stgsna.f(3)

NAME
stgsna.f -

SYNOPSIS

  Functions/Subroutines

subroutine stgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
STGSNA

Function/Subroutine Documentation

  subroutine stgsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real, dimension( ldvl, * )VL, integerLDVL, real, dimension( ldvr, * )VR, integerLDVR, real, dimension( * )S, real, dimension( * )DIF, integerMM, integerM, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)

STGSNA

Purpose:

     STGSNA 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 SGGES),
     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 REAL 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 REAL 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 REAL 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 STGEVC.
              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 REAL 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 STGEVC.
              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. 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 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. 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 REAL 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 SLATDF), then the parameter DIFDRI (see below) should be
         changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
         See STGSYL 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
         SLATDF), then the parameter DIFDRI (see below) should be changed
         from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
         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 stgsna.f.

Author

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