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RedHat 9 (Linux i386) - man page for dtgsna (redhat section l)

DTGSNA(l)					)					DTGSNA(l)

NAME
       DTGSNA  - estimate reciprocal condition numbers for specified eigenvalues and/or eigenvec-
       tors of a matrix pair (A, B) in generalized real Schur canonical form (or  of  any  matrix
       pair  (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where Z' denotes the transpose of
       Z

SYNOPSIS
       SUBROUTINE DTGSNA( 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( * )

	   DOUBLE	  PRECISION  A(  LDA,  * ), B( LDB, * ), DIF( * ), S( * ), VL( LDVL, * ),
			  VR( LDVR, * ), WORK( * )

PURPOSE
       DTGSNA estimates reciprocal condition numbers for specified eigenvalues	and/or	eigenvec-
       tors  of  a  matrix pair (A, B) in generalized real Schur canonical form (or of any matrix
       pair (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where Z' denotes the transpose  of
       Z.   (A, B) must be in generalized real Schur form (as returned by DGGES), i.e. A is block
       upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.

ARGUMENTS
       JOB     (input) CHARACTER*1
	       Specifies whether condition numbers are required for eigenvalues (S) or	eigenvec-
	       tors (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 specified by the array
	       SELECT.

       SELECT  (input) LOGICAL array, dimension (N)
	       If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition  numbers  are
	       required.  To  select  condition numbers for the eigenpair corresponding to a real
	       eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select condition numbers cor-
	       responding  to  a  complex  conjugate  pair of eigenvalues w(j) and w(j+1), either
	       SELECT(j) or SELECT(j+1) or both, must be set to .TRUE..  If HOWMNY = 'A',  SELECT
	       is not referenced.

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

       A       (input) DOUBLE PRECISION array, dimension (LDA,N)
	       The upper quasi-triangular matrix A in the pair (A,B).

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

       B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DTGEVC.  If JOB = 'V', VL is not refer-
	       enced.

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

       VR      (input) DOUBLE PRECISION 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 ov VR, as returned by DTGEVC.  If JOB = 'V', VR is not ref-
	       erenced.

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

       S       (output) 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. For a complex conjugate pair  of  ei-
	       genvalues  two  consecutive  elements  of  S are set to the same value. Thus S(j),
	       DIF(j), and the j-th columns of VL and VR all correspond  to  the  same	eigenpair
	       (but  not  in general the j-th eigenpair, unless all eigenpairs are selected).  If
	       JOB = 'V', S is not referenced.

       DIF     (output) 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. For a complex eigenvec-
	       tor two consecutive elements of DIF are set to the same value. 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.	If JOB = 'E', DIF is  not  refer-
	       enced.

       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 condi-
	       tion numbers; for each selected real eigenvalue one element is used, and for  each
	       selected  complex conjugate pair of eigenvalues, two elements are used.	If HOWMNY
	       = 'A', M is set to N.

       WORK    (workspace/output) DOUBLE PRECISION 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  >=  N.  If JOB = 'V' or 'B' LWORK >=
	       2*N*(N+2)+16.

	       If LWORK = -1, then a workspace query is assumed; the routine only calculates  the
	       optimal	size of the WORK array, returns this value as the first entry of the WORK
	       array, and no error message related to LWORK is issued by XERBLA.

       IWORK   (workspace) INTEGER array, dimension (N + 6)
	       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 a generalized eigenvalue w = (a, b)  is  defined
       as

	    S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v))

       where  u  and  v  are  the  left  and right 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 (= u'Av/u'Bv) 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 DIF(i) of right eigenvector u and left eigenvector
       v corresponding to the generalized eigenvalue w is defined as follows:

       a) If the i-th eigenvalue w = (a,b) is real

	  Suppose U and V are orthogonal transformations such that

		     U'*(A, B)*V  = (S, T) = ( a   *  ) ( b  *	)  1
					     ( 0  S22 ),( 0 T22 )  n-1
					       1  n-1	  1 n-1

	  Then the reciprocal condition number DIF(i) is

		     Difl((a, b), (S22, T22)) = sigma-min( Zl ),

	  where sigma-min(Zl) denotes the smallest singular value of the
	  2(n-1)-by-2(n-1) matrix

	      Zl = [ kron(a, In-1)  -kron(1, S22) ]
		   [ kron(b, In-1)  -kron(1, T22) ] .

	  Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
	  Kronecker product between the matrices X and Y.

	  Note that if the default method for computing DIF(i) is wanted
	  (see DLATDF), then the parameter DIFDRI (see below) should be
	  changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
	  See DTGSYL for more details.

       b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,

	  Suppose U and V are orthogonal transformations such that

		     U'*(A, B)*V = (S, T) = ( S11  *   ) ( T11	*  )  2
					    ( 0    S22 ),( 0	T22) n-2
					      2    n-2	   2	n-2

	  and (S11, T11) corresponds to the complex conjugate eigenvalue
	  pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
	  that

	      U1'*S11*V1 = ( s11 s12 )	 and U1'*T11*V1 = ( t11 t12 )
			   (  0  s22 )			  (  0	t22 )

	  where the generalized eigenvalues w = s11/t11 and
	  conjg(w) = s22/t22.

	  Then the reciprocal condition number DIF(i) is bounded by

	      min( d1, max( 1, |real(s11)/real(s22)| )*d2 )

	  where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
	  Z1 is the complex 2-by-2 matrix

		   Z1 =  [ s11	-s22 ]
			 [ t11	-t22 ],

	  This is done by computing (using real arithmetic) the
	  roots of the characteristical polynomial det(Z1' * Z1 - lambda I),
	  where Z1' denotes the conjugate transpose of Z1 and det(X) denotes
	  the determinant of X.

	  and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
	  upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)

		   Z2 = [ kron(S11', In-2)  -kron(I2, S22) ]
			[ kron(T11', In-2)  -kron(I2, T22) ]

	  Note that if the default method for computing DIF is wanted (see
	  DLATDF), then the parameter DIFDRI (see below) should be changed
	  from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
	  for more details.

       For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate
       of Difl.

       An approximate error bound for the i-th 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 				DTGSNA(l)


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