LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ ctzrqf()

 subroutine ctzrqf ( integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tau, integer info )

CTZRQF

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``` This routine is deprecated and has been replaced by routine CTZRZF.

CTZRQF 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 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 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```
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 137 of file ctzrqf.f.

138*
139* -- LAPACK computational routine --
140* -- LAPACK is a software package provided by Univ. of Tennessee, --
141* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
142*
143* .. Scalar Arguments ..
144 INTEGER INFO, LDA, M, N
145* ..
146* .. Array Arguments ..
147 COMPLEX A( LDA, * ), TAU( * )
148* ..
149*
150* =====================================================================
151*
152* .. Parameters ..
153 COMPLEX CONE, CZERO
154 parameter( cone = ( 1.0e+0, 0.0e+0 ),
155 \$ czero = ( 0.0e+0, 0.0e+0 ) )
156* ..
157* .. Local Scalars ..
158 INTEGER I, K, M1
159 COMPLEX ALPHA
160* ..
161* .. Intrinsic Functions ..
162 INTRINSIC conjg, max, min
163* ..
164* .. External Subroutines ..
165 EXTERNAL caxpy, ccopy, cgemv, cgerc, clacgv, clarfg,
166 \$ xerbla
167* ..
168* .. Executable Statements ..
169*
170* Test the input parameters.
171*
172 info = 0
173 IF( m.LT.0 ) THEN
174 info = -1
175 ELSE IF( n.LT.m ) THEN
176 info = -2
177 ELSE IF( lda.LT.max( 1, m ) ) THEN
178 info = -4
179 END IF
180 IF( info.NE.0 ) THEN
181 CALL xerbla( 'CTZRQF', -info )
182 RETURN
183 END IF
184*
185* Perform the factorization.
186*
187 IF( m.EQ.0 )
188 \$ RETURN
189 IF( m.EQ.n ) THEN
190 DO 10 i = 1, n
191 tau( i ) = czero
192 10 CONTINUE
193 ELSE
194 m1 = min( m+1, n )
195 DO 20 k = m, 1, -1
196*
197* Use a Householder reflection to zero the kth row of A.
198* First set up the reflection.
199*
200 a( k, k ) = conjg( a( k, k ) )
201 CALL clacgv( n-m, a( k, m1 ), lda )
202 alpha = a( k, k )
203 CALL clarfg( n-m+1, alpha, a( k, m1 ), lda, tau( k ) )
204 a( k, k ) = alpha
205 tau( k ) = conjg( tau( k ) )
206*
207 IF( tau( k ).NE.czero .AND. k.GT.1 ) THEN
208*
209* We now perform the operation A := A*P( k )**H.
210*
211* Use the first ( k - 1 ) elements of TAU to store a( k ),
212* where a( k ) consists of the first ( k - 1 ) elements of
213* the kth column of A. Also let B denote the first
214* ( k - 1 ) rows of the last ( n - m ) columns of A.
215*
216 CALL ccopy( k-1, a( 1, k ), 1, tau, 1 )
217*
218* Form w = a( k ) + B*z( k ) in TAU.
219*
220 CALL cgemv( 'No transpose', k-1, n-m, cone, a( 1, m1 ),
221 \$ lda, a( k, m1 ), lda, cone, tau, 1 )
222*
223* Now form a( k ) := a( k ) - conjg(tau)*w
224* and B := B - conjg(tau)*w*z( k )**H.
225*
226 CALL caxpy( k-1, -conjg( tau( k ) ), tau, 1, a( 1, k ),
227 \$ 1 )
228 CALL cgerc( k-1, n-m, -conjg( tau( k ) ), tau, 1,
229 \$ a( k, m1 ), lda, a( 1, m1 ), lda )
230 END IF
231 20 CONTINUE
232 END IF
233*
234 RETURN
235*
236* End of CTZRQF
237*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine caxpy(n, ca, cx, incx, cy, incy)
CAXPY
Definition caxpy.f:88
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
subroutine cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
CGEMV
Definition cgemv.f:160
subroutine cgerc(m, n, alpha, x, incx, y, incy, a, lda)
CGERC
Definition cgerc.f:130
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
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