 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

◆ ctgsna()

 subroutine ctgsna ( character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) S, real, dimension( * ) DIF, integer MM, integer M, complex, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO )

CTGSNA

Purpose:
CTGSNA 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 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 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 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 CTGEVC. 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 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 CTGEVC. 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 REAL 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 REAL 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 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 CLATDF.

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:
 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.

 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.

 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 ctgsna.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  REAL DIF( * ), S( * )
324  COMPLEX A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
325  \$ VR( LDVR, * ), WORK( * )
326 * ..
327 *
328 * =====================================================================
329 *
330 * .. Parameters ..
331  REAL ZERO, ONE
332  INTEGER IDIFJB
333  parameter( zero = 0.0e+0, one = 1.0e+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  REAL BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
339  COMPLEX YHAX, YHBX
340 * ..
341 * .. Local Arrays ..
342  COMPLEX DUMMY( 1 ), DUMMY1( 1 )
343 * ..
344 * .. External Functions ..
345  LOGICAL LSAME
346  REAL SCNRM2, SLAMCH, SLAPY2
347  COMPLEX CDOTC
348  EXTERNAL lsame, scnrm2, slamch, slapy2, cdotc
349 * ..
350 * .. External Subroutines ..
351  EXTERNAL cgemv, clacpy, ctgexc, ctgsyl, slabad, xerbla
352 * ..
353 * .. Intrinsic Functions ..
354  INTRINSIC abs, cmplx, 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( 'CTGSNA', -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 = slamch( 'P' )
429  smlnum = slamch( 'S' ) / eps
430  bignum = one / smlnum
431  CALL slabad( 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 = scnrm2( n, vr( 1, ks ), 1 )
451  lnrm = scnrm2( n, vl( 1, ks ), 1 )
452  CALL cgemv( 'N', n, n, cmplx( one, zero ), a, lda,
453  \$ vr( 1, ks ), 1, cmplx( zero, zero ), work, 1 )
454  yhax = cdotc( n, work, 1, vl( 1, ks ), 1 )
455  CALL cgemv( 'N', n, n, cmplx( one, zero ), b, ldb,
456  \$ vr( 1, ks ), 1, cmplx( zero, zero ), work, 1 )
457  yhbx = cdotc( n, work, 1, vl( 1, ks ), 1 )
458  cond = slapy2( 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 ) = slapy2( 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 clacpy( 'Full', n, n, a, lda, work, n )
478  CALL clacpy( 'Full', n, n, b, ldb, work( n*n+1 ), n )
479  ifst = k
480  ilst = 1
481 *
482  CALL ctgexc( .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 ctgsyl( '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 CTGSNA
515 *
real function slapy2(X, Y)
SLAPY2 returns sqrt(x2+y2).
Definition: slapy2.f:63
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
complex function cdotc(N, CX, INCX, CY, INCY)
CDOTC
Definition: cdotc.f:83
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:158
subroutine ctgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO)
CTGEXC
Definition: ctgexc.f:200
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine ctgsyl(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
CTGSYL
Definition: ctgsyl.f:295
real(wp) function scnrm2(n, x, incx)
SCNRM2
Definition: scnrm2.f90:90
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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