1*> \brief \b SGET52
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 SGET52( LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR,
12*                          ALPHAI, BETA, WORK, RESULT )
13*
14*       .. Scalar Arguments ..
15*       LOGICAL            LEFT
16*       INTEGER            LDA, LDB, LDE, N
17*       ..
18*       .. Array Arguments ..
19*       REAL               A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
20*      $                   B( LDB, * ), BETA( * ), E( LDE, * ),
21*      $                   RESULT( 2 ), WORK( * )
22*       ..
23*
24*
25*> \par Purpose:
26*  =============
27*>
28*> \verbatim
29*>
30*> SGET52  does an eigenvector check for the generalized eigenvalue
31*> problem.
32*>
33*> The basic test for right eigenvectors is:
34*>
35*>                           | b(j) A E(j) -  a(j) B E(j) |
36*>         RESULT(1) = max   -------------------------------
37*>                      j    n ulp max( |b(j) A|, |a(j) B| )
38*>
39*> using the 1-norm.  Here, a(j)/b(j) = w is the j-th generalized
40*> eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th
41*> generalized eigenvalue of m A - B.
42*>
43*> For real eigenvalues, the test is straightforward.  For complex
44*> eigenvalues, E(j) and a(j) are complex, represented by
45*> Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that
46*> eigenvector becomes
47*>
48*>                 max( |Wr|, |Wi| )
49*>     --------------------------------------------
50*>     n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )
51*>
52*> where
53*>
54*>     Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)
55*>
56*>     Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)
57*>
58*>                         T   T  _
59*> For left eigenvectors, A , B , a, and b  are used.
60*>
61*> SGET52 also tests the normalization of E.  Each eigenvector is
62*> supposed to be normalized so that the maximum "absolute value"
63*> of its elements is 1, where in this case, "absolute value"
64*> of a complex value x is  |Re(x)| + |Im(x)| ; let us call this
65*> maximum "absolute value" norm of a vector v  M(v).
66*> if a(j)=b(j)=0, then the eigenvector is set to be the jth coordinate
67*> vector.  The normalization test is:
68*>
69*>         RESULT(2) =      max       | M(v(j)) - 1 | / ( n ulp )
70*>                    eigenvectors v(j)
71*> \endverbatim
72*
73*  Arguments:
74*  ==========
75*
76*> \param[in] LEFT
77*> \verbatim
78*>          LEFT is LOGICAL
79*>          =.TRUE.:  The eigenvectors in the columns of E are assumed
80*>                    to be *left* eigenvectors.
81*>          =.FALSE.: The eigenvectors in the columns of E are assumed
82*>                    to be *right* eigenvectors.
83*> \endverbatim
84*>
85*> \param[in] N
86*> \verbatim
87*>          N is INTEGER
88*>          The size of the matrices.  If it is zero, SGET52 does
89*>          nothing.  It must be at least zero.
90*> \endverbatim
91*>
92*> \param[in] A
93*> \verbatim
94*>          A is REAL array, dimension (LDA, N)
95*>          The matrix A.
96*> \endverbatim
97*>
98*> \param[in] LDA
99*> \verbatim
100*>          LDA is INTEGER
101*>          The leading dimension of A.  It must be at least 1
102*>          and at least N.
103*> \endverbatim
104*>
105*> \param[in] B
106*> \verbatim
107*>          B is REAL array, dimension (LDB, N)
108*>          The matrix B.
109*> \endverbatim
110*>
111*> \param[in] LDB
112*> \verbatim
113*>          LDB is INTEGER
114*>          The leading dimension of B.  It must be at least 1
115*>          and at least N.
116*> \endverbatim
117*>
118*> \param[in] E
119*> \verbatim
120*>          E is REAL array, dimension (LDE, N)
121*>          The matrix of eigenvectors.  It must be O( 1 ).  Complex
122*>          eigenvalues and eigenvectors always come in pairs, the
123*>          eigenvalue and its conjugate being stored in adjacent
124*>          elements of ALPHAR, ALPHAI, and BETA.  Thus, if a(j)/b(j)
125*>          and a(j+1)/b(j+1) are a complex conjugate pair of
126*>          generalized eigenvalues, then E(,j) contains the real part
127*>          of the eigenvector and E(,j+1) contains the imaginary part.
128*>          Note that whether E(,j) is a real eigenvector or part of a
129*>          complex one is specified by whether ALPHAI(j) is zero or not.
130*> \endverbatim
131*>
132*> \param[in] LDE
133*> \verbatim
134*>          LDE is INTEGER
135*>          The leading dimension of E.  It must be at least 1 and at
136*>          least N.
137*> \endverbatim
138*>
139*> \param[in] ALPHAR
140*> \verbatim
141*>          ALPHAR is REAL array, dimension (N)
142*>          The real parts of the values a(j) as described above, which,
143*>          along with b(j), define the generalized eigenvalues.
144*>          Complex eigenvalues always come in complex conjugate pairs
145*>          a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent
146*>          elements in ALPHAR, ALPHAI, and BETA.  Thus, if the j-th
147*>          and (j+1)-st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1)
148*>          is assumed to be equal to ALPHAR(j)/BETA(j).
149*> \endverbatim
150*>
151*> \param[in] ALPHAI
152*> \verbatim
153*>          ALPHAI is REAL array, dimension (N)
154*>          The imaginary parts of the values a(j) as described above,
155*>          which, along with b(j), define the generalized eigenvalues.
156*>          If ALPHAI(j)=0, then the eigenvalue is real, otherwise it
157*>          is part of a complex conjugate pair.  Complex eigenvalues
158*>          always come in complex conjugate pairs a(j)/b(j) and
159*>          a(j+1)/b(j+1), which are stored in adjacent elements in
160*>          ALPHAR, ALPHAI, and BETA.  Thus, if the j-th and (j+1)-st
161*>          eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to
162*>          be equal to  -ALPHAI(j)/BETA(j).  Also, nonzero values in
163*>          ALPHAI are assumed to always come in adjacent pairs.
164*> \endverbatim
165*>
166*> \param[in] BETA
167*> \verbatim
168*>          BETA is REAL array, dimension (N)
169*>          The values b(j) as described above, which, along with a(j),
170*>          define the generalized eigenvalues.
171*> \endverbatim
172*>
173*> \param[out] WORK
174*> \verbatim
175*>          WORK is REAL array, dimension (N**2+N)
176*> \endverbatim
177*>
178*> \param[out] RESULT
179*> \verbatim
180*>          RESULT is REAL array, dimension (2)
181*>          The values computed by the test described above.  If A E or
182*>          B E is likely to overflow, then RESULT(1:2) is set to
183*>          10 / ulp.
184*> \endverbatim
185*
186*  Authors:
187*  ========
188*
189*> \author Univ. of Tennessee
190*> \author Univ. of California Berkeley
191*> \author Univ. of Colorado Denver
192*> \author NAG Ltd.
193*
194*> \date November 2011
195*
196*> \ingroup single_eig
197*
198*  =====================================================================
199      SUBROUTINE SGET52( LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR,
200     $                   ALPHAI, BETA, WORK, RESULT )
201*
202*  -- LAPACK test routine (version 3.4.0) --
203*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
204*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
205*     November 2011
206*
207*     .. Scalar Arguments ..
208      LOGICAL            LEFT
209      INTEGER            LDA, LDB, LDE, N
210*     ..
211*     .. Array Arguments ..
212      REAL               A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
213     $                   B( LDB, * ), BETA( * ), E( LDE, * ),
214     $                   RESULT( 2 ), WORK( * )
215*     ..
216*
217*  =====================================================================
218*
219*     .. Parameters ..
220      REAL               ZERO, ONE, TEN
221      PARAMETER          ( ZERO = 0.0, ONE = 1.0, TEN = 10.0 )
222*     ..
223*     .. Local Scalars ..
224      LOGICAL            ILCPLX
225      CHARACTER          NORMAB, TRANS
226      INTEGER            J, JVEC
227      REAL               ABMAX, ACOEF, ALFMAX, ANORM, BCOEFI, BCOEFR,
228     $                   BETMAX, BNORM, ENORM, ENRMER, ERRNRM, SAFMAX,
229     $                   SAFMIN, SALFI, SALFR, SBETA, SCALE, TEMP1, ULP
230*     ..
231*     .. External Functions ..
232      REAL               SLAMCH, SLANGE
233      EXTERNAL           SLAMCH, SLANGE
234*     ..
235*     .. External Subroutines ..
236      EXTERNAL           SGEMV
237*     ..
238*     .. Intrinsic Functions ..
239      INTRINSIC          ABS, MAX, REAL
240*     ..
241*     .. Executable Statements ..
242*
243      RESULT( 1 ) = ZERO
244      RESULT( 2 ) = ZERO
245      IF( N.LE.0 )
246     $   RETURN
247*
248      SAFMIN = SLAMCH( 'Safe minimum' )
249      SAFMAX = ONE / SAFMIN
250      ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' )
251*
252      IF( LEFT ) THEN
253         TRANS = 'T'
254         NORMAB = 'I'
255      ELSE
256         TRANS = 'N'
257         NORMAB = 'O'
258      END IF
259*
260*     Norm of A, B, and E:
261*
262      ANORM = MAX( SLANGE( NORMAB, N, N, A, LDA, WORK ), SAFMIN )
263      BNORM = MAX( SLANGE( NORMAB, N, N, B, LDB, WORK ), SAFMIN )
264      ENORM = MAX( SLANGE( 'O', N, N, E, LDE, WORK ), ULP )
265      ALFMAX = SAFMAX / MAX( ONE, BNORM )
266      BETMAX = SAFMAX / MAX( ONE, ANORM )
267*
268*     Compute error matrix.
269*     Column i = ( b(i) A - a(i) B ) E(i) / max( |a(i) B| |b(i) A| )
270*
271      ILCPLX = .FALSE.
272      DO 10 JVEC = 1, N
273         IF( ILCPLX ) THEN
274*
275*           2nd Eigenvalue/-vector of pair -- do nothing
276*
277            ILCPLX = .FALSE.
278         ELSE
279            SALFR = ALPHAR( JVEC )
280            SALFI = ALPHAI( JVEC )
281            SBETA = BETA( JVEC )
282            IF( SALFI.EQ.ZERO ) THEN
283*
284*              Real eigenvalue and -vector
285*
286               ABMAX = MAX( ABS( SALFR ), ABS( SBETA ) )
287               IF( ABS( SALFR ).GT.ALFMAX .OR. ABS( SBETA ).GT.
288     $             BETMAX .OR. ABMAX.LT.ONE ) THEN
289                  SCALE = ONE / MAX( ABMAX, SAFMIN )
290                  SALFR = SCALE*SALFR
291                  SBETA = SCALE*SBETA
292               END IF
293               SCALE = ONE / MAX( ABS( SALFR )*BNORM,
294     $                 ABS( SBETA )*ANORM, SAFMIN )
295               ACOEF = SCALE*SBETA
296               BCOEFR = SCALE*SALFR
297               CALL SGEMV( TRANS, N, N, ACOEF, A, LDA, E( 1, JVEC ), 1,
298     $                     ZERO, WORK( N*( JVEC-1 )+1 ), 1 )
299               CALL SGEMV( TRANS, N, N, -BCOEFR, B, LDA, E( 1, JVEC ),
300     $                     1, ONE, WORK( N*( JVEC-1 )+1 ), 1 )
301            ELSE
302*
303*              Complex conjugate pair
304*
305               ILCPLX = .TRUE.
306               IF( JVEC.EQ.N ) THEN
307                  RESULT( 1 ) = TEN / ULP
308                  RETURN
309               END IF
310               ABMAX = MAX( ABS( SALFR )+ABS( SALFI ), ABS( SBETA ) )
311               IF( ABS( SALFR )+ABS( SALFI ).GT.ALFMAX .OR.
312     $             ABS( SBETA ).GT.BETMAX .OR. ABMAX.LT.ONE ) THEN
313                  SCALE = ONE / MAX( ABMAX, SAFMIN )
314                  SALFR = SCALE*SALFR
315                  SALFI = SCALE*SALFI
316                  SBETA = SCALE*SBETA
317               END IF
318               SCALE = ONE / MAX( ( ABS( SALFR )+ABS( SALFI ) )*BNORM,
319     $                 ABS( SBETA )*ANORM, SAFMIN )
320               ACOEF = SCALE*SBETA
321               BCOEFR = SCALE*SALFR
322               BCOEFI = SCALE*SALFI
323               IF( LEFT ) THEN
324                  BCOEFI = -BCOEFI
325               END IF
326*
327               CALL SGEMV( TRANS, N, N, ACOEF, A, LDA, E( 1, JVEC ), 1,
328     $                     ZERO, WORK( N*( JVEC-1 )+1 ), 1 )
329               CALL SGEMV( TRANS, N, N, -BCOEFR, B, LDA, E( 1, JVEC ),
330     $                     1, ONE, WORK( N*( JVEC-1 )+1 ), 1 )
331               CALL SGEMV( TRANS, N, N, BCOEFI, B, LDA, E( 1, JVEC+1 ),
332     $                     1, ONE, WORK( N*( JVEC-1 )+1 ), 1 )
333*
334               CALL SGEMV( TRANS, N, N, ACOEF, A, LDA, E( 1, JVEC+1 ),
335     $                     1, ZERO, WORK( N*JVEC+1 ), 1 )
336               CALL SGEMV( TRANS, N, N, -BCOEFI, B, LDA, E( 1, JVEC ),
337     $                     1, ONE, WORK( N*JVEC+1 ), 1 )
338               CALL SGEMV( TRANS, N, N, -BCOEFR, B, LDA, E( 1, JVEC+1 ),
339     $                     1, ONE, WORK( N*JVEC+1 ), 1 )
340            END IF
341         END IF
342   10 CONTINUE
343*
344      ERRNRM = SLANGE( 'One', N, N, WORK, N, WORK( N**2+1 ) ) / ENORM
345*
346*     Compute RESULT(1)
347*
348      RESULT( 1 ) = ERRNRM / ULP
349*
350*     Normalization of E:
351*
352      ENRMER = ZERO
353      ILCPLX = .FALSE.
354      DO 40 JVEC = 1, N
355         IF( ILCPLX ) THEN
356            ILCPLX = .FALSE.
357         ELSE
358            TEMP1 = ZERO
359            IF( ALPHAI( JVEC ).EQ.ZERO ) THEN
360               DO 20 J = 1, N
361                  TEMP1 = MAX( TEMP1, ABS( E( J, JVEC ) ) )
362   20          CONTINUE
363               ENRMER = MAX( ENRMER, TEMP1-ONE )
364            ELSE
365               ILCPLX = .TRUE.
366               DO 30 J = 1, N
367                  TEMP1 = MAX( TEMP1, ABS( E( J, JVEC ) )+
368     $                    ABS( E( J, JVEC+1 ) ) )
369   30          CONTINUE
370               ENRMER = MAX( ENRMER, TEMP1-ONE )
371            END IF
372         END IF
373   40 CONTINUE
374*
375*     Compute RESULT(2) : the normalization error in E.
376*
377      RESULT( 2 ) = ENRMER / ( REAL( N )*ULP )
378*
379      RETURN
380*
381*     End of SGET52
382*
383      END
384