 LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine zlatrz ( integer M, integer N, integer L, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK )

ZLATRZ factors an upper trapezoidal matrix by means of unitary transformations.

Purpose:
``` ZLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix
[ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z by means
of unitary transformations, where  Z is an (M+L)-by-(M+L) unitary
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 COMPLEX*16 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 unitary 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 COMPLEX*16 array, dimension (M) The scalar factors of the elementary reflectors.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (M)`
Date
September 2012
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 )**H,   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 142 of file zlatrz.f.

142 *
143 * -- LAPACK computational routine (version 3.4.2) --
144 * -- LAPACK is a software package provided by Univ. of Tennessee, --
145 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146 * September 2012
147 *
148 * .. Scalar Arguments ..
149  INTEGER l, lda, m, n
150 * ..
151 * .. Array Arguments ..
152  COMPLEX*16 a( lda, * ), tau( * ), work( * )
153 * ..
154 *
155 * =====================================================================
156 *
157 * .. Parameters ..
158  COMPLEX*16 zero
159  parameter ( zero = ( 0.0d+0, 0.0d+0 ) )
160 * ..
161 * .. Local Scalars ..
162  INTEGER i
163  COMPLEX*16 alpha
164 * ..
165 * .. External Subroutines ..
166  EXTERNAL zlacgv, zlarfg, zlarz
167 * ..
168 * .. Intrinsic Functions ..
169  INTRINSIC dconjg
170 * ..
171 * .. Executable Statements ..
172 *
173 * Quick return if possible
174 *
175  IF( m.EQ.0 ) THEN
176  RETURN
177  ELSE IF( m.EQ.n ) THEN
178  DO 10 i = 1, n
179  tau( i ) = zero
180  10 CONTINUE
181  RETURN
182  END IF
183 *
184  DO 20 i = m, 1, -1
185 *
186 * Generate elementary reflector H(i) to annihilate
187 * [ A(i,i) A(i,n-l+1:n) ]
188 *
189  CALL zlacgv( l, a( i, n-l+1 ), lda )
190  alpha = dconjg( a( i, i ) )
191  CALL zlarfg( l+1, alpha, a( i, n-l+1 ), lda, tau( i ) )
192  tau( i ) = dconjg( tau( i ) )
193 *
194 * Apply H(i) to A(1:i-1,i:n) from the right
195 *
196  CALL zlarz( 'Right', i-1, n-i+1, l, a( i, n-l+1 ), lda,
197  \$ dconjg( tau( i ) ), a( 1, i ), lda, work )
198  a( i, i ) = dconjg( alpha )
199 *
200  20 CONTINUE
201 *
202  RETURN
203 *
204 * End of ZLATRZ
205 *
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:108
subroutine zlarz(SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK)
ZLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
Definition: zlarz.f:149
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76

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