Today’s class Roots of equation Finish up incremental search Open methods Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

False Position Method 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 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

False Position Method Numerical Methods, Prof. Jinbo Bi CSE, UConn Lecture 5 Prof. Jinbo Bi CSE, UConn

False Position Method Modified False Position Method Detect when you get stuck and use a bisection method Can get you to convergence faster Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Incremental Searches 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 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Incremental Searches Numerical Methods, Prof. Jinbo Bi CSE, UConn Lecture 5 Prof. Jinbo Bi CSE, UConn

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 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Open Methods Numerical Methods, Prof. Jinbo Bi CSE, UConn Lecture 5 As shown in this Figure, Bracket methods shrinks the search interval, by repeating this process, it always results in better estimate of true root. This means convergence. Now for open method, we start from one single estimate, or perhaps two estimates that do not necessarily bracket the root. Then we apply a formula to get next estimate. This does not guarantee convergence. Look at this figure. Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

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 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Fixed-point iteration The function transformation allows us to use g(x) to calculate a new guess of x Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example 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 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example true value of the root: 0.56714329 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example Numerical Methods, Prof. Jinbo Bi CSE, UConn Lecture 5 Here are the two graphical methods to determine the root of the function f(x). One method is the standard way that we draw the curve that corresponds to the function itself, then we look at where this curve intersects with the x-axis to find a root. The other method is to examine the function so we find finding the root of f(x) is the same as finding the value of x so that this e(-x) = x, which means we look at the intersection of the two functions, one is e(-x), the other is f(x) = x. So the fixed point method follows the idea of the second graphical method. Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Fixed-point iteration Convergence properties If converge, much faster than bracketing methods May not converge Depends on the curve characteristics Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Fixed-point iteration Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Fixed-point iteration Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Fixed-point iteration Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Fixed-point iteration Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Convergence Analysis Assume xr is the true root Combine with the iterative relationship Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Fixed-point iteration 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. Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Similar idea to False Position Method Use tangent to guide you to the root Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example Find root of f(x)=e-x-x Start with an estimate of x0=0 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example true value of the root: 0.56714329 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Convergence analysis First-order Taylor series expansion At root Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Newton-Raphson method is quadratically convergent If Newton-Raphson method does converge, The error at each iteration is roughly proportional to the square of the previous error. This means that the number of correct decimal places approximately doubles with each iteration. Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Problems and Pitfalls Slow convergence when initial guess is not close enough May not converge at all Problems with multiple roots Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method Algorithm should guard against slow convergence or divergence If slow convergence or divergence detected, use another method Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Secant method Newton-Raphson method requires calculation of the derivative Instead, approximate the derivative using backward finite divided difference Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Secant method From Newton-Raphson method Replace with backward finite difference approximation Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example Find root of f(x)=e-x-x Start with an estimate of x-1=0 and x0=1 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example true value of the root: 0.56714329 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Secant Method vs. False-Position Method False-Position method always brackets the root False-Position will always converge Secant method may not converge Secant method usually converges much faster Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Secant Method vs. False-Position Method Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Modified Secant Method Instead of using backward finite difference to estimate the derivative, use a small delta Substitute back into Newton-Raphson formula Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example Find root of f(x)=e-x-x Start with an estimate of x0=1 and δ=0.01 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Example true value of the root: 0.56714329 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn

Next class Polynomial roots Read Chapter 7 Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn