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*> \brief \b SGEQPF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> Download SGEQPF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgeqpf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgeqpf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgeqpf.f">
*> [TXT]</a>
*
* Definition
* ==========
*
* SUBROUTINE SGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* REAL A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> This routine is deprecated and has been replaced by routine SGEQP3.
*>
*> SGEQPF computes a QR factorization with column pivoting of a
*> real M-by-N matrix A: A*P = Q*R.
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \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 REAL array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the upper triangle of the array contains the
*> min(M,N)-by-N upper triangular matrix R; the elements
*> below the diagonal, together with the array TAU,
*> represent the orthogonal matrix Q as a product of
*> min(m,n) elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
*> to the front of A*P (a leading column); if JPVT(i) = 0,
*> the i-th column of A is a free column.
*> On exit, if JPVT(i) = k, then the i-th column of A*P
*> was the k-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is REAL array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (3*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 November 2011
*
*> \ingroup realGEcomputational
*
*
* Further Details
* ===============
*>\details \b Further \b Details
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(n)
*>
*> Each H(i) has the form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).
*>
*> The matrix P is represented in jpvt as follows: If
*> jpvt(j) = i
*> then the jth column of P is the ith canonical unit vector.
*>
*> Partial column norm updating strategy modified by
*> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
*> University of Zagreb, Croatia.
*> -- April 2011 --
*> For more details see LAPACK Working Note 176.
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
*
* -- LAPACK computational routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
REAL A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, ITEMP, J, MA, MN, PVT
REAL AII, TEMP, TEMP2, TOL3Z
* ..
* .. External Subroutines ..
EXTERNAL SGEQR2, SLARF, SLARFG, SORM2R, SSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. External Functions ..
INTEGER ISAMAX
REAL SLAMCH, SNRM2
EXTERNAL ISAMAX, SLAMCH, SNRM2
* ..
* .. 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( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGEQPF', -INFO )
RETURN
END IF
*
MN = MIN( M, N )
TOL3Z = SQRT(SLAMCH('Epsilon'))
*
* Move initial columns up front
*
ITEMP = 1
DO 10 I = 1, N
IF( JPVT( I ).NE.0 ) THEN
IF( I.NE.ITEMP ) THEN
CALL SSWAP( M, A( 1, I ), 1, A( 1, ITEMP ), 1 )
JPVT( I ) = JPVT( ITEMP )
JPVT( ITEMP ) = I
ELSE
JPVT( I ) = I
END IF
ITEMP = ITEMP + 1
ELSE
JPVT( I ) = I
END IF
10 CONTINUE
ITEMP = ITEMP - 1
*
* Compute the QR factorization and update remaining columns
*
IF( ITEMP.GT.0 ) THEN
MA = MIN( ITEMP, M )
CALL SGEQR2( M, MA, A, LDA, TAU, WORK, INFO )
IF( MA.LT.N ) THEN
CALL SORM2R( 'Left', 'Transpose', M, N-MA, MA, A, LDA, TAU,
$ A( 1, MA+1 ), LDA, WORK, INFO )
END IF
END IF
*
IF( ITEMP.LT.MN ) THEN
*
* Initialize partial column norms. The first n elements of
* work store the exact column norms.
*
DO 20 I = ITEMP + 1, N
WORK( I ) = SNRM2( M-ITEMP, A( ITEMP+1, I ), 1 )
WORK( N+I ) = WORK( I )
20 CONTINUE
*
* Compute factorization
*
DO 40 I = ITEMP + 1, MN
*
* Determine ith pivot column and swap if necessary
*
PVT = ( I-1 ) + ISAMAX( N-I+1, WORK( I ), 1 )
*
IF( PVT.NE.I ) THEN
CALL SSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
ITEMP = JPVT( PVT )
JPVT( PVT ) = JPVT( I )
JPVT( I ) = ITEMP
WORK( PVT ) = WORK( I )
WORK( N+PVT ) = WORK( N+I )
END IF
*
* Generate elementary reflector H(i)
*
IF( I.LT.M ) THEN
CALL SLARFG( M-I+1, A( I, I ), A( I+1, I ), 1, TAU( I ) )
ELSE
CALL SLARFG( 1, A( M, M ), A( M, M ), 1, TAU( M ) )
END IF
*
IF( I.LT.N ) THEN
*
* Apply H(i) to A(i:m,i+1:n) from the left
*
AII = A( I, I )
A( I, I ) = ONE
CALL SLARF( 'LEFT', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK( 2*N+1 ) )
A( I, I ) = AII
END IF
*
* Update partial column norms
*
DO 30 J = I + 1, N
IF( WORK( J ).NE.ZERO ) THEN
*
* NOTE: The following 4 lines follow from the analysis in
* Lapack Working Note 176.
*
TEMP = ABS( A( I, J ) ) / WORK( J )
TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
TEMP2 = TEMP*( WORK( J ) / WORK( N+J ) )**2
IF( TEMP2 .LE. TOL3Z ) THEN
IF( M-I.GT.0 ) THEN
WORK( J ) = SNRM2( M-I, A( I+1, J ), 1 )
WORK( N+J ) = WORK( J )
ELSE
WORK( J ) = ZERO
WORK( N+J ) = ZERO
END IF
ELSE
WORK( J ) = WORK( J )*SQRT( TEMP )
END IF
END IF
30 CONTINUE
*
40 CONTINUE
END IF
RETURN
*
* End of SGEQPF
*
END
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