 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dlatrz()

 subroutine dlatrz ( integer M, integer N, integer L, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK )

DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.

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.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in] L ``` L is INTEGER The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= L >= 0.``` [in,out] A ``` A is 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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] TAU ``` TAU is DOUBLE PRECISION array, dimension (M) The scalar factors of the elementary reflectors.``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (M)`
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
```  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 form

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

where

T( k ) = I - tau*u( k )*u( k )**T,   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 ).```

Definition at line 139 of file dlatrz.f.

140*
141* -- LAPACK computational routine --
142* -- LAPACK is a software package provided by Univ. of Tennessee, --
143* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
144*
145* .. Scalar Arguments ..
146 INTEGER L, LDA, M, N
147* ..
148* .. Array Arguments ..
149 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
150* ..
151*
152* =====================================================================
153*
154* .. Parameters ..
155 DOUBLE PRECISION ZERO
156 parameter( zero = 0.0d+0 )
157* ..
158* .. Local Scalars ..
159 INTEGER I
160* ..
161* .. External Subroutines ..
162 EXTERNAL dlarfg, dlarz
163* ..
164* .. Executable Statements ..
165*
166* Test the input arguments
167*
168* Quick return if possible
169*
170 IF( m.EQ.0 ) THEN
171 RETURN
172 ELSE IF( m.EQ.n ) THEN
173 DO 10 i = 1, n
174 tau( i ) = zero
175 10 CONTINUE
176 RETURN
177 END IF
178*
179 DO 20 i = m, 1, -1
180*
181* Generate elementary reflector H(i) to annihilate
182* [ A(i,i) A(i,n-l+1:n) ]
183*
184 CALL dlarfg( l+1, a( i, i ), a( i, n-l+1 ), lda, tau( i ) )
185*
186* Apply H(i) to A(1:i-1,i:n) from the right
187*
188 CALL dlarz( 'Right', i-1, n-i+1, l, a( i, n-l+1 ), lda,
189 \$ tau( i ), a( 1, i ), lda, work )
190*
191 20 CONTINUE
192*
193 RETURN
194*
195* End of DLATRZ
196*
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
subroutine dlarz(SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK)
DLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
Definition: dlarz.f:145
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