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CTGSNA(l)			       )			     CTGSNA(l)

NAME
       CTGSNA  - estimate reciprocal condition numbers for specified eigenval‐
       ues and/or eigenvectors of a matrix pair (A, B)

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

	   CHARACTER	  HOWMNY, JOB

	   INTEGER	  INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N

	   LOGICAL	  SELECT( * )

	   INTEGER	  IWORK( * )

	   REAL		  DIF( * ), S( * )

	   COMPLEX	  A( LDA, * ), B( LDB, * ), VL( LDVL, * ), VR( LDVR, *
			  ), WORK( * )

PURPOSE
       CTGSNA estimates reciprocal condition numbers for specified eigenvalues
       and/or  eigenvectors of a matrix pair (A, B).  (A, B) must be in gener‐
       alized Schur canonical form, that is, A and B are both  upper  triangu‐
       lar.

ARGUMENTS
       JOB     (input) CHARACTER*1
	       Specifies  whether condition numbers are required for eigenval‐
	       ues (S) or eigenvectors (DIF):
	       = 'E': for eigenvalues only (S);
	       = 'V': for eigenvectors only (DIF);
	       = 'B': for both eigenvalues and eigenvectors (S and DIF).

       HOWMNY  (input) CHARACTER*1
	       = 'A': compute condition numbers for all eigenpairs;
	       = 'S': compute condition numbers for selected eigenpairs speci‐
	       fied by the array SELECT.

       SELECT  (input) LOGICAL array, dimension (N)
	       If HOWMNY = 'S', SELECT specifies the eigenpairs for which con‐
	       dition 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	refer‐
	       enced.

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

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

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

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

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

       VL      (input) 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	refer‐
	       enced.

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

       VR      (input) 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 refer‐
	       enced.

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

       S       (output) 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     (output) 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      (input) INTEGER
	       The number of elements in the arrays S and DIF. MM >= M.

       M       (output) 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    (workspace/output) COMPLEX array, dimension (LWORK)
	       If  JOB	= 'E', WORK is not referenced.	Otherwise, on exit, if
	       INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK  (input) INTEGER
	      The dimension of the array WORK. LWORK >= 1.  If JOB  =  'V'  or
	      'B', LWORK >= 2*N*N.

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

       INFO    (output) 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 eigen‐
       value w = (a, b) is defined as

	       S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v))

       where u and v are the right and left eigenvectors of (A, B) correspond‐
       ing  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'Au/v'Bu) 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 eigenval‐
       ue 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' is the conjugate
       transpose of X. kron(X, Y) is the Kronecker product between the	matri‐
       ces 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.

       Based on contributions by
	  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.

LAPACK version 3.0		 15 June 2000			     CTGSNA(l)
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