1: *> \brief \b DTZRQF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
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7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, M, N
25: * ..
26: * .. Array Arguments ..
27: * DOUBLE PRECISION 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 This routine is deprecated and has been replaced by routine DTZRZF.
37: *>
38: *> DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
39: *> to upper triangular form by means of orthogonal 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 orthogonal 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 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 M+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] 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 doubleOTHERcomputational
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 ), which is used to introduce zeros into
111: *> 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 )**T, 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 DTZRQF( 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: DOUBLE PRECISION A( LDA, * ), TAU( * )
148: * ..
149: *
150: * =====================================================================
151: *
152: * .. Parameters ..
153: DOUBLE PRECISION ONE, ZERO
154: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
155: * ..
156: * .. Local Scalars ..
157: INTEGER I, K, M1
158: * ..
159: * .. Intrinsic Functions ..
160: INTRINSIC MAX, MIN
161: * ..
162: * .. External Subroutines ..
163: EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA
164: * ..
165: * .. Executable Statements ..
166: *
167: * Test the input parameters.
168: *
169: INFO = 0
170: IF( M.LT.0 ) THEN
171: INFO = -1
172: ELSE IF( N.LT.M ) THEN
173: INFO = -2
174: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
175: INFO = -4
176: END IF
177: IF( INFO.NE.0 ) THEN
178: CALL XERBLA( 'DTZRQF', -INFO )
179: RETURN
180: END IF
181: *
182: * Perform the factorization.
183: *
184: IF( M.EQ.0 )
185: $ RETURN
186: IF( M.EQ.N ) THEN
187: DO 10 I = 1, N
188: TAU( I ) = ZERO
189: 10 CONTINUE
190: ELSE
191: M1 = MIN( M+1, N )
192: DO 20 K = M, 1, -1
193: *
194: * Use a Householder reflection to zero the kth row of A.
195: * First set up the reflection.
196: *
197: CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) )
198: *
199: IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN
200: *
201: * We now perform the operation A := A*P( k ).
202: *
203: * Use the first ( k - 1 ) elements of TAU to store a( k ),
204: * where a( k ) consists of the first ( k - 1 ) elements of
205: * the kth column of A. Also let B denote the first
206: * ( k - 1 ) rows of the last ( n - m ) columns of A.
207: *
208: CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 )
209: *
210: * Form w = a( k ) + B*z( k ) in TAU.
211: *
212: CALL DGEMV( 'No transpose', K-1, N-M, ONE, A( 1, M1 ),
213: $ LDA, A( K, M1 ), LDA, ONE, TAU, 1 )
214: *
215: * Now form a( k ) := a( k ) - tau*w
216: * and B := B - tau*w*z( k )**T.
217: *
218: CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )
219: CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA,
220: $ A( 1, M1 ), LDA )
221: END IF
222: 20 CONTINUE
223: END IF
224: *
225: RETURN
226: *
227: * End of DTZRQF
228: *
229: END
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