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