Basic Mathematical Tools

Here is a collection of nine tools which can be used when trying to solve various mathematical problems.

• The Plot Applet can display the graphs of 1 to 5 functions simultaneously. The functions must have the Cartesian form y=f(x) or the parametric form x=f(t) and y=g(t). Polar equations of the form r=f(t) can be plotted by setting x=f(t)*cos(t) and y=f(t)*sin(t). Functions must be separated by a semiclon (;) in the function input field. Clicking on a point on a graph will display the (x,y) coordinates of that point.
• The Root-finding Applet implements Bisection, Secant, Newton, Modified Newton, Mueller and Inverse Interpolation methods to solve nonlinear equations in a single variable of the form f(x)=0. It implements Picard iteration and Steffensen's algorithm to solve fixed point problems of the form f(x)=x.
• The Linear Systems Applet can be used to solve small systems of n linear equations in n unknowns using gaussian elimination with naive, partial, or scaled partial pivoting. It can be used to find the Doolittle LU decomposition of a square matrix with and without pivoting and it can find the inverse of a square matrix.
• The Numerical Integration Applet approximates the value of a definite integral using the Right-endpoint Rule, the Midpoint Rule, the Left-endpoint Rule, the Trapezoid Rule, Simpson's Rule, an Adaptive method based on Simpson's Rule, Romberg Integration, and Gaussian quadrature.
• The First Order Applet can be used to generate the numerical solution of initial value problems of the form y'=f(t,y), y(a)=y0, a<=t<=b. The applet implements most of the the standard numerical algorithms for solving such problems, including variable step-size methods.
• The Systems Applet can be used to generate the numerical solution of initial value problems of the form x'=f(t,x,y,z), y'=g(t,x,y,z), z'=h(t,x,y,z), x(a)=x0, y(a)=y0, z(a)=z0, a<=t<=b. The applet implements most of the the standard numerical algorithms for solving such problems, including variable step-size methods.

Be aware of the following rules when entering a function expression.

1. Supported mathematical operations (from highest to lowest precedence).

   ^ * / + -

2. Supported mathematical functions. These are all of the functions included in Java's Math class.

   sin, cos, tan, asin, acos, atan, exp, log, sqrt, pow, abs

3. Grouping. Spaces are ignored when the string is parsed and terms may be grouped with any of

   (), {}, []

All operations must be explicitly written, so the expression 2x must be written as 2*x.