# Variations of Newtons method

1. Jun 11, 2010

### defunc

Are there any variations of newtons method, say where you use higher derivatives?

2. Jun 11, 2010

### HallsofIvy

I don't know of any that are commonly used but it's not hard to extend the basic idea.

In "Newton' method", we approximate the function by its tangent line at a given point, and determine where that line crosses y= 0.

We could, as well, approximate the function by a parabola that (1) passes through the given point, (2) has the same slope their, and (3) has the same second derivative.

That is, to solve f(x)= 0, start with some given point $(x_0, f(x_0))$, at which f(x) has derivative $f'(x_0)$ and second derivative $f''(x_0)$.

We can write a parabola through $(x_0, f(x_0))$ as $y= a(x- x_0)^2+ b(x- x_0)+ f(x_0)$. That has derivative, at $x= x_0$, $y'= b$ and second dervative $y''= 2a$ so that the "approximating parabola" is $f''(x_0)(x- x_0)^2/2+ f'(x_0)(x- x_0)+ f(x_0)$. Set that equal to 0 and solve for x to get the next x at which to approximate.

But the fact is that there are just some "approximating" methods that are just so good, any improvement by a "better" approximation just isn't worth the additional work. Newton's method is one of those!

3. Jun 11, 2010

### Studiot

As usually presented Newton's method is intended for a single unknown variable.
Hamming gives an improved method, sensitive to the difference between the current approximation and the last.

There is an equivalent method when you have a system of equations usng the Jacobian in place of the first derivative.

Last edited: Jun 11, 2010
4. Jun 11, 2010

### Shaybay92

Isnt this method the basic idea behind Taylor Polynomials?

5. Jun 11, 2010

### hotvette

Yep, Halley's method uses 2nd derivatives in addition to first derivatives. If Newton's method converges quickly, Halley's method is turbo charged.

http://en.wikipedia.org/wiki/Halley's_method

6. Jun 13, 2010

### Count Iblis

The following method is often used when the evaluating the function is very time consuming. You first find a few starting values. E.g. if y(x) is your finction, you find two points x1 and x2 such that y changes siign and then you do, say, two bisection steps, ending up with four points. Then you do an interpolation using the four points, but not of y asa function of x, but instead of x as a function of y. In that interpolating polynoimial, x(y) you simply put y equal to zero and you obtain a very accurate estimate of the zero of y.

You then throw away the first point and construct a new interpolating polynomial using the last four points, insert y = 0 in there and then iterate this process.