approx.sq

Polynomial approximation

At least in a limited range, continuous functions can be approximated by polynomials. A higher degree provides more freedom to approximate the function and permits to reduce the error. SQ file approx.sq permits to compare functions with two kinds of approximations: Taylor and Chebyshev.

First contact

The function which is approximated is a sine. It is displayed in black in the upper left plot, and its Taylor approximation in red. Taylor approximations are based on the successive derivatives of the approximated function at a single point, represented by the blue vertical line which you can drag. In the upper right plot, a slider permits to change the degree of the polynomial. The lower left plot represents the error between the function and its approximation.

Figures

Approximation

The approximated function is drawn in black, the polynomial approximation in red, and the center of the approximation as a blue vertical line which can be manipulated. For the Chebyshev approximation, the range is represented by two green vertical lines which can also be manipulated.

Error

The error between the approximation and the approximated function is displayed.

Degree

A slider shows the degree of the polynomial approximation. It can be changed interactively.

Settings

Taylor/Chebyshev Approximation

The Taylor approximation is a polynomial whose first derivatives are the same as the approximated function's for a given value of x. The Chebyshev approximation is defined over a range; at some fixed points which correspond to the zeros of a Chebyshev polynomial, the function and its approximation are equal.

Exponential/Sine

Two functions can be approximated in approx.sq: an exponential and a sine. Both are continuous functions whose Taylor and Chebyshev approximations converge to their value when the degree and the approximated range (for the Chebyshev approximation) grow toward infinity.