dtgsna.f

来自「famous linear algebra library (LAPACK) p」· F 代码 · 共 581 行 · 第 1/2 页

      SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
     $                   LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
     $                   IWORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          HOWMNY, JOB
      INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
     $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
*     ..
*
*  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', Q*B*Z') with orthogonal matrices Q and Z, where
*  Z' 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.
*
*
*  Arguments
*  =========
*
*  JOB     (input) 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  (input) CHARACTER*1
*          = 'A': compute condition numbers for all eigenpairs;
*          = 'S': compute condition numbers for selected eigenpairs
*                 specified by the array SELECT.
*
*  SELECT  (input) 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       (input) INTEGER
*          The order of the square matrix pair (A, B). N >= 0.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
*          The upper quasi-triangular matrix A in the pair (A,B).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,N).
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
*          The upper triangular matrix B in the pair (A,B).
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B. LDB >= max(1,N).
*
*  VL      (input) 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    (input) INTEGER
*          The leading dimension of the array VL. LDVL >= 1.
*          If JOB = 'E' or 'B', LDVL >= N.
*
*  VR      (input) 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    (input) INTEGER
*          The leading dimension of the array VR. LDVR >= 1.
*          If JOB = 'E' or 'B', LDVR >= N.
*
*  S       (output) 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     (output) 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      (input) INTEGER
*          The number of elements in the arrays S and DIF. MM >= M.
*
*  M       (output) 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    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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   (workspace) INTEGER array, dimension (N + 6)
*          If JOB = 'E', IWORK is not referenced.
*
*  INFO    (output) INTEGER
*          =0: Successful exit
*          <0: If INFO = -i, the i-th argument had an illegal value
*
*
*  Further Details
*  ===============
*
*  The reciprocal of the condition number of a generalized eigenvalue
*  w = (a, b) is defined as
*
*       S(w) = (|u'Av|**2 + |u'Bv|**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'Av/u'Bv)
*  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'*(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'*(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'*S11*V1 = ( s11 s12 )   and U1'*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' * Z1 - lambda I),
*     where Z1' denotes the conjugate 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', In-2)  -kron(I2, S22) ]
*                   [ kron(T11', 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.
*
*  Based on contributions by
*     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.
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            DIFDRI
      PARAMETER          ( DIFDRI = 3 )
      DOUBLE PRECISION   ZERO, ONE, TWO, FOUR
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
     $                   FOUR = 4.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS
      INTEGER            I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2