dlatrz (l) - Linux Manuals

dlatrz: factors the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means of orthogonal transformations

NAME

DLATRZ - factors the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means of orthogonal transformations

SYNOPSIS

SUBROUTINE DLATRZ(
M, N, L, A, LDA, TAU, WORK )

    
INTEGER L, LDA, M, N

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

PURPOSE

DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal matrix and, R and A1 are M-by-M upper triangular matrices.

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.
L (input) INTEGER
The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= L >= 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 N-L+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) DOUBLE PRECISION array, dimension (M)

FURTHER DETAILS

Based on contributions by

  A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA The factorization is obtained by Householderaqs 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 form

Z( k ) = ( I     0   ),

      ( 0  T( k ) )
where

T( k ) = I - tau*u( k )*u( k )aq,   u( k ) = (   1    ),
                                         (   0    )
                                         ( z( k ) ) tau is a scalar and z( k ) is an l element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of A2. The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of A2, such that the elements of z( k ) are in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of A1.
Z is given by

Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

Pages related to dlatrz

  • dlatrz (3)
  • dlatrd (l) - reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Qaq * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A
  • dlatrs (l) - solves one of the triangular systems A *x = s*b or Aaq*x = s*b with scaling to prevent overflow
  • dlat2s (l) - converts a DOUBLE PRECISION triangular matrix, SA, to a SINGLE PRECISION triangular matrix, A
  • dlatbs (l) - solves one of the triangular systems A *x = s*b or Aaq*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular band matrix
  • dlatdf (l) - uses the LU factorization of the n-by-n matrix Z computed by DGETC2 and computes a contribution to the reciprocal Dif-estimate by solving Z * x = b for x, and choosing the r.h.s
  • dlatps (l) - solves one of the triangular systems A *x = s*b or Aaq*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form
  • dlatzm (l) - routine i deprecated and has been replaced by routine DORMRZ
  • dla_gbamv (l) - performs one of the matrix-vector operations y := alpha*abs(A)*abs(x) + beta*abs(y),