Numerical AnalysisFault analysis: 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

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

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

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

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

Condition

•Definitions:

Condition

Condition

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

error on the data▫Absolute condition kA and relative condition

kR

Condition

•If F(g) is a differentiable function:

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

Condition : example

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

•Using the formula from the previous section:

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.

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

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).”

Numerical stability : forward stability

•Given by:

Numerical stability : forward stability

Numerical stability : weak stability

Numerical stability : weak 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:

Numerical stability : backward 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

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

function f:

Numerical stability : example

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

Numerical stability : example

•Can you evaluate algorithm B?

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.