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.

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

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)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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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