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

 subroutine ztgsna ( character job, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) dif, integer mm, integer m, complex*16, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info )

ZTGSNA

Purpose:
``` ZTGSNA estimates reciprocal condition numbers for specified
eigenvalues and/or eigenvectors of a matrix pair (A, B).

(A, B) must be in generalized Schur canonical form, that is, A and
B are both upper triangular.```
Parameters
 [in] JOB ``` JOB is CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (DIF): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (DIF); = 'B': for both eigenvalues and eigenvectors (S and DIF).``` [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 corresponding j-th eigenvalue and/or eigenvector, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced.``` [in] N ``` N is INTEGER The order of the square matrix pair (A, B). N >= 0.``` [in] A ``` A is COMPLEX*16 array, dimension (LDA,N) The upper triangular matrix A in the pair (A,B).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] B ``` B is COMPLEX*16 array, dimension (LDB,N) The upper triangular matrix B in the pair (A, B).``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [in] VL ``` VL is COMPLEX*16 array, dimension (LDVL,M) IF JOB = 'E' or 'B', VL must contain left eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by ZTGEVC. 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 (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by ZTGEVC. If JOB = 'V', VR is not referenced.``` [in] LDVR ``` LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1; 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. If JOB = 'V', S is not referenced.``` [out] DIF ``` DIF 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 the eigenvalues cannot be reordered to compute DIF(j), DIF(j) is set to 0; this can only occur when the true value would be very small anyway. For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate of Difl. If JOB = 'E', DIF is not referenced.``` [in] MM ``` MM is INTEGER The number of elements in the arrays S and DIF. MM >= M.``` [out] M ``` M is INTEGER The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected eigenvalue one element is used. If HOWMNY = 'A', M is set to N.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N). If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).``` [out] IWORK ``` IWORK is INTEGER array, dimension (N+2) If JOB = 'E', IWORK 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 the i-th generalized
eigenvalue w = (a, b) is defined as

S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))

where u and v are the right and left eigenvectors of (A, B)
corresponding to w; |z| denotes the absolute value of the complex
number, and norm(u) denotes the 2-norm of the vector u. The pair
(a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the
matrix pair (A, B). If both a and b equal zero, then (A,B) is
singular and S(I) = -1 is returned.

An approximate error bound on the chordal distance between the i-th
computed generalized eigenvalue w and the corresponding exact
eigenvalue lambda is

chord(w, lambda) <=   EPS * norm(A, B) / S(I),

where EPS is the machine precision.

The reciprocal of the condition number of the right eigenvector u
and left eigenvector v corresponding to the generalized eigenvalue w
is defined as follows. Suppose

(A, B) = ( a   *  ) ( b  *  )  1
( 0  A22 ),( 0 B22 )  n-1
1  n-1     1 n-1

Then the reciprocal condition number DIF(I) is

Difl[(a, b), (A22, B22)]  = sigma-min( Zl )

where sigma-min(Zl) denotes the smallest singular value of

Zl = [ kron(a, In-1) -kron(1, A22) ]
[ kron(b, In-1) -kron(1, B22) ].

Here In-1 is the identity matrix of size n-1 and X**H is the conjugate
transpose of X. kron(X, Y) is the Kronecker product between the
matrices X and Y.

We approximate the smallest singular value of Zl with an upper
bound. This is done by ZLATDF.

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

EPS * norm(A, B) / DIF(i).

See ref. [2-3] for more details and further references.```
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
References:
```  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software, Report
UMINF - 94.04, Department of Computing Science, Umea University,
S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
To appear in Numerical Algorithms, 1996.

[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75.
To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.```

Definition at line 308 of file ztgsna.f.

311*
312* -- LAPACK computational routine --
313* -- LAPACK is a software package provided by Univ. of Tennessee, --
314* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
315*
316* .. Scalar Arguments ..
317 CHARACTER HOWMNY, JOB
318 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
319* ..
320* .. Array Arguments ..
321 LOGICAL SELECT( * )
322 INTEGER IWORK( * )
323 DOUBLE PRECISION DIF( * ), S( * )
324 COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
325 \$ VR( LDVR, * ), WORK( * )
326* ..
327*
328* =====================================================================
329*
330* .. Parameters ..
331 DOUBLE PRECISION ZERO, ONE
332 INTEGER IDIFJB
333 parameter( zero = 0.0d+0, one = 1.0d+0, idifjb = 3 )
334* ..
335* .. Local Scalars ..
336 LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS
337 INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2
338 DOUBLE PRECISION BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
339 COMPLEX*16 YHAX, YHBX
340* ..
341* .. Local Arrays ..
342 COMPLEX*16 DUMMY( 1 ), DUMMY1( 1 )
343* ..
344* .. External Functions ..
345 LOGICAL LSAME
346 DOUBLE PRECISION DLAMCH, DLAPY2, DZNRM2
347 COMPLEX*16 ZDOTC
348 EXTERNAL lsame, dlamch, dlapy2, dznrm2, zdotc
349* ..
350* .. External Subroutines ..
351 EXTERNAL xerbla, zgemv, zlacpy, ztgexc, ztgsyl
352* ..
353* .. Intrinsic Functions ..
354 INTRINSIC abs, dcmplx, max
355* ..
356* .. Executable Statements ..
357*
358* Decode and test the input parameters
359*
360 wantbh = lsame( job, 'B' )
361 wants = lsame( job, 'E' ) .OR. wantbh
362 wantdf = lsame( job, 'V' ) .OR. wantbh
363*
364 somcon = lsame( howmny, 'S' )
365*
366 info = 0
367 lquery = ( lwork.EQ.-1 )
368*
369 IF( .NOT.wants .AND. .NOT.wantdf ) THEN
370 info = -1
371 ELSE IF( .NOT.lsame( howmny, 'A' ) .AND. .NOT.somcon ) THEN
372 info = -2
373 ELSE IF( n.LT.0 ) THEN
374 info = -4
375 ELSE IF( lda.LT.max( 1, n ) ) THEN
376 info = -6
377 ELSE IF( ldb.LT.max( 1, n ) ) THEN
378 info = -8
379 ELSE IF( wants .AND. ldvl.LT.n ) THEN
380 info = -10
381 ELSE IF( wants .AND. ldvr.LT.n ) THEN
382 info = -12
383 ELSE
384*
385* Set M to the number of eigenpairs for which condition numbers
386* are required, and test MM.
387*
388 IF( somcon ) THEN
389 m = 0
390 DO 10 k = 1, n
391 IF( SELECT( k ) )
392 \$ m = m + 1
393 10 CONTINUE
394 ELSE
395 m = n
396 END IF
397*
398 IF( n.EQ.0 ) THEN
399 lwmin = 1
400 ELSE IF( lsame( job, 'V' ) .OR. lsame( job, 'B' ) ) THEN
401 lwmin = 2*n*n
402 ELSE
403 lwmin = n
404 END IF
405 work( 1 ) = lwmin
406*
407 IF( mm.LT.m ) THEN
408 info = -15
409 ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
410 info = -18
411 END IF
412 END IF
413*
414 IF( info.NE.0 ) THEN
415 CALL xerbla( 'ZTGSNA', -info )
416 RETURN
417 ELSE IF( lquery ) THEN
418 RETURN
419 END IF
420*
421* Quick return if possible
422*
423 IF( n.EQ.0 )
424 \$ RETURN
425*
426* Get machine constants
427*
428 eps = dlamch( 'P' )
429 smlnum = dlamch( 'S' ) / eps
430 bignum = one / smlnum
431 ks = 0
432 DO 20 k = 1, n
433*
434* Determine whether condition numbers are required for the k-th
435* eigenpair.
436*
437 IF( somcon ) THEN
438 IF( .NOT.SELECT( k ) )
439 \$ GO TO 20
440 END IF
441*
442 ks = ks + 1
443*
444 IF( wants ) THEN
445*
446* Compute the reciprocal condition number of the k-th
447* eigenvalue.
448*
449 rnrm = dznrm2( n, vr( 1, ks ), 1 )
450 lnrm = dznrm2( n, vl( 1, ks ), 1 )
451 CALL zgemv( 'N', n, n, dcmplx( one, zero ), a, lda,
452 \$ vr( 1, ks ), 1, dcmplx( zero, zero ), work, 1 )
453 yhax = zdotc( n, work, 1, vl( 1, ks ), 1 )
454 CALL zgemv( 'N', n, n, dcmplx( one, zero ), b, ldb,
455 \$ vr( 1, ks ), 1, dcmplx( zero, zero ), work, 1 )
456 yhbx = zdotc( n, work, 1, vl( 1, ks ), 1 )
457 cond = dlapy2( abs( yhax ), abs( yhbx ) )
458 IF( cond.EQ.zero ) THEN
459 s( ks ) = -one
460 ELSE
461 s( ks ) = cond / ( rnrm*lnrm )
462 END IF
463 END IF
464*
465 IF( wantdf ) THEN
466 IF( n.EQ.1 ) THEN
467 dif( ks ) = dlapy2( abs( a( 1, 1 ) ), abs( b( 1, 1 ) ) )
468 ELSE
469*
470* Estimate the reciprocal condition number of the k-th
471* eigenvectors.
472*
473* Copy the matrix (A, B) to the array WORK and move the
474* (k,k)th pair to the (1,1) position.
475*
476 CALL zlacpy( 'Full', n, n, a, lda, work, n )
477 CALL zlacpy( 'Full', n, n, b, ldb, work( n*n+1 ), n )
478 ifst = k
479 ilst = 1
480*
481 CALL ztgexc( .false., .false., n, work, n, work( n*n+1 ),
482 \$ n, dummy, 1, dummy1, 1, ifst, ilst, ierr )
483*
484 IF( ierr.GT.0 ) THEN
485*
486* Ill-conditioned problem - swap rejected.
487*
488 dif( ks ) = zero
489 ELSE
490*
491* Reordering successful, solve generalized Sylvester
492* equation for R and L,
493* A22 * R - L * A11 = A12
494* B22 * R - L * B11 = B12,
495* and compute estimate of Difl[(A11,B11), (A22, B22)].
496*
497 n1 = 1
498 n2 = n - n1
499 i = n*n + 1
500 CALL ztgsyl( 'N', idifjb, n2, n1, work( n*n1+n1+1 ),
501 \$ n, work, n, work( n1+1 ), n,
502 \$ work( n*n1+n1+i ), n, work( i ), n,
503 \$ work( n1+i ), n, scale, dif( ks ), dummy,
504 \$ 1, iwork, ierr )
505 END IF
506 END IF
507 END IF
508*
509 20 CONTINUE
510 work( 1 ) = lwmin
511 RETURN
512*
513* End of ZTGSNA
514*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
complex *16 function zdotc(n, zx, incx, zy, incy)
ZDOTC
Definition zdotc.f:83
subroutine zgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
ZGEMV
Definition zgemv.f:160
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
double precision function dlapy2(x, y)
DLAPY2 returns sqrt(x2+y2).
Definition dlapy2.f:63
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition dznrm2.f90:90
subroutine ztgexc(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, info)
ZTGEXC
Definition ztgexc.f:200
subroutine ztgsyl(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
ZTGSYL
Definition ztgsyl.f:295
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