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SUBROUTINE <a name="DTZRQF.1"></a><a href="dtzrqf.f.html#DTZRQF.1">DTZRQF</a>( M, N, A, LDA, TAU, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> INTEGER INFO, LDA, M, N
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> DOUBLE PRECISION A( LDA, * ), TAU( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This routine is deprecated and has been replaced by routine <a name="DTZRZF.17"></a><a href="dtzrzf.f.html#DTZRZF.1">DTZRZF</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DTZRQF.19"></a><a href="dtzrqf.f.html#DTZRQF.1">DTZRQF</a> reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
</span><span class="comment">*</span><span class="comment"> to upper triangular form by means of orthogonal transformations.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The upper trapezoidal matrix A is factored as
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A = ( R 0 ) * Z,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
</span><span class="comment">*</span><span class="comment"> triangular matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> M (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of rows of the matrix A. M >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of columns of the matrix A. N >= M.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment"> On entry, the leading M-by-N upper trapezoidal part of the
</span><span class="comment">*</span><span class="comment"> array A must contain the matrix to be factorized.
</span><span class="comment">*</span><span class="comment"> On exit, the leading M-by-M upper triangular part of A
</span><span class="comment">*</span><span class="comment"> contains the upper triangular matrix R, and elements M+1 to
</span><span class="comment">*</span><span class="comment"> N of the first M rows of A, with the array TAU, represent the
</span><span class="comment">*</span><span class="comment"> orthogonal matrix Z as a product of M elementary reflectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TAU (output) DOUBLE PRECISION array, dimension (M)
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INFO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: successful exit
</span><span class="comment">*</span><span class="comment"> < 0: if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The factorization is obtained by Householder's method. The kth
</span><span class="comment">*</span><span class="comment"> transformation matrix, Z( k ), which is used to introduce zeros into
</span><span class="comment">*</span><span class="comment"> the ( m - k + 1 )th row of A, is given in the form
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z( k ) = ( I 0 ),
</span><span class="comment">*</span><span class="comment"> ( 0 T( k ) )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
</span><span class="comment">*</span><span class="comment"> ( 0 )
</span><span class="comment">*</span><span class="comment"> ( z( k ) )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> tau is a scalar and z( k ) is an ( n - m ) element vector.
</span><span class="comment">*</span><span class="comment"> tau and z( k ) are chosen to annihilate the elements of the kth row
</span><span class="comment">*</span><span class="comment"> of X.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The scalar tau is returned in the kth element of TAU and the vector
</span><span class="comment">*</span><span class="comment"> u( k ) in the kth row of A, such that the elements of z( k ) are
</span><span class="comment">*</span><span class="comment"> in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
</span><span class="comment">*</span><span class="comment"> the upper triangular part of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z is given by
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> INTEGER I, K, M1
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC MAX, MIN
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL DAXPY, DCOPY, DGEMV, DGER, <a name="DLARFG.98"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>, <a name="XERBLA.98"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Test the input parameters.
</span><span class="comment">*</span><span class="comment">
</span> INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL <a name="XERBLA.113"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="DTZRQF.113"></a><a href="dtzrqf.f.html#DTZRQF.1">DTZRQF</a>'</span>, -INFO )
RETURN
END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Perform the factorization.
</span><span class="comment">*</span><span class="comment">
</span> IF( M.EQ.0 )
$ RETURN
IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = ZERO
10 CONTINUE
ELSE
M1 = MIN( M+1, N )
DO 20 K = M, 1, -1
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Use a Householder reflection to zero the kth row of A.
</span><span class="comment">*</span><span class="comment"> First set up the reflection.
</span><span class="comment">*</span><span class="comment">
</span> CALL <a name="DLARFG.132"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) )
<span class="comment">*</span><span class="comment">
</span> IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> We now perform the operation A := A*P( k ).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Use the first ( k - 1 ) elements of TAU to store a( k ),
</span><span class="comment">*</span><span class="comment"> where a( k ) consists of the first ( k - 1 ) elements of
</span><span class="comment">*</span><span class="comment"> the kth column of A. Also let B denote the first
</span><span class="comment">*</span><span class="comment"> ( k - 1 ) rows of the last ( n - m ) columns of A.
</span><span class="comment">*</span><span class="comment">
</span> CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Form w = a( k ) + B*z( k ) in TAU.
</span><span class="comment">*</span><span class="comment">
</span> CALL DGEMV( <span class="string">'No transpose'</span>, K-1, N-M, ONE, A( 1, M1 ),
$ LDA, A( K, M1 ), LDA, ONE, TAU, 1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Now form a( k ) := a( k ) - tau*w
</span><span class="comment">*</span><span class="comment"> and B := B - tau*w*z( k )'.
</span><span class="comment">*</span><span class="comment">
</span> CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )
CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA,
$ A( 1, M1 ), LDA )
END IF
20 CONTINUE
END IF
<span class="comment">*</span><span class="comment">
</span> RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> End of <a name="DTZRQF.162"></a><a href="dtzrqf.f.html#DTZRQF.1">DTZRQF</a>
</span><span class="comment">*</span><span class="comment">
</span> END
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