17.2.2  Rational minimax approximation

The minimax command finds the rational function which approximates a continuous function f:[a,b]→ℝ most closely in the sense of ℓ^∞ norm. It operates exclusively in floating-point arithmetic.

• minimax takes three mandatory argument and a sequence of optional arguments:
  • expr, a univariate expression representing the function f.
  • x=a..b, a variable with bounds.
  • deg, which is either a positive integer n or a list [n,m] where n is a negative and m a positive integer, specifying the degrees of the numerator (and denominator) of the approximant (by default, m=0).
  • Optionally, opts, a sequence of options each of which is one of:
    • limit=maxiter, where maxiter is a positive integer specifying the number of iterations of the Remez algorithm (by default, maxiter=100).
    • rand⟨=N⟩, where N is a positive integer specifying the number of tries with initial node skewing (by default, N=1 and no skewing is performed).
    • threshold=tol, where tol is a real number greater than 1 specifying the terminating criterion max_x∈[a,b] e(x)<tol·min_x∈[a,b] e(x) for the Remez algorithm, where e(x)=|f(x)−r(x)| is the error function in the current iteration (by default, tol=1.02).
• minimax(expr,x=a..b,deg ⟨,limit=maxiter ⟩) uses the Remez method (see e.g. here) to find the rational function r(x)=p(x)/q(x), where p is a polynomial of degree n and q a polynomial of degree m normalized such that its trailing coefficient is equal to 1, which minimizes the ℓ^∞ error when approximating expr on [a,b]. If r has no singularities in [a,b], then a list containing ||f−r||_∞ and r(x) is returned. Otherwise, an approximation of 1/b−a∫_a^b|f(x)−r(x)|dx, computed by using Simpson’s rule with 20(n+m+2) steps, is returned as the first element (this may serve as a rough estimation of fitness outside poles) and a warning is printed in the message area.
• By specifying only the parameter n, you get an approximation of f by a polynomial. This is faster and the result is always well-defined, but rational approximants provide more flexibility and can be significantly more accurate even for smaller values of n and m (these should be chosen so that r has no singularities in [a,b], however).
• With the rand⟨=N⟩ option, minimax skews the initial (Chebyshev) nodes randomly before proceeding to the first Remez step. This may help the algorithm to converge to a well-defined result when m>0. If N is specified, then the entire computation is performed N times and the best result is selected. This will be a pole-free function when possible. The rand option is usually not needed when m=0, i.e. when the minimax polynomial is sought.

Example

Find a function of form r(x)=a₂x²+a₁x+a₀/b₂x²+b₁x+b₀ which is an optimal ℓ^∞-approximation of f(x)=ln(x+1)/2+sinx on the segment [0,7]. Define f by entering:

Now compute the approximation by entering:

To display the approximation, enter:

(Here, r(x) is drawn in blue.) You can conclude that max_{0≤ x≤ 7}|f(x)−r(x)|=tol. Indeed:

The maximal absolute error is attained at x=3.45532968966.

As another example, consider the function f(x)=xe^sinx/10+7cosx for x∈[0,10]. To show the graph, enter:

In the examples that follow, approximation functions are not output to save space.

To obtain the ℓ^∞ error of the 15th order polynomial approximation, enter:

By requesting a rational approximant, you can achieve significantly better accuracy with polynomials of lower degrees:

Sometimes you get poles in the approximant, for example:

Warning: the result is undefined at point(s) 3.03555290496

By using the rand option, you obtain a pole-free result:

Evaluation time: 6.11