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ECE 595, Section 10Numerical Simulations

Lecture 2: Problems in Numerical

Computing

Prof. Peter Bermel

January 9, 2013

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Outline

Overall Goals

Finding Special Values

Fourier Transforms Eigenproblems

Ordinary Differential Equations

Partial Differential Equations

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Recap: Goals for This Class

Learn/review key mathematics

Learn widely-used numerical

techniques

Become a capable user of thissoftware

Appreciate strengths andweaknesses of competing

algorithms Convey your research results to an

audience of colleagues

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RW Hamming (left),

developing error-

correcting codes (AT&T)

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Finding Special Values

Finding zeros

1D versus multidimensional

Speed versus certainty

Finding minima and maxima (optimization)

Convex vs. non-convex problems

Global vs. local search

Derivative vs. non-derivative search

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Finding Zeros

Key concept: bracketing

Bisection continuouslyhalve intervals

Brents method adds

inverse quadraticinterpolation

Newton-Raphson method uses tangent

Laguerres method assumespacing of roots at a and b:

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Finding Minima (or Maxima)

Golden Section Search

Brents Method

Downhill Simplex Conjugate gradient

methods

Multiple level, singlelinkage (MLSL)

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These and further images from Numerical

Recipes, by WH Press et al.

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Recap: Fourier Transforms

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J.W. Cooley (IEEE

Global HistoryNetwork)

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Cooley-Tukey Algorithm

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Recap: Eigenproblems

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Eigenproblems

Direct method: solve det(A-1)=0

Similarity transformations: A Z-1AZ

Atomic transformations: construct each Z explicitly

Factorization methods: QR and QL methods

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Atomic Transformations in

Eigenproblems

Jacobi

Householder

Keep iterating until off-diagonal elements aresmall, or use factorization approach

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Factorization in Eigenproblems

Most common approach known as QR method

Can also do the same with Slow in general, but fast in certain cases:

Tridiagonal matrices

Hessenberg matrix

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Ordinary Differential Equations

Euler method nave rearrangement of ODE

Runge-Kutta methods match multiple Euler

steps to a higher-order Taylor expansion

Richardson extrapolation extrapolate

computed value to 0 step size

Predictor-corrector methods store solution

to extrapolate next point, and then correct it

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ODE Boundary Value Problems

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Boundary Value MethodShooting Method

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Partial Differential Equations

Classes: parabolic, hyperbolic, and elliptic

Initial value vs. boundary value problems

Finite difference

Finite element methods

Monte-Carlo

Spectral

Variational methods

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Next Class

Introduction to computational complexity Please read Chapter 1 of Computational

Complexity: A Modern Approach by Arora& Barak

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http://www.cs.princeton.edu/theory/complexity/modelchap.pdfhttp://www.cs.princeton.edu/theory/complexity/modelchap.pdfhttp://www.cs.princeton.edu/theory/complexity/modelchap.pdfhttp://www.cs.princeton.edu/theory/complexity/modelchap.pdfhttp://www.cs.princeton.edu/theory/complexity/modelchap.pdfhttp://www.cs.princeton.edu/theory/complexity/modelchap.pdfhttp://www.cs.princeton.edu/theory/complexity/modelchap.pdf