LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ clatrz()

 subroutine clatrz ( integer M, integer N, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK )

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

Purpose:
``` CLATRZ 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 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 array, dimension (M) The scalar factors of the elementary reflectors.``` [out] WORK ` WORK is COMPLEX 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 clatrz.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 A( LDA, * ), TAU( * ), WORK( * )
150 * ..
151 *
152 * =====================================================================
153 *
154 * .. Parameters ..
155  COMPLEX ZERO
156  parameter( zero = ( 0.0e+0, 0.0e+0 ) )
157 * ..
158 * .. Local Scalars ..
159  INTEGER I
160  COMPLEX ALPHA
161 * ..
162 * .. External Subroutines ..
163  EXTERNAL clacgv, clarfg, clarz
164 * ..
165 * .. Intrinsic Functions ..
166  INTRINSIC conjg
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 clacgv( l, a( i, n-l+1 ), lda )
187  alpha = conjg( a( i, i ) )
188  CALL clarfg( l+1, alpha, a( i, n-l+1 ), lda, tau( i ) )
189  tau( i ) = conjg( tau( i ) )
190 *
191 * Apply H(i) to A(1:i-1,i:n) from the right
192 *
193  CALL clarz( 'Right', i-1, n-i+1, l, a( i, n-l+1 ), lda,
194  \$ conjg( tau( i ) ), a( 1, i ), lda, work )
195  a( i, i ) = conjg( alpha )
196 *
197  20 CONTINUE
198 *
199  RETURN
200 *
201 * End of CLATRZ
202 *
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:74
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
subroutine clarz(SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK)
CLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
Definition: clarz.f:147
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