*> \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
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*> \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 tpqrt2
*
*> \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