1*> \brief \b SGQRTS
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 SGQRTS( N, M, P, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
12*                          BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
13*
14*       .. Scalar Arguments ..
15*       INTEGER            LDA, LDB, LWORK, M, P, N
16*       ..
17*       .. Array Arguments ..
18*       REAL               A( LDA, * ), AF( LDA, * ), R( LDA, * ),
19*      $                   Q( LDA, * ), B( LDB, * ), BF( LDB, * ),
20*      $                   T( LDB, * ), Z( LDB, * ), BWK( LDB, * ),
21*      $                   TAUA( * ), TAUB( * ), RESULT( 4 ),
22*      $                   RWORK( * ), WORK( LWORK )
23*       ..
24*
25*
26*> \par Purpose:
27*  =============
28*>
29*> \verbatim
30*>
31*> SGQRTS tests SGGQRF, which computes the GQR factorization of an
32*> N-by-M matrix A and a N-by-P matrix B: A = Q*R and B = Q*T*Z.
33*> \endverbatim
34*
35*  Arguments:
36*  ==========
37*
38*> \param[in] N
39*> \verbatim
40*>          N is INTEGER
41*>          The number of rows of the matrices A and B.  N >= 0.
42*> \endverbatim
43*>
44*> \param[in] M
45*> \verbatim
46*>          M is INTEGER
47*>          The number of columns of the matrix A.  M >= 0.
48*> \endverbatim
49*>
50*> \param[in] P
51*> \verbatim
52*>          P is INTEGER
53*>          The number of columns of the matrix B.  P >= 0.
54*> \endverbatim
55*>
56*> \param[in] A
57*> \verbatim
58*>          A is REAL array, dimension (LDA,M)
59*>          The N-by-M matrix A.
60*> \endverbatim
61*>
62*> \param[out] AF
63*> \verbatim
64*>          AF is REAL array, dimension (LDA,N)
65*>          Details of the GQR factorization of A and B, as returned
66*>          by SGGQRF, see SGGQRF for further details.
67*> \endverbatim
68*>
69*> \param[out] Q
70*> \verbatim
71*>          Q is REAL array, dimension (LDA,N)
72*>          The M-by-M orthogonal matrix Q.
73*> \endverbatim
74*>
75*> \param[out] R
76*> \verbatim
77*>          R is REAL array, dimension (LDA,MAX(M,N))
78*> \endverbatim
79*>
80*> \param[in] LDA
81*> \verbatim
82*>          LDA is INTEGER
83*>          The leading dimension of the arrays A, AF, R and Q.
84*>          LDA >= max(M,N).
85*> \endverbatim
86*>
87*> \param[out] TAUA
88*> \verbatim
89*>          TAUA is REAL array, dimension (min(M,N))
90*>          The scalar factors of the elementary reflectors, as returned
91*>          by SGGQRF.
92*> \endverbatim
93*>
94*> \param[in] B
95*> \verbatim
96*>          B is REAL array, dimension (LDB,P)
97*>          On entry, the N-by-P matrix A.
98*> \endverbatim
99*>
100*> \param[out] BF
101*> \verbatim
102*>          BF is REAL array, dimension (LDB,N)
103*>          Details of the GQR factorization of A and B, as returned
104*>          by SGGQRF, see SGGQRF for further details.
105*> \endverbatim
106*>
107*> \param[out] Z
108*> \verbatim
109*>          Z is REAL array, dimension (LDB,P)
110*>          The P-by-P orthogonal matrix Z.
111*> \endverbatim
112*>
113*> \param[out] T
114*> \verbatim
115*>          T is REAL array, dimension (LDB,max(P,N))
116*> \endverbatim
117*>
118*> \param[out] BWK
119*> \verbatim
120*>          BWK is REAL array, dimension (LDB,N)
121*> \endverbatim
122*>
123*> \param[in] LDB
124*> \verbatim
125*>          LDB is INTEGER
126*>          The leading dimension of the arrays B, BF, Z and T.
127*>          LDB >= max(P,N).
128*> \endverbatim
129*>
130*> \param[out] TAUB
131*> \verbatim
132*>          TAUB is REAL array, dimension (min(P,N))
133*>          The scalar factors of the elementary reflectors, as returned
134*>          by SGGRQF.
135*> \endverbatim
136*>
137*> \param[out] WORK
138*> \verbatim
139*>          WORK is REAL array, dimension (LWORK)
140*> \endverbatim
141*>
142*> \param[in] LWORK
143*> \verbatim
144*>          LWORK is INTEGER
145*>          The dimension of the array WORK, LWORK >= max(N,M,P)**2.
146*> \endverbatim
147*>
148*> \param[out] RWORK
149*> \verbatim
150*>          RWORK is REAL array, dimension (max(N,M,P))
151*> \endverbatim
152*>
153*> \param[out] RESULT
154*> \verbatim
155*>          RESULT is REAL array, dimension (4)
156*>          The test ratios:
157*>            RESULT(1) = norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP)
158*>            RESULT(2) = norm( T*Z - Q'*B ) / (MAX(P,N)*norm(B)*ULP)
159*>            RESULT(3) = norm( I - Q'*Q ) / ( M*ULP )
160*>            RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
161*> \endverbatim
162*
163*  Authors:
164*  ========
165*
166*> \author Univ. of Tennessee
167*> \author Univ. of California Berkeley
168*> \author Univ. of Colorado Denver
169*> \author NAG Ltd.
170*
171*> \date December 2016
172*
173*> \ingroup single_eig
174*
175*  =====================================================================
176      SUBROUTINE SGQRTS( N, M, P, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
177     $                   BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
178*
179*  -- LAPACK test routine (version 3.7.0) --
180*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
181*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
182*     December 2016
183*
184*     .. Scalar Arguments ..
185      INTEGER            LDA, LDB, LWORK, M, P, N
186*     ..
187*     .. Array Arguments ..
188      REAL               A( LDA, * ), AF( LDA, * ), R( LDA, * ),
189     $                   Q( LDA, * ), B( LDB, * ), BF( LDB, * ),
190     $                   T( LDB, * ), Z( LDB, * ), BWK( LDB, * ),
191     $                   TAUA( * ), TAUB( * ), RESULT( 4 ),
192     $                   RWORK( * ), WORK( LWORK )
193*     ..
194*
195*  =====================================================================
196*
197*     .. Parameters ..
198      REAL               ZERO, ONE
199      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
200      REAL               ROGUE
201      PARAMETER          ( ROGUE = -1.0E+10 )
202*     ..
203*     .. Local Scalars ..
204      INTEGER            INFO
205      REAL               ANORM, BNORM, ULP, UNFL, RESID
206*     ..
207*     .. External Functions ..
208      REAL               SLAMCH, SLANGE, SLANSY
209      EXTERNAL           SLAMCH, SLANGE, SLANSY
210*     ..
211*     .. External Subroutines ..
212      EXTERNAL           SGEMM, SLACPY, SLASET, SORGQR,
213     $                   SORGRQ, SSYRK
214*     ..
215*     .. Intrinsic Functions ..
216      INTRINSIC          MAX, MIN, REAL
217*     ..
218*     .. Executable Statements ..
219*
220      ULP = SLAMCH( 'Precision' )
221      UNFL = SLAMCH( 'Safe minimum' )
222*
223*     Copy the matrix A to the array AF.
224*
225      CALL SLACPY( 'Full', N, M, A, LDA, AF, LDA )
226      CALL SLACPY( 'Full', N, P, B, LDB, BF, LDB )
227*
228      ANORM = MAX( SLANGE( '1', N, M, A, LDA, RWORK ), UNFL )
229      BNORM = MAX( SLANGE( '1', N, P, B, LDB, RWORK ), UNFL )
230*
231*     Factorize the matrices A and B in the arrays AF and BF.
232*
233      CALL SGGQRF( N, M, P, AF, LDA, TAUA, BF, LDB, TAUB, WORK,
234     $             LWORK, INFO )
235*
236*     Generate the N-by-N matrix Q
237*
238      CALL SLASET( 'Full', N, N, ROGUE, ROGUE, Q, LDA )
239      CALL SLACPY( 'Lower', N-1, M, AF( 2,1 ), LDA, Q( 2,1 ), LDA )
240      CALL SORGQR( N, N, MIN( N, M ), Q, LDA, TAUA, WORK, LWORK, INFO )
241*
242*     Generate the P-by-P matrix Z
243*
244      CALL SLASET( 'Full', P, P, ROGUE, ROGUE, Z, LDB )
245      IF( N.LE.P ) THEN
246         IF( N.GT.0 .AND. N.LT.P )
247     $      CALL SLACPY( 'Full', N, P-N, BF, LDB, Z( P-N+1, 1 ), LDB )
248         IF( N.GT.1 )
249     $      CALL SLACPY( 'Lower', N-1, N-1, BF( 2, P-N+1 ), LDB,
250     $                    Z( P-N+2, P-N+1 ), LDB )
251      ELSE
252         IF( P.GT.1)
253     $      CALL SLACPY( 'Lower', P-1, P-1, BF( N-P+2, 1 ), LDB,
254     $                    Z( 2, 1 ), LDB )
255      END IF
256      CALL SORGRQ( P, P, MIN( N, P ), Z, LDB, TAUB, WORK, LWORK, INFO )
257*
258*     Copy R
259*
260      CALL SLASET( 'Full', N, M, ZERO, ZERO, R, LDA )
261      CALL SLACPY( 'Upper', N, M, AF, LDA, R, LDA )
262*
263*     Copy T
264*
265      CALL SLASET( 'Full', N, P, ZERO, ZERO, T, LDB )
266      IF( N.LE.P ) THEN
267         CALL SLACPY( 'Upper', N, N, BF( 1, P-N+1 ), LDB, T( 1, P-N+1 ),
268     $                LDB )
269      ELSE
270         CALL SLACPY( 'Full', N-P, P, BF, LDB, T, LDB )
271         CALL SLACPY( 'Upper', P, P, BF( N-P+1, 1 ), LDB, T( N-P+1, 1 ),
272     $                LDB )
273      END IF
274*
275*     Compute R - Q'*A
276*
277      CALL SGEMM( 'Transpose', 'No transpose', N, M, N, -ONE, Q, LDA, A,
278     $            LDA, ONE, R, LDA )
279*
280*     Compute norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP ) .
281*
282      RESID = SLANGE( '1', N, M, R, LDA, RWORK )
283      IF( ANORM.GT.ZERO ) THEN
284         RESULT( 1 ) = ( ( RESID / REAL( MAX(1,M,N) ) ) / ANORM ) / ULP
285      ELSE
286         RESULT( 1 ) = ZERO
287      END IF
288*
289*     Compute T*Z - Q'*B
290*
291      CALL SGEMM( 'No Transpose', 'No transpose', N, P, P, ONE, T, LDB,
292     $            Z, LDB, ZERO, BWK, LDB )
293      CALL SGEMM( 'Transpose', 'No transpose', N, P, N, -ONE, Q, LDA,
294     $            B, LDB, ONE, BWK, LDB )
295*
296*     Compute norm( T*Z - Q'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
297*
298      RESID = SLANGE( '1', N, P, BWK, LDB, RWORK )
299      IF( BNORM.GT.ZERO ) THEN
300         RESULT( 2 ) = ( ( RESID / REAL( MAX(1,P,N ) ) )/BNORM ) / ULP
301      ELSE
302         RESULT( 2 ) = ZERO
303      END IF
304*
305*     Compute I - Q'*Q
306*
307      CALL SLASET( 'Full', N, N, ZERO, ONE, R, LDA )
308      CALL SSYRK( 'Upper', 'Transpose', N, N, -ONE, Q, LDA, ONE, R,
309     $            LDA )
310*
311*     Compute norm( I - Q'*Q ) / ( N * ULP ) .
312*
313      RESID = SLANSY( '1', 'Upper', N, R, LDA, RWORK )
314      RESULT( 3 ) = ( RESID / REAL( MAX( 1, N ) ) ) / ULP
315*
316*     Compute I - Z'*Z
317*
318      CALL SLASET( 'Full', P, P, ZERO, ONE, T, LDB )
319      CALL SSYRK( 'Upper', 'Transpose', P, P, -ONE, Z, LDB, ONE, T,
320     $            LDB )
321*
322*     Compute norm( I - Z'*Z ) / ( P*ULP ) .
323*
324      RESID = SLANSY( '1', 'Upper', P, T, LDB, RWORK )
325      RESULT( 4 ) = ( RESID / REAL( MAX( 1, P ) ) ) / ULP
326*
327      RETURN
328*
329*     End of SGQRTS
330*
331      END
332