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DTZRZF(l)					)					DTZRZF(l)

       DTZRZF  -  reduce  the M-by-N ( M<=N ) real upper trapezoidal matrix A to upper triangular
       form by means of orthogonal transformations



	   DOUBLE	  PRECISION A( LDA, * ), TAU( * ), WORK( * )

       DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix  A  to	upper  triangular
       form  by  means of orthogonal transformations.  The upper trapezoidal matrix A is factored

	  A = ( R  0 ) * Z,

       where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix.

       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 leading M-by-N upper trapezoidal part of the array  A  must  contain
	       the matrix to be factorized.  On exit, the leading M-by-M upper triangular part of
	       A contains the upper triangular matrix R, and elements M+1 to N	of  the  first	M
	       rows of A, with the array TAU, represent the orthogonal matrix Z as a product of M
	       elementary reflectors.

       LDA     (input) INTEGER
	       The leading dimension of the array A.  LDA >= max(1,M).

       TAU     (output) DOUBLE PRECISION array, dimension (M)
	       The scalar factors of the elementary reflectors.

       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,M).	For  optimum  performance
	       LWORK >= M*NB, where NB is the optimal blocksize.

	       If  LWORK = -1, then a workspace query is assumed; the routine only calculates the
	       optimal size of the WORK array, returns this value as the first entry of the  WORK
	       array, and no error message related to LWORK is issued by XERBLA.

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value

       Based on contributions by
	 A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

       The  factorization is obtained by Householder's method.	The kth transformation matrix, Z(
       k ), which is used to introduce zeros into the ( m - k + 1 )th row of A, is given  in  the

	  Z( k ) = ( I	   0   ),
		   ( 0	T( k ) )


	  T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
						      (   0    )
						      ( z( k ) )

       tau  is	a scalar and z( k ) is an ( n - m ) element vector.  tau and z( k ) are chosen to
       annihilate the elements of the kth row of X.

       The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth  row
       of A, such that the elements of z( k ) are in  a( k, m + 1 ), ..., a( k, n ). The elements
       of R are returned in the upper triangular part of A.

       Z is given by

	  Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

LAPACK version 3.0			   15 June 2000 				DTZRZF(l)
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