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 trapezoid in 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 simpson in 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.
GLint can 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/Int scheme for numerical integration.
- Maple's numerical integration procedures available through evalf/Int
- An interface for the various numerical integration XXint routines: 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