LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ 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```
Date
December 2016
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 313 of file ztgsna.f.

313 *
314 * -- LAPACK computational routine (version 3.7.0) --
315 * -- LAPACK is a software package provided by Univ. of Tennessee, --
316 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
317 * December 2016
318 *
319 * .. Scalar Arguments ..
320  CHARACTER howmny, job
321  INTEGER info, lda, ldb, ldvl, ldvr, lwork, m, mm, n
322 * ..
323 * .. Array Arguments ..
324  LOGICAL select( * )
325  INTEGER iwork( * )
326  DOUBLE PRECISION dif( * ), s( * )
327  COMPLEX*16 a( lda, * ), b( ldb, * ), vl( ldvl, * ),
328  \$ vr( ldvr, * ), work( * )
329 * ..
330 *
331 * =====================================================================
332 *
333 * .. Parameters ..
334  DOUBLE PRECISION zero, one
335  INTEGER idifjb
336  parameter( zero = 0.0d+0, one = 1.0d+0, idifjb = 3 )
337 * ..
338 * .. Local Scalars ..
339  LOGICAL lquery, somcon, wantbh, wantdf, wants
340  INTEGER i, ierr, ifst, ilst, k, ks, lwmin, n1, n2
341  DOUBLE PRECISION bignum, cond, eps, lnrm, rnrm, scale, smlnum
342  COMPLEX*16 yhax, yhbx
343 * ..
344 * .. Local Arrays ..
345  COMPLEX*16 dummy( 1 ), dummy1( 1 )
346 * ..
347 * .. External Functions ..
348  LOGICAL lsame
349  DOUBLE PRECISION dlamch, dlapy2, dznrm2
350  COMPLEX*16 zdotc
351  EXTERNAL lsame, dlamch, dlapy2, dznrm2, zdotc
352 * ..
353 * .. External Subroutines ..
354  EXTERNAL dlabad, xerbla, zgemv, zlacpy, ztgexc, ztgsyl
355 * ..
356 * .. Intrinsic Functions ..
357  INTRINSIC abs, dcmplx, max
358 * ..
359 * .. Executable Statements ..
360 *
361 * Decode and test the input parameters
362 *
363  wantbh = lsame( job, 'B' )
364  wants = lsame( job, 'E' ) .OR. wantbh
365  wantdf = lsame( job, 'V' ) .OR. wantbh
366 *
367  somcon = lsame( howmny, 'S' )
368 *
369  info = 0
370  lquery = ( lwork.EQ.-1 )
371 *
372  IF( .NOT.wants .AND. .NOT.wantdf ) THEN
373  info = -1
374  ELSE IF( .NOT.lsame( howmny, 'A' ) .AND. .NOT.somcon ) THEN
375  info = -2
376  ELSE IF( n.LT.0 ) THEN
377  info = -4
378  ELSE IF( lda.LT.max( 1, n ) ) THEN
379  info = -6
380  ELSE IF( ldb.LT.max( 1, n ) ) THEN
381  info = -8
382  ELSE IF( wants .AND. ldvl.LT.n ) THEN
383  info = -10
384  ELSE IF( wants .AND. ldvr.LT.n ) THEN
385  info = -12
386  ELSE
387 *
388 * Set M to the number of eigenpairs for which condition numbers
389 * are required, and test MM.
390 *
391  IF( somcon ) THEN
392  m = 0
393  DO 10 k = 1, n
394  IF( SELECT( k ) )
395  \$ m = m + 1
396  10 CONTINUE
397  ELSE
398  m = n
399  END IF
400 *
401  IF( n.EQ.0 ) THEN
402  lwmin = 1
403  ELSE IF( lsame( job, 'V' ) .OR. lsame( job, 'B' ) ) THEN
404  lwmin = 2*n*n
405  ELSE
406  lwmin = n
407  END IF
408  work( 1 ) = lwmin
409 *
410  IF( mm.LT.m ) THEN
411  info = -15
412  ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
413  info = -18
414  END IF
415  END IF
416 *
417  IF( info.NE.0 ) THEN
418  CALL xerbla( 'ZTGSNA', -info )
419  RETURN
420  ELSE IF( lquery ) THEN
421  RETURN
422  END IF
423 *
424 * Quick return if possible
425 *
426  IF( n.EQ.0 )
427  \$ RETURN
428 *
429 * Get machine constants
430 *
431  eps = dlamch( 'P' )
432  smlnum = dlamch( 'S' ) / eps
433  bignum = one / smlnum
434  CALL dlabad( smlnum, bignum )
435  ks = 0
436  DO 20 k = 1, n
437 *
438 * Determine whether condition numbers are required for the k-th
439 * eigenpair.
440 *
441  IF( somcon ) THEN
442  IF( .NOT.SELECT( k ) )
443  \$ GO TO 20
444  END IF
445 *
446  ks = ks + 1
447 *
448  IF( wants ) THEN
449 *
450 * Compute the reciprocal condition number of the k-th
451 * eigenvalue.
452 *
453  rnrm = dznrm2( n, vr( 1, ks ), 1 )
454  lnrm = dznrm2( n, vl( 1, ks ), 1 )
455  CALL zgemv( 'N', n, n, dcmplx( one, zero ), a, lda,
456  \$ vr( 1, ks ), 1, dcmplx( zero, zero ), work, 1 )
457  yhax = zdotc( n, work, 1, vl( 1, ks ), 1 )
458  CALL zgemv( 'N', n, n, dcmplx( one, zero ), b, ldb,
459  \$ vr( 1, ks ), 1, dcmplx( zero, zero ), work, 1 )
460  yhbx = zdotc( n, work, 1, vl( 1, ks ), 1 )
461  cond = dlapy2( abs( yhax ), abs( yhbx ) )
462  IF( cond.EQ.zero ) THEN
463  s( ks ) = -one
464  ELSE
465  s( ks ) = cond / ( rnrm*lnrm )
466  END IF
467  END IF
468 *
469  IF( wantdf ) THEN
470  IF( n.EQ.1 ) THEN
471  dif( ks ) = dlapy2( abs( a( 1, 1 ) ), abs( b( 1, 1 ) ) )
472  ELSE
473 *
474 * Estimate the reciprocal condition number of the k-th
475 * eigenvectors.
476 *
477 * Copy the matrix (A, B) to the array WORK and move the
478 * (k,k)th pair to the (1,1) position.
479 *
480  CALL zlacpy( 'Full', n, n, a, lda, work, n )
481  CALL zlacpy( 'Full', n, n, b, ldb, work( n*n+1 ), n )
482  ifst = k
483  ilst = 1
484 *
485  CALL ztgexc( .false., .false., n, work, n, work( n*n+1 ),
486  \$ n, dummy, 1, dummy1, 1, ifst, ilst, ierr )
487 *
488  IF( ierr.GT.0 ) THEN
489 *
490 * Ill-conditioned problem - swap rejected.
491 *
492  dif( ks ) = zero
493  ELSE
494 *
495 * Reordering successful, solve generalized Sylvester
496 * equation for R and L,
497 * A22 * R - L * A11 = A12
498 * B22 * R - L * B11 = B12,
499 * and compute estimate of Difl[(A11,B11), (A22, B22)].
500 *
501  n1 = 1
502  n2 = n - n1
503  i = n*n + 1
504  CALL ztgsyl( 'N', idifjb, n2, n1, work( n*n1+n1+1 ),
505  \$ n, work, n, work( n1+1 ), n,
506  \$ work( n*n1+n1+i ), n, work( i ), n,
507  \$ work( n1+i ), n, scale, dif( ks ), dummy,
508  \$ 1, iwork, ierr )
509  END IF
510  END IF
511  END IF
512 *
513  20 CONTINUE
514  work( 1 ) = lwmin
515  RETURN
516 *
517 * End of ZTGSNA
518 *
subroutine ztgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO)
ZTGEXC
Definition: ztgexc.f:202
double precision function dlapy2(X, Y)
DLAPY2 returns sqrt(x2+y2).
Definition: dlapy2.f:65
double precision function dznrm2(N, X, INCX)
DZNRM2
Definition: dznrm2.f:77
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
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:297
complex *16 function zdotc(N, ZX, INCX, ZY, INCY)
ZDOTC
Definition: zdotc.f:85
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
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