1DLATRZ(1)                LAPACK routine (version 3.1)                DLATRZ(1)
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NAME

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

11       SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
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13           INTEGER        L, LDA, M, N
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15           DOUBLE         PRECISION A( LDA, * ), TAU( * ), WORK( * )
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PURPOSE

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

25       M       (input) INTEGER
26               The number of rows of the matrix A.  M >= 0.
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28       N       (input) INTEGER
29               The number of columns of the matrix A.  N >= 0.
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31       L       (input) INTEGER
32               The number of columns of the matrix A containing the meaningful
33               part of the Householder vectors. N-M >= L >= 0.
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35       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
36               On  entry,  the  leading  M-by-N  upper trapezoidal part of the
37               array A must contain the matrix to be factorized.  On exit, the
38               leading  M-by-M  upper  triangular part of A contains the upper
39               triangular matrix R, and elements N-L+1 to N  of  the  first  M
40               rows  of A, with the array TAU, represent the orthogonal matrix
41               Z as a product of M elementary reflectors.
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43       LDA     (input) INTEGER
44               The leading dimension of the array A.  LDA >= max(1,M).
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46       TAU     (output) DOUBLE PRECISION array, dimension (M)
47               The scalar factors of the elementary reflectors.
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49       WORK    (workspace) DOUBLE PRECISION array, dimension (M)
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FURTHER DETAILS

52       Based on contributions by
53         A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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55       The factorization is obtained by Householder's method.  The kth  trans‐
56       formation matrix, Z( k ), which is used to introduce zeros into the ( m
57       - k + 1 )th row of A, is given in the form
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59          Z( k ) = ( I     0   ),
60                   ( 0  T( k ) )
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62       where
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64          T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
65                                                      (   0    )
66                                                      ( z( k ) )
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68       tau is a scalar and z( k ) is an l element vector. tau and z( k  )  are
69       chosen to annihilate the elements of the kth row of A2.
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71       The  scalar tau is returned in the kth element of TAU and the vector u(
72       k ) in the kth row of A2, such that the elements of z( k ) are  in   a(
73       k, l + 1 ), ..., a( k, n ). The elements of R are returned in the upper
74       triangular part of A1.
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76       Z is given by
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78          Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
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83 LAPACK routine (version 3.1)    November 2006                       DLATRZ(1)
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