Maple worksheets on approximation of functions |

Numerical methods topics:

- Introduction - errors
- Root-finding
- Interpolation
- Numerical integration
- 1st order differential equations
- 2nd order differential equations
- Linear systems
- Finite difference methods
- The Duffing equation
- Approximation of functions
- The numerical evaluation of mathematical functions
- Special inverse functions
- The derivation of Runge-Kutta schemes
- Interpolation for Runge-Kutta schemes

The following Maple worksheets can be downloaded.

They are all compatible with

Classic Worksheet Maple 10.

The method of moments- moment.mws

- An introduction to the moment scheme for constructing approximating polynomials
- The general moment scheme
- A procedure for constructing moment polynomials -
momentpoly

Local Taylor series approximation of functions- loctaylor.mws

- Defining procedures and viewing Maple library code
- Local Taylor series approximation for functions
- A procedure for constructing local Taylor series approximations -
loctaylor

Chebyshev polynomials and Chebyshev series- chebfit.mws

- Definition of Chebyshev polynomials
- Properties of Chebyshev polynomials - orthogonality relations
- Expressing a polynomial as a Chebyshev sum
- An alternative method for calculating Chebyshev coefficients
- Chebyshev series
- Example: the Chebyshev series for exp(x)
- A procedure for computing Chebyshev polynomial:
chebseries

Using interpolating polynomials to approximate functions- interpoly.mws

- A procedure constructing an interpolating polynomial approximation -
interpolyinterpoly: examples with evenly spaced nodes- Using an interpolating polynomial to emulate a finite Chebyshev series
interpoly: general examples

Jacobi polynomials and interpolating polynomials- jacobi.mws

- Jacobi polynomials
- Zeros of the Jacobi polynomials
interpoly: examples with nodes spaced in the pattern of the zeros of Jacobi polynomials

The Remez algorithm for constructing minimax polynomial approximations- minimax.mws

- The minimax polynomial approximation for a continuous function on a closed interval
- The calculation of a minimax polynomial - introduction to the Remez algorithm
- An error estimate for the minimax polynomial
- The calculation of a minimax polynomial for exp(x) on [-1,1]

The Remez algorithm for constructing minimax rational approximations: version I -- ratminmax.mws

using an iterative method for obtaining the minimax error

- The calculation of a minimax rational approximation for ln(1+x) on [0,1]
- A utility routine for calculating the critical points of a function -
critpts- The calculation of a minimax rational approximation for exp(x) on [-1,1]
- Comparison of polynomial and rational minimax approximations

The Remez algorithm for constructing minimax rational approximations: version II -- ratminmax2.mws

solving a rational equation to obtain the minimax error

- The calculation of a minimax rational approximation for ln(1+x) on [0,1]
- A utility routine for calculating the critical points of a function -
critpts- The calculation of a minimax rational approximation for exp(x) on [-1,1]
- Comparison of polynomial and rational minimax approximations

A procedure implementing the Remez algorithm- remez.mws

- A procedure for constructing minimax polynomial and rational approximations via the Remez algorithm -
remezremez: examples

The Remez algorithm: standard and non-standard error curves for rational approximations to an even function- RZeven.mws

- Standard and non-standard error curves
- A minimax rational approximation for cos(Pi/4*x) on [-1,1]
- A minimax rational approximation for cosh(x) on [-1,1]

Using the Remez algorithm for constructing polynomial approximations for sin(x) and cos(x) on [-Pi/4,Pi/4]- RZsincos.mws

More examples of minimax rational approximations- RZexamp.mws

Testing the Remez algorithm with "badly behaved" functions- RZexamp2.mws

Function approximation procedures- fcnapprx.zip