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SUBROUTINE DGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( LWORK )
* ..
*
* Purpose
* =======
*
* DGEQLF computes a QL factorization of a real M-by-N matrix A:
* A = Q * L.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit,
* if m >= n, the lower triangle of the subarray
* A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
* if m <= n, the elements on and below the (n-m)-th
* superdiagonal contain the M-by-N lower trapezoidal matrix L;
* the remaining elements, with the array TAU, represent the
* orthogonal matrix Q as a product of elementary reflectors
* (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* For optimum performance LWORK >= N*NB, where NB is the
* optimal blocksize.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(k) . . . H(2) H(1), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
* A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, MU, NB,
$ NBMIN, NU, NX
* ..
* .. External Subroutines ..
EXTERNAL DGEQL2, DLARFB, DLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. 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
ELSE IF( LWORK.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQLF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
K = MIN( M, N )
IF( K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Determine the block size.
*
NB = ILAENV( 1, 'DGEQLF', ' ', M, N, -1, -1 )
NBMIN = 2
NX = 1
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGEQLF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGEQLF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code initially.
* The last kk columns are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
IB = MIN( K-I+1, NB )
*
* Compute the QL factorization of the current block
* A(1:m-k+i+ib-1,n-k+i:n-k+i+ib-1)
*
CALL DGEQL2( M-K+I+IB-1, IB, A( 1, N-K+I ), LDA, TAU( I ),
$ WORK, IINFO )
IF( N-K+I.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
$ A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H' to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
*
CALL DLARFB( 'Left', 'Transpose', 'Backward',
$ 'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
$ A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
$ WORK( IB+1 ), LDWORK )
END IF
10 CONTINUE
MU = M - K + I + NB - 1
NU = N - K + I + NB - 1
ELSE
MU = M
NU = N
END IF
*
* Use unblocked code to factor the last or only block
*
IF( MU.GT.0 .AND. NU.GT.0 )
$ CALL DGEQL2( MU, NU, A, LDA, TAU, WORK, IINFO )
*
WORK( 1 ) = IWS
RETURN
*
* End of DGEQLF
*
END
SUBROUTINE DGEQL2( M, N, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGEQL2 computes a QL factorization of a real m by n matrix A:
* A = Q * L.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the m by n matrix A.
* On exit, if m >= n, the lower triangle of the subarray
* A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
* if m <= n, the elements on and below the (n-m)-th
* superdiagonal contain the m by n lower trapezoidal matrix L;
* the remaining elements, with the array TAU, represent the
* orthogonal matrix Q as a product of elementary reflectors
* (see Further Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) DOUBLE PRECISION array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(k) . . . H(2) H(1), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
* A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K
DOUBLE PRECISION AII
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DLARFG, 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( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQL2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO 10 I = K, 1, -1
*
* Generate elementary reflector H(i) to annihilate
* A(1:m-k+i-1,n-k+i)
*
CALL DLARFG( M-K+I, A( M-K+I, N-K+I ), A( 1, N-K+I ), 1,
$ TAU( I ) )
*
* Apply H(i) to A(1:m-k+i,1:n-k+i-1) from the left
*
AII = A( M-K+I, N-K+I )
A( M-K+I, N-K+I ) = ONE
CALL DLARF( 'Left', M-K+I, N-K+I-1, A( 1, N-K+I ), 1, TAU( I ),
$ A, LDA, WORK )
A( M-K+I, N-K+I ) = AII
10 CONTINUE
RETURN
*
* End of DGEQL2
*
END
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