LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
stzrqf.f
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1 *> \brief \b STZRQF
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE STZRQF( M, N, A, LDA, TAU, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, M, N
25 * ..
26 * .. Array Arguments ..
27 * REAL 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 This routine is deprecated and has been replaced by routine STZRZF.
37 *>
38 *> STZRQF 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 REAL 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 REAL 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 realOTHERcomputational
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 stzrqf( 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  REAL A( LDA, * ), TAU( * )
148 * ..
149 *
150 * =====================================================================
151 *
152 * .. Parameters ..
153  REAL ONE, ZERO
154  parameter( one = 1.0e+0, zero = 0.0e+0 )
155 * ..
156 * .. Local Scalars ..
157  INTEGER I, K, M1
158 * ..
159 * .. Intrinsic Functions ..
160  INTRINSIC max, min
161 * ..
162 * .. External Subroutines ..
163  EXTERNAL saxpy, scopy, sgemv, sger, slarfg, 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( 'STZRQF', -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 slarfg( 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 scopy( k-1, a( 1, k ), 1, tau, 1 )
209 *
210 * Form w = a( k ) + B*z( k ) in TAU.
211 *
212  CALL sgemv( '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 saxpy( k-1, -tau( k ), tau, 1, a( 1, k ), 1 )
219  CALL sger( 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 STZRQF
228 *
229  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine stzrqf(M, N, A, LDA, TAU, INFO)
STZRQF
Definition: stzrqf.f:138
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine sger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
SGER
Definition: sger.f:130
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156