1: *> \brief \b DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLATRZ + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrz.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrz.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrz.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER L, LDA, M, N
25: * ..
26: * .. Array Arguments ..
27: * DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
28: * ..
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
37: *> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
38: *> of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
39: *> matrix and, R and A1 are M-by-M upper triangular matrices.
40: *> \endverbatim
41: *
42: * Arguments:
43: * ==========
44: *
45: *> \param[in] M
46: *> \verbatim
47: *> M is INTEGER
48: *> The number of rows of the matrix A. M >= 0.
49: *> \endverbatim
50: *>
51: *> \param[in] N
52: *> \verbatim
53: *> N is INTEGER
54: *> The number of columns of the matrix A. N >= 0.
55: *> \endverbatim
56: *>
57: *> \param[in] L
58: *> \verbatim
59: *> L is INTEGER
60: *> The number of columns of the matrix A containing the
61: *> meaningful part of the Householder vectors. N-M >= L >= 0.
62: *> \endverbatim
63: *>
64: *> \param[in,out] A
65: *> \verbatim
66: *> A is DOUBLE PRECISION 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 N-L+1 to
71: *> N of the first M rows of A, with the array TAU, represent the
72: *> orthogonal 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 DOUBLE PRECISION array, dimension (M)
84: *> The scalar factors of the elementary reflectors.
85: *> \endverbatim
86: *>
87: *> \param[out] WORK
88: *> \verbatim
89: *> WORK is DOUBLE PRECISION array, dimension (M)
90: *> \endverbatim
91: *
92: * Authors:
93: * ========
94: *
95: *> \author Univ. of Tennessee
96: *> \author Univ. of California Berkeley
97: *> \author Univ. of Colorado Denver
98: *> \author NAG Ltd.
99: *
100: *> \ingroup doubleOTHERcomputational
101: *
102: *> \par Contributors:
103: * ==================
104: *>
105: *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
106: *
107: *> \par Further Details:
108: * =====================
109: *>
110: *> \verbatim
111: *>
112: *> The factorization is obtained by Householder's method. The kth
113: *> transformation matrix, Z( k ), which is used to introduce zeros into
114: *> the ( m - k + 1 )th row of A, is given in the form
115: *>
116: *> Z( k ) = ( I 0 ),
117: *> ( 0 T( k ) )
118: *>
119: *> where
120: *>
121: *> T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
122: *> ( 0 )
123: *> ( z( k ) )
124: *>
125: *> tau is a scalar and z( k ) is an l element vector. tau and z( k )
126: *> are chosen to annihilate the elements of the kth row of A2.
127: *>
128: *> The scalar tau is returned in the kth element of TAU and the vector
129: *> u( k ) in the kth row of A2, such that the elements of z( k ) are
130: *> in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
131: *> the upper triangular part of A1.
132: *>
133: *> Z is given by
134: *>
135: *> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
136: *> \endverbatim
137: *>
138: * =====================================================================
139: SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
140: *
141: * -- LAPACK computational routine --
142: * -- LAPACK is a software package provided by Univ. of Tennessee, --
143: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
144: *
145: * .. Scalar Arguments ..
146: INTEGER L, LDA, M, N
147: * ..
148: * .. Array Arguments ..
149: DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
150: * ..
151: *
152: * =====================================================================
153: *
154: * .. Parameters ..
155: DOUBLE PRECISION ZERO
156: PARAMETER ( ZERO = 0.0D+0 )
157: * ..
158: * .. Local Scalars ..
159: INTEGER I
160: * ..
161: * .. External Subroutines ..
162: EXTERNAL DLARFG, DLARZ
163: * ..
164: * .. Executable Statements ..
165: *
166: * Test the input arguments
167: *
168: * Quick return if possible
169: *
170: IF( M.EQ.0 ) THEN
171: RETURN
172: ELSE IF( M.EQ.N ) THEN
173: DO 10 I = 1, N
174: TAU( I ) = ZERO
175: 10 CONTINUE
176: RETURN
177: END IF
178: *
179: DO 20 I = M, 1, -1
180: *
181: * Generate elementary reflector H(i) to annihilate
182: * [ A(i,i) A(i,n-l+1:n) ]
183: *
184: CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
185: *
186: * Apply H(i) to A(1:i-1,i:n) from the right
187: *
188: CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
189: $ TAU( I ), A( 1, I ), LDA, WORK )
190: *
191: 20 CONTINUE
192: *
193: RETURN
194: *
195: * End of DLATRZ
196: *
197: END
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