LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
ctzrqf.f
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1 *> \brief \b CTZRQF
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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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 This routine is deprecated and has been replaced by routine CTZRZF.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 complexOTHERcomputational
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)
XERBLA
Definition: xerbla.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:158
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
subroutine ctzrqf(M, N, A, LDA, TAU, INFO)
CTZRQF
Definition: ctzrqf.f:138