 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ ztzrqf()

 subroutine ztzrqf ( integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, integer INFO )

ZTZRQF

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

Purpose:
``` This routine is deprecated and has been replaced by routine ZTZRZF.

ZTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
to upper triangular form by means of unitary transformations.

The upper trapezoidal matrix A is factored as

A = ( R  0 ) * Z,

where Z is an N-by-N unitary matrix and R is an M-by-M upper
triangular matrix.```
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 >= M.``` [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 M+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] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date
December 2016
Further Details:
```  The  factorization is obtained by Householder's method.  The kth
transformation matrix, Z( k ), whose conjugate transpose 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 ( n - m ) element vector.
tau and z( k ) are chosen to annihilate the elements of the kth row
of X.

The scalar tau is returned in the kth element of TAU and the vector
u( k ) in the kth row of A, such that the elements of z( k ) are
in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
the upper triangular part of A.

Z is given by

Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).```

Definition at line 140 of file ztzrqf.f.

140 *
141 * -- LAPACK computational routine (version 3.7.0) --
142 * -- LAPACK is a software package provided by Univ. of Tennessee, --
143 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
144 * December 2016
145 *
146 * .. Scalar Arguments ..
147  INTEGER info, lda, m, n
148 * ..
149 * .. Array Arguments ..
150  COMPLEX*16 a( lda, * ), tau( * )
151 * ..
152 *
153 * =====================================================================
154 *
155 * .. Parameters ..
156  COMPLEX*16 cone, czero
157  parameter( cone = ( 1.0d+0, 0.0d+0 ),
158  \$ czero = ( 0.0d+0, 0.0d+0 ) )
159 * ..
160 * .. Local Scalars ..
161  INTEGER i, k, m1
162  COMPLEX*16 alpha
163 * ..
164 * .. Intrinsic Functions ..
165  INTRINSIC dconjg, max, min
166 * ..
167 * .. External Subroutines ..
168  EXTERNAL xerbla, zaxpy, zcopy, zgemv, zgerc, zlacgv,
169  \$ zlarfg
170 * ..
171 * .. Executable Statements ..
172 *
173 * Test the input parameters.
174 *
175  info = 0
176  IF( m.LT.0 ) THEN
177  info = -1
178  ELSE IF( n.LT.m ) THEN
179  info = -2
180  ELSE IF( lda.LT.max( 1, m ) ) THEN
181  info = -4
182  END IF
183  IF( info.NE.0 ) THEN
184  CALL xerbla( 'ZTZRQF', -info )
185  RETURN
186  END IF
187 *
188 * Perform the factorization.
189 *
190  IF( m.EQ.0 )
191  \$ RETURN
192  IF( m.EQ.n ) THEN
193  DO 10 i = 1, n
194  tau( i ) = czero
195  10 CONTINUE
196  ELSE
197  m1 = min( m+1, n )
198  DO 20 k = m, 1, -1
199 *
200 * Use a Householder reflection to zero the kth row of A.
201 * First set up the reflection.
202 *
203  a( k, k ) = dconjg( a( k, k ) )
204  CALL zlacgv( n-m, a( k, m1 ), lda )
205  alpha = a( k, k )
206  CALL zlarfg( n-m+1, alpha, a( k, m1 ), lda, tau( k ) )
207  a( k, k ) = alpha
208  tau( k ) = dconjg( tau( k ) )
209 *
210  IF( tau( k ).NE.czero .AND. k.GT.1 ) THEN
211 *
212 * We now perform the operation A := A*P( k )**H.
213 *
214 * Use the first ( k - 1 ) elements of TAU to store a( k ),
215 * where a( k ) consists of the first ( k - 1 ) elements of
216 * the kth column of A. Also let B denote the first
217 * ( k - 1 ) rows of the last ( n - m ) columns of A.
218 *
219  CALL zcopy( k-1, a( 1, k ), 1, tau, 1 )
220 *
221 * Form w = a( k ) + B*z( k ) in TAU.
222 *
223  CALL zgemv( 'No transpose', k-1, n-m, cone, a( 1, m1 ),
224  \$ lda, a( k, m1 ), lda, cone, tau, 1 )
225 *
226 * Now form a( k ) := a( k ) - conjg(tau)*w
227 * and B := B - conjg(tau)*w*z( k )**H.
228 *
229  CALL zaxpy( k-1, -dconjg( tau( k ) ), tau, 1, a( 1, k ),
230  \$ 1 )
231  CALL zgerc( k-1, n-m, -dconjg( tau( k ) ), tau, 1,
232  \$ a( k, m1 ), lda, a( 1, m1 ), lda )
233  END IF
234  20 CONTINUE
235  END IF
236 *
237  RETURN
238 *
239 * End of ZTZRQF
240 *
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76
subroutine zgerc(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
ZGERC
Definition: zgerc.f:132
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:90
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:83
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:108
Here is the call graph for this function:
Here is the caller graph for this function: