CS 221 Lecture 11 Numerical Integration Tuesday, 22 November 2011 - - PowerPoint PPT Presentation

CS 221 Lecture 11 Numerical Integration

Tuesday, 22 November 2011

Today’s Agenda

• 1. Announcements
• 2. Numerical Integration
• 3. Class Quiz 3

Announcements

• Last In-Class Quiz is today.
• No Labs Thursday – Happy Thanksgiving!

– Remember to be thankful.

• Lab Quiz 3 next Thursday, 1 December

– Coverage: simultaneous equation-solving, programming – More details in next Lecture

– –

Numerical Integration

• Sometimes it’s difficult or impossible to integrate

a function symbolically

– Example:

The antiderivative of f(x) = exp(-x2) can’t be expressed in elementary form.

– Example:

The function f(x) may have been obtained by measurements (sampling), so there’s no symbolic definition to work with.

• Numerical integration provides a way to

determine the definite integral (area under a given region of the curve of f(x))

Built-in Integration Functions

• Quadrature: numerical integration of a function of
• ne variable
• quad(fun,a,b) computes the integral of fun(x)

from a to b

• Uses adaptive Simpson quadrature

– To within error tolerance of 10-6 – fun(x) must be vectorizable (uses only vector’able ops)

• Some functions are not vectorizable
• quad() silently gives the wrong answer!
• Other variations: quadl(), quadgk()

Example

function y = fooey(x) if x < 1 y = -x + 2; else y = (x-1)^2 + 1; end end Want to compute the definite integral of fooey(x) from 0 to 3. This function is not vectorizable (because of the “if-else”.

– Correct answer: 6.1667

Implementing Trapezoidal Integration

• Compute trapezoid width:

(upper – lower)/(number of slices)

• Compute vertices of all the trapezoids

– linspace(lower,upper,n):

Creates a vector of n equally-spaced elements with first element = lower, last element = upper

• Step through the vector of vertices, computing

the area of each trapezoid and adding it to the total area

– for loop