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In the solution of beam b=vibration and buckling problems the transcendental question may result for the natural frequencies of the buckling load.
The smallest positive root of tan x -lemda (x) = 0.
The smallest positive root , x , of the equation for different values of lemda are listed below.

lemda - x

1.5-0.96740

2.0-1.16556

2.5-1.26440

3.0-1.32419

3.5-1.36437

4.0-1.39325

4.5-1.41502

5.0-1.43203

From the values at 3.0 , 3.5 and 4.0 use quadratic interpolation to find the value of x for lemda=3.25he following methods:

a) use newton's methods for finding roots

b) use didderence table and 1) interpolate using lagrange quadratic form 2) interpolate using newtons quadratic form

c) explain any difference using the two interpolation forms.

d) explain any difference using interpolation versus root solving methods.
Which is more accurate?
When would you prefer one over the other?
Solution
lembda=3.25;

f=@(x) tan(x)-lembda*x;
df=@(x) (1/cos(x))^2-lembda;



x=1.36437;

tol=1.e-6;
done=0;

while ~done
    xnew=x-f(x)/df(x);
    if abs(xnew-x)<tol
       done=1;
    end
    x=xnew;
end
xnewton=x;
disp(['Newton Method Solution for lembda=3.25 = ' num2str(xnewton,8) ]);...

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