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

 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.

Download ZLATRZ + dependencies [TGZ] [ZIP] [TXT]

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)`
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 139 of file zlatrz.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 COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
150* ..
151*
152* =====================================================================
153*
154* .. Parameters ..
155 COMPLEX*16 ZERO
156 parameter( zero = ( 0.0d+0, 0.0d+0 ) )
157* ..
158* .. Local Scalars ..
159 INTEGER I
160 COMPLEX*16 ALPHA
161* ..
162* .. External Subroutines ..
163 EXTERNAL zlacgv, zlarfg, zlarz
164* ..
165* .. Intrinsic Functions ..
166 INTRINSIC dconjg
167* ..
168* .. Executable Statements ..
169*
170* Quick return if possible
171*
172 IF( m.EQ.0 ) THEN
173 RETURN
174 ELSE IF( m.EQ.n ) THEN
175 DO 10 i = 1, n
176 tau( i ) = zero
177 10 CONTINUE
178 RETURN
179 END IF
180*
181 DO 20 i = m, 1, -1
182*
183* Generate elementary reflector H(i) to annihilate
184* [ A(i,i) A(i,n-l+1:n) ]
185*
186 CALL zlacgv( l, a( i, n-l+1 ), lda )
187 alpha = dconjg( a( i, i ) )
188 CALL zlarfg( l+1, alpha, a( i, n-l+1 ), lda, tau( i ) )
189 tau( i ) = dconjg( tau( i ) )
190*
191* Apply H(i) to A(1:i-1,i:n) from the right
192*
193 CALL zlarz( 'Right', i-1, n-i+1, l, a( i, n-l+1 ), lda,
194 \$ dconjg( tau( i ) ), a( 1, i ), lda, work )
195 a( i, i ) = dconjg( alpha )
196*
197 20 CONTINUE
198*
199 RETURN
200*
201* End of ZLATRZ
202*
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106
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:147
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