LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ dget52()

subroutine dget52 ( logical left,
integer n,
double precision, dimension( lda, * ) a,
integer lda,
double precision, dimension( ldb, * ) b,
integer ldb,
double precision, dimension( lde, * ) e,
integer lde,
double precision, dimension( * ) alphar,
double precision, dimension( * ) alphai,
double precision, dimension( * ) beta,
double precision, dimension( * ) work,
double precision, dimension( 2 ) result )

DGET52

Purpose:
!>
!> DGET52  does an eigenvector check for the generalized eigenvalue
!> problem.
!>
!> The basic test for right eigenvectors is:
!>
!>                           | b(j) A E(j) -  a(j) B E(j) |
!>         RESULT(1) = max   -------------------------------
!>                      j    n ulp max( |b(j) A|, |a(j) B| )
!>
!> using the 1-norm.  Here, a(j)/b(j) = w is the j-th generalized
!> eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th
!> generalized eigenvalue of m A - B.
!>
!> For real eigenvalues, the test is straightforward.  For complex
!> eigenvalues, E(j) and a(j) are complex, represented by
!> Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that
!> eigenvector becomes
!>
!>                 max( |Wr|, |Wi| )
!>     --------------------------------------------
!>     n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )
!>
!> where
!>
!>     Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)
!>
!>     Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)
!>
!>                         T   T  _
!> For left eigenvectors, A , B , a, and b  are used.
!>
!> DGET52 also tests the normalization of E.  Each eigenvector is
!> supposed to be normalized so that the maximum 
!> of its elements is 1, where in this case, 
!> of a complex value x is  |Re(x)| + |Im(x)| ; let us call this
!> maximum  norm of a vector v  M(v).
!> if a(j)=b(j)=0, then the eigenvector is set to be the jth coordinate
!> vector.  The normalization test is:
!>
!>         RESULT(2) =      max       | M(v(j)) - 1 | / ( n ulp )
!>                    eigenvectors v(j)
!>
Parameters
[in]LEFT
!>          LEFT is LOGICAL
!>          =.TRUE.:  The eigenvectors in the columns of E are assumed
!>                    to be *left* eigenvectors.
!>          =.FALSE.: The eigenvectors in the columns of E are assumed
!>                    to be *right* eigenvectors.
!>
[in]N
!>          N is INTEGER
!>          The size of the matrices.  If it is zero, DGET52 does
!>          nothing.  It must be at least zero.
!>
[in]A
!>          A is DOUBLE PRECISION array, dimension (LDA, N)
!>          The matrix A.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of A.  It must be at least 1
!>          and at least N.
!>
[in]B
!>          B is DOUBLE PRECISION array, dimension (LDB, N)
!>          The matrix B.
!>
[in]LDB
!>          LDB is INTEGER
!>          The leading dimension of B.  It must be at least 1
!>          and at least N.
!>
[in]E
!>          E is DOUBLE PRECISION array, dimension (LDE, N)
!>          The matrix of eigenvectors.  It must be O( 1 ).  Complex
!>          eigenvalues and eigenvectors always come in pairs, the
!>          eigenvalue and its conjugate being stored in adjacent
!>          elements of ALPHAR, ALPHAI, and BETA.  Thus, if a(j)/b(j)
!>          and a(j+1)/b(j+1) are a complex conjugate pair of
!>          generalized eigenvalues, then E(,j) contains the real part
!>          of the eigenvector and E(,j+1) contains the imaginary part.
!>          Note that whether E(,j) is a real eigenvector or part of a
!>          complex one is specified by whether ALPHAI(j) is zero or not.
!>
[in]LDE
!>          LDE is INTEGER
!>          The leading dimension of E.  It must be at least 1 and at
!>          least N.
!>
[in]ALPHAR
!>          ALPHAR is DOUBLE PRECISION array, dimension (N)
!>          The real parts of the values a(j) as described above, which,
!>          along with b(j), define the generalized eigenvalues.
!>          Complex eigenvalues always come in complex conjugate pairs
!>          a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent
!>          elements in ALPHAR, ALPHAI, and BETA.  Thus, if the j-th
!>          and (j+1)-st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1)
!>          is assumed to be equal to ALPHAR(j)/BETA(j).
!>
[in]ALPHAI
!>          ALPHAI is DOUBLE PRECISION array, dimension (N)
!>          The imaginary parts of the values a(j) as described above,
!>          which, along with b(j), define the generalized eigenvalues.
!>          If ALPHAI(j)=0, then the eigenvalue is real, otherwise it
!>          is part of a complex conjugate pair.  Complex eigenvalues
!>          always come in complex conjugate pairs a(j)/b(j) and
!>          a(j+1)/b(j+1), which are stored in adjacent elements in
!>          ALPHAR, ALPHAI, and BETA.  Thus, if the j-th and (j+1)-st
!>          eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to
!>          be equal to  -ALPHAI(j)/BETA(j).  Also, nonzero values in
!>          ALPHAI are assumed to always come in adjacent pairs.
!>
[in]BETA
!>          BETA is DOUBLE PRECISION array, dimension (N)
!>          The values b(j) as described above, which, along with a(j),
!>          define the generalized eigenvalues.
!>
[out]WORK
!>          WORK is DOUBLE PRECISION array, dimension (N**2+N)
!>
[out]RESULT
!>          RESULT is DOUBLE PRECISION array, dimension (2)
!>          The values computed by the test described above.  If A E or
!>          B E is likely to overflow, then RESULT(1:2) is set to
!>          10 / ulp.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 197 of file dget52.f.

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 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
210 $ B( LDB, * ), BETA( * ), E( LDE, * ),
211 $ RESULT( 2 ), WORK( * )
212* ..
213*
214* =====================================================================
215*
216* .. Parameters ..
217 DOUBLE PRECISION ZERO, ONE, TEN
218 parameter( zero = 0.0d0, one = 1.0d0, ten = 10.0d0 )
219* ..
220* .. Local Scalars ..
221 LOGICAL ILCPLX
222 CHARACTER NORMAB, TRANS
223 INTEGER J, JVEC
224 DOUBLE PRECISION 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 DOUBLE PRECISION DLAMCH, DLANGE
230 EXTERNAL dlamch, dlange
231* ..
232* .. External Subroutines ..
233 EXTERNAL dgemv
234* ..
235* .. Intrinsic Functions ..
236 INTRINSIC abs, dble, max
237* ..
238* .. Executable Statements ..
239*
240 result( 1 ) = zero
241 result( 2 ) = zero
242 IF( n.LE.0 )
243 $ RETURN
244*
245 safmin = dlamch( 'Safe minimum' )
246 safmax = one / safmin
247 ulp = dlamch( 'Epsilon' )*dlamch( '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( dlange( normab, n, n, a, lda, work ), safmin )
260 bnorm = max( dlange( normab, n, n, b, ldb, work ), safmin )
261 enorm = max( dlange( '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 dgemv( trans, n, n, acoef, a, lda, e( 1, jvec ), 1,
295 $ zero, work( n*( jvec-1 )+1 ), 1 )
296 CALL dgemv( trans, n, n, -bcoefr, b, ldb, 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 dgemv( trans, n, n, acoef, a, lda, e( 1, jvec ), 1,
325 $ zero, work( n*( jvec-1 )+1 ), 1 )
326 CALL dgemv( trans, n, n, -bcoefr, b, ldb, e( 1, jvec ),
327 $ 1, one, work( n*( jvec-1 )+1 ), 1 )
328 CALL dgemv( trans, n, n, bcoefi, b, ldb, e( 1, jvec+1 ),
329 $ 1, one, work( n*( jvec-1 )+1 ), 1 )
330*
331 CALL dgemv( trans, n, n, acoef, a, lda, e( 1, jvec+1 ),
332 $ 1, zero, work( n*jvec+1 ), 1 )
333 CALL dgemv( trans, n, n, -bcoefi, b, ldb, e( 1, jvec ),
334 $ 1, one, work( n*jvec+1 ), 1 )
335 CALL dgemv( trans, n, n, -bcoefr, b, ldb, e( 1, jvec+1 ),
336 $ 1, one, work( n*jvec+1 ), 1 )
337 END IF
338 END IF
339 10 CONTINUE
340*
341 errnrm = dlange( '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 / ( dble( n )*ulp )
375*
376 RETURN
377*
378* End of DGET52
379*
lde
logical function lde(ri, rj, lr)
Definition dblat2.f:2970
dgemv
subroutine dgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
DGEMV
Definition dgemv.f:158
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
dlange
double precision function dlange(norm, m, n, a, lda, work)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition dlange.f:112
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