LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ ztrevc()

 subroutine ztrevc ( character side, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info )

ZTREVC

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

Purpose:
``` ZTREVC 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 ZHSEQR.

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*16 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*16 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 ZHSEQR). 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*16 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 ZHSEQR). 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*16 array, dimension (2*N)` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
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 ztrevc.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 DOUBLE PRECISION RWORK( * )
230 COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
231 \$ WORK( * )
232* ..
233*
234* =====================================================================
235*
236* .. Parameters ..
237 DOUBLE PRECISION ZERO, ONE
238 parameter( zero = 0.0d+0, one = 1.0d+0 )
239 COMPLEX*16 CMZERO, CMONE
240 parameter( cmzero = ( 0.0d+0, 0.0d+0 ),
241 \$ cmone = ( 1.0d+0, 0.0d+0 ) )
242* ..
243* .. Local Scalars ..
244 LOGICAL ALLV, BOTHV, LEFTV, OVER, RIGHTV, SOMEV
245 INTEGER I, II, IS, J, K, KI
246 DOUBLE PRECISION OVFL, REMAX, SCALE, SMIN, SMLNUM, ULP, UNFL
247 COMPLEX*16 CDUM
248* ..
249* .. External Functions ..
250 LOGICAL LSAME
251 INTEGER IZAMAX
252 DOUBLE PRECISION DLAMCH, DZASUM
253 EXTERNAL lsame, izamax, dlamch, dzasum
254* ..
255* .. External Subroutines ..
256 EXTERNAL xerbla, zcopy, zdscal, zgemv, zlatrs
257* ..
258* .. Intrinsic Functions ..
259 INTRINSIC abs, dble, dcmplx, dconjg, dimag, max
260* ..
261* .. Statement Functions ..
262 DOUBLE PRECISION CABS1
263* ..
264* .. Statement Function definitions ..
265 cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( 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( 'ZTREVC', -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 = dlamch( 'Safe minimum' )
321 ovfl = one / unfl
322 ulp = dlamch( '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 ) = dzasum( 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 zlatrs( '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 zcopy( ki, work( 1 ), 1, vr( 1, is ), 1 )
380*
381 ii = izamax( ki, vr( 1, is ), 1 )
382 remax = one / cabs1( vr( ii, is ) )
383 CALL zdscal( 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 zgemv( 'N', n, ki-1, cmone, vr, ldvr, work( 1 ),
391 \$ 1, dcmplx( scale ), vr( 1, ki ), 1 )
392*
393 ii = izamax( n, vr( 1, ki ), 1 )
394 remax = one / cabs1( vr( ii, ki ) )
395 CALL zdscal( 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 ) = -dconjg( 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 zlatrs( '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 zcopy( n-ki+1, work( ki ), 1, vl( ki, is ), 1 )
449*
450 ii = izamax( n-ki+1, vl( ki, is ), 1 ) + ki - 1
451 remax = one / cabs1( vl( ii, is ) )
452 CALL zdscal( 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 zgemv( 'N', n, n-ki, cmone, vl( 1, ki+1 ), ldvl,
460 \$ work( ki+1 ), 1, dcmplx( scale ),
461 \$ vl( 1, ki ), 1 )
462*
463 ii = izamax( n, vl( 1, ki ), 1 )
464 remax = one / cabs1( vl( ii, ki ) )
465 CALL zdscal( 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 ZTREVC
481*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
double precision function dzasum(n, zx, incx)
DZASUM
Definition dzasum.f:72
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
subroutine zgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
ZGEMV
Definition zgemv.f:160
integer function izamax(n, zx, incx)
IZAMAX
Definition izamax.f:71
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
subroutine zlatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
ZLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
Definition zlatrs.f:239
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine zdscal(n, da, zx, incx)
ZDSCAL
Definition zdscal.f:78
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