Loading...
Searching...
No Matches

## stgsna()

subroutine stgsna | ( | character | job,
|  | character | howmny,
|  | logical, dimension( * ) | select,
|  | integer | n,
|  | real, dimension( lda, * ) | a,
|  | integer | lda,
|  | real, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | real, dimension( ldvl, * ) | vl,
|  | integer | ldvl,
|  | real, dimension( ldvr, * ) | vr,
|  | integer | ldvr,
|  | real, dimension( * ) | s,
|  | real, dimension( * ) | dif,
|  | integer | mm,
|  | integer | m,
|  | real, dimension( * ) | work,
|  | integer | lwork,
|  | integer, dimension( * ) | iwork,
|  | integer | info )

STGSNA

Download STGSNA + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  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).
      !>

  [in]

  HOWMNY

      !>          HOWMNY is CHARACTER*1
      !>          = 'A': compute condition numbers for all eigenpairs;
      !>          = 'S': compute condition numbers for selected eigenpairs
      !>                 specified by the array SELECT.
      !>

  [in]

  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.
      !>

  [in]

  N

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

  [in]

  A

      !>          A is REAL array, dimension (LDA,N)
      !>          The upper quasi-triangular matrix A in the pair (A,B).
      !>

  [in]

  LDA

      !>          LDA is INTEGER
      !>          The leading dimension of the array A. LDA >= max(1,N).
      !>

  [in]

  B

      !>          B is REAL array, dimension (LDB,N)
      !>          The upper triangular matrix B in the pair (A,B).
      !>

  [in]

  LDB

      !>          LDB is INTEGER
      !>          The leading dimension of the array B. LDB >= max(1,N).
      !>

  [in]

  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.
      !>

  [in]

  LDVL

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

  [in]

  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.
      !>

  [in]

  LDVR

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

  [out]

  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.
      !>

  [out]

  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.
      !>

  [in]

  MM

      !>          MM is INTEGER
      !>          The number of elements in the arrays S and DIF. MM >= M.
      !>

  [out]

  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.
      !>

  [out]

  WORK

      !>          WORK is REAL array, dimension (MAX(1,LWORK))
      !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
      !>

  [in]

  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.
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (N + 6)
      !>          If JOB = 'E', IWORK is not referenced.
      !>

  [out]

  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.

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.
      !>