Index

## 1. Purpose of this website

In this website, I explain how to find real or complex solutions of one or more equations by computing successive approximations in a spreadsheet. I wrote the website mainly for people who need to solve one or more equations of as many variables, and who need a quick-to-learn, easy-to-use method without a lot of theory.

## 2. Strategy of successive approximations

In this website, you'll learn the strategy called "Newton's method," and it involves the following basic steps:

1. You'll design "approximation functions" using the equations you need to solve.
2. You'll estimate a reasonable solution of the equations by guessing.
3. You'll process your estimate with the approximation functions, and produce a better estimate of the solution.
4. You'll repeat Step 3 until the approximations converge on a solution (this step is often called "recursion").

## 3. Equations that you can solve by this method

You can use this method to solve equations that have the following properties:

• The number of equations equals the number of unknown quantities.
• Each equation can be arranged to equal zero. An example is this set of three equations and three variables: 7x4y2 - x3z4 + 19 = 0
6x2z2 + 5y3z - 7 = 0
x2z5 - 3x5z3 - 5 = 0
• The number of solutions is finite. If the number is infinite, you can still use this method to find solutions, but it will take forever to find them all!

## 4. Required skills

You don't need a math degree in order to use successive approximations. In fact, you probably learned most of the necessary math skills in high-school algebra and pre-calculus. If you are solving one equation of one unknown quantity, then these are some skills that you will need:

If you are solving a system of equations, then these are some additional skills you must have:

• Computing the partial derivatives of a function
• Expressing the differential of a function in terms of its partial derivatives and the differentials of its variables.
• Knowing the basic rules of matrix multiplication and inversion. However, you won't need to invert or multiply matrices yourself, because your spreadsheet can do that for you.

## 4. Required software

The spreadsheet you need depends on how many equations you're solving and the type of solutions you want.