LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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ctzrqf.f
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1*> \brief \b CTZRQF
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CTZRQF + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ctzrqf.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ctzrqf.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ctzrqf.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CTZRQF( M, N, A, LDA, TAU, INFO )
22*
23* .. Scalar Arguments ..
24* INTEGER INFO, LDA, M, N
25* ..
26* .. Array Arguments ..
27* COMPLEX A( LDA, * ), TAU( * )
28* ..
29*
30*
31*> \par Purpose:
32* =============
33*>
34*> \verbatim
35*>
36*> This routine is deprecated and has been replaced by routine CTZRZF.
37*>
38*> CTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
39*> to upper triangular form by means of unitary transformations.
40*>
41*> The upper trapezoidal matrix A is factored as
42*>
43*> A = ( R 0 ) * Z,
44*>
45*> where Z is an N-by-N unitary matrix and R is an M-by-M upper
46*> triangular matrix.
47*> \endverbatim
48*
49* Arguments:
50* ==========
51*
52*> \param[in] M
53*> \verbatim
54*> M is INTEGER
55*> The number of rows of the matrix A. M >= 0.
56*> \endverbatim
57*>
58*> \param[in] N
59*> \verbatim
60*> N is INTEGER
61*> The number of columns of the matrix A. N >= M.
62*> \endverbatim
63*>
64*> \param[in,out] A
65*> \verbatim
66*> A is COMPLEX array, dimension (LDA,N)
67*> On entry, the leading M-by-N upper trapezoidal part of the
68*> array A must contain the matrix to be factorized.
69*> On exit, the leading M-by-M upper triangular part of A
70*> contains the upper triangular matrix R, and elements M+1 to
71*> N of the first M rows of A, with the array TAU, represent the
72*> unitary matrix Z as a product of M elementary reflectors.
73*> \endverbatim
74*>
75*> \param[in] LDA
76*> \verbatim
77*> LDA is INTEGER
78*> The leading dimension of the array A. LDA >= max(1,M).
79*> \endverbatim
80*>
81*> \param[out] TAU
82*> \verbatim
83*> TAU is COMPLEX array, dimension (M)
84*> The scalar factors of the elementary reflectors.
85*> \endverbatim
86*>
87*> \param[out] INFO
88*> \verbatim
89*> INFO is INTEGER
90*> = 0: successful exit
91*> < 0: if INFO = -i, the i-th argument had an illegal value
92*> \endverbatim
93*
94* Authors:
95* ========
96*
97*> \author Univ. of Tennessee
98*> \author Univ. of California Berkeley
99*> \author Univ. of Colorado Denver
100*> \author NAG Ltd.
101*
102*> \ingroup tzrqf
103*
104*> \par Further Details:
105* =====================
106*>
107*> \verbatim
108*>
109*> The factorization is obtained by Householder's method. The kth
110*> transformation matrix, Z( k ), whose conjugate transpose is used to
111*> introduce zeros into the (m - k + 1)th row of A, is given in the form
112*>
113*> Z( k ) = ( I 0 ),
114*> ( 0 T( k ) )
115*>
116*> where
117*>
118*> T( k ) = I - tau*u( k )*u( k )**H, u( k ) = ( 1 ),
119*> ( 0 )
120*> ( z( k ) )
121*>
122*> tau is a scalar and z( k ) is an ( n - m ) element vector.
123*> tau and z( k ) are chosen to annihilate the elements of the kth row
124*> of X.
125*>
126*> The scalar tau is returned in the kth element of TAU and the vector
127*> u( k ) in the kth row of A, such that the elements of z( k ) are
128*> in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
129*> the upper triangular part of A.
130*>
131*> Z is given by
132*>
133*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
134*> \endverbatim
135*>
136* =====================================================================
137 SUBROUTINE ctzrqf( M, N, A, LDA, TAU, INFO )
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*
238 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine ctzrqf(m, n, a, lda, tau, info)
CTZRQF
Definition ctzrqf.f:138
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