MATLAB: Need help with newton’s theorem (matlab)

MATLABnewtonnewton's method

newton script: http://pastebin.com/RF0ehxFi

function i use: http://pastebin.com/V88WLe5s

After running script i put inittial starting point -3, tolerance parameter = 1e-5 , max iteration = 35,

however there is something wrong with the newton script i need to know what it is please thank you. i think the output after plotting fplot(function, [-15,1]) there is something wrong but i am not sure .

any help will be appreicated thanks

Best Answer

It works fine - you're just using a (contrived!) nasty example. Plot your function between -3 and 20 and see where the first Newton step takes you

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MATLAB: Finding multiple roots using newton raphson

Newton's method cannot be used to find multiple roots. You basically find one root, and you are done.

Then you can start at a new point. Hopefully Newton's method will converge to another root. Or it might just find the first one.

So a simple approach for 1-d problem is to evaluate the function at a bunch of points, looking to see where there might be zero crossings. A root MUST exist in the area of any zero crossing. But you might miss two roots, if they are close to each other. So sample sufficiently tightly so this will not happen. But wherever your tests see a crossing, start Newton's method there, and you SHOULD probably find another root.

There is no magic in the above - just careful programming.

Now, could you use homotopy (continuation) methods to find multiple roots? Well yes, in some cases, yes. But that is wildly beyond just using Newton's method. And it involves an ODE solver, which is NOT at all Newton's method. I don't think this is the goal of your assignment, unless you wanted to invest a lot of effort.

Ok, there are other tricks. You can use a root deflation scheme, so as you find a root, you modify the function, so the root you just found is no longer a root. This can get tricky too, if you are not careful.

The point is, you cannot simply just modify Newton's method to find multiple roots. The simple approach is as I suggested, just look for sign changes in a sequence of function evaluations.

yfun = @(x) 4*exp(-0.3*x).*(sin((3*x)+0.25));
xtest = linspace(0,4,1000);
ytest = yfun(xtest);
indp2n = strfind(sign(ytest),[1,-1]);
indn2p = strfind(sign(ytest),[-1,1]);
xstart = (xtest([indp2n,indn2p]) + xtest([indp2n,indn2p]+1))/2;

Note my use of .* in yfun. That will be necessary.

ezplot(yfun,[0 4])
grid on
hold on
plot(xstart,yfun(xstart),'ro')

Now, just start your Newton scheme at each of those points. It will converge in only a few iterations from each start point, since Newton is quadratically convergent, and those are darn good starting values.

MATLAB: Newton’s Method in Matlab

First, consider if you are trying to solve the wrong problem.

In your question, you state it as e^2*sin(x) - x = 0

However, in your code, you write exp(2*sin(x)) - x. You do realize there is a difference? What I don't know is if you have miswritten your question, or is it your code?

fun = @(x) exp(2)*sin(x) - x;
fplot(fun,[0,3])
yline(0);
fzero(fun,3)
ans =
       2.7589

So the function you claim to want to solve has a zero around x==2.76, which is inconsistent with your claim of where the root lies. Next, look at the function you wrote code for:

fun = @(x) exp(2*sin(x)) - x;
fplot(fun,[-5,3])
yline(0);

Indeed, this does seem to have a root near 2.6357. So, just possibly, you really do want to solve the problem exp(2*sin(x))-x==0.

fzero(fun,3)
ans =
       2.6357

But now, look at the plot! What happens when you start Newton's method at x==1? THINK! Where will the first iteration go? On which side of that hump is x==1?

The point is, Newton's method tries to drive the functino to zero. But if you start the iterations BELOW x==1.5028, which direction will Newton's method push you?

fminbnd(@(x) -fun(x),1,3)
ans =
       1.5028

Think about the meaning of the iterations of Newton's method. What is the goal? What will happen?

Best Answer

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MATLAB: Finding multiple roots using newton raphson

MATLAB: Newton’s Method in Matlab

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