slaqp2()

subroutine slaqp2	(	integer	m,
		integer	n,
		integer	offset,
		real, dimension( lda, * )	a,
		integer	lda,
		integer, dimension( * )	jpvt,
		real, dimension( * )	tau,
		real, dimension( * )	vn1,
		real, dimension( * )	vn2,
		real, dimension( * )	work )

SLAQP2 computes a QR factorization with column pivoting of the matrix block.

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Purpose:

!>
!> SLAQP2 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 REAL 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 REAL array, dimension (min(M,N)) !> The scalar factors of the elementary reflectors. !>
[in,out]	VN1	!> VN1 is REAL array, dimension (N) !> The vector with the partial column norms. !>
[in,out]	VN2	!> VN2 is REAL array, dimension (N) !> The vector with the exact column norms. !>
[out]	WORK	!> WORK is REAL 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 slaqp2.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( * )
      REAL               A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ITEMP, J, MN, OFFPI, PVT
      REAL               TEMP, TEMP2, TOL3Z
*     ..
*     .. External Subroutines ..
      EXTERNAL           slarf1f, slarfg, sswap
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, sqrt
*     ..
*     .. External Functions ..
      INTEGER            ISAMAX
      REAL               SLAMCH, SNRM2
      EXTERNAL           isamax, slamch, snrm2
*     ..
*     .. Executable Statements ..
*
      mn = min( m-offset, n )
      tol3z = sqrt(slamch('Epsilon'))
*
*     Compute factorization.
*
      DO 20 i = 1, mn
*
         offpi = offset + i
*
*        Determine ith pivot column and swap if necessary.
*
         pvt = ( i-1 ) + isamax( n-i+1, vn1( 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
            vn1( pvt ) = vn1( i )
            vn2( pvt ) = vn2( i )
         END IF
*
*        Generate elementary reflector H(i).
*
         IF( offpi.LT.m ) THEN
            CALL slarfg( m-offpi+1, a( offpi, i ), a( offpi+1, i ),
     $                   1,
     $                   tau( i ) )
         ELSE
            CALL slarfg( 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 slarf1f( '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 ) = snrm2( 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 SLAQP2
*

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