MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante IMSP, UPLB 1st Sem AY 2012-2013

Other Method: Bairstow’s Method A special problem associated with polynomials Pn(x) is the possibility of complex roots. Newton's method, the secant method, and Muller's method all can find complex roots if complex arithmetic is used and/or complex initial approximations are specified. Fortunately, complex arithmetic is available in several programming languages, such as FORTRAN and SCILAB. However, hand calculation using complex arithmetic is tedious and time consuming. When polynomials with real coefficients have complex roots, they occur in conjugate pairs, which corresponds to a quadratic factor of the polynomial Pn(x). Bairstow's method extracts quadratic factors from a polynomial using only real arithmetic.

Other Method: Bairstow’s Method Given a polynomial: We can factor a quadratic factor: Quotient term Remainder term

Other Method: Bairstow’s Method What we want to do is make the remainder=0, and find the correct r and s. Then use quadratic formula!!! After getting the correct factor, do synthetic division or long division to get the other factors (of course you can do Bairstow’s again if necessary.)

Other Method: Bairstow’s Method FORMULA: (Derivation omitted) User input:

Other Method: Bairstow’s Method

Other Method: Bairstow’s Method Stopping criteria: (choose one, or you can use both) See Scilab syntax for example (in MS Word file).

Other Methods: Laguerre’s Method Jenkins-Traub Method Brent’s Method - uses a superlinear method (i.e., inverse quadratic interpolation) and monitors its behavior to ensure that it is behaving properly. If not, some interval halving steps are used to ensure at least linear behavior until the root is approached more closely, at which time the procedure reverts to the superlinear method. Brent's method does not require evaluation of the derivative. This approach combines the efficiency of open methods with the robustness of closed methods.

Other Methods: Graeffe’s Method - an algorithm for finding all of the roots of a polynomial. Eigenvalues of Companion Matrix Etc. etc. etc…. NOW, let’s discuss Wilkinson’s Polynomial…

Wilkinson’s Polynomial Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of the roots can be very sensitive to perturbations in the coefficients of the polynomial. In 1984, he described the personal impact of this discovery: “Speaking for myself I regard it as the most traumatic experience in my career as a numerical analyst.”

Wilkinson’s Polynomial So what are the roots?

Wilkinson’s Polynomial

Wilkinson’s Polynomial If the coefficient of x19 is changed from −210 to −210.0000001192, then the polynomial value w(20) decreases from 0 to −6.25×1017, and the root at x = 20 grows to x ≈ 20.8. Actually, other roots are also greatly displaced. The change to the coefficient is tiny, and this ill- conditioning is rampant when using computers. Stability Analysis must be done to check if the function is well-conditioned or ill-conditioned (stability analysis in root-finding will not be discussed).

Wilkinson’s Polynomial

END OF CHAPTER 1 Next topics: (these are optional topics in our course outline) -Solving Nonlinear Systems (using Newton’s Method) and -Numerical Optimization (using Golden- section Search and Newton’s Method)