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y = @(x)((10^-9)*x*(cosh(x)*sin(x)-sinh(x)*cos(x)) - ((10^-9)-1*(x^4)*(1+cosh(x)*cos(x))))

Suppose I have a kind of equation similar to the one above:

'y' is a function in variable 'x', it is a nonlinear, non-transcendental eqaution.

I have tried 'fzero', and I hae also plotted the function. So that plot gives me a shot regarding approximate location of the point whereas the function crosses y =0 line (corresponding x point is the answer). This is how I found the initial guess (eyeballing) and found following values for x:

0.149524, 1.875236, 4.694101

But wenenver I apply these numbers into the main equaiton to find the value of function at the mentioned points using 'subs' function in matlab, it is NOT equal to zero!!!!! (however we know value of function at the solution points should be equal to zero!!!!). Of course, using funciton 'round' yields zero. In short, I am a bit hessitant if my roots are correct or not.

Does anybody havea recommendation to find the roots? (logically snice I plot the function, I can estimate the first root should be pretty close to 0.15, and applying this initial guess into the 'fzero' function, yields 0.149524.) I don't know should trust it or not. (another reasin which makes me hesitant is the fact that physically the system obtaining such kind of nonlinear non-transcendental eqaution should have the first root close to 1.87).

Does anybody has a recommendation to solve such an enigma?

Thanks a lot

Matt J
on 16 Mar 2020

Edited: Matt J
on 16 Mar 2020

fzero is a numerical root finder. One doesn't expect y(x) to be exactly zero at the roots that it finds, but it should be pretty close. One way to evaluate the result is to plot the function in a tight interval around the supposed root and see how close to zero it is compared to its neighbors. The three roots you've indicated look pretty good to me.

fun = @(x)((10^-9)*x*(cosh(x)*sin(x)-sinh(x)*cos(x)) - ((10^-9)-1*(x^4)*(1+cosh(x)*cos(x))));

[x1,y1]=fzero(fun,[0,0.2])

[x2,y2]=fzero(fun,[1,2])

[x3,y3]=fzero(fun,[4.2,5])

close all

figure;

fplot(fun, x1+[-1,1]*.0001);

hold on; plot(x1,y1,'ro','MarkerSize',6); hold off

figure;

fplot(fun, x2+[-1,1]*.0001);

hold on; plot(x2,y2,'ro','MarkerSize',6); hold off

figure;

fplot(fun, x3+[-1,1]*.0001);

hold on; plot(x3,y3,'ro','MarkerSize',6); hold off

Walter Roberson
on 17 Mar 2020

syms laL

>> pretty(y1(laL))

4

laL (cos(laL) cosh(laL) + 1) laL (cos(laL) sinh(laL) - cosh(laL) sin(laL)) 1

----------------------------- - --------------------------------------------- - -------

1000 1000000000 1000000

>> pretty(y(laL))

4 laL (cos(laL) sinh(laL) - cosh(laL) sin(laL)) 1

laL (cos(laL) cosh(laL) + 1) - --------------------------------------------- - ----------

1000000000 1000000000

They are not the same equation. The y one has first term multiplied by 1000 and the third term divided by 1000 compared to the y1 equation.

Walter Roberson
on 18 Mar 2020

You are working with two different equations. You need to figure out which of the two equations is right.

I suspect that when you built

y = @(x)((10^-9)*x*(cosh(x)*sin(x)-sinh(x)*cos(x)) - ((10^-9)-1*(x^4)*(1+cosh(x)*cos(x))))

that you made a mistake.

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