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ctgsna.f(3)				      LAPACK				      ctgsna.f(3)

NAME
       ctgsna.f -

SYNOPSIS
   Functions/Subroutines
       subroutine ctgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM,
	   M, WORK, LWORK, IWORK, INFO)
	   CTGSNA

Function/Subroutine Documentation
   subroutine ctgsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN,
       complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB,
       complex, dimension( ldvl, * )VL, integerLDVL, complex, dimension( ldvr, * )VR,
       integerLDVR, real, dimension( * )S, real, dimension( * )DIF, integerMM, integerM, complex,
       dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)
       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:
	   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).

	   HOWMNY

		     HOWMNY is CHARACTER*1
		     = 'A': compute condition numbers for all eigenpairs;
		     = 'S': compute condition numbers for selected eigenpairs
			    specified by the array SELECT.

	   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.

	   N

		     N is INTEGER
		     The order of the square matrix pair (A, B). N >= 0.

	   A

		     A is COMPLEX array, dimension (LDA,N)
		     The upper triangular matrix A in the pair (A,B).

	   LDA

		     LDA is INTEGER
		     The leading dimension of the array A. LDA >= max(1,N).

	   B

		     B is COMPLEX array, dimension (LDB,N)
		     The upper triangular matrix B in the pair (A, B).

	   LDB

		     LDB is INTEGER
		     The leading dimension of the array B. LDB >= max(1,N).

	   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.

	   LDVL

		     LDVL is INTEGER
		     The leading dimension of the array VL. LDVL >= 1; and
		     If JOB = 'E' or 'B', LDVL >= N.

	   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.

	   LDVR

		     LDVR is INTEGER
		     The leading dimension of the array VR. LDVR >= 1;
		     If JOB = 'E' or 'B', LDVR >= N.

	   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.

	   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.

	   MM

		     MM is INTEGER
		     The number of elements in the arrays S and DIF. MM >= M.

	   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.

	   WORK

		     WORK is COMPLEX array, dimension (MAX(1,LWORK))
		     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

	   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).

	   IWORK

		     IWORK is INTEGER array, dimension (N+2)
		     If JOB = 'E', IWORK is not referenced.

	   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.

       Date:
	   November 2011

       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:

	     [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 310 of file ctgsna.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      ctgsna.f(3)
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