1*> \brief \b ZQLT01
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 ZQLT01( M, N, A, AF, Q, L, LDA, TAU, WORK, LWORK,
12*                          RWORK, RESULT )
13*
14*       .. Scalar Arguments ..
15*       INTEGER            LDA, LWORK, M, N
16*       ..
17*       .. Array Arguments ..
18*       DOUBLE PRECISION   RESULT( * ), RWORK( * )
19*       COMPLEX*16         A( LDA, * ), AF( LDA, * ), L( LDA, * ),
20*      $                   Q( LDA, * ), TAU( * ), WORK( LWORK )
21*       ..
22*
23*
24*> \par Purpose:
25*  =============
26*>
27*> \verbatim
28*>
29*> ZQLT01 tests ZGEQLF, which computes the QL factorization of an m-by-n
30*> matrix A, and partially tests ZUNGQL which forms the m-by-m
31*> orthogonal matrix Q.
32*>
33*> ZQLT01 compares L with Q'*A, 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 COMPLEX*16 array, dimension (LDA,N)
54*>          The m-by-n matrix A.
55*> \endverbatim
56*>
57*> \param[out] AF
58*> \verbatim
59*>          AF is COMPLEX*16 array, dimension (LDA,N)
60*>          Details of the QL factorization of A, as returned by ZGEQLF.
61*>          See ZGEQLF for further details.
62*> \endverbatim
63*>
64*> \param[out] Q
65*> \verbatim
66*>          Q is COMPLEX*16 array, dimension (LDA,M)
67*>          The m-by-m orthogonal matrix Q.
68*> \endverbatim
69*>
70*> \param[out] L
71*> \verbatim
72*>          L is COMPLEX*16 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 R.
79*>          LDA >= max(M,N).
80*> \endverbatim
81*>
82*> \param[out] TAU
83*> \verbatim
84*>          TAU is COMPLEX*16 array, dimension (min(M,N))
85*>          The scalar factors of the elementary reflectors, as returned
86*>          by ZGEQLF.
87*> \endverbatim
88*>
89*> \param[out] WORK
90*> \verbatim
91*>          WORK is COMPLEX*16 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 DOUBLE PRECISION array, dimension (M)
103*> \endverbatim
104*>
105*> \param[out] RESULT
106*> \verbatim
107*>          RESULT is DOUBLE PRECISION array, dimension (2)
108*>          The test ratios:
109*>          RESULT(1) = norm( L - Q'*A ) / ( M * norm(A) * EPS )
110*>          RESULT(2) = norm( I - Q'*Q ) / ( M * 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*> \date December 2016
122*
123*> \ingroup complex16_lin
124*
125*  =====================================================================
126      SUBROUTINE ZQLT01( M, N, A, AF, Q, L, LDA, TAU, WORK, LWORK,
127     $                   RWORK, RESULT )
128*
129*  -- LAPACK test routine (version 3.7.0) --
130*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
131*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
132*     December 2016
133*
134*     .. Scalar Arguments ..
135      INTEGER            LDA, LWORK, M, N
136*     ..
137*     .. Array Arguments ..
138      DOUBLE PRECISION   RESULT( * ), RWORK( * )
139      COMPLEX*16         A( LDA, * ), AF( LDA, * ), L( LDA, * ),
140     $                   Q( LDA, * ), TAU( * ), WORK( LWORK )
141*     ..
142*
143*  =====================================================================
144*
145*     .. Parameters ..
146      DOUBLE PRECISION   ZERO, ONE
147      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
148      COMPLEX*16         ROGUE
149      PARAMETER          ( ROGUE = ( -1.0D+10, -1.0D+10 ) )
150*     ..
151*     .. Local Scalars ..
152      INTEGER            INFO, MINMN
153      DOUBLE PRECISION   ANORM, EPS, RESID
154*     ..
155*     .. External Functions ..
156      DOUBLE PRECISION   DLAMCH, ZLANGE, ZLANSY
157      EXTERNAL           DLAMCH, ZLANGE, ZLANSY
158*     ..
159*     .. External Subroutines ..
160      EXTERNAL           ZGEMM, ZGEQLF, ZHERK, ZLACPY, ZLASET, ZUNGQL
161*     ..
162*     .. Intrinsic Functions ..
163      INTRINSIC          DBLE, DCMPLX, MAX, MIN
164*     ..
165*     .. Scalars in Common ..
166      CHARACTER*32       SRNAMT
167*     ..
168*     .. Common blocks ..
169      COMMON             / SRNAMC / SRNAMT
170*     ..
171*     .. Executable Statements ..
172*
173      MINMN = MIN( M, N )
174      EPS = DLAMCH( 'Epsilon' )
175*
176*     Copy the matrix A to the array AF.
177*
178      CALL ZLACPY( 'Full', M, N, A, LDA, AF, LDA )
179*
180*     Factorize the matrix A in the array AF.
181*
182      SRNAMT = 'ZGEQLF'
183      CALL ZGEQLF( M, N, AF, LDA, TAU, WORK, LWORK, INFO )
184*
185*     Copy details of Q
186*
187      CALL ZLASET( 'Full', M, M, ROGUE, ROGUE, Q, LDA )
188      IF( M.GE.N ) THEN
189         IF( N.LT.M .AND. N.GT.0 )
190     $      CALL ZLACPY( 'Full', M-N, N, AF, LDA, Q( 1, M-N+1 ), LDA )
191         IF( N.GT.1 )
192     $      CALL ZLACPY( 'Upper', N-1, N-1, AF( M-N+1, 2 ), LDA,
193     $                   Q( M-N+1, M-N+2 ), LDA )
194      ELSE
195         IF( M.GT.1 )
196     $      CALL ZLACPY( 'Upper', M-1, M-1, AF( 1, N-M+2 ), LDA,
197     $                   Q( 1, 2 ), LDA )
198      END IF
199*
200*     Generate the m-by-m matrix Q
201*
202      SRNAMT = 'ZUNGQL'
203      CALL ZUNGQL( M, M, MINMN, Q, LDA, TAU, WORK, LWORK, INFO )
204*
205*     Copy L
206*
207      CALL ZLASET( 'Full', M, N, DCMPLX( ZERO ), DCMPLX( ZERO ), L,
208     $             LDA )
209      IF( M.GE.N ) THEN
210         IF( N.GT.0 )
211     $      CALL ZLACPY( 'Lower', N, N, AF( M-N+1, 1 ), LDA,
212     $                   L( M-N+1, 1 ), LDA )
213      ELSE
214         IF( N.GT.M .AND. M.GT.0 )
215     $      CALL ZLACPY( 'Full', M, N-M, AF, LDA, L, LDA )
216         IF( M.GT.0 )
217     $      CALL ZLACPY( 'Lower', M, M, AF( 1, N-M+1 ), LDA,
218     $                   L( 1, N-M+1 ), LDA )
219      END IF
220*
221*     Compute L - Q'*A
222*
223      CALL ZGEMM( 'Conjugate transpose', 'No transpose', M, N, M,
224     $            DCMPLX( -ONE ), Q, LDA, A, LDA, DCMPLX( ONE ), L,
225     $            LDA )
226*
227*     Compute norm( L - Q'*A ) / ( M * norm(A) * EPS ) .
228*
229      ANORM = ZLANGE( '1', M, N, A, LDA, RWORK )
230      RESID = ZLANGE( '1', M, N, L, LDA, RWORK )
231      IF( ANORM.GT.ZERO ) THEN
232         RESULT( 1 ) = ( ( RESID / DBLE( MAX( 1, M ) ) ) / ANORM ) / EPS
233      ELSE
234         RESULT( 1 ) = ZERO
235      END IF
236*
237*     Compute I - Q'*Q
238*
239      CALL ZLASET( 'Full', M, M, DCMPLX( ZERO ), DCMPLX( ONE ), L, LDA )
240      CALL ZHERK( 'Upper', 'Conjugate transpose', M, M, -ONE, Q, LDA,
241     $            ONE, L, LDA )
242*
243*     Compute norm( I - Q'*Q ) / ( M * EPS ) .
244*
245      RESID = ZLANSY( '1', 'Upper', M, L, LDA, RWORK )
246*
247      RESULT( 2 ) = ( RESID / DBLE( MAX( 1, M ) ) ) / EPS
248*
249      RETURN
250*
251*     End of ZQLT01
252*
253      END
254