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DTGSNA(1)		 LAPACK routine (version 3.2)		     DTGSNA(1)

NAME
       DTGSNA - estimates reciprocal condition numbers for specified eigenval‐
       ues and/or eigenvectors 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  eigenvectors  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  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  eigenpair  corresponding  to  a  real   eigenvalue	 w(j),
	       SELECT(j)  must	be  set to .TRUE.. To select condition numbers
	       corresponding 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	refer‐
	       enced.

       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 eigenvalues two consecu‐
	       tive  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	refer‐
	       enced.

       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 eigenvector 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 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 real	eigen‐
	       value one element is used, and for each selected complex conju‐
	       gate pair of eigenvalues, two elements are used.	 If  HOWMNY  =
	       'A', M is set to N.

       WORK	  (workspace/output)   DOUBLE	PRECISION   array,   dimension
       (MAX(1,LWORK))
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The dimension of the array WORK. LWORK >= max(1,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  sin‐
       gular 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 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 routine (version 3.2)	 November 2008			     DTGSNA(1)
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