Function Intersection using Newton's Method

Hi everyone, I'm trying to write a function that finds a point at

which two functions f(x) and g(x) intersect, i.e., f(x) = g(x). I'm using

Newton's Method and making and h(x)=f(x)-g(x) and h'(x) as well, but I'm not getting the right x-value. Please help me debug my code!

 % function x = fgIntersect(f, df, g, dg, x0, tol, maxIter)
h=f(x0)-g(x0)
dh=df(x0)-dg(x0)
k=1;
while k<=maxIter
x=x0-h/dh;
if abs(x-x0)<tol*abs(x)
return
end
x0=x;
k=k+1;
end

end

[Matt J]

Give us example data that let's us reproduce the failure.

[SB]

format compact; format long;

f = @(x) exp(x) - 3;

df = @(x) exp(x);

g = @(x) sqrt(x);

dg = @(x) .5*x^(-.5);

x = fgIntersect(f, df, g, dg, 1, 1e-6, 50)

x should equal 1.434542442506692

Another case:

format compact; format long;

p1 = [1 -2 3 -8];

p2 = [1 -3 2 -4];

f = @(x) polyval(p1,x);

df = @(x) polyval(polyder(p1),x);

g = @(x) polyval(p2,x);

dg = @(x) polyval(polyder(p2),x);

x = fgIntersect(f, df, g, dg, 2, 1e-6, 50)

x should be 1.561552842846145

Answer 1

[Matt J]

You're not updating h and dh within your loop.

[SB]

Oh,how would I update them? Should I just put the definitions inside the loop for h and dh?

[Matt J]

As new points x are generated in the loop, you want h and dh to be evaluated at those points.

[SB]

Thank you so much! I got it, I can't believe I forgot to update them within my loop.

[Torsten]

f = @(x) exp(x) - 3;
df = @(x) exp(x);
g = @(x) sqrt(x);
dg = @(x) .5*x^(-.5);
x = fgIntersect(f, df, g, dg, 1, 1e-6, 50)

x = 1.4345

p1 = [1 -2 3 -8];
p2 = [1 -3 2 -4];
f = @(x) polyval(p1,x);
df = @(x) polyval(polyder(p1),x);
g = @(x) polyval(p2,x);
dg = @(x) polyval(polyder(p2),x);
x = fgIntersect(f, df, g, dg, 2, 1e-6, 50)

x = 1.5616

function x = fgIntersect(f, df, g, dg, x0, tol, maxIter)
  h=f(x0)-g(x0);
  dh=df(x0)-dg(x0);
  k=1;
  while k<=maxIter
    x=x0-h/dh;
    if abs(x-x0)<tol*abs(x)
        return
    end
    x0=x;
    h = f(x0)-g(x0);
    dh = df(x0)-dg(x0);
    k=k+1;
  end
end