tgsna - Man Page

subroutine ctgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)
CTGSNA
subroutine dtgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)
DTGSNA
subroutine stgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)
STGSNA
subroutine ztgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)
ZTGSNA

CTGSNA  

Purpose:

 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.

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 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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  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.

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 308 of file ctgsna.f.

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.

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 378 of file dtgsna.f.

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.

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 378 of file stgsna.f.

ZTGSNA  

Purpose:

 ZTGSNA 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.

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 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*16 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*16 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*16 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 ZTGEVC.
          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*16 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 ZTGEVC.
          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.
          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.
          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*16 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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  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 ZLATDF.

  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.

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 308 of file ztgsna.f.