Function Repository Resource:

MiniMaxApproximation

Find a mini-max approximation of a function

Contributed by: Wolfram Research

ResourceFunction["MiniMaxApproximation"][expr,{x,{x₀,x₁},m,n}]

finds the rational polynomial function of x, with numerator order m and denominator order n, that gives a mini-max approximation to expr on the interval x₀ to x₁.

ResourceFunction["MiniMaxApproximation"][expr,approx,{x,{x₀,x₁},m,n}]

finds the mini-max approximation to expr, starting the iterative algorithm with approx.

Details and Options

ResourceFunction["MiniMaxApproximation"] minimizes the maximum value of the relative error between the approximation and expr.

ResourceFunction["MiniMaxApproximation"] returns {abscissa,{approximation,maxerror}}, where abscissa is a list of the abscissas where the relative error is a local maximum, approximation is the rational approximant and maxerror is the global maximum of the relative error.

When an approximation is given as the second argument of ResourceFunction["MiniMaxApproximation"], it must have the same form as the result returned by ResourceFunction["MiniMaxApproximation"].

The following options can be given:

"Bias"

bias in the automatic choice of interpolation points

"Brake"

{5,5}

braking to apply on the iterative algorithm

"Derivatives"

Automatic

function to use for derivatives

MaxIterations

maximum number of iterations to use

"PlotFlag"

False

whether to plot relative error

"PrintFlag"

False

whether to print status information

WorkingPrecision

MachinePrecision

precision to use in internal computations

The option "Bias" is used to control the initial rational approximation in the same way it is used in RationalInterpolation.

Examples

Basic Examples (3)

A list containing the points where the maximum error occurs and the desired interpolation, along with the value of the error:

In[1]:=

$ResourceFunction["MiniMaxApproximation"][E^x, {x, {0, 2}, 2, 4}]$

Out[1]=

The rational approximation:

In[2]:=

Out[2]=

The relative error in the approximation over the interval:

In[3]:=

$Plot[1 - %/E^x, {x, 0, 2}]$

Out[3]=

Options (3)

Brake (2)

The convergence process does not finish within a small number of iterations without braking:

In[4]:=

$ResourceFunction["MiniMaxApproximation"][ Sin[x], {x, {.01 \[Pi], .99 \[Pi]}, 2, 3}, "Brake" -> {0, 0}, MaxIterations -> 2]$

Out[4]=

With the default brake, the conversion succeeds:

In[5]:=

$ResourceFunction["MiniMaxApproximation"][ Sin[x], {x, {.01 \[Pi], .99 \[Pi]}, 2, 3}, "Brake" -> {5, 5}, MaxIterations -> 2]$

Out[5]=

Derivatives (1)

Supply a function that gives a list of derivatives of the function to be approximated, evaluated at numeric points:

In[6]:=

$derivs[x_?NumberQ] := Block[{exp = E^x}, {exp, exp, exp}]$

In[7]:=

$ResourceFunction["MiniMaxApproximation"][E^x, {x, {-1, 1}, 2, 2}, "Derivatives" -> derivs[x]]$

Out[7]=

Possible Issues (3)

Because MiniMaxApproximation tries to minimize the maximum of the relative error, it is not possible to find a minimax approximation to a function that has a zero in the interval in question:

In[8]:=