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

 subroutine ztrsna ( character job, 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, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, complex*16, dimension( ldwork, * ) work, integer ldwork, double precision, dimension( * ) rwork, integer info )

ZTRSNA

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Purpose:
``` ZTRSNA estimates reciprocal condition numbers for specified
eigenvalues and/or right eigenvectors of a complex upper triangular
matrix T (or of any matrix Q*T*Q**H with Q unitary).```
Parameters
 [in] JOB ``` JOB is CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (SEP); = 'B': for both eigenvalues and eigenvectors (S and SEP).``` [in] HOWMNY ``` HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT.``` [in] SELECT ``` SELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the j-th eigenpair, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced.``` [in] N ``` N is INTEGER The order of the matrix T. N >= 0.``` [in] T ``` T is COMPLEX*16 array, dimension (LDT,N) The upper triangular matrix T.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N).``` [in] VL ``` VL is COMPLEX*16 array, dimension (LDVL,M) If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by ZHSEIN or ZTREVC. If JOB = 'V', VL is not referenced.``` [in] LDVL ``` LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.``` [in] VR ``` VR is COMPLEX*16 array, dimension (LDVR,M) If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by ZHSEIN or ZTREVC. If JOB = 'V', VR is not referenced.``` [in] LDVR ``` LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.``` [out] S ``` S is DOUBLE PRECISION array, dimension (MM) If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. Thus S(j), SEP(j), and the j-th columns of VL and VR all correspond to the same eigenpair (but not in general the j-th eigenpair, unless all eigenpairs are selected). If JOB = 'V', S is not referenced.``` [out] SEP ``` SEP is DOUBLE PRECISION array, dimension (MM) If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If JOB = 'E', SEP is not referenced.``` [in] MM ``` MM is INTEGER The number of elements in the arrays S (if JOB = 'E' or 'B') and/or SEP (if JOB = 'V' or 'B'). MM >= M.``` [out] M ``` M is INTEGER The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = 'A', M is set to N.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension (LDWORK,N+6) If JOB = 'E', WORK is not referenced.``` [in] LDWORK ``` LDWORK is INTEGER The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.``` [out] RWORK ``` RWORK is DOUBLE PRECISION array, dimension (N) If JOB = 'E', RWORK is not referenced.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  The reciprocal of the condition number of an eigenvalue lambda is
defined as

S(lambda) = |v**H*u| / (norm(u)*norm(v))

where u and v are the right and left eigenvectors of T corresponding
to lambda; v**H denotes the conjugate transpose of v, and norm(u)
denotes the Euclidean norm. These reciprocal condition numbers always
lie between zero (very badly conditioned) and one (very well
conditioned). If n = 1, S(lambda) is defined to be 1.

An approximate error bound for a computed eigenvalue W(i) is given by

EPS * norm(T) / S(i)

where EPS is the machine precision.

The reciprocal of the condition number of the right eigenvector u
corresponding to lambda is defined as follows. Suppose

T = ( lambda  c  )
(   0    T22 )

Then the reciprocal condition number is

SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )

where sigma-min denotes the smallest singular value. We approximate
the smallest singular value by the reciprocal of an estimate of the
one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
defined to be abs(T(1,1)).

An approximate error bound for a computed right eigenvector VR(i)
is given by

EPS * norm(T) / SEP(i)```

Definition at line 246 of file ztrsna.f.

249*
250* -- LAPACK computational routine --
251* -- LAPACK is a software package provided by Univ. of Tennessee, --
252* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
253*
254* .. Scalar Arguments ..
255 CHARACTER HOWMNY, JOB
256 INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
257* ..
258* .. Array Arguments ..
259 LOGICAL SELECT( * )
260 DOUBLE PRECISION RWORK( * ), S( * ), SEP( * )
261 COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
262 \$ WORK( LDWORK, * )
263* ..
264*
265* =====================================================================
266*
267* .. Parameters ..
268 DOUBLE PRECISION ZERO, ONE
269 parameter( zero = 0.0d+0, one = 1.0d0+0 )
270* ..
271* .. Local Scalars ..
272 LOGICAL SOMCON, WANTBH, WANTS, WANTSP
273 CHARACTER NORMIN
274 INTEGER I, IERR, IX, J, K, KASE, KS
275 DOUBLE PRECISION BIGNUM, EPS, EST, LNRM, RNRM, SCALE, SMLNUM,
276 \$ XNORM
277 COMPLEX*16 CDUM, PROD
278* ..
279* .. Local Arrays ..
280 INTEGER ISAVE( 3 )
281 COMPLEX*16 DUMMY( 1 )
282* ..
283* .. External Functions ..
284 LOGICAL LSAME
285 INTEGER IZAMAX
286 DOUBLE PRECISION DLAMCH, DZNRM2
287 COMPLEX*16 ZDOTC
288 EXTERNAL lsame, izamax, dlamch, dznrm2, zdotc
289* ..
290* .. External Subroutines ..
291 EXTERNAL xerbla, zdrscl, zlacn2, zlacpy, zlatrs, ztrexc
292* ..
293* .. Intrinsic Functions ..
294 INTRINSIC abs, dble, dimag, max
295* ..
296* .. Statement Functions ..
297 DOUBLE PRECISION CABS1
298* ..
299* .. Statement Function definitions ..
300 cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
301* ..
302* .. Executable Statements ..
303*
304* Decode and test the input parameters
305*
306 wantbh = lsame( job, 'B' )
307 wants = lsame( job, 'E' ) .OR. wantbh
308 wantsp = lsame( job, 'V' ) .OR. wantbh
309*
310 somcon = lsame( howmny, 'S' )
311*
312* Set M to the number of eigenpairs for which condition numbers are
313* to be computed.
314*
315 IF( somcon ) THEN
316 m = 0
317 DO 10 j = 1, n
318 IF( SELECT( j ) )
319 \$ m = m + 1
320 10 CONTINUE
321 ELSE
322 m = n
323 END IF
324*
325 info = 0
326 IF( .NOT.wants .AND. .NOT.wantsp ) THEN
327 info = -1
328 ELSE IF( .NOT.lsame( howmny, 'A' ) .AND. .NOT.somcon ) THEN
329 info = -2
330 ELSE IF( n.LT.0 ) THEN
331 info = -4
332 ELSE IF( ldt.LT.max( 1, n ) ) THEN
333 info = -6
334 ELSE IF( ldvl.LT.1 .OR. ( wants .AND. ldvl.LT.n ) ) THEN
335 info = -8
336 ELSE IF( ldvr.LT.1 .OR. ( wants .AND. ldvr.LT.n ) ) THEN
337 info = -10
338 ELSE IF( mm.LT.m ) THEN
339 info = -13
340 ELSE IF( ldwork.LT.1 .OR. ( wantsp .AND. ldwork.LT.n ) ) THEN
341 info = -16
342 END IF
343 IF( info.NE.0 ) THEN
344 CALL xerbla( 'ZTRSNA', -info )
345 RETURN
346 END IF
347*
348* Quick return if possible
349*
350 IF( n.EQ.0 )
351 \$ RETURN
352*
353 IF( n.EQ.1 ) THEN
354 IF( somcon ) THEN
355 IF( .NOT.SELECT( 1 ) )
356 \$ RETURN
357 END IF
358 IF( wants )
359 \$ s( 1 ) = one
360 IF( wantsp )
361 \$ sep( 1 ) = abs( t( 1, 1 ) )
362 RETURN
363 END IF
364*
365* Get machine constants
366*
367 eps = dlamch( 'P' )
368 smlnum = dlamch( 'S' ) / eps
369 bignum = one / smlnum
370*
371 ks = 1
372 DO 50 k = 1, n
373*
374 IF( somcon ) THEN
375 IF( .NOT.SELECT( k ) )
376 \$ GO TO 50
377 END IF
378*
379 IF( wants ) THEN
380*
381* Compute the reciprocal condition number of the k-th
382* eigenvalue.
383*
384 prod = zdotc( n, vr( 1, ks ), 1, vl( 1, ks ), 1 )
385 rnrm = dznrm2( n, vr( 1, ks ), 1 )
386 lnrm = dznrm2( n, vl( 1, ks ), 1 )
387 s( ks ) = abs( prod ) / ( rnrm*lnrm )
388*
389 END IF
390*
391 IF( wantsp ) THEN
392*
393* Estimate the reciprocal condition number of the k-th
394* eigenvector.
395*
396* Copy the matrix T to the array WORK and swap the k-th
397* diagonal element to the (1,1) position.
398*
399 CALL zlacpy( 'Full', n, n, t, ldt, work, ldwork )
400 CALL ztrexc( 'No Q', n, work, ldwork, dummy, 1, k, 1, ierr )
401*
402* Form C = T22 - lambda*I in WORK(2:N,2:N).
403*
404 DO 20 i = 2, n
405 work( i, i ) = work( i, i ) - work( 1, 1 )
406 20 CONTINUE
407*
408* Estimate a lower bound for the 1-norm of inv(C**H). The 1st
409* and (N+1)th columns of WORK are used to store work vectors.
410*
411 sep( ks ) = zero
412 est = zero
413 kase = 0
414 normin = 'N'
415 30 CONTINUE
416 CALL zlacn2( n-1, work( 1, n+1 ), work, est, kase, isave )
417*
418 IF( kase.NE.0 ) THEN
419 IF( kase.EQ.1 ) THEN
420*
421* Solve C**H*x = scale*b
422*
423 CALL zlatrs( 'Upper', 'Conjugate transpose',
424 \$ 'Nonunit', normin, n-1, work( 2, 2 ),
425 \$ ldwork, work, scale, rwork, ierr )
426 ELSE
427*
428* Solve C*x = scale*b
429*
430 CALL zlatrs( 'Upper', 'No transpose', 'Nonunit',
431 \$ normin, n-1, work( 2, 2 ), ldwork, work,
432 \$ scale, rwork, ierr )
433 END IF
434 normin = 'Y'
435 IF( scale.NE.one ) THEN
436*
437* Multiply by 1/SCALE if doing so will not cause
438* overflow.
439*
440 ix = izamax( n-1, work, 1 )
441 xnorm = cabs1( work( ix, 1 ) )
442 IF( scale.LT.xnorm*smlnum .OR. scale.EQ.zero )
443 \$ GO TO 40
444 CALL zdrscl( n, scale, work, 1 )
445 END IF
446 GO TO 30
447 END IF
448*
449 sep( ks ) = one / max( est, smlnum )
450 END IF
451*
452 40 CONTINUE
453 ks = ks + 1
454 50 CONTINUE
455 RETURN
456*
457* End of ZTRSNA
458*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
complex *16 function zdotc(n, zx, incx, zy, incy)
ZDOTC
Definition zdotc.f:83
integer function izamax(n, zx, incx)
IZAMAX
Definition izamax.f:71
subroutine zlacn2(n, v, x, est, kase, isave)
ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition zlacn2.f:133
subroutine zlacpy(uplo, m, n, a, lda, b, ldb)
ZLACPY copies all or part of one two-dimensional array to another.
Definition zlacpy.f:103
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
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition dznrm2.f90:90
subroutine zdrscl(n, sa, sx, incx)
ZDRSCL multiplies a vector by the reciprocal of a real scalar.
Definition zdrscl.f:84
subroutine ztrexc(compq, n, t, ldt, q, ldq, ifst, ilst, info)
ZTREXC
Definition ztrexc.f:126
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