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Predictive iteration method 
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#1
Jan1511, 07:45 PM

P: 3,967

Hi,
I came up with a predictive iterative method that converges very rapidly, when writing a java simulation. It turns out that this method can be used for a wide variety of problems and is very simple because unlike the bijective method it does not require upper and lower bounds to be found initially and unlike the NewtonRaphson method it does not require that the derivative of the equation be found first. This predictive method is conceptually simple, so I am almost certain this method is already well known and has a name, but here it is anyway in Liberty basic (free):
{EDIT} It may be that the above method is in fact the NewtonRaphson method in disguise. Would anyone agree? 


#2
Jan1711, 02:48 PM

P: 644

Yes, I believe it is the same thing, using a discrete derivative. It basically constructs a straight line, and then jumps to the xvalue where that line crosses the function value T.



#3
Jan1711, 03:39 PM

Engineering
Sci Advisor
HW Helper
Thanks
P: 7,284

The standard name for this is the secant method.
There are several versons of exactly how to do it, depending on exactly which points you use to estimate the derivative. For example you can use the last two you calculated, or the two that have the smallest function value, or keep the same value for every iteration, etc. It behaves very similar to Newton's method, but when the two points are close together you can lose accuracy because the slope is calculated from the difference of numbers that are almost equal. If you find the derivative directly as in Newton's method, you avoid that problem. 


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