CTGSNA estimates reciprocal condition numbers for specified


 eigenvalues and/or eigenvectors of a matrix pair (A, B).


 (A, B) must be in generalized Schur canonical form, that is, A and


 B are both 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 corresponding j-th eigenvalue and/or eigenvector,


          SELECT(j) 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 COMPLEX array, dimension (LDA,N)


          The upper 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 COMPLEX 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 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 referenced.

LDVL

          LDVL is INTEGER


          The leading dimension of the array VL. LDVL >= 1; and


          If JOB = 'E' or 'B', LDVL >= N.

VR

          VR is 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 referenced.

LDVR

          LDVR is INTEGER


          The leading dimension of the array VR. LDVR >= 1;


          If JOB = 'E' or 'B', LDVR >= N.

S

          S is 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

          DIF is 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

          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 eigenvalue


          one element is used. If HOWMNY = 'A', M is set to N.

WORK

          WORK is COMPLEX 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 >= max(1,2*N*N).

IWORK

          IWORK is INTEGER array, dimension (N+2)


          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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  The reciprocal of the condition number of the i-th generalized


  eigenvalue w = (a, b) is defined as


          S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))


  where u and v are the right and left 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 (= v**HAu/v**HBu) 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 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**H is the conjugate


  transpose of X. kron(X, Y) is the Kronecker product between the


  matrices 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.

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.