Computer applications in Mathematics-Lect8

13MAA160 Computer Applications in Mathematics
Week 8 Lecture (Maple 6): Turning
algorithms into programs
6.1 Lecture overview
6.1.1 Objectives
* To present simple algorithms to perform important mathematical
tasks.
* To demonstrate by example how algorithms can be turned into
computer programs.
6.1.2 Commands (old and new)
Command so far covered:
* Maple packages: with, plots (1), LinearAlgebra (2).
* Functions and numbers: ":=", ?, common functions, Digits, evalf, Float, Pi, restart, "->", piecewise, subs, unapply, rand, abs.
* Basic Plotting: plot, plot3d, animate (1), contourplot (1), display (1), implicitplot (1).
* Algebra: expand, factor, simplify, convert, combine, solve, fsolve.
* Calculus: diff, implicitdiff, int, series, sum.
* Sets: union, intersect, minus.
* Linear Algebra: "<,>", Matrix, Norm (2), DotProduct (2), CrossProduct
(2), Multiply (2), Determinant (2), MatrixInverse (2).
* Output commands: print, printf.
* Loops: for (from, to, do, od), while (do, od).
* Conditional Statements: if (then, else, elif, fi).
* Commenting commands: #.
* Procedures: proc (local, end proc), return, error.
New commands:
* None.
> > > >
> restart;6.2 Root finding
The Maple commands solve and fsolve are used to find solutions of algebraic equations. If solve doesn't work then we have to use fsolve (often with the plot command) to find approximate values to the roots.
Three basic methods for finding roots will be demonstrated in this Lecture:
1) Bisection
2) A "simple" fixed-point method
3) The Newton-Raphson method
The function used in the examples is
f(x) = exp(-x) - x
We want to find x such that f(x) =0
f:=x->exp(-x)-x;plot(f(x),x=-2..2,labels=[x,"f(x)"]);
There is a root at x* ≈ 0.5, .... and using fsolve :
Maple_answer_1:=fsolve(f(x),x=0..1);
.... is the correct answer to 10 decimal places.
6.2.1 Bisection method
6.2.2 The "simple" fixed-point method
6.2.3 The Newton-Raphson (fixed-point) method
6.3 Numerical Integration
6.3.1 Numerical integration using Maple
6.3.2 The (left) Riemann Sum method。