Muller’s Method

1. Muller’s Method By: Matt Carpenter

2. What is it? • Muller’s Method is a generalization of the Secant Method in a different direction. Instead of intersecting the line through two previous points with the x-axis, we use three previous points X0, X1, X2, draw the parabola y = p(x) through them, and intersect the parabola with the x-axis.

3. What if there are two points? • With a parabola, it is possible for it to cross the x-axis at two different places. So to find out which one we would use, we just find out which one is closer to the X2 point that we picked. To do that, just take the X2 minus the root, and take the smallest of the absolute values. • It is also possible for there to be no roots if it is above the x-axis. • So with having those three points, we are able to make an equation out of it and us the quadratic formula to find the roots.

4. How to do it • To start, you would be given an equation and the three initial guesses. For example: • The equation could be 3x2 + 4x +7 and then the initial guesses could be 0, 1, and 2. To get the y coordinate for these, you would simply just plug them into the equation. • For 0 it would be 7 • For 1 it would be 14 • And for 2 it would be 27

5. The coordinates that we have now are (0,7), (1,14), and (2,27). Once we have these, we then use the Lagrange Interpolation. As a reminder, this is what it looks like: • Simplifying it a little would look like this:

6. From that, we can get our a, b, and c needed for the quadratic formula. So,

7. With our example, our answers would be: • a = 3 • b = 8 • c = 7

8. Quadratic Formula: • Root 1: • Root 2:

9. Advantages? • You can see here the difference between the two methods. Muller’s method will find the root faster than the secant method.

10. Disadvantages? • If there is a negative discriminant for the quadratic formula, then you can’t find the root very easily without a program. • They can take a while to do by hand.