I recently discovered the Computing Skillset site (via open-struct-engineer), which has good information on the basis and use of the Newton-Raphson method:

I have added an example from these papers to my ItSolve spreadsheet, illustrating some advantages of Brent’s Method for problems that may be difficult to solve using the Newton-Raphson method. The updated spreadsheet (including full open-source code) may be downloaded from:

ItSolve.zip

The example finds solutions to the function shown below:

Using the Newton-Raphson Method a solution may not be found:

The screen-shots below show all four solutions between -7 and 7 using Brent’s method:

Note that:

• The values specified for the search range must return a positive and negative function result. If both are positive, or both negative an error message is returned.
• The Quadbrent function works with functions entered as text on the spreadsheet as shown, or (by default) will call a named VBA function. The QuadbrentT function is a front-end for Quadbrent with defaults set to use text on the spreadsheet, as shown above.
• The text function must be entered in a format recognisable by Excel, for instance the value Pi is entered as Pi().