Cents and ratios | Ratios with factors | Mean tone in cents | Under / Over | UO, non octave & scale tree | Quintic | Music and virtual flowers |
You can start the iteration at any value close to the expected solution, and it also works well to choose a large value such as 1000. Try using other starting values to find the other solution(s).
If you have a solution, the Value at solution will be 0, or very small. Note that the method used need not find a solution - with some starting points it might not get started at all.
When you use the button to search for a start value, the idea is to go through a sequence of values, e.g. from -10 to 10 in steps of 0.1, and look for any change in sign of the function or its derivative. Whenever such a change is found, you search for a nearby root using Newton's method.
When it shows x.xxxx e-15, or the like this means it is very small - e-15 means that there are 14 zeroes before the first significant figure.
If it doesn't find one, try other start values. If that doesn't work, it probably means that there are no solutions to find. This means that the only solutions are ones that need square roots of negative numbers (complex numbers) as this applet doesn't look for those. This can sometimes happen with a quartic or quadratic. Quintics and cubics will always have at least one solution.
If you are interested in the numerical values, these solutions are often as accurate on a computer as the exact solutions obtained using sqrt etc even when such solutions are available. That's because of the tiny rounding errors involved in evaluating mathematical functions on a computer using the normal level of precision.
It's possible to find an algebraic formulae for the general solutions for the cubic and quartic. Here is a page I did which you can use to compare algebraic and iterative solutions of the cubic: Solution of cubic .
For the cubic, see this page by Carsten Magnusson . For another solution to the cubic by the algebraic method, see his javascript applet here . Another useful link is this web page by Songi Du,
For c-code (if you are a programmer) see http://jtull.rutgers.edu/~rhoads/Code/code.html
The java script for this page uses Newton's method:
Find x2 = x - f(x)/f'(x) Then set x = x2 and repeat the process. Keep going until the new value for x is very close to the previous one.
The java script for this page keeps going until the step is less than 10 ^{ -15 } .
Newton's method can be used to generate fractals when you use it with complex numbers (involving square roots of negative numbers). The idea is to choose various starting values and colour the points according to which root you get to - you get a really intricate patterns at the boundaries between the regions - see this web page Newton's method .