## AB13FD

### Computing the distance from a real matrix to the nearest complex matrix with an eigenvalue on the imaginary axis, using SVD

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To compute beta(A), the 2-norm distance from a real matrix A to
      the nearest complex matrix with an eigenvalue on the imaginary
      axis. If A is stable in the sense that all eigenvalues of A lie
      in the open left half complex plane, then beta(A) is the complex
      stability radius, i.e., the distance to the nearest unstable
      complex matrix. The value of beta(A) is the minimum of the
      smallest singular value of (A - jwI), taken over all real w.
      The value of w corresponding to the minimum is also computed.

Specification

          SUBROUTINE AB13FD( N, A, LDA, BETA, OMEGA, TOL, DWORK, LDWORK,
         $                   CWORK, LCWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, LCWORK, LDA, LDWORK, N
          DOUBLE PRECISION  BETA, OMEGA, TOL
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), DWORK(*)
          COMPLEX*16        CWORK(*)

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrix A.  N >= 0.

      A       (input) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N part of this array must contain the
              matrix A.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,N).

      BETA    (output) DOUBLE PRECISION
              The computed value of beta(A), which actually is an upper
              bound.

      OMEGA   (output) DOUBLE PRECISION
              The value of w such that the smallest singular value of
              (A - jwI) equals beta(A).

Tolerances

      TOL     DOUBLE PRECISION
              Specifies the accuracy with which beta(A) is to be
              calculated. (See the Numerical Aspects section below.)
              If the user sets TOL to be less than EPS, where EPS is the
              machine precision (see LAPACK Library Routine DLAMCH),
              then the tolerance is taken to be EPS.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.
              If DWORK(1) is not needed, the first 2*N*N entries of
              DWORK may overlay CWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX( 1, 3*N*(N+2) ).
              For optimum performance LDWORK should be larger.

      CWORK   COMPLEX*16 array, dimension (LCWORK)
              On exit, if INFO = 0, CWORK(1) returns the optimal value
              of LCWORK.
              If CWORK(1) is not needed, the first N*N entries of
              CWORK may overlay DWORK.

      LCWORK  INTEGER
              The length of the array CWORK.
              LCWORK >= MAX( 1, N*(N+3) ).
              For optimum performance LCWORK should be larger.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the routine fails to compute beta(A) within the
                    specified tolerance. Nevertheless, the returned
                    value is an upper bound on beta(A);
              = 2:  either the QR or SVD algorithm (LAPACK Library
                    routines DHSEQR, DGESVD or ZGESVD) fails to
                    converge; this error is very rare.

Method

      AB13FD combines the methods of [1] and [2] into a provably
      reliable, quadratically convergent algorithm. It uses the simple
      bisection strategy of [1] to find an interval which contains
      beta(A), and then switches to the modified bisection strategy of
      [2] which converges quadratically to a minimizer. Note that the
      efficiency of the strategy degrades if there are several local
      minima that are near or equal the global minimum.

References

      [1] Byers, R.
          A bisection method for measuring the distance of a stable
          matrix to the unstable matrices.
          SIAM J. Sci. Stat. Comput., Vol. 9, No. 5, pp. 875-880, 1988.

      [2] Boyd, S. and Balakrishnan, K.
          A regularity result for the singular values of a transfer
          matrix and a quadratically convergent algorithm for computing
          its L-infinity norm.
          Systems and Control Letters, Vol. 15, pp. 1-7, 1990.

Numerical Aspects

      In the presence of rounding errors, the computed function value
      BETA  satisfies

            beta(A) <= BETA + epsilon,

            BETA/(1+TOL) - delta <= MAX(beta(A), SQRT(2*N*EPS)*norm(A)),

      where norm(A) is the Frobenius norm of A,

            epsilon = p(N) * EPS * norm(A),
      and
            delta   = p(N) * SQRT(EPS) * norm(A),

      and p(N) is a low degree polynomial. It is recommended to choose
      TOL greater than SQRT(EPS). Although rounding errors can cause
      AB13FD to fail for smaller values of TOL, nevertheless, it usually
      succeeds. Regardless of success or failure, the first inequality
      holds.

Further Comments

      None

Example

Program Text

    *     AB13FD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 20 )
          INTEGER          LDA
          PARAMETER        ( LDA = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 3*NMAX*( NMAX + 2 ) )
          INTEGER          LCWORK
          PARAMETER        ( LCWORK = NMAX*( NMAX + 3 ) )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, N
          DOUBLE PRECISION BETA, OMEGA, TOL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK)
          COMPLEX*16       CWORK(LCWORK)
    *     .. External Subroutines ..
          EXTERNAL         AB13FD, UD01MD
    *     ..
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
    *     Read N, TOL and next A (row wise).
          READ ( NIN, FMT = * ) N, TOL
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) N
          ELSE
             DO 10 I = 1, N
                READ ( NIN, FMT = * ) ( A(I,J), J = 1, N )
       10    CONTINUE
    *
             WRITE ( NOUT, FMT = 99998 ) N, TOL
             CALL UD01MD( N, N, 5, NOUT, A, LDA, 'A', INFO )
    *
             CALL AB13FD( N, A, LDA, BETA, OMEGA, TOL, DWORK, LDWORK, CWORK,
         $                LCWORK, INFO )
    *
             IF ( INFO.NE.0 )
         $      WRITE ( NOUT, FMT = 99996 ) INFO
             WRITE ( NOUT, FMT = 99997 ) BETA, OMEGA
          END IF
    *
    99999 FORMAT (' AB13FD EXAMPLE PROGRAM RESULTS', /1X)
    99998 FORMAT (' N =', I2, 3X, 'TOL =', D10.3)
    99997 FORMAT (' Stability radius :', D18.11, /
         *        ' Minimizing omega :', D18.11)
    99996 FORMAT (' INFO on exit from AB13FD = ', I2)
    99995 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

    AB13FD EXAMPLE PROGRAM DATA
    4   0.0D-00   0.0D-00
         246.500        242.500        202.500       -197.500
        -252.500       -248.500       -207.500        202.500
        -302.500       -297.500       -248.500        242.500
        -307.500       -302.500       -252.500        246.500

Program Results

     AB13FD EXAMPLE PROGRAM RESULTS

     N = 4   TOL = 0.000D+00
     A ( 4X 4)

                1              2              3              4
      1    0.2465000D+03  0.2425000D+03  0.2025000D+03 -0.1975000D+03
      2   -0.2525000D+03 -0.2485000D+03 -0.2075000D+03  0.2025000D+03
      3   -0.3025000D+03 -0.2975000D+03 -0.2485000D+03  0.2425000D+03
      4   -0.3075000D+03 -0.3025000D+03 -0.2525000D+03  0.2465000D+03

     Stability radius : 0.39196472317D-02
     Minimizing omega : 0.98966520430D+00

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