1*> \brief \b SRQT01
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*  Definition:
9*  ===========
10*
11*       SUBROUTINE SRQT01( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK,
12*                          RWORK, RESULT )
13*
14*       .. Scalar Arguments ..
15*       INTEGER            LDA, LWORK, M, N
16*       ..
17*       .. Array Arguments ..
18*       REAL               A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
19*      $                   R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
20*      $                   WORK( LWORK )
21*       ..
22*
23*
24*> \par Purpose:
25*  =============
26*>
27*> \verbatim
28*>
29*> SRQT01 tests SGERQF, which computes the RQ factorization of an m-by-n
30*> matrix A, and partially tests SORGRQ which forms the n-by-n
31*> orthogonal matrix Q.
32*>
33*> SRQT01 compares R with A*Q', and checks that Q is orthogonal.
34*> \endverbatim
35*
36*  Arguments:
37*  ==========
38*
39*> \param[in] M
40*> \verbatim
41*>          M is INTEGER
42*>          The number of rows of the matrix A.  M >= 0.
43*> \endverbatim
44*>
45*> \param[in] N
46*> \verbatim
47*>          N is INTEGER
48*>          The number of columns of the matrix A.  N >= 0.
49*> \endverbatim
50*>
51*> \param[in] A
52*> \verbatim
53*>          A is REAL array, dimension (LDA,N)
54*>          The m-by-n matrix A.
55*> \endverbatim
56*>
57*> \param[out] AF
58*> \verbatim
59*>          AF is REAL array, dimension (LDA,N)
60*>          Details of the RQ factorization of A, as returned by SGERQF.
61*>          See SGERQF for further details.
62*> \endverbatim
63*>
64*> \param[out] Q
65*> \verbatim
66*>          Q is REAL array, dimension (LDA,N)
67*>          The n-by-n orthogonal matrix Q.
68*> \endverbatim
69*>
70*> \param[out] R
71*> \verbatim
72*>          R is REAL array, dimension (LDA,max(M,N))
73*> \endverbatim
74*>
75*> \param[in] LDA
76*> \verbatim
77*>          LDA is INTEGER
78*>          The leading dimension of the arrays A, AF, Q and L.
79*>          LDA >= max(M,N).
80*> \endverbatim
81*>
82*> \param[out] TAU
83*> \verbatim
84*>          TAU is REAL array, dimension (min(M,N))
85*>          The scalar factors of the elementary reflectors, as returned
86*>          by SGERQF.
87*> \endverbatim
88*>
89*> \param[out] WORK
90*> \verbatim
91*>          WORK is REAL array, dimension (LWORK)
92*> \endverbatim
93*>
94*> \param[in] LWORK
95*> \verbatim
96*>          LWORK is INTEGER
97*>          The dimension of the array WORK.
98*> \endverbatim
99*>
100*> \param[out] RWORK
101*> \verbatim
102*>          RWORK is REAL array, dimension (max(M,N))
103*> \endverbatim
104*>
105*> \param[out] RESULT
106*> \verbatim
107*>          RESULT is REAL array, dimension (2)
108*>          The test ratios:
109*>          RESULT(1) = norm( R - A*Q' ) / ( N * norm(A) * EPS )
110*>          RESULT(2) = norm( I - Q*Q' ) / ( N * EPS )
111*> \endverbatim
112*
113*  Authors:
114*  ========
115*
116*> \author Univ. of Tennessee
117*> \author Univ. of California Berkeley
118*> \author Univ. of Colorado Denver
119*> \author NAG Ltd.
120*
121*> \ingroup single_lin
122*
123*  =====================================================================
124      SUBROUTINE SRQT01( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK,
125     $                   RWORK, RESULT )
126*
127*  -- LAPACK test routine --
128*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
129*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
130*
131*     .. Scalar Arguments ..
132      INTEGER            LDA, LWORK, M, N
133*     ..
134*     .. Array Arguments ..
135      REAL               A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
136     $                   R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
137     $                   WORK( LWORK )
138*     ..
139*
140*  =====================================================================
141*
142*     .. Parameters ..
143      REAL               ZERO, ONE
144      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
145      REAL               ROGUE
146      PARAMETER          ( ROGUE = -1.0E+10 )
147*     ..
148*     .. Local Scalars ..
149      INTEGER            INFO, MINMN
150      REAL               ANORM, EPS, RESID
151*     ..
152*     .. External Functions ..
153      REAL               SLAMCH, SLANGE, SLANSY
154      EXTERNAL           SLAMCH, SLANGE, SLANSY
155*     ..
156*     .. External Subroutines ..
157      EXTERNAL           SGEMM, SGERQF, SLACPY, SLASET, SORGRQ, SSYRK
158*     ..
159*     .. Intrinsic Functions ..
160      INTRINSIC          MAX, MIN, REAL
161*     ..
162*     .. Scalars in Common ..
163      CHARACTER*32       SRNAMT
164*     ..
165*     .. Common blocks ..
166      COMMON             / SRNAMC / SRNAMT
167*     ..
168*     .. Executable Statements ..
169*
170      MINMN = MIN( M, N )
171      EPS = SLAMCH( 'Epsilon' )
172*
173*     Copy the matrix A to the array AF.
174*
175      CALL SLACPY( 'Full', M, N, A, LDA, AF, LDA )
176*
177*     Factorize the matrix A in the array AF.
178*
179      SRNAMT = 'SGERQF'
180      CALL SGERQF( M, N, AF, LDA, TAU, WORK, LWORK, INFO )
181*
182*     Copy details of Q
183*
184      CALL SLASET( 'Full', N, N, ROGUE, ROGUE, Q, LDA )
185      IF( M.LE.N ) THEN
186         IF( M.GT.0 .AND. M.LT.N )
187     $      CALL SLACPY( 'Full', M, N-M, AF, LDA, Q( N-M+1, 1 ), LDA )
188         IF( M.GT.1 )
189     $      CALL SLACPY( 'Lower', M-1, M-1, AF( 2, N-M+1 ), LDA,
190     $                   Q( N-M+2, N-M+1 ), LDA )
191      ELSE
192         IF( N.GT.1 )
193     $      CALL SLACPY( 'Lower', N-1, N-1, AF( M-N+2, 1 ), LDA,
194     $                   Q( 2, 1 ), LDA )
195      END IF
196*
197*     Generate the n-by-n matrix Q
198*
199      SRNAMT = 'SORGRQ'
200      CALL SORGRQ( N, N, MINMN, Q, LDA, TAU, WORK, LWORK, INFO )
201*
202*     Copy R
203*
204      CALL SLASET( 'Full', M, N, ZERO, ZERO, R, LDA )
205      IF( M.LE.N ) THEN
206         IF( M.GT.0 )
207     $      CALL SLACPY( 'Upper', M, M, AF( 1, N-M+1 ), LDA,
208     $                   R( 1, N-M+1 ), LDA )
209      ELSE
210         IF( M.GT.N .AND. N.GT.0 )
211     $      CALL SLACPY( 'Full', M-N, N, AF, LDA, R, LDA )
212         IF( N.GT.0 )
213     $      CALL SLACPY( 'Upper', N, N, AF( M-N+1, 1 ), LDA,
214     $                   R( M-N+1, 1 ), LDA )
215      END IF
216*
217*     Compute R - A*Q'
218*
219      CALL SGEMM( 'No transpose', 'Transpose', M, N, N, -ONE, A, LDA, Q,
220     $            LDA, ONE, R, LDA )
221*
222*     Compute norm( R - Q'*A ) / ( N * norm(A) * EPS ) .
223*
224      ANORM = SLANGE( '1', M, N, A, LDA, RWORK )
225      RESID = SLANGE( '1', M, N, R, LDA, RWORK )
226      IF( ANORM.GT.ZERO ) THEN
227         RESULT( 1 ) = ( ( RESID / REAL( MAX( 1, N ) ) ) / ANORM ) / EPS
228      ELSE
229         RESULT( 1 ) = ZERO
230      END IF
231*
232*     Compute I - Q*Q'
233*
234      CALL SLASET( 'Full', N, N, ZERO, ONE, R, LDA )
235      CALL SSYRK( 'Upper', 'No transpose', N, N, -ONE, Q, LDA, ONE, R,
236     $            LDA )
237*
238*     Compute norm( I - Q*Q' ) / ( N * EPS ) .
239*
240      RESID = SLANSY( '1', 'Upper', N, R, LDA, RWORK )
241*
242      RESULT( 2 ) = ( RESID / REAL( MAX( 1, N ) ) ) / EPS
243*
244      RETURN
245*
246*     End of SRQT01
247*
248      END
249