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.

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

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  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.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

  [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.