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Numerical Methods
- 1. Numerical Methods ESUG, Gent 1999 Didier Besset Independent Consultant
- 14. Successive Approximations Begin Compute or choose initial object Compute next object Object sufficient? Yes End No
- 15. Iterative Processes NewtonZeroFinder TrapezeIntegrator SimpsonIntegrator RombergIntegrator EigenValues MaximumLikelihoodFit LeastSquareFit GeneticAlgorithm SimplexAlgorithm HillClimbing IterativeProcess FunctionalIterator ClusterAnalyzer InfiniteSeries ContinuedFraction
- 16. Implementation Begin iterations := 0 Yes Perform clean-up Compute next object iterations := iterations + 1 No iterations < maximumIterations Compute or choose initial object End No Yes precision < desiredPrecision
- 17. Methods evaluate finalizeIteration hasConverged evaluateIteration initializeIteration Compute or choose initial object Yes No Compute next object iterations := iterations + 1 Perform clean-up No Yes iterations := 0 iterations < maximumIterations Begin End precision < desiredPrecision
- 18. Simple example of use | iterativeProcess result | iterativeProcess := <a subclass of IterativeProcess> new. result := iterativeProcess evaluate. iterativeProcess hasConverged ifFalse:[ <special case processing> ].
- 19. Advanced example of use | iterativeProcess result precision | iterativeProcess := < a subclass of DhbIterativeProcess > new. iterativeProcess desiredPrecision: 1.0e-6; maximumIterations: 25. result := iterativeProcess evaluate. iterativeProcess hasConverged ifTrue: [ Transcript nextPutAll: ‘Result obtained after ‘. iterativeProcess iteration printOn: Transcript. Transcript nextPutAll: ‘iterations. Attained precision is ‘. iterativeProcess precision printOn: Transcript. ] ifFalse:[ Transcript nextPutAll: ‘Process did not converge’]. Transcript cr.
- 20. Class Diagram IterativeProcess evaluate initializeIteration hasConverged evaluateIteration finalizeIteration precisionOf:relativeTo: desiredPrecision (g,s) maximumIterations (g,s) precision (g) iterations (g) result (g)
- 22. Example of use | iterativeProcess result | iterativeProcess := <a subclass of FunctionalIterator> function: ( DhbPolynomial new: #(1 2 3). result := iterativeProcess evaluate. iterativeProcess hasConverged ifFalse:[ <special case processing> ].
- 23. Class Diagram initializeIteration computeInitialValues relativePrecision setFunction: functionBlock (s) FunctionalIterator IterativeProcess value: AbstractFunction
- 27. Newton’s Zero Finding x 1 f(x) x 2 x 3 y = f(x 1 ) + f’(x 1 ) · (x- x 1 )
- 28. Example of use | zeroFinder result | zeroFinder:= DhbNewtonZeroFinder function: [ :x | x errorFunction - 0.9] derivative: [ :x | x normal]. zeroFinder initialValue: 1. result := zeroFinder evaluate. zeroFinder hasConverged ifFalse:[ <special case processing> ].
- 29. Class Diagram evaluateIteration computeInitialValues (o) defaultDerivativeBlock initialValue: setFunction: (o) NewtonZeroFinder IterativeProcess FunctionalIterator value: AbstractFunction derivativeBlock (s)
- 30. Newton’s Zero Finding x 1 Effect of a nearly 0 derivative during iterations x 2 x 3
- 39. Example of use | matrix jacobi eigenvalues transform | matrix := DhbMatrix rows: #( ( 3 -2 0) (-2 7 1) (0 1 5)). jacobi := DhbJacobiTransformation new: matrix. jacobi evaluate. iterativeProcess hasConverged ifTrue:[ eigenvalues := jacobi result. transform := jacobi transform. ] ifFalse:[ <special case processing> ].
- 40. Class Diagram evaluateIteration largestOffDiagonalIndices transformAt:and: finalizeIteration sortEigenValues exchangeAt: printOn: JacobiTransformation IterativeProcess lowerRows (i) transform (g)
- 47. Example of use | server finder | server := < a subclass of DhbAbstractClusterDataServer > new. finder := DhbClusterFinder new: 15 server: server type: DhbEuclidianMetric. finder evaluate. finder tally
- 48. Class Diagram evaluateIteration accumulate: indexOfNearestCluster: initializeIteration finalizeIteration ClusterFinder IterativeProcess dataServer (i) clusters (i,g) accumulate: distance: isEmpty: reset Cluster openDataStream closeDataStream dimension seedClusterIn: accumulateDataIn: ClusterDataServer accumulator (i) metric (i) count (g)
- 50. Questions ? To reach me: [email_address]
Notas do Editor
- At that time, the radius of the earth was not known exactly. The radius of the circle corresponding to the error quoted by Jamshid Masud (0.7mm) is 20 times larger than the half axis of the orbit of Pluto.
- DhbFloatingPointMachine is a class which determines automatically the parameters of the floating point representation on the current machine.
- In this talk, only the classes in “pink” boxes will be discussed.
- Tasks in green boxes are the responsibility of the subclasses.
- Methods “evaluate” and “hasConverged” are methods of the super class. Methods “initializeIterations”, “evaluateIterations” and “finalizeIterations” are the responsibility of the subclasses.
- This is the abstract class at the top of the hierarchy of any iterative process. The class diagram shows the class name at top (without prefix), the methods (other than “get” and “set” of instance variables) in the middle and the instance variables at bottom. Instance variables have a corresponding “get” methods when tagged with a “g” and a corresponding “set” method when tagged with a “s”. The code is available in Listing 1.
- The code is available in Listing 2.
- The code is available in Listing 3.
- The code is available in Listing 4.
- The code is available in Listing 5.