Lecture05 Complete

7/27/2019 Lecture05 Complete

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Lecture05

Lecture 05: Rootfinding cont'dsolving non-linear equations f(x)=0

Outline

1) Classic Algorithms for solving f(x)=0

Bisection: review and finding a bracket

Newton's Method

Examples: f(x)=x-exp(-x)

Convergence of Newton's Method

Failure of Newton's Method

Secant Methods

2) Comparison of convergence

f(x)=x-exp(-x)

f(x)=A-mP/x*[(1+x/m)*mn-1]

3) Fail-safe hybrid methods

NewtSafe (Newton + Bisection: Numerical Recipes)

Brent's method (Secant+IQI+Bisection)

Matlab's fzero (Brent's method)


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Rootfinding: solve f(x)=0

f(r) = A - m*P/r [ (1-r/m)mn

-1] =0

Classical Methods:

Bisection (linear convergence)Newton's Method (Quadratic)Secant Method (super-linear)

Combined Methods

RootSafe (Newton + Bisection)Brent's method (fzero in matlab)

Bisection:

Given a bracket [a,b]:cut the interval in half and find the newbracket. Repeat until the bracket width is < tol. Given an initialbracket, bisection is guaranteed to converge.


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Finding a bracket:

1) Plot your function!2) Guess and Grow:

given x0, GROWTH_FACTOR:set dx

a=x0, b=x0, fa=f(a), fb=f(b)

while (not bracket(a,b))a=a-dxtestbracket(a,b)b=b+dx;testbracket(a,b)dx=(b-a)*GROWTH_FACTOR

end

Faster Methods: Newton's method

Points:

1) Given a bracket, Bisection is guaranteed to converge linearly to a root2) However: Bisection uses almost no information about f(x) beyond itssign3) Can we do better?

Newton's method: (Newton-Raphson)

Simple idea: given f(x) and f'(x), pretend f is locally linear and predictthe zero crossing:

Algebraically:


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Newton's method: the geometric picture

Newton's method:

Example f(x) = x-exp(-x)


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Asymptotic Convergence of Newton's method:

2) multiple root e.g. f(x)=(x-1)2

(this is a mess...I need to clean these notes up)

Fragility and Failure of Newton's method:

Example: f(x)=sin(2x)


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Fragility and Failure of Newton's method:

Example: f(x)=sin(2x)

xk+1

Other Issues with Newton's method:

Need to supply both f(x) and f'(x)

Example: FTV equation f(r) = A - m*P/r*[(1+r/m)mn

-1]

(Cheat: symbolic differentiation, matlab symbolic toolbox (Maple),Mathematica)

syms A m P r nf = A-m*P/r*((1+r/m)^(m*n)-1)diff(f,r)

Point: would like rootfinding method with convergence behavior ofNewton but without derivatives


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Secant Methods:

Idea: replace f'(x) in Newton with finite difference approximation

Given xk

and xk-1

let f'(x) ~

or choose the next point as

Secant Methods:Geometric Picture


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Secant Methods:The Algorithm

Given f(x) and bracket [a,b]

initialize x1

=a, x2=b f

1=f(x

1), f

2=f(x

2)

Equivalence of Secant method and

linear interpolation


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Higher order Secant Methods:Inverse Quadratic Interpolation

Idea: Rather than use linear interpolation, use quadratic interpolation

Secant Methods: Comments

Similar to Newton's Method

1) Convergence is Superlinear (but not quite quadratic)

2) Secant methods are not guaranteed to converge

3) Secant methods do not preserve brackets

But with good guess can be almost as good as Newton's method withoutderivatives


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Hybrid Methods:

Design Goals:

1) Robustness: Given a bracket [a,b], maintain the bracket

2) Efficiency: Use superlinear convergent methods when possible

Some Options:

Have derivatives:NewtSafe (or RootSafe, Numerical Recipes)

Newton's method within a bracket, Bisection otherwise

No Derivatives:

Brent's Algorithm (zbrent Numerical Recipes, fzero Matlab)Returns minimum bracket using combination ofBisectionSecant methodInverse Quadratic Interpolation

NewtSafe Algorithm:


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NewtSafe Algorithm:

Lecture05 Complete

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