Upload
joris-schelfaut
View
221
Download
1
Tags:
Embed Size (px)
DESCRIPTION
Slides based on "Inleiding tot de numerieke wiskunde" by Prof. Dr. Adhemar Bultheel and course notes by Prof. Dr. Marc Van Barel.
Citation preview
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.