TAILIEUCHUNG - A textbook of Computer Based Numerical and Statiscal Techniques part 11

A textbook of Computer Based Numerical and Statiscal Techniques part 11. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 86 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES x6 - . Thus the approximation value to the root is correct up to five decimals. METHODS FOR COMPLEX ROOTS We now consider methods for determining complex roots of non-linear equations. Even if all coefficients of a non-linear equation are real the equation can have complex roots. The iterative methods like the Secant method or the Newton-Raphson method are applicable to complex roots also provided complex arithmetic is used. Starting with the complex initial approximation if the iteration converges to a complex root then the asymptotic convergence rate is the same as that for a real root. The problem of finding a complex root off z 0 where z is a complex variable is equivalent to finding real values x and y such that f z f x iy u x y iv x y 0 Where u and v are real functions. This problem is equivalent to solving a system of two non-linear equations in two real unknowns x and y u x y 0 v x y 0 Which can be solved using the methods discussed in previous section. Example 6. Find all roots of the equation f x x3 2x2 - x 5 using Newton-Raphson method. Use initial approximations x0 - 3 for real root and x0 1 i for complex root. Sol. Given f x x3 2x2 - x 5 f x 3x2 4x - 1 Newton-Raphson formula is given by x x - fM -1 f x L For real root Taken initial approximation as x0 - 3. x _ IM 0 f xo L - 3 - IT x - fx 1 f x1 L First approximation Second approximation x1 x1 x2 - x2 - Third approximation x2 - X3 x - fxl x2 - f x 87 ALGEBRAIC AND TRANSCENDENTAL EQUATION x3 - v 97846797 x3 - Since the second and third approximations are same for five decimals hence the real root is correct up to five decimals. For complex root Initial approximation is x0 1 i First approximation x - X - fxl 1 0 f X0 1 i 3 2 1 i 2 - 1 i 5 X 1 i 1 3 1 i 4 1 i - 1 x1 53 1 114 i 1 109 Thus i is .

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