sgeqrt()

subroutine sgeqrt	(	integer	m,
		integer	n,
		integer	nb,
		real, dimension( lda, * )	a,
		integer	lda,
		real, dimension( ldt, * )	t,
		integer	ldt,
		real, dimension( * )	work,
		integer	info
	)

SGEQRT

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Purpose:

 SGEQRT computes a blocked QR factorization of a real M-by-N matrix A
 using the compact WY representation of Q.

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]	NB	NB is INTEGER The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
[in,out]	A	A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal are the columns of V.
[in]	LDA	LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
[out]	T	T is REAL array, dimension (LDT,MIN(M,N)) The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See below for further details.
[in]	LDT	LDT is INTEGER The leading dimension of the array T. LDT >= NB.
[out]	WORK	WORK is REAL array, dimension (NB*N)
[out]	INFO	INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  The matrix V stores the elementary reflectors H(i) in the i-th column
  below the diagonal. For example, if M=5 and N=3, the matrix V is

               V = (  1       )
                   ( v1  1    )
                   ( v1 v2  1 )
                   ( v1 v2 v3 )
                   ( v1 v2 v3 )

  where the vi's represent the vectors which define H(i), which are returned
  in the matrix A.  The 1's along the diagonal of V are not stored in A.

  Let K=MIN(M,N).  The number of blocks is B = ceiling(K/NB), where each
  block is of order NB except for the last block, which is of order
  IB = K - (B-1)*NB.  For each of the B blocks, a upper triangular block
  reflector factor is computed: T1, T2, ..., TB.  The NB-by-NB (and IB-by-IB
  for the last block) T's are stored in the NB-by-K matrix T as

               T = (T1 T2 ... TB).

Definition at line 140 of file sgeqrt.f.

*
*  -- 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, LDT, M, N, NB
*     ..
*     .. Array Arguments ..
      REAL A( LDA, * ), T( LDT, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     ..
*     .. Local Scalars ..
      INTEGER    I, IB, IINFO, K
      LOGICAL    USE_RECURSIVE_QR
      parameter( use_recursive_qr=.true. )
*     ..
*     .. External Subroutines ..
      EXTERNAL   sgeqrt2, sgeqrt3, slarfb, xerbla
*     ..
*     .. 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( nb.LT.1 .OR. ( nb.GT.min(m,n) .AND. min(m,n).GT.0 ) )THEN
         info = -3
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -5
      ELSE IF( ldt.LT.nb ) THEN
         info = -7
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SGEQRT', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      k = min( m, n )
      IF( k.EQ.0 ) RETURN
*
*     Blocked loop of length K
*
      DO i = 1, k,  nb
         ib = min( k-i+1, nb )
*
*     Compute the QR factorization of the current block A(I:M,I:I+IB-1)
*
         IF( use_recursive_qr ) THEN
            CALL sgeqrt3( m-i+1, ib, a(i,i), lda, t(1,i), ldt, iinfo )
         ELSE
            CALL sgeqrt2( m-i+1, ib, a(i,i), lda, t(1,i), ldt, iinfo )
         END IF
         IF( i+ib.LE.n ) THEN
*
*     Update by applying H**T to A(I:M,I+IB:N) from the left
*
            CALL slarfb( 'L', 'T', 'F', 'C', m-i+1, n-i-ib+1, ib,
     $                   a( i, i ), lda, t( 1, i ), ldt,
     $                   a( i, i+ib ), lda, work , n-i-ib+1 )
         END IF
      END DO
      RETURN
*
*     End of SGEQRT
*

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