2/9/2014Bairstow Method Next: Curve Fitting Up: Main Previous: Fixed point Iteration: Bairstow Method Bairstow Method is an iterative method used to find both the real and complex roots of a polynomial. It is based on the idea of synthetic division of the given polynomial by a quadratic function and can be used to find all the roots of a polynomial. Given a polynomial say, (B.1) Bairstow's method divides the polynomial by a quadratic function. (B.2) Now the quotient will be a polynomial (B.3) and the remainder is a linear function , i.e. (B.4) Since the quotient division the co-efficients and the remainder are obtained by standard synthetic can be obtained by the following recurrence relation. (B.5a) http://nptel.ac.in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/Rathish-kumar/ratish-1/f3node9.html 1/7 in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/Rathish-kumar/ratish-1/f3node9.(B.ac.html 2/7 . as: (B. So Bairstow's method reduces to is zero.7a) (B. second and higher order terms may be the improvement over guess value may be obtained by equating (B.e.5b) for If is an exact factor of real/complex roots of (B. terms i. Since both and are functions of r and s we can have Taylor series expansion of .2/9/2014 Bairstow Method (B. For finding such values Bairstow's method uses a strategy similar to Newton Raphson's method.6a) (B. we need the partial derivatives of http://nptel.6b) to zero i.e. (B.5c) then the remainder are the roots of considered based on some guess values for determining the values of r and s such that is zero and the .7b) To solve the system of equations .6a). so that . It may be noted that is .6b) For neglected. r and s. Now we can calculate the percentage of approximate errors in (r.10a) (B.html 3/7 .10b) These equations can be solved for to and turn be used to improve guess value .8a) (B.2/9/2014 Bairstow Method w.7b) may be written as.11) http://nptel. which amounts to using the recurrence relation replacing with and with i. (B.7a)-(B.e.ac.t.8b) (B.s) by (B. Bairstow has shown that these partial derivatives can be obtained by synthetic division of . (B.in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/Rathish-kumar/ratish-1/f3node9.r.8c) for where (B.9) The system of equations (B. The previous values of can serve as the starting guesses for this application. then we repeat the process with the new guess i.html 4/7 .2/9/2014 Bairstow Method If or . 2.in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/Rathish-kumar/ratish-1/f3node9.5a)-(B. If the quotient polynomial is a linear function say then the remaining single root is given by Example: Find all the roots of the polynomial by Bairstow method . Otherwise the roots of can be determined by (B.e.12) to obtain the . 3. With the initial values Solution: Set iteration=1 Using the recurrence relations (B.8a)-(B.5c) and (B. If the quotient polynomial is a third (or higher) order polynomial then we can again apply the Bairstow's method to the quotient polynomial.8c) we get http://nptel. If the quotient polynomial remaining two roots of is a quadratic function then use (B.12) If we want to find all the roots of then at this point we have the following three possibilities: 1. . where is the iteration stopping error.ac. 2/9/2014 Bairstow Method the simultaneous equations for and are: on solving we get and Set iteration=2 now we have to solve On solving we get http://nptel.ac.html 5/7 .in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/Rathish-kumar/ratish-1/f3node9. in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/Rathish-kumar/ratish-1/f3node9.e Exercises: (1) Use initial approximation form to find a quadratic factor of the of the polynomial equation using Bairstow method and hence find all its roots.ac.html 6/7 .2/9/2014 Bairstow Method Now proceeding in the above manner in about ten iteration we get Now on using with we get So at this point Quotient is a quadratic equation Roots of are: Roots are i. (2) Use initial approximaton to find a quadratic factor of the form of the polynomial equation http://nptel. ac.2/9/2014 Bairstow Method using Bairstow method and hence find all the roots.html 7/7 . Next: Curve Fitting Up: Main Previous: Fixed point iteration: http://nptel.in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/Rathish-kumar/ratish-1/f3node9.