CS 205A class_8 notes

Scientific Computing

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
CS205 – Class 8 Covered In Class : 1, 3, 4, 5, 6 Reading : Heath Chapter 6 1. Optimization – given an objective function f , find relative maxima or minima. Note that since max f = min f it is enough to only consider minima. a. We’ll start with scalar functions f of one variable for now. b. unconstained – any x ± R is acceptable c. constrained – minimize f on a subset S R d. usually find local minima, since global minima are hard to find i. one option is to find many local minima and compare them to find a global minimum e. Not equivalent to solving for f(x) = 0. There might exist no such x or the minimum may be attained somewhere f(x) < 0. f. poorly conditioned since '( ) 0 fx = at a minimum, i.e. locally flat (similar to a multiple root) – error tolerance should be more like ε as opposed to g. given a critical point where '( ) 0 = , we can use the sign of the second derivative to determine whether we have a local minimum, a local maximum, or an inflection point i. if ''( ) 0 > , concave up, minimum ii. if ''( ) 0 < , concave down, maximum iii. otherwise when the second derivative vanishes, we have an inflection point, i.e. neither a minimum nor a maximum -2 0 2 0 2 -15 -10 -5 0 0 2 0 2 0 5 10 15 -2.5 0 2.5 5 0 5 -20 0 20 Local maxima ''( ) 0 > Local minima 0 < Saddle 0 = h. unimodal [,* ] ax is monotonically decreasing and [* ,] x b is monotonically increasing * x is the minimum – most schemes need a unimodal interval in order to converge 2.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.

Page1 / 3

CS 205A class_8 notes - CS205 Class 8 Covered In Class: 1,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online