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One of the most important problems in mathematics is to find the values of the *n* unknowns *x*_{1}, *x*_{2}, ..., *x*_{n} that satisfy the system of *n* equations

That is, we want to solve for ** x** in

*A* is commonly referred to as the *coefficient* matrix. In order to save space, we usually write column vectors in coordinate form, ** x** = (

Suppose we knew the true values of *x*_{2}, *x*_{3}, … , *x*_{n} .*x*_{1}*x*_{1}*x*_{2}, *x*_{3}, … , *x*_{n} ,*x*_{1}.*x*_{i}*x*_{1}, … , *x*_{i-1}, *x*_{i+1}, … , *x*_{n}.** x** = (

The fundamental idea of an iterative method is to use * x^{current}*,

*Iterate* means *repeat*, so an iterative method repeats this process over and over, each time using the current approximation to produce a better approximation for the true solution, until the current approximation is sufficiently close to the true solution -- or until you realize that the sequence of approximations resulting from these iterations is not converging to the true solution.

So given an initial guess or approximation **x**^{(0)}* x*, we use

Since we don’t actually have the true solution * x* (if we did, why would we be trying to find it?), we can’t check to see how close our current approximation

Another way to check the accuracy of our current approximation is by looking at the differences in successive approximations, **x**^{(k)} − **x**^{(k-1)}.**x**^{(k)}* x* if

This idea of iteration is not unique to linear algebra. An example that may be familiar to you is Newton’s Method , which finds approximations to the root *x* of *f*(*x*) = 0.*x*^{(k)}*x*^{(k+1)}

Under the right conditions, *x*^{(k)} *x* [that is, *f* (*x ^{k}*) 0] as

David M. Strong, "Iterative Methods for Solving [i]Ax[/i] = [i]b[/i] - Introduction to the Iterative Methods," *Loci* (July 2005)

Journal of Online Mathematics and its Applications