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ECE 595, Section 10Numerical Simulations

Lecture 6: Finding Special Values

Prof. Peter Bermel

January 18, 2013

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Outline

Recap from Wednesday

Root Finding Bisection

Newton-Raphson method Brents method

Optimization Golden Section Search

Brents Method Downhill Simplex

Conjugate gradient methods

Multiple level, single linkage (MLSL)

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Recap from Wednesday

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Crouts algorithm

Band diagonal

sparse matrix

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Finding Zeros

Relevance in micro & nanoresearch

Key concept: bracketing

Bisection continuously halveintervals

Newton-Raphson method uses tangent

Laguerres method forpolynomials

Brents method adds inversequadratic interpolation

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Importance of Bracketing

Critically important for both root finding and

optimization

Can always guarantee at least one solution for

continuous functions with sign change in 1D

If more than one solution present, may not be

able to guarantee which one is reached

method-dependent

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Bisection

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Newton-Raphson Method

Key assumption: system

is nearly linear in region

between starting point

and root

When sufficiently close,

converge quadratically on

correct value (from Taylorexpansion)

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NR Method Failures

Getting stuck in a limit

cycle is possible

Can even get worse

certain locally flat

curves can send you

into outer space!

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Laguerres Method

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Brents Method: Finding Roots

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Optimization

Relevance in micro & nanoresearch

Convexity

Search classifications

Techniques: Brents Method

Golden Section Search

Downhill Simplex

Conjugate gradientmethods

Multiple level, singlelinkage (MLSL)

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These and further images from Numerical

Recipes, by WH Press et al.

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Convexity

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Search Types

Local assumes convex/concave problem

Global uses heuristics to deal with multiple

optima

Non-derivative based no specific

assumptions about best search direction

Derivative based incorporates derivatives to

determine search direction

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Brents Method: Finding Optima

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Next Class

Is on Wednesday, Jan. 23 (because of Martin LutherKing, Jr. Day)

Discussion of optimization and eigenproblems

Please read Chapter 11 of Numerical Recipes byW.H. Press et al.

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