Introduction to Successive Approximations
in Your Spreadsheet
1. Purpose of this website
2. Strategy of successive approximations
3. Equations that you can solve by this method
4. Required skills
4. Required software
5. Next page menu
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.
In this website, you'll learn the strategy called "Newton's method," and it involves the following basic steps:
- You'll design "approximation functions" using the equations you need to solve.
- You'll estimate a reasonable solution of the equations by guessing.
- You'll process your estimate with the approximation functions, and produce a better estimate of the solution.
- You'll repeat Step 3 until the approximations converge on a solution (this step is often called "recursion").
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!
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:
- Manipulating equations into different forms using addition, multiplication, and possibly variable-substitution.
- Computing first-derivatives of functions, or else finding them in the Derivative Table. You can review slopes and derivatives at the Slopes page.
- Working with complex numbers and functions (only if you want to find complex solutions). You can review complex numbers at the Complex Numbers page.
- Using a spreadsheet program such as Microsoft Excel 95 (or higher version), Quattro, Lotus-123, or similar.
- Typing and copying spreadsheet-formulas that use cell references as variables.
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.
The spreadsheet you need depends on how many equations you're solving and the type of solutions you want.
- If you're finding real solutions of one equation, then any spreadsheet program will work, because no special functions are required for this.
- If you're finding complex solutions of one equation, then your spreadsheet must have conditional functions (if/then statements), or it must have a function that computes the imaginary argument from real and imaginary coordinates. Quattro, Lotus and Excel all have conditional functions or imaginary-argument functions or both.
- If you're finding real solutions of several equations, then your spreadsheet must have functions that compute the elements of inverted matrices and matrix products. I know that Excel has these matrix functions, and I would assume that other programs do too.
- If you're finding complex solutions of several equations, then your spreadsheet must have conditional functions or an imaginary argument functions, and also matrix inverse and matrix product functions. Excel has all of these functions.
Your choice of the next page appears below; my recommendation is the Procedures page at the top of the menu. This menu also appears in the Index whose link appears at the top of every page in this website.
Unpublished Work. © Copyright 2001 Pat Russell. Updated April 17, 2009.
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