Final Year Project Presentation

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Final Year Project Presentation

• Bsc(Hons) Mathematics for Finance and Management,
  University of Portsmouth
• Emmanouil Christinidis
• Project Title Numerical Integration with
  Gauss-Turan Formulae and Matlab programming
• Supervisor Dr. A. Makroglou
• 2nd Assessor Prof. M. Tamiz

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Aim of the project

• To study some of the theory of Gauss-type
  integration rules and their extensions known with
  the name Gauss-Turan rules,
• To develop computer programs, implementing the
  corresponding algorithms,
• To apply the methods for solving an application
  in the area of numerical least squares
  approximation.

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• Programming was done in Matlab
• Main References
• The research papers by
• C. A. Micchelli, T. J. Rivlin, Turan Formulae and
  Highest Precision Quadrature Rules for Chebyshev
  Coefficients, IBM J. Res. Develop., 1972, p.
  372-379
• B. Bojanov, On a quadrature formula of Micchelli
  and Rivlin, J. Comp. Applied Mathematics, 70
  (1996), p. 349-356.

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Outline of the talk

• Form of the problem
• Applications - Motivation
• Gauss-Chebyshev quadrature an introduction
• Numerical results
• Conclusions and further research

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Form of the problem

• Types of integrals
• where Chebyshev polynomial of degree
  n given by

-1 x 1.
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2. Applications - Motivation

• Chebyshev polynomials, and Gauss-type
  integration have important application in
  numerical analysis
• Differential equations (ODEs, PDEs)
• Integral equations
• Approximation theory
• and in various applied sciences like economics
  and finance, probability theory.

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2. Applications - Motivation

• An example from approximation theory (Best
  Approx.)
• Given
• Find
• where

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2. Applications Motivation ctd.

• s.t. in the least square
  sense.
• Solution

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3. Gauss-Chebyshev Quadrature - an introduction

• Special case of integrals of the form
• with

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3. Gauss-Chebyshev Quadrature - an introduction
ctd.

• General form of rule
• Characteristic of Gauss-type rules
• Both the nodes and the weights have to
  be estimated.
• Weights are all equal.

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3. Gauss-Chebyshev Quadrature - an introduction,
ctd.

• Derivation an overview
• Two methods used in the literature.
• Based on the use of orthogonal polynomials,
• Based on the idea of trying to make the rule
  exact for special choices of
• An example using (2) method A 2-point rule.

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3. Gauss-Chebyshev Quadrature an introduction,
ctd.

• The formula
• Error 0 for a polynomial of degree up to
  ( degree of precision).

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4. Numerical Results

• Method 3-point Gauss-Chebyshev rule
• Choices of

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4. Numerical Results, ctd.

• Results for the integral of
• using a 3-point Gauss-Chebyshev method
• f(x) Computed value exact_value
  error
• x 1 0.0000000000 0.0000000000
  1.1102230246e-016
• x 2 1.5707963268 1.5707963268
  2.2204460493e-016
• x 3 0.0000000000 0.0000000000
  0.0000000000e000
• x 4 1.1780972451 1.1780972451
  2.2204460493e-016
• x 5 0.0000000000 0.0000000000
  0.0000000000e000
• x 6 0.8835729338 0.9817477042
  9.8174770425e-002
• x 7 0.0000000000 0.0000000000
  0.0000000000e000
• x 8 0.6626797004 0.8590292412
  1.9634954085e-001

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4. Numerical Results, ctd.

• results for the integral of
• using the Gauss-Chebyshev method
• n Computed value
• 3 3.9773219601
• 4 3.9774626347
• 5 3.9774632588
• 6 3.9774632605
• 7 3.9774632605
• 8 3.9774632605
• 9 3.9774632605
• 10 3.9774632605

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4. Numerical Results, ctd.
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5. Conclusions and further research

• Summarizing Numerical results were presented for
  the
• evaluation of integrals of the form
• using
  Gauss-Chebyshev rules.
• An application in numerical approximation.
• Some of the results obtained using Matlab were
  verified by using Maple.
• The degree of precision of the rules were also
  verified computationally.

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5. Conclusions and further research, ctd.

• The approximation to was good even though
  3 terms were only included.
• Note The involved integrals were evaluated
  w.r.t. a 10-point Gauss-Chebyshev rule.
• More results are to be presented in the project
  involving more test problems and some Gauss-Turan
  rules.

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5. Conclusions and further research, ctd.

• Ideas for further research
• Study convergence analysis (a research paper was
  found, which is to be published in the Journal
  Mathematics of Computation not out yet).
• More applications
• Program more general Gauss-Turan rules from the
  literature.

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Additional References

• Ph. J. Davis and Ph. Rabinowitz, Methods of
  Numerical Integration, Academic Press, 1975
• G. P. Weeg and G. B. Reed, Introduction to
  Numerical Analysis, Blaisdell Publishing Co,
  1966.