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Maple worksheets on numerical root-finding |
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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 bisection method - bisect.mws
- A introduction to root-finding using graphs in conjunction with Maple's fsolve procedure.
- A procedure for finding a zero of a function by the bisection method: bisect.
Newton's method - newton.mws
- Explanation of Newton's method.
- A procedure for graphing successive steps of Newton's method: newton_step
- A procedure for finding a zero of a function by Newton's method: newton.
The secant method - secant.mws
- Explanation of the secant method for root finding.
- A procedure for graphing successive steps of the secant method: secant_step
- A procedure for finding a zero of a function by the secant method: secant.
Halley's method - halley.mws
- The iterative formula for Halley's method.
- Osculating hyperbolas and the Halley iteration formula.
- A procedure for graphing successive steps of the secant method: halley_step
- A procedure implementing Halley's method for root-finding: halley
The improved "leap-frog" Newton method for root-finding - LFnewton.mws
- The iterative formula for the improved "leap-frog" Newton method.
- A procedure for graphing successive steps of the improved "leap-frog" Newton method: impnewton_step
- A procedure for finding a zero of a function by the improved "leap-frog" Newton method: impnewton.
Brent's method for root-finding - brent.mws
- Inverse quadratic interpolation and Brent's method.
- A procedure for finding a zero of a function by Brent's method: brent.
General fixed point iteration for root-finding - itersolve.mws
- Hero's formula for calculating square roots
- Iterative formulas for solving equations
- A procedure for finding a fixed point of a real function: fixedpoint.
- Graphical illustration of iterative solutions
- Oscillating values from an iterative formula
A procedure for finding all roots in an interval where a sign change occurs - allbisect.mws
- A procedure for finding all zeros of a function in an interval where a sign change occurs: allbisect.
A bisection method is first used to pin down each root to a small interval.A procedure for finding all roots in an interval - allroots.mws
- A procedure for finding all the zeros of a function in an interval: allroots.
It is a modification of the procedure allsolve written by Robert Israel from U.B.C.
It makes use of range arithmetic via Maple's evalr.Numerical determination of a local maximum or minimum - maxmin.mws
- Golden section search for a maximum or minimum.
- Pictorial example of golden section search.
- Procedures which search for a local maximum or minimum: goldmax/goldmin.
- Searching for a maximum or minimum using parabolic interpolation.
- Pictorial example using parabolic interpolation.
- Procedures which search for a local maximum or minimum: findmax/findmin.
The logistic family of functions and chaos - chaos.mws
- The logistic family of functions.
- Iterating members of the logistic family.
- Bifurcation
- A procedure for displaying the bifurcation diagram of the logistic family: logisticdisplay.
- The bifurcation diagram of the logistic family.
- Chaos and islands of regularity.
- Finding a parameter value for the 3-cycle.
- Finding a parameter value for the 1st 5-cycle.
- Finding parameter values for the other two 5-cycles.
Root-finding procedures - roots.zip
