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LECTURE 4: Algebraic Equations

Solutions of nonlinear algebraic equations--BASIS for DIFFERENTIAL EQ!

Kepler's equation: x - a sinx = b

1. Bisection Method

For a given function f(x) with two known initial points x=a and x=b, we can find a 3^rd point c=(a+b)/2.

2. Newton's Method---very popular

Pros:   fast, one initial point

Cons: 

(1) no convergent (sometimes)

(2) OSC

(3) Derivative f’(x)=0 causes the method to diverge

It's a quadratic convergent method!

e.g. how to solve equation?

--We can derive an interesting formula.

1. Secant Method

Explain the method both by formula and graphs: similar to Newton, but avoids computing derivatives

--This method converges faster than Newton.

3. Polynomial Interpolation:

(a, f(a)) and (b, f(b))

g(x) = (b-x)/(b-a) f(a) + (x-a)/(b-a) f(b)

Now, if you have N+1 points

(x0, f0),

(x1, f1),

... ...

(xN, fN),

Solve equations for (a0, a1, ..., aN)

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1. Solutions of nonlinear algebraic equations--BASIS for DIFFERENTIAL EQ!

Kepler's equation:  x - a sinx = b

How do we solve this equation?

1. bisection method

Explain the method

advantage: easy, must know the interval where the root lies

disadvantage: slow

c = (a+b)/2

if f(c)f(a) <0 then a <-- c 

if f(c)f(b) < 0 then b <-- c 

Error:

Result:

We can estimate the steps needed for certain errors.