. LDA >= max(1,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] B
|
|
*> \verbatim
|
|
*> B is DOUBLE PRECISION array, dimension (LDB,N)
|
|
*> On entry, the pentagonal M-by-N matrix B. The first M-L rows
|
|
*> are rectangular, and the last L rows are upper trapezoidal.
|
|
*> On exit, B contains the pentagonal matrix V. See Further Details.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDB
|
|
*> \verbatim
|
|
*> LDB is INTEGER
|
|
*> The leading dimension of the array B. LDB >= max(1,M).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] T
|
|
*> \verbatim
|
|
*> T is DOUBLE PRECISION array, dimension (LDT,N)
|
|
*> The N-by-N upper triangular factor T of the block reflector.
|
|
*> See 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.
|
|
*
|
|
*> \ingroup doubleOTHERcomputational
|
|
*
|
|
*> \par Further Details:
|
|
* =====================
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> The input matrix C is a (N+M)-by-N matrix
|
|
*>
|
|
*> C = [ A ]
|
|
*> [ B ]
|
|
*>
|
|
*> where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
|
|
*> matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
|
|
*> upper trapezoidal matrix B2:
|
|
*>
|
|
*> B = [ B1 ] <- (M-L)-by-N rectangular
|
|
*> [ B2 ] <- L-by-N upper trapezoidal.
|
|
*>
|
|
*> The upper trapezoidal matrix B2 consists of the first L rows of a
|
|
*> N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
|
|
*> B is rectangular M-by-N; if M=L=N, B is upper triangular.
|
|
*>
|
|
*> The matrix W stores the elementary reflectors H(i) in the i-th column
|
|
*> below the diagonal (of A) in the (N+M)-by-N input matrix C
|
|
*>
|
|
*> C = [ A ] <- upper triangular N-by-N
|
|
*> [ B ] <- M-by-N pentagonal
|
|
*>
|
|
*> so that W can be represented as
|
|
*>
|
|
*> W = [ I ] <- identity, N-by-N
|
|
*> [ V ] <- M-by-N, same form as B.
|
|
*>
|
|
*> Thus, all of information needed for W is contained on exit in B, which
|
|
*> we call V above. Note that V has the same form as B; that is,
|
|
*>
|
|
*> V = [ V1 ] <- (M-L)-by-N rectangular
|
|
*> [ V2 ] <- L-by-N upper trapezoidal.
|
|
*>
|
|
*> The columns of V represent the vectors which define the H(i)'s.
|
|
*> The (M+N)-by-(M+N) block reflector H is then given by
|
|
*>
|
|
*> H = I - W * T * W**T
|
|
*>
|
|
*> where W^H is the conjugate transpose of W and T is the upper triangular
|
|
*> factor of the block reflector.
|
|
*> \endverbatim
|
|
*>
|
|
* =====================================================================
|
|
SUBROUTINE DTPQRT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
|
|
*
|
|
* -- LAPACK computational routine --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
*
|
|
* .. Scalar Arguments ..
|
|
INTEGER INFO, LDA, LDB, LDT, N, M, L
|
|
* ..
|
|
* .. Array Arguments ..
|
|
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ONE, ZERO
|
|
PARAMETER( ONE = 1.0, ZERO = 0.0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
INTEGER I, J, P, MP, NP
|
|
DOUBLE PRECISION ALPHA
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DLARFG, DGEMV, DGER, DTRMV, XERBLA
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC MAX, MIN
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input arguments
|
|
*
|
|
INFO = 0
|
|
IF( M.LT.0 ) THEN
|
|
INFO = -1
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( L.LT.0 .OR. L.GT.MIN(M,N) ) THEN
|
|
INFO = -3
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
|
INFO = -5
|
|
ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
|
|
INFO = -7
|
|
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
|
|
INFO = -9
|
|
END IF
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'DTPQRT2', -INFO )
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
IF( N.EQ.0 .OR. M.EQ.0 ) RETURN
|
|
*
|
|
DO I = 1, N
|
|
*
|
|
* Generate elementary reflector H(I) to annihilate B(:,I)
|
|
*
|
|
P = M-L+MIN( L, I )
|
|
CALL DLARFG( P+1, A( I, I ), B( 1, I ), 1, T( I, 1 ) )
|
|
IF( I.LT.N ) THEN
|
|
*
|
|
* W(1:N-I) := C(I:M,I+1:N)^H * C(I:M,I) [use W = T(:,N)]
|
|
*
|
|
DO J = 1, N-I
|
|
T( J, N ) = (A( I, I+J ))
|
|
END DO
|
|
CALL DGEMV( 'T', P, N-I, ONE, B( 1, I+1 ), LDB,
|
|
$ B( 1, I ), 1, ONE, T( 1, N ), 1 )
|
|
*
|
|
* C(I:M,I+1:N) = C(I:m,I+1:N) + alpha*C(I:M,I)*W(1:N-1)^H
|
|
*
|
|
ALPHA = -(T( I, 1 ))
|
|
DO J = 1, N-I
|
|
A( I, I+J ) = A( I, I+J ) + ALPHA*(T( J, N ))
|
|
END DO
|
|
CALL DGER( P, N-I, ALPHA, B( 1, I ), 1,
|
|
$ T( 1, N ), 1, B( 1, I+1 ), LDB )
|
|
END IF
|
|
END DO
|
|
*
|
|
DO I = 2, N
|
|
*
|
|
* T(1:I-1,I) := C(I:M,1:I-1)^H * (alpha * C(I:M,I))
|
|
*
|
|
ALPHA = -T( I, 1 )
|
|
|
|
DO J = 1, I-1
|
|
T( J, I ) = ZERO
|
|
END DO
|
|
P = MIN( I-1, L )
|
|
MP = MIN( M-L+1, M )
|
|
NP = MIN( P+1, N )
|
|
*
|
|
* Triangular part of B2
|
|
*
|
|
DO J = 1, P
|
|
T( J, I ) = ALPHA*B( M-L+J, I )
|
|
END DO
|
|
CALL DTRMV( 'U', 'T', 'N', P, B( MP, 1 ), LDB,
|
|
$ T( 1, I ), 1 )
|
|
*
|
|
* Rectangular part of B2
|
|
*
|
|
CALL DGEMV( 'T', L, I-1-P, ALPHA, B( MP, NP ), LDB,
|
|
$ B( MP, I ), 1, ZERO, T( NP, I ), 1 )
|
|
*
|
|
* B1
|
|
*
|
|
CALL DGEMV( 'T', M-L, I-1, ALPHA, B, LDB, B( 1, I ), 1,
|
|
$ ONE, T( 1, I ), 1 )
|
|
*
|
|
* T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
|
|
*
|
|
CALL DTRMV( 'U', 'N', 'N', I-1, T, LDT, T( 1, I ), 1 )
|
|
*
|
|
* T(I,I) = tau(I)
|
|
*
|
|
T( I, I ) = T( I, 1 )
|
|
T( I, 1 ) = ZERO
|
|
END DO
|
|
|
|
*
|
|
* End of DTPQRT2
|
|
*
|
|
END
|
|
|