Maple worksheets on numerical integration |

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 trapezoidal rule- trapezoid.mws

- Description of the trapezoidal rule.
- A procedure for illustrating the trapezoidal rule graphically:
drawtrap.- Use of the procedure
trapezoidin the "student" package.- A procedure for performing the trapezoidal rule iteratively:
trap.

Simpson's rule- simpson.mws

- Derivation of Simpson's rule.
- Use of the procedure
simpsonin the "student" package.- A procedure for performing Simpson's rule iteratively:
simp.

Applying Simpson's rule adaptively- adaptsimp.mws

- A preliminary adaptive procedure using Simpson's rule
- A procedure for performing Simpson's rule adaptively:
SPint

Error analysis for the trapezoidal rule and Simpson's rule- simperr.mws

- An estimate for the error of the trapezoidal rule
- An estimate for the error of Simpson's rule.

Simpson's rule for unequally spaced data points- simpint.mws

- A version of Simpson's rule for unequally spaced data points.
- A procedure for applying Simpson's rule to numerical data:
gensimp- "Indefinite" numerical integration via parabolic interpolation.
- A procedure to perform "indefinite integration" for numerical data:
simpinterp

Newton-Cotes rules- NCint.mws

- Constructing Newton-Cotes Integration formulas
- Compound Newton-Cotes formulas
- A procedure for performing Newton-Cotes rules adaptively:
NCint

There are "built-in" coefficients for rules with an even number of intervals (odd number of points).

There is an option to display the points used in evaluating the integral along a graph of the integrand.

Gauss-Legendre quadrature I- gauss.mws

- An introduction to Gauss integration formulas
- The Gauss 3-point rule
- Legendre polynomials
- Legendre polynomials as a "coordinate system" for polynomials
- Gauss integration formulas via Legendre polynomials
- A more convenient formula for the weights
- Computation of Gauss-Legendre nodes and weights

Gauss-Legendre quadrature II- GLint.mws

- A procedure for performing Gauss-Legendre quadrature adaptively:
GLint

There are built in abscissas and weights, which can be used up to a precision of 40 digits.

GLintcan refine the abscissas and weights to a higher precision if required.

There is an option to display the points used in evaluating the integral along a graph of the integrand.

Gauss-Kronrod quadrature- kronrod.mws

- The basic idea of Kronrod extensions
- Extension of Gauss-Legendre integration formulas
- Calculation of nodes and weights for the 7 to15 node Gauss-Kronrod extension
- A preliminary routine for 7-15 Gauss-Kronrod integration
- Procedures for constructing Gauss-Kronrod nodes and weights

A procedure for performing Gauss-Kronrod quadrature- GKint.mws

- The basic idea of adaptive Gauss-Kronrod integration
- A procedure for performing Gauss-Kronrod quadrature adaptively:
GKint

There are built in abscissas and weights, which can be used up to a precision of 40 digits.

There is an option to display the points used in evaluating the integral along a graph of the integrand- Solving equations involving integrals.
- An arc length example

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

Clenshaw-Curtis quadrature- CCint.mws

- Clenshaw-Curtis integration formula
- A basic procedure for Clenshaw-Curtis quadrature using Chebyshev series of fixed length
- A procedure for performing Clenshaw-Curtis quadrature iteratively:
CCint

Romberg integration- romberg.mws

- Iterated application of the trapezoidal rule
- The Romberg triangle . . preliminary procedures for Romberg integration
- A procedure for performing Romberg integration:
RBint

Comparison of methods of numerical integration- compare.mws

- Using Maple's
evalf/Intscheme for numerical integration.- Maple's numerical integration procedures available through
evalf/Int- An interface for the various numerical integration
XXintroutines:quad/int

Numerical integration of functions of varying "smoothness"- smooth.mws

- Integrating a succession of progressively smoother functions.

Numerical integration procedures- intg.zip

Data for the Gauss-Kronrod integration procedure GKint- GKdata.zip