LAPACK 3.11.0
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

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

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
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 dlabad, 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 CALL dlabad( smlnum, bignum )
432 ks = 0
433 DO 20 k = 1, n
434*
435* Determine whether condition numbers are required for the k-th
436* eigenpair.
437*
438 IF( somcon ) THEN
439 IF( .NOT.SELECT( k ) )
440 $ GO TO 20
441 END IF
442*
443 ks = ks + 1
444*
445 IF( wants ) THEN
446*
447* Compute the reciprocal condition number of the k-th
448* eigenvalue.
449*
450 rnrm = dznrm2( n, vr( 1, ks ), 1 )
451 lnrm = dznrm2( n, vl( 1, ks ), 1 )
452 CALL zgemv( 'N', n, n, dcmplx( one, zero ), a, lda,
453 $ vr( 1, ks ), 1, dcmplx( zero, zero ), work, 1 )
454 yhax = zdotc( n, work, 1, vl( 1, ks ), 1 )
455 CALL zgemv( 'N', n, n, dcmplx( one, zero ), b, ldb,
456 $ vr( 1, ks ), 1, dcmplx( zero, zero ), work, 1 )
457 yhbx = zdotc( n, work, 1, vl( 1, ks ), 1 )
458 cond = dlapy2( abs( yhax ), abs( yhbx ) )
459 IF( cond.EQ.zero ) THEN
460 s( ks ) = -one
461 ELSE
462 s( ks ) = cond / ( rnrm*lnrm )
463 END IF
464 END IF
465*
466 IF( wantdf ) THEN
467 IF( n.EQ.1 ) THEN
468 dif( ks ) = dlapy2( abs( a( 1, 1 ) ), abs( b( 1, 1 ) ) )
469 ELSE
470*
471* Estimate the reciprocal condition number of the k-th
472* eigenvectors.
473*
474* Copy the matrix (A, B) to the array WORK and move the
475* (k,k)th pair to the (1,1) position.
476*
477 CALL zlacpy( 'Full', n, n, a, lda, work, n )
478 CALL zlacpy( 'Full', n, n, b, ldb, work( n*n+1 ), n )
479 ifst = k
480 ilst = 1
481*
482 CALL ztgexc( .false., .false., n, work, n, work( n*n+1 ),
483 $ n, dummy, 1, dummy1, 1, ifst, ilst, ierr )
484*
485 IF( ierr.GT.0 ) THEN
486*
487* Ill-conditioned problem - swap rejected.
488*
489 dif( ks ) = zero
490 ELSE
491*
492* Reordering successful, solve generalized Sylvester
493* equation for R and L,
494* A22 * R - L * A11 = A12
495* B22 * R - L * B11 = B12,
496* and compute estimate of Difl[(A11,B11), (A22, B22)].
497*
498 n1 = 1
499 n2 = n - n1
500 i = n*n + 1
501 CALL ztgsyl( 'N', idifjb, n2, n1, work( n*n1+n1+1 ),
502 $ n, work, n, work( n1+1 ), n,
503 $ work( n*n1+n1+i ), n, work( i ), n,
504 $ work( n1+i ), n, scale, dif( ks ), dummy,
505 $ 1, iwork, ierr )
506 END IF
507 END IF
508 END IF
509*
510 20 CONTINUE
511 work( 1 ) = lwmin
512 RETURN
513*
514* End of ZTGSNA
515*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
double precision function dlapy2(X, Y)
DLAPY2 returns sqrt(x2+y2).
Definition: dlapy2.f:63
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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:158
subroutine ztgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO)
ZTGEXC
Definition: ztgexc.f:200
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
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
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition: dznrm2.f90:90
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