chapter 1 scientific computing approximation in scientific computing (1.2) january 12, 2010

23
Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Upload: clementine-knight

Post on 27-Dec-2015

235 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Chapter 1Scientific Computing

• Approximation in Scientific Computing (1.2)

January 12, 2010

Page 2: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Absolute and Relative Errors

Page 3: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Example: Approximations

Floating-point number system

Irrational number has infinite digits in decimal expansion

Model Earth as an ellipsoid?

Page 4: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

General Strategy in Scientific Computing

Page 5: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Sources of Approximation

Page 6: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Computational and Data Errors

Page 7: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Truncation and Rounding Errors

Page 8: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Example: Finite Difference Approximation

By Taylor Expansion

Truncation Error

Page 9: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Example: Finite Difference Approximation

Minimizing mh/2 + 2epsilon /h

Rounding Error

Page 10: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010
Page 11: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Forward and Backward Errors

Page 12: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Example

(relative) backward error is about twice the forward error

Page 13: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Example: Backward Error Analysis

Page 14: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Example, cont.

(relative) forward and backward errors are similar.

Page 15: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Example -Sensitivity

Page 16: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Sensitivity and Conditioning

Page 17: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Condition Number

Page 18: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Example

Page 19: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Examples

1. What is the condition number of f (x) = sin(x) at x =0, pi/2 and pi?

cond# = | x cot (x) |

2. What is the condition number of f (x) = x2 + 2x at x =0, 1 and 10? For sufficiently large x?

Page 20: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Stability

Page 21: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Accuracy

Page 22: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Review Problems

• Homework One is out and it is due next Thursday.

(1.2) What are the approximate absolute and relative erros in approximating pi by a) 3 and b) 3.14?

(1.5) Consider the function f(x, y) = x–y. Measure the size of the input (x, y) by | x | + | y |, and assuming that | x | + |y | ~ 1 and x – y ~ ε show that cond(f) ~ 1 / ε. What can you conclude about the sensitivity of substration

Page 23: Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

(1.7) Let (b, p, U, L) be the four integers that characterize a floating-point number system. Given b= 10, what are the smallest values of p and U, and largest value of L such that both 2365.27 and 0.0000512 can be represented exactly in a normalized floating-point system?

(1.17) Let x be a given nonzero floating-point number in a normalized system and let y be an adjacent floating-point number, also nonzero.

a) What is the minimum possible spacing between x and y? b) What is the maximum possible spacing between x and y?

(1.12) In floating-point arithmetic, which expressions can be evaluated more accurately?

x2 –y2 or (x – y ) ( x + y)Example: x = 3469, y= 3451 b=10, p=3, chopping

Exact value = 124560, and …