Week 7 Notes – Adam Ata

Numerical Integration

Trapezoidal rule

A1=(f(x0)+f(x1))*h/2

A2=(f(x1)+f(x2))*h/2

A3=(f(x2)+f(x3))*h/2

In general:

+

Example: : with m = 4 and h = .5

=

Example: with m = 3 and h = pi/6

=77048

Exact solution is 1

Simpson rule

Approximates the function using a quadratic polynomial every 3 points.

Letting

When [a,b] is divided into 2M subintervals, h = (b-a)/2M

Example: 2M = 6

In general:

Example: with 2M = 4,

Exact solution is 1

With 2M =2,

Numerical Optimization:

Used to obtain the minimum or maximum of a function

Definitions:

Local maximum at x=p exists if f(x)<=f(p) for all x near p

Local minimum at x=p exists if f(x)>=f(p) for all x near p

If there is a maximum or a minimum at x=p then f ’(p)=0

Obtaining a maximum or minimum can be done by solving f ‘(x) = 0

When a maximum exists at x=p, f’’(p)<0

When a maximum exists at x=p, f’’(p)>0

Minimuzation using the steepest descent or Gradient Method:

Assume we want to minimize f(x) at N variables where

points in the direction of greater increase.

For maximization: where h is the step size

For minimization:

Example:

Finding the minimum:

….

Solutions of Differential Equations:

Consider the equation:

By separating the variables and integrating,

Some differential equations are impossible to solve analytically though.

Euler’s Method

Let [a, b] be the interval over which we want to find the solution y ‘ = f(t, y) with

We will find a set of points, that approximate

Divide [a, b] into M equal subintervals with the step size,

Solve

Using the first two terms of the taylor series expansion of y(t),

Now to obtain

In general

Example:

Heun’s Method

Solve

Use fundamental theorem of calculus to integrate y’(t) over

and use numerical integration to approximate it.

Trapezoidal rule:

But we still need to know.

Use Euler’s Approximation:

In general:

Example:

Taylor Series Method

Want to solve y’=f(t, y)

We can obtain

Example: