by Lale Yurttas, Texas A&M UniversityChapter 2*Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.Chapter 2ROOTS OF EQUATIONS:POLYNOMIALS

Chapter 2

by Lale Yurttas, Texas A&M UniversityChapter 2*Roots of PolynomialsThe roots of polynomials such as

Follow these rules:For an nth order equation, there are n real or complex roots.If n is odd, there is at least one real root.If complex root exist in conjugate pairs (that is, l+mi and l-mi), where i=-1.

Chapter 2

by Lale Yurttas, Texas A&M UniversityChapter 2*Conventional MethodsIf only real roots exist, bracketing and open methods could be used. However finding good initial guesses complicates both the open and bracketing methods, also the open methods could be susceptible to divergence.To find the real and complex roots of polynomials Mller and Bairstow methods.

Chapter 2

by Lale Yurttas, Texas A&M UniversityChapter 2*MLLERS METHODObtains a root estimate by projecting a parabola to the x-axis through three function values.

Figure 7.3Secant methodMllers method

Chapter 2

Proceduresby Lale Yurttas, Texas A&M UniversityChapter 2*Choose three initial guesses: x0, x1 and x2 Find coefficient of a, b and c by using the following equations:

Chapter 2

by Lale Yurttas, Texas A&M UniversityChapter 2*Roots can be found by applying an alternative form of quadratic formula:

term yields two roots, the sign is chosen to agree with b. This will result in a largest denominator, and will give root estimate that is closest to x2.

The error can be calculated as

Chapter 2

by Lale Yurttas, Texas A&M UniversityChapter 2*Once x3 is determined, the process is repeated using the following guidelines:If only real roots are being located, choose the two original points that are nearest the new root estimate, x3.If both real and complex roots are estimated, employ a sequential approach just like in secant method, x1, x2, and x3 to replace xo, x1, and x2.

Chapter 2

Example 1Use Mllers method with guesses of xo, x1 and x2 = 4.5, 5.5, and 5, respectively, to determine a root of equation f(x) = x3 13x 12. Note that the roots of this equation are -3, -1 and 4. by Lale Yurttas, Texas A&M UniversityChapter 2*

Chapter 2

by Lale Yurttas, Texas A&M UniversityChapter 2*BAIRSTOWS METHODAn iterative approach loosely related to both Mller and Newton-Raphson methods.

Chapter 2

ProceduresChoose initial guesses of r & s:Find the value of bo, b1, b2, , bn by using the following equation:

Find the value of c1, c2, c3, , cn by using the following equation:

by Lale Yurttas, Texas A&M UniversityChapter 7*

Chapter 7

by Lale Yurttas, Texas A&M UniversityChapter 2*Then

At each step the error can be estimated as

Solved for Dr and Ds, in turn are employed to improve the initial guesses.

Chapter 2

by Lale Yurttas, Texas A&M UniversityChapter 2*

The values of the roots are determined byAt this point three possibilities exist:The quotient is a third-order polynomial or greater. The previous values of r and s serve as initial guesses and Bairstows method is applied to the quotient to evaluate new r and s values.The quotient is quadratic. The remaining two roots are evaluated directly, using the above eqn.The quotient is a 1st order polynomial. The remaining single root can be evaluated simply as x=-s/r.

Chapter 2