LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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sget52.f
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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*> \ingroup single_eig
195*
196* =====================================================================
197 SUBROUTINE sget52( LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR,
198 $ ALPHAI, BETA, WORK, RESULT )
199*
200* -- LAPACK test routine --
201* -- LAPACK is a software package provided by Univ. of Tennessee, --
202* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
203*
204* .. Scalar Arguments ..
205 LOGICAL LEFT
206 INTEGER LDA, LDB, LDE, N
207* ..
208* .. Array Arguments ..
209 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
210 $ b( ldb, * ), beta( * ), e( lde, * ),
211 $ result( 2 ), work( * )
212* ..
213*
214* =====================================================================
215*
216* .. Parameters ..
217 REAL ZERO, ONE, TEN
218 parameter( zero = 0.0, one = 1.0, ten = 10.0 )
219* ..
220* .. Local Scalars ..
221 LOGICAL ILCPLX
222 CHARACTER NORMAB, TRANS
223 INTEGER J, JVEC
224 REAL ABMAX, ACOEF, ALFMAX, ANORM, BCOEFI, BCOEFR,
225 $ betmax, bnorm, enorm, enrmer, errnrm, safmax,
226 $ safmin, salfi, salfr, sbeta, scale, temp1, ulp
227* ..
228* .. External Functions ..
229 REAL SLAMCH, SLANGE
230 EXTERNAL slamch, slange
231* ..
232* .. External Subroutines ..
233 EXTERNAL sgemv
234* ..
235* .. Intrinsic Functions ..
236 INTRINSIC abs, max, real
237* ..
238* .. Executable Statements ..
239*
240 result( 1 ) = zero
241 result( 2 ) = zero
242 IF( n.LE.0 )
243 $ RETURN
244*
245 safmin = slamch( 'Safe minimum' )
246 safmax = one / safmin
247 ulp = slamch( 'Epsilon' )*slamch( 'Base' )
248*
249 IF( left ) THEN
250 trans = 'T'
251 normab = 'I'
252 ELSE
253 trans = 'N'
254 normab = 'O'
255 END IF
256*
257* Norm of A, B, and E:
258*
259 anorm = max( slange( normab, n, n, a, lda, work ), safmin )
260 bnorm = max( slange( normab, n, n, b, ldb, work ), safmin )
261 enorm = max( slange( 'O', n, n, e, lde, work ), ulp )
262 alfmax = safmax / max( one, bnorm )
263 betmax = safmax / max( one, anorm )
264*
265* Compute error matrix.
266* Column i = ( b(i) A - a(i) B ) E(i) / max( |a(i) B|, |b(i) A| )
267*
268 ilcplx = .false.
269 DO 10 jvec = 1, n
270 IF( ilcplx ) THEN
271*
272* 2nd Eigenvalue/-vector of pair -- do nothing
273*
274 ilcplx = .false.
275 ELSE
276 salfr = alphar( jvec )
277 salfi = alphai( jvec )
278 sbeta = beta( jvec )
279 IF( salfi.EQ.zero ) THEN
280*
281* Real eigenvalue and -vector
282*
283 abmax = max( abs( salfr ), abs( sbeta ) )
284 IF( abs( salfr ).GT.alfmax .OR. abs( sbeta ).GT.
285 $ betmax .OR. abmax.LT.one ) THEN
286 scale = one / max( abmax, safmin )
287 salfr = scale*salfr
288 sbeta = scale*sbeta
289 END IF
290 scale = one / max( abs( salfr )*bnorm,
291 $ abs( sbeta )*anorm, safmin )
292 acoef = scale*sbeta
293 bcoefr = scale*salfr
294 CALL sgemv( trans, n, n, acoef, a, lda, e( 1, jvec ), 1,
295 $ zero, work( n*( jvec-1 )+1 ), 1 )
296 CALL sgemv( trans, n, n, -bcoefr, b, lda, e( 1, jvec ),
297 $ 1, one, work( n*( jvec-1 )+1 ), 1 )
298 ELSE
299*
300* Complex conjugate pair
301*
302 ilcplx = .true.
303 IF( jvec.EQ.n ) THEN
304 result( 1 ) = ten / ulp
305 RETURN
306 END IF
307 abmax = max( abs( salfr )+abs( salfi ), abs( sbeta ) )
308 IF( abs( salfr )+abs( salfi ).GT.alfmax .OR.
309 $ abs( sbeta ).GT.betmax .OR. abmax.LT.one ) THEN
310 scale = one / max( abmax, safmin )
311 salfr = scale*salfr
312 salfi = scale*salfi
313 sbeta = scale*sbeta
314 END IF
315 scale = one / max( ( abs( salfr )+abs( salfi ) )*bnorm,
316 $ abs( sbeta )*anorm, safmin )
317 acoef = scale*sbeta
318 bcoefr = scale*salfr
319 bcoefi = scale*salfi
320 IF( left ) THEN
321 bcoefi = -bcoefi
322 END IF
323*
324 CALL sgemv( trans, n, n, acoef, a, lda, e( 1, jvec ), 1,
325 $ zero, work( n*( jvec-1 )+1 ), 1 )
326 CALL sgemv( trans, n, n, -bcoefr, b, lda, e( 1, jvec ),
327 $ 1, one, work( n*( jvec-1 )+1 ), 1 )
328 CALL sgemv( trans, n, n, bcoefi, b, lda, e( 1, jvec+1 ),
329 $ 1, one, work( n*( jvec-1 )+1 ), 1 )
330*
331 CALL sgemv( trans, n, n, acoef, a, lda, e( 1, jvec+1 ),
332 $ 1, zero, work( n*jvec+1 ), 1 )
333 CALL sgemv( trans, n, n, -bcoefi, b, lda, e( 1, jvec ),
334 $ 1, one, work( n*jvec+1 ), 1 )
335 CALL sgemv( trans, n, n, -bcoefr, b, lda, e( 1, jvec+1 ),
336 $ 1, one, work( n*jvec+1 ), 1 )
337 END IF
338 END IF
339 10 CONTINUE
340*
341 errnrm = slange( 'One', n, n, work, n, work( n**2+1 ) ) / enorm
342*
343* Compute RESULT(1)
344*
345 result( 1 ) = errnrm / ulp
346*
347* Normalization of E:
348*
349 enrmer = zero
350 ilcplx = .false.
351 DO 40 jvec = 1, n
352 IF( ilcplx ) THEN
353 ilcplx = .false.
354 ELSE
355 temp1 = zero
356 IF( alphai( jvec ).EQ.zero ) THEN
357 DO 20 j = 1, n
358 temp1 = max( temp1, abs( e( j, jvec ) ) )
359 20 CONTINUE
360 enrmer = max( enrmer, abs( temp1-one ) )
361 ELSE
362 ilcplx = .true.
363 DO 30 j = 1, n
364 temp1 = max( temp1, abs( e( j, jvec ) )+
365 $ abs( e( j, jvec+1 ) ) )
366 30 CONTINUE
367 enrmer = max( enrmer, abs( temp1-one ) )
368 END IF
369 END IF
370 40 CONTINUE
371*
372* Compute RESULT(2) : the normalization error in E.
373*
374 result( 2 ) = enrmer / ( real( n )*ulp )
375*
376 RETURN
377*
378* End of SGET52
379*
380 END
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
subroutine sget52(LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR, ALPHAI, BETA, WORK, RESULT)
SGET52
Definition: sget52.f:199