Annotation of rpl/lapack/lapack/ztzrqf.f, revision 1.13
1.10 bertrand 1: *> \brief \b ZTZRQF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZTZRQF + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztzrqf.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, M, N
25: * ..
26: * .. Array Arguments ..
27: * COMPLEX*16 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 This routine is deprecated and has been replaced by routine ZTZRZF.
37: *>
38: *> ZTZRQF 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*16 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*16 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: *> \date November 2011
103: *
104: *> \ingroup complex16OTHERcomputational
105: *
106: *> \par Further Details:
107: * =====================
108: *>
109: *> \verbatim
110: *>
111: *> The factorization is obtained by Householder's method. The kth
112: *> transformation matrix, Z( k ), whose conjugate transpose is used to
113: *> introduce zeros into the (m - k + 1)th row of A, is given in the form
114: *>
115: *> Z( k ) = ( I 0 ),
116: *> ( 0 T( k ) )
117: *>
118: *> where
119: *>
120: *> T( k ) = I - tau*u( k )*u( k )**H, u( k ) = ( 1 ),
121: *> ( 0 )
122: *> ( z( k ) )
123: *>
124: *> tau is a scalar and z( k ) is an ( n - m ) element vector.
125: *> tau and z( k ) are chosen to annihilate the elements of the kth row
126: *> of X.
127: *>
128: *> The scalar tau is returned in the kth element of TAU and the vector
129: *> u( k ) in the kth row of A, such that the elements of z( k ) are
130: *> in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
131: *> the upper triangular part of A.
132: *>
133: *> Z is given by
134: *>
135: *> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
136: *> \endverbatim
137: *>
138: * =====================================================================
1.1 bertrand 139: SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
140: *
1.10 bertrand 141: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 142: * -- LAPACK is a software package provided by Univ. of Tennessee, --
143: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.10 bertrand 144: * November 2011
1.1 bertrand 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,
1.5 bertrand 169: $ ZLARFG
1.1 bertrand 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 )
1.5 bertrand 206: CALL ZLARFG( N-M+1, ALPHA, A( K, M1 ), LDA, TAU( K ) )
1.1 bertrand 207: A( K, K ) = ALPHA
208: TAU( K ) = DCONJG( TAU( K ) )
209: *
210: IF( TAU( K ).NE.CZERO .AND. K.GT.1 ) THEN
211: *
1.9 bertrand 212: * We now perform the operation A := A*P( k )**H.
1.1 bertrand 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
1.9 bertrand 227: * and B := B - conjg(tau)*w*z( k )**H.
1.1 bertrand 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: *
241: END
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