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### 2.10 Function polyfitinf for polynomial fitting.

#### Helptext:

    function [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT,REF0)

      Best polynomial approximation in discrete uniform norm

      INPUT VARIABLES:

      M       : degree of the fitting polynomial
      N       : number of data points
      X(N)    : x-coordinates of data points
      Y(N)    : y-coordinates of data points
      K       : character of the polynomial:
                      K = 0 : mixed parity polynomial
                      K = 1 : odd polynomial  ( X(1) must be >  0 )
                      K = 2 : even polynomial ( X(1) must be >= 0 )
      EPSH    : tolerance for leveling. A useful value for 24-bit
                mantissa is EPSH = 2.0E-7
      MAXIT   : upper limit for number of exchange steps
      REF0(M2): initial alternating set ( N-vector ). This is an
                OPTIONAL argument. The length M2 is given by:
                      M2 = M + 2                      , if K = 0
                      M2 = integer part of (M+3)/2    , if K = 1
                      M2 = 2 + M/2 (M must be even)   , if K = 2

      OUTPUT VARIABLES:

      A       : polynomial coefficients of the best approximation
                in order of increasing powers:
                      p*(x) = A(1) + A(2)*x + A(3)*x^2 + ...
      REF     : selected alternating set of points
      HMAX    : maximum deviation ( uniform norm of p* - f )
      H       : pointwise approximation errors
    	R		: total number of iterations
      EQUAL   : success of failure of algorithm
                      EQUAL=1 :  succesful
                      EQUAL=0 :  convergence not acheived
                      EQUAL=-1:  input error
                      EQUAL=-2:  algorithm failure

      Relies on function EXCH, provided below.

      Example:
      M = 5; N = 10000; K = 0; EPSH = 10^-12; MAXIT = 10;
      X = linspace(-1,1,N);   % uniformly spaced nodes on [-1,1]
      k=1; Y = abs(X).^k;     % the function Y to approximate
      [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT);
      p = polyval(A,X); plot(X,Y,X,p) % p is the best approximation

      Note: using an even value of M, e.g., M=2, in the example above, makes
      the algorithm to fail with EQUAL=-2, because of collocation, which
      appears because both the appriximating function and the polynomial are
      even functions. The way aroung it is to approximate only the right half
      of the function, setting K = 2 : even polynomial. For example:

    N = 10000; K = 2; EPSH = 10^-12; MAXIT = 10;  X = linspace(0,1,N);
    for i = 1:2
        k = 2*i-1; Y = abs(X).^k;
        for j = 1:4
            M = 2^j;
            [~,~,HMAX] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT);
            approxerror(i,j) = HMAX;
        end
    end
    disp('Table 3.1 from Approximation theory and methods, M.J.D.POWELL, p. 27');
    disp(' ');
    disp('            n          K=1          K=3');
    disp(' '); format short g;
    disp([(2.^(1:4))' approxerror']);

      ALGORITHM:

      Computation of the polynomial that best approximates the data (X,Y)
      in the discrete uniform norm, i.e. the polynomial with the  minimum
      value of max{ | p(x_i) - y_i | , x_i in X } . That polynomial, also
      known as minimax polynomial, is obtained by the exchange algorithm,
      a finite iterative process requiring, at most,
         n
       (   ) iterations ( usually p = M + 2. See also function EXCH ).
         p
      since this number can be very large , the routine  may not converge
      within MAXIT iterations . The  other possibility of  failure occurs
      when there is insufficient floating point precision  for  the input
      data chosen.

      CREDITS: This routine was developed and modified as
      computer assignments in Approximation Theory courses by
      Prof. Andrew Knyazev, University of Colorado Denver, USA.

      Team Fall 98 (Revision 1.0):
              Chanchai Aniwathananon
              Crhistopher Mehl
              David A. Duran
              Saulo P. Oliveira

      Team Spring 11 (Revision 1.1): Manuchehr Aminian

      The algorithm and the comments are based on a FORTRAN code written
      by Joseph C. Simpson. The code is available on Netlib repository:
      http://www.netlib.org/toms/501
      See also: Communications of the ACM, V14, pp.355-356(1971)

      NOTES:

      1) A may contain the collocation polynomial
      2) If MAXIT is exceeded, REF contains a new reference set
      3) M, EPSH and REF can be altered during the execution
      4) To keep consistency to the original code , EPSH can be
      negative. However, the use of REF0 is *NOT* determined by
      EPSH< 0, but only by its inclusion as an input parameter.

      Some parts of the code can still take advantage of vectorization.

      Revision 1.0 from 1998 is a direct human translation of
      the FORTRAN code http://www.netlib.org/toms/501
      Revision 1.1 is a clean-up and technical update.
      Tested on MATLAB Version 7.11.0.584 (R2010b) and
      GNU Octave Version 3.2.4

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