Numerical Analysis 2. Condition and Stability

27
Numerical Analysis Fault analysis: condition and stability

description

Slides based on "Inleiding tot de numerieke wiskunde" by Prof. Dr. Adhemar Bultheel and course notes by Prof. Dr. Marc Van Barel.

Transcript of Numerical Analysis 2. Condition and Stability

Page 1: Numerical Analysis 2. Condition and Stability

Numerical AnalysisFault analysis: condition and stability

Page 2: Numerical Analysis 2. Condition and Stability

Overview

•Description of a numerical problem•Condition•Condition : example•Numerical stability•Numerical stability : forward stability•Numerical stability : weak stability•Numerical stability : backward stability•Numerical stability : example

Page 3: Numerical Analysis 2. Condition and Stability

Overview

•Description of a numerical problem•Condition•Condition : example•Numerical stability•Numerical stability : forward stability•Numerical stability : weak stability•Numerical stability : backward stability•Numerical stability : example

Page 4: Numerical Analysis 2. Condition and Stability

Description of a numerical problem

•A relation F between data g and results rr = F(g)

▫F is an exact mathematical description of the relation

▫Different methods may apply to the same description

Page 5: Numerical Analysis 2. Condition and Stability

Overview

•Description of a numerical problem•Condition•Condition : example•Numerical stability•Numerical stability : forward stability•Numerical stability : weak stability•Numerical stability : backward stability•Numerical stability : example

Page 6: Numerical Analysis 2. Condition and Stability

Condition

•Definition:“the condition of a numerical problem

indicates how much the result r is being influenced if the data g are altered”

•Exact relationship•Characteristic to a certain problem•Independent of the method

Page 7: Numerical Analysis 2. Condition and Stability

Condition

•Definitions:

Page 8: Numerical Analysis 2. Condition and Stability

Condition

Page 9: Numerical Analysis 2. Condition and Stability

Condition

•Condition number:▫Ratio of the error on the result and the

error on the data▫Absolute condition kA and relative condition

kR

Page 10: Numerical Analysis 2. Condition and Stability

Condition

•If F(g) is a differentiable function:

Page 11: Numerical Analysis 2. Condition and Stability

Overview

•Description of a numerical problem•Condition•Condition : example•Numerical stability•Numerical stability : forward stability•Numerical stability : weak stability•Numerical stability : backward stability•Numerical stability : example

Page 12: Numerical Analysis 2. Condition and Stability

Condition : example

•What is the condition of the evaluation of the function f :

•Using the formula from the previous section:

Page 13: Numerical Analysis 2. Condition and Stability

Condition : example

•What can we conclude?▫De denominator approaches zero for values

{x1 = –1; x2 = 3/2}

▫For these values the function is ill-conditioned, as the relative error becomes very large.

Page 14: Numerical Analysis 2. Condition and Stability

Overview

•Description of a numerical problem•Condition•Condition : example•Numerical stability•Numerical stability : forward stability•Numerical stability : weak stability•Numerical stability : backward stability•Numerical stability : example

Page 15: Numerical Analysis 2. Condition and Stability

Numerical stability

•Implementing an exact relation F is usually not feasable:▫Discretization▫Rounding errorF*

• Definition:• “numerical stability measures the deviation

of F* (the approximation) from F (the exact result).”

Page 16: Numerical Analysis 2. Condition and Stability

Numerical stability : forward stability

•Given by:

Page 17: Numerical Analysis 2. Condition and Stability

Numerical stability : forward stability

Page 18: Numerical Analysis 2. Condition and Stability

Numerical stability : weak stability

Page 19: Numerical Analysis 2. Condition and Stability

Numerical stability : weak stability

Page 20: Numerical Analysis 2. Condition and Stability

Numerical stability : backward stability

•The idea is the following:▫Consider the result r* = F(g) to be the

exact result▫Find data g*’ corresponding to r*▫Measure the stability with the following:

Page 21: Numerical Analysis 2. Condition and Stability

Numerical stability : backward stability

Page 22: Numerical Analysis 2. Condition and Stability

Overview

•Description of a numerical problem•Condition•Condition : example•Numerical stability•Numerical stability : forward stability•Numerical stability : weak stability•Numerical stability : backward stability•Numerical stability : example

Page 23: Numerical Analysis 2. Condition and Stability

Numerical stability : example• Investigate the stability of algorithms A and B for the

function f:

Page 24: Numerical Analysis 2. Condition and Stability

Numerical stability : example

Page 25: Numerical Analysis 2. Condition and Stability

Numerical stability : example

•Resulting formula:

for x1 = 0, the relative error is large, and the condition is small:Unstable

but for -3/2 the problem is also ill-conditionedStability is weak

Page 26: Numerical Analysis 2. Condition and Stability

Numerical stability : example

•Can you evaluate algorithm B?

Page 27: Numerical Analysis 2. Condition and Stability

Sources• “Inleiding tot de numerieke wiskunde”, A.

Bultheel, 2007, Acco• http://en.wikipedia.org/wiki/Numerical_analysis• http://en.wikipedia.org/wiki/Condition_number

By knowledgedriver, 2012.