Numerical Analysis 2. Condition and Stability
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Transcript of Numerical Analysis 2. Condition and Stability
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.