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

Absolute and Relative Errors

Example: Approximations

Floating-point number system

Irrational number has infinite digits in decimal expansion

Model Earth as an ellipsoid?

General Strategy in Scientific Computing

Sources of Approximation

Computational and Data Errors

Truncation and Rounding Errors

Example: Finite Difference Approximation

By Taylor Expansion

Truncation Error

Example: Finite Difference Approximation

Minimizing mh/2 + 2epsilon /h

Rounding Error

Forward and Backward Errors

Example

(relative) backward error is about twice the forward error

Example: Backward Error Analysis

Example, cont.

(relative) forward and backward errors are similar.

Example -Sensitivity

Sensitivity and Conditioning

Condition Number

Example

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?

Stability

Accuracy

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

(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 …