LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ ctrevc()

subroutine ctrevc ( character  side,
character  howmny,
logical, dimension( * )  select,
integer  n,
complex, dimension( ldt, * )  t,
integer  ldt,
complex, dimension( ldvl, * )  vl,
integer  ldvl,
complex, dimension( ldvr, * )  vr,
integer  ldvr,
integer  mm,
integer  m,
complex, dimension( * )  work,
real, dimension( * )  rwork,
integer  info 
)

CTREVC

Download CTREVC + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CTREVC computes some or all of the right and/or left eigenvectors of
 a complex upper triangular matrix T.
 Matrices of this type are produced by the Schur factorization of
 a complex general matrix:  A = Q*T*Q**H, as computed by CHSEQR.

 The right eigenvector x and the left eigenvector y of T corresponding
 to an eigenvalue w are defined by:

              T*x = w*x,     (y**H)*T = w*(y**H)

 where y**H denotes the conjugate transpose of the vector y.
 The eigenvalues are not input to this routine, but are read directly
 from the diagonal of T.

 This routine returns the matrices X and/or Y of right and left
 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
 input matrix.  If Q is the unitary factor that reduces a matrix A to
 Schur form T, then Q*X and Q*Y are the matrices of right and left
 eigenvectors of A.
Parameters
[in]SIDE
          SIDE is CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors.
[in]HOWMNY
          HOWMNY is CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed using the matrices supplied in
                  VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT.
[in]SELECT
          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed.
          The eigenvector corresponding to the j-th eigenvalue is
          computed if SELECT(j) = .TRUE..
          Not referenced if HOWMNY = 'A' or 'B'.
[in]N
          N is INTEGER
          The order of the matrix T. N >= 0.
[in,out]T
          T is COMPLEX array, dimension (LDT,N)
          The upper triangular matrix T.  T is modified, but restored
          on exit.
[in]LDT
          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).
[in,out]VL
          VL is COMPLEX array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by CHSEQR).
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VL, in the same order as their
                           eigenvalues.
          Not referenced if SIDE = 'R'.
[in]LDVL
          LDVL is INTEGER
          The leading dimension of the array VL.  LDVL >= 1, and if
          SIDE = 'L' or 'B', LDVL >= N.
[in,out]VR
          VR is COMPLEX array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by CHSEQR).
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VR, in the same order as their
                           eigenvalues.
          Not referenced if SIDE = 'L'.
[in]LDVR
          LDVR is INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          SIDE = 'R' or 'B'; LDVR >= N.
[in]MM
          MM is INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.
[out]M
          M is INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
          is set to N.  Each selected eigenvector occupies one
          column.
[out]WORK
          WORK is COMPLEX array, dimension (2*N)
[out]RWORK
          RWORK is REAL array, dimension (N)
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow.

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|.

Definition at line 216 of file ctrevc.f.

218*
219* -- LAPACK computational routine --
220* -- LAPACK is a software package provided by Univ. of Tennessee, --
221* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
222*
223* .. Scalar Arguments ..
224 CHARACTER HOWMNY, SIDE
225 INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
226* ..
227* .. Array Arguments ..
228 LOGICAL SELECT( * )
229 REAL RWORK( * )
230 COMPLEX T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
231 $ WORK( * )
232* ..
233*
234* =====================================================================
235*
236* .. Parameters ..
237 REAL ZERO, ONE
238 parameter( zero = 0.0e+0, one = 1.0e+0 )
239 COMPLEX CMZERO, CMONE
240 parameter( cmzero = ( 0.0e+0, 0.0e+0 ),
241 $ cmone = ( 1.0e+0, 0.0e+0 ) )
242* ..
243* .. Local Scalars ..
244 LOGICAL ALLV, BOTHV, LEFTV, OVER, RIGHTV, SOMEV
245 INTEGER I, II, IS, J, K, KI
246 REAL OVFL, REMAX, SCALE, SMIN, SMLNUM, ULP, UNFL
247 COMPLEX CDUM
248* ..
249* .. External Functions ..
250 LOGICAL LSAME
251 INTEGER ICAMAX
252 REAL SCASUM, SLAMCH
253 EXTERNAL lsame, icamax, scasum, slamch
254* ..
255* .. External Subroutines ..
256 EXTERNAL ccopy, cgemv, clatrs, csscal, xerbla
257* ..
258* .. Intrinsic Functions ..
259 INTRINSIC abs, aimag, cmplx, conjg, max, real
260* ..
261* .. Statement Functions ..
262 REAL CABS1
263* ..
264* .. Statement Function definitions ..
265 cabs1( cdum ) = abs( real( cdum ) ) + abs( aimag( cdum ) )
266* ..
267* .. Executable Statements ..
268*
269* Decode and test the input parameters
270*
271 bothv = lsame( side, 'B' )
272 rightv = lsame( side, 'R' ) .OR. bothv
273 leftv = lsame( side, 'L' ) .OR. bothv
274*
275 allv = lsame( howmny, 'A' )
276 over = lsame( howmny, 'B' )
277 somev = lsame( howmny, 'S' )
278*
279* Set M to the number of columns required to store the selected
280* eigenvectors.
281*
282 IF( somev ) THEN
283 m = 0
284 DO 10 j = 1, n
285 IF( SELECT( j ) )
286 $ m = m + 1
287 10 CONTINUE
288 ELSE
289 m = n
290 END IF
291*
292 info = 0
293 IF( .NOT.rightv .AND. .NOT.leftv ) THEN
294 info = -1
295 ELSE IF( .NOT.allv .AND. .NOT.over .AND. .NOT.somev ) THEN
296 info = -2
297 ELSE IF( n.LT.0 ) THEN
298 info = -4
299 ELSE IF( ldt.LT.max( 1, n ) ) THEN
300 info = -6
301 ELSE IF( ldvl.LT.1 .OR. ( leftv .AND. ldvl.LT.n ) ) THEN
302 info = -8
303 ELSE IF( ldvr.LT.1 .OR. ( rightv .AND. ldvr.LT.n ) ) THEN
304 info = -10
305 ELSE IF( mm.LT.m ) THEN
306 info = -11
307 END IF
308 IF( info.NE.0 ) THEN
309 CALL xerbla( 'CTREVC', -info )
310 RETURN
311 END IF
312*
313* Quick return if possible.
314*
315 IF( n.EQ.0 )
316 $ RETURN
317*
318* Set the constants to control overflow.
319*
320 unfl = slamch( 'Safe minimum' )
321 ovfl = one / unfl
322 ulp = slamch( 'Precision' )
323 smlnum = unfl*( n / ulp )
324*
325* Store the diagonal elements of T in working array WORK.
326*
327 DO 20 i = 1, n
328 work( i+n ) = t( i, i )
329 20 CONTINUE
330*
331* Compute 1-norm of each column of strictly upper triangular
332* part of T to control overflow in triangular solver.
333*
334 rwork( 1 ) = zero
335 DO 30 j = 2, n
336 rwork( j ) = scasum( j-1, t( 1, j ), 1 )
337 30 CONTINUE
338*
339 IF( rightv ) THEN
340*
341* Compute right eigenvectors.
342*
343 is = m
344 DO 80 ki = n, 1, -1
345*
346 IF( somev ) THEN
347 IF( .NOT.SELECT( ki ) )
348 $ GO TO 80
349 END IF
350 smin = max( ulp*( cabs1( t( ki, ki ) ) ), smlnum )
351*
352 work( 1 ) = cmone
353*
354* Form right-hand side.
355*
356 DO 40 k = 1, ki - 1
357 work( k ) = -t( k, ki )
358 40 CONTINUE
359*
360* Solve the triangular system:
361* (T(1:KI-1,1:KI-1) - T(KI,KI))*X = SCALE*WORK.
362*
363 DO 50 k = 1, ki - 1
364 t( k, k ) = t( k, k ) - t( ki, ki )
365 IF( cabs1( t( k, k ) ).LT.smin )
366 $ t( k, k ) = smin
367 50 CONTINUE
368*
369 IF( ki.GT.1 ) THEN
370 CALL clatrs( 'Upper', 'No transpose', 'Non-unit', 'Y',
371 $ ki-1, t, ldt, work( 1 ), scale, rwork,
372 $ info )
373 work( ki ) = scale
374 END IF
375*
376* Copy the vector x or Q*x to VR and normalize.
377*
378 IF( .NOT.over ) THEN
379 CALL ccopy( ki, work( 1 ), 1, vr( 1, is ), 1 )
380*
381 ii = icamax( ki, vr( 1, is ), 1 )
382 remax = one / cabs1( vr( ii, is ) )
383 CALL csscal( ki, remax, vr( 1, is ), 1 )
384*
385 DO 60 k = ki + 1, n
386 vr( k, is ) = cmzero
387 60 CONTINUE
388 ELSE
389 IF( ki.GT.1 )
390 $ CALL cgemv( 'N', n, ki-1, cmone, vr, ldvr, work( 1 ),
391 $ 1, cmplx( scale ), vr( 1, ki ), 1 )
392*
393 ii = icamax( n, vr( 1, ki ), 1 )
394 remax = one / cabs1( vr( ii, ki ) )
395 CALL csscal( n, remax, vr( 1, ki ), 1 )
396 END IF
397*
398* Set back the original diagonal elements of T.
399*
400 DO 70 k = 1, ki - 1
401 t( k, k ) = work( k+n )
402 70 CONTINUE
403*
404 is = is - 1
405 80 CONTINUE
406 END IF
407*
408 IF( leftv ) THEN
409*
410* Compute left eigenvectors.
411*
412 is = 1
413 DO 130 ki = 1, n
414*
415 IF( somev ) THEN
416 IF( .NOT.SELECT( ki ) )
417 $ GO TO 130
418 END IF
419 smin = max( ulp*( cabs1( t( ki, ki ) ) ), smlnum )
420*
421 work( n ) = cmone
422*
423* Form right-hand side.
424*
425 DO 90 k = ki + 1, n
426 work( k ) = -conjg( t( ki, k ) )
427 90 CONTINUE
428*
429* Solve the triangular system:
430* (T(KI+1:N,KI+1:N) - T(KI,KI))**H*X = SCALE*WORK.
431*
432 DO 100 k = ki + 1, n
433 t( k, k ) = t( k, k ) - t( ki, ki )
434 IF( cabs1( t( k, k ) ).LT.smin )
435 $ t( k, k ) = smin
436 100 CONTINUE
437*
438 IF( ki.LT.n ) THEN
439 CALL clatrs( 'Upper', 'Conjugate transpose', 'Non-unit',
440 $ 'Y', n-ki, t( ki+1, ki+1 ), ldt,
441 $ work( ki+1 ), scale, rwork, info )
442 work( ki ) = scale
443 END IF
444*
445* Copy the vector x or Q*x to VL and normalize.
446*
447 IF( .NOT.over ) THEN
448 CALL ccopy( n-ki+1, work( ki ), 1, vl( ki, is ), 1 )
449*
450 ii = icamax( n-ki+1, vl( ki, is ), 1 ) + ki - 1
451 remax = one / cabs1( vl( ii, is ) )
452 CALL csscal( n-ki+1, remax, vl( ki, is ), 1 )
453*
454 DO 110 k = 1, ki - 1
455 vl( k, is ) = cmzero
456 110 CONTINUE
457 ELSE
458 IF( ki.LT.n )
459 $ CALL cgemv( 'N', n, n-ki, cmone, vl( 1, ki+1 ), ldvl,
460 $ work( ki+1 ), 1, cmplx( scale ),
461 $ vl( 1, ki ), 1 )
462*
463 ii = icamax( n, vl( 1, ki ), 1 )
464 remax = one / cabs1( vl( ii, ki ) )
465 CALL csscal( n, remax, vl( 1, ki ), 1 )
466 END IF
467*
468* Set back the original diagonal elements of T.
469*
470 DO 120 k = ki + 1, n
471 t( k, k ) = work( k+n )
472 120 CONTINUE
473*
474 is = is + 1
475 130 CONTINUE
476 END IF
477*
478 RETURN
479*
480* End of CTREVC
481*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
real function scasum(n, cx, incx)
SCASUM
Definition scasum.f:72
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
subroutine cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
CGEMV
Definition cgemv.f:160
integer function icamax(n, cx, incx)
ICAMAX
Definition icamax.f:71
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
subroutine clatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
Definition clatrs.f:239
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine csscal(n, sa, cx, incx)
CSSCAL
Definition csscal.f:78
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