dlaqp2()

subroutine dlaqp2	(	integer	m,
		integer	n,
		integer	offset,
		double precision, dimension( lda, * )	a,
		integer	lda,
		integer, dimension( * )	jpvt,
		double precision, dimension( * )	tau,
		double precision, dimension( * )	vn1,
		double precision, dimension( * )	vn2,
		double precision, dimension( * )	work )

DLAQP2 computes a QR factorization with column pivoting of the matrix block.

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Purpose:

!>
!> DLAQP2 computes a QR factorization with column pivoting of
!> the block A(OFFSET+1:M,1:N).
!> The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
!> 

Parameters

[in]	M	!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns of the matrix A. N >= 0. !>
[in]	OFFSET	!> OFFSET is INTEGER !> The number of rows of the matrix A that must be pivoted !> but no factorized. OFFSET >= 0. !>
[in,out]	A	!> A is DOUBLE PRECISION array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the upper triangle of block A(OFFSET+1:M,1:N) is !> the triangular factor obtained; the elements in block !> A(OFFSET+1:M,1:N) below the diagonal, together with the !> array TAU, represent the orthogonal matrix Q as a product of !> elementary reflectors. Block A(1:OFFSET,1:N) has been !> accordingly pivoted, but no factorized. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
[in,out]	JPVT	!> 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. !>
[out]	TAU	!> TAU is DOUBLE PRECISION array, dimension (min(M,N)) !> The scalar factors of the elementary reflectors. !>
[in,out]	VN1	!> VN1 is DOUBLE PRECISION array, dimension (N) !> The vector with the partial column norms. !>
[in,out]	VN2	!> VN2 is DOUBLE PRECISION array, dimension (N) !> The vector with the exact column norms. !>
[out]	WORK	!> WORK is DOUBLE PRECISION array, dimension (N) !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA
Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:

LAPACK Working Note 176 [PDF]

Definition at line 143 of file dlaqp2.f.

*
*  -- LAPACK auxiliary 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            LDA, M, N, OFFSET
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ITEMP, J, MN, OFFPI, PVT
      DOUBLE PRECISION   TEMP, TEMP2, TOL3Z
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlarf, dlarfg, dswap
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, sqrt
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DNRM2
      EXTERNAL           idamax, dlamch, dnrm2
*     ..
*     .. Executable Statements ..
*
      mn = min( m-offset, n )
      tol3z = sqrt(dlamch('Epsilon'))
*
*     Compute factorization.
*
      DO 20 i = 1, mn
*
         offpi = offset + i
*
*        Determine ith pivot column and swap if necessary.
*
         pvt = ( i-1 ) + idamax( n-i+1, vn1( i ), 1 )
*
         IF( pvt.NE.i ) THEN
            CALL dswap( m, a( 1, pvt ), 1, a( 1, i ), 1 )
            itemp = jpvt( pvt )
            jpvt( pvt ) = jpvt( i )
            jpvt( i ) = itemp
            vn1( pvt ) = vn1( i )
            vn2( pvt ) = vn2( i )
         END IF
*
*        Generate elementary reflector H(i).
*
         IF( offpi.LT.m ) THEN
            CALL dlarfg( m-offpi+1, a( offpi, i ), a( offpi+1, i ),
     $                   1,
     $                   tau( i ) )
         ELSE
            CALL dlarfg( 1, a( m, i ), a( m, i ), 1, tau( i ) )
         END IF
*
         IF( i.LT.n ) THEN
*
*           Apply H(i)**T to A(offset+i:m,i+1:n) from the left.
*
            CALL dlarf1f( 'Left', m-offpi+1, n-i, a( offpi, i ), 1,
     $                  tau( i ), a( offpi, i+1 ), lda, work( 1 ) )
         END IF
*
*        Update partial column norms.
*
         DO 10 j = i + 1, n
            IF( vn1( j ).NE.zero ) THEN
*
*              NOTE: The following 4 lines follow from the analysis in
*              Lapack Working Note 176.
*
               temp = one - ( abs( a( offpi, j ) ) / vn1( j ) )**2
               temp = max( temp, zero )
               temp2 = temp*( vn1( j ) / vn2( j ) )**2
               IF( temp2 .LE. tol3z ) THEN
                  IF( offpi.LT.m ) THEN
                     vn1( j ) = dnrm2( m-offpi, a( offpi+1, j ), 1 )
                     vn2( j ) = vn1( j )
                  ELSE
                     vn1( j ) = zero
                     vn2( j ) = zero
                  END IF
               ELSE
                  vn1( j ) = vn1( j )*sqrt( temp )
               END IF
            END IF
   10    CONTINUE
*
   20 CONTINUE
*
      RETURN
*
*     End of DLAQP2
*

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