today’s class roots of equation finish up incremental search open methods numerical methods,...

Today’s class

• Roots of equation• Finish up incremental search • Open methods

Numerical Methods, Lecture 5 1

Prof. Jinbo Bi CSE, UConn

• Although the interval [a,b] where the root becomes iteratively closer with the false position method, unlike the bisection method, the size of the interval does not necessarily converge to zero.

• Sometimes it can cause the false position to converge slower than bisection

False Position Method



• Modified False Position Method• Detect when you get stuck and use a

bisection method• Can get you to convergence faster




• Dependent on knowing the bracket in which the root falls

• Can use bracketed incremental search to speed up exhaustive search• How big a bracket or increment can

determine how long the search will take• Too small increment and it will take too long• Too big increment may miss roots, in

partular, the multiple roots

Incremental Searches



Incremental Searches



Open Methods

• Bracket methods depend on knowing the interval in which the root resides

• What if you don’t know the upper and lower bound on the root?

• Open methods• Use a single estimate of the root• Use two starting points but not bracketing the

root• May not converge on root



Open Methods



Open Methods

• Fixed-Point Iteration• One-point iteration• Successive substitution

• Start with equation f(x) = 0 and rearrange so x is on left hand side.

• If algebraic manipulation doesn’t work, just add x to both sides



Fixed-point iteration

• The function transformation allows us to use g(x) to calculate a new guess of x



• Find root of f(x)=e-x-x• Transform f(x)=0 to x=g(x)=e-x

• Start with an estimate of x0=0

• x1=g(x0)=e-0=1

Example



Example

• true value of the root: 0.56714329



Example




• Convergence properties• If converge, much faster than bracketing

methods• May not converge• Depends on the curve characteristics




Numerical Methods,Lecture 5 17


• Assume xr is the true root

• Combine with the iterative relationship

Convergence Analysis



• Use derivative mean-value theorem

• If the derivative is less than 1, the error will get smaller with each iteration (monotonic or oscillating).

• If the derivative is greater than 1, the error will get larger with each iteration.




• Similar idea to False Position Method• Use tangent to guide you to the root

Newton-Raphson Method



• Find root of f(x)=e-x-x

• Start with an estimate of x0=0

Example



Example




• Convergence analysis• First-order Taylor series expansion

• At root




• Newton-Raphson method is quadratically convergent


2,1, )('2

)("it

r

rit E

xf

xfE



• Problems and Pitfalls• Slow convergence when initial guess is not

close enough• May not converge at all• Problems with multiple roots




• Algorithm should guard against slow convergence or divergence

• If slow convergence or divergence detected, use another method




• Newton-Raphson method requires calculation of the derivative

• Instead, approximate the derivative using backward finite divided difference

Secant method



• From Newton-Raphson method

• Replace with backward finite difference approximation

Secant method




• Start with an estimate of x-1=0 and x0=1

Example



Example




• False-Position method always brackets the root

• False-Position will always converge• Secant method may not converge• Secant method usually converges much

faster

Secant Method vs. False-Position Method



Secant Method vs. False-Position Method



• Instead of using backward finite difference to estimate the derivative, use a small delta

• Substitute back into Newton-Raphson formula

Modified Secant Method




• Start with an estimate of x0=1 and δ=0.01

Example



Example




• Polynomial roots• Read Chapter 7

Next class



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