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Maple worksheets on approximation of functions |
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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 - interpoly
- interpoly: 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 -
using an iterative method for obtaining the minimax error - ratminmax.mws
- 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 -
solving a rational equation to obtain the minimax error - ratminmax2.mws
- 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 - remez
- remez: 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
