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dtgsna.f(3)				      LAPACK				      dtgsna.f(3)

NAME
       dtgsna.f -

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

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

       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**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
	    Z**T 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.

       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 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

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

	   A

		     A is DOUBLE PRECISION array, dimension (LDA,N)
		     The upper quasi-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 DOUBLE PRECISION 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 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 referenced.

	   LDVL

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

	   VR

		     VR is 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 referenced.

	   LDVR

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

	   S

		     S is 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
		     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

		     DIF is 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

		     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 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

		     WORK is DOUBLE PRECISION 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 >= 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

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

		  S(w) = (|u**TAv|**2 + |u**TBv|**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**TAv/u**TBv)
	     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**T*(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**T*(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**T*S11*V1 = ( s11 s12 ) and U1**T*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**T * Z1 - lambda I),
		where Z1**T denotes the 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**T, In-2)  -kron(I2, S22) ]
			      [ kron(T11**T, 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.

       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 380 of file dtgsna.f.

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

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