A New Technique to Compute Complex Roots

In this paper we modified Newton – Raphson and Secant methods to Complex form, then from that we get new iterative formula to them, named Newton (Secant) real iteration formula and Newton (Secant) imaginary iteration formula, real iteration used to find real roots while imaginary iteration one will be used to find complex roots of nonlinear equations such as ( 0 2   a x ). To find real roots, we start with a real initial point in real iteration getting a sequence of real numbers, to find complex roots start with a complex initial point in complex formula getting a sequence of complex numbers, also we write a new algorithm for this technique and write the program by using Matlab application system version 7.8 for this new method such that it will determine the real roots when we enter the initial real point but when we enter the initial complex point it will determine the complex roots. Introduction Complex analysis can roughly be though of as a subject that applies the theory of calculus to imaginary numbers. But what is exactly are imaginary numbers ? we can take the square root of a negative number, but let pretend we can and being by using the symbol 1   i . Also numerical analysis is concerned with the mathematical derivation description and analysis of methods of obtaining numerical solutions of mathematical problems; such that find complex roots of linear and non – linear equations by using iteration methods. (Abbas & Sasan, 2009) If R is the real field and C is the Complex field then consider the following definitions: Newton – Raphson method can be used to approximate the root of any nonlinear equation of any degree by the following iterative formula (Atkinson , 1998)


Introduction
Complex analysis can roughly be though of as a subject that applies the theory of calculus to imaginary numbers.But what is exactly are imaginary numbers ?we can take the square root of a negative number, but let pretend we can and being by using the symbol 1   i . Also numerical analysis is concerned with the mathematical derivation description and analysis of methods of obtaining numerical solutions of mathematical problems; such that find complex roots of linear and nonlinear equations by using iteration methods.(Abbas & Sasan, 2009) If R is the real field and C is the Complex field then consider the following definitions: Newton -Raphson method can be used to approximate the root of any nonlinear equation of any degree by the following iterative formula (Atkinson , 1998) ) ( has no derivative then we have difference approximation in this case, Newton -Raphson method is called Secant method (Steven, 2008).know we will show comparison between Newton and Secant method (Abbas&Sasan, 2008): Need for one initial point .

Secant method
2. Need for two initial points.

Stable .
4. No need to derivative.

Complex iterative formula 1-Complex Newton -Raphson formula (Complex Iteration ) :
In Newton formula Replace the variable x with the variable z in both side then we get the following form in the above form we get the formula know equate the real and the imaginary parts in (4) we get two iterative formula named the real iterative formula(Newton-Raphson ) (R.I.F)(5)and the imaginary iterative formula (I.I.F)(6): R.I.F and I.I.F were used to evaluate real and imaginary roots of nonlinear equations respectively.

2-Complex Secant formula:
In secant method if we replace the variable x by the variable z then we get the complex secant formula which can separated into two formulas, the first called the real iterative formula starts with real initial point ) ( 0 x the second called is the imaginary iterative formula starts with complex initial point ) ( 0 y , and where used to find real and imaginary roots of nonlinear equations respectively.Now in the above form we get the formula Now equating the real and the imaginary parts of both sides we get that Equation ( 11) called the real secant iterative starting with two real initial points . ,...
Equation ( 12) called the imaginary secant iterative starting with two imaginary initial points .

3-A new modified Newton -Raphson algorithm :
For computing complex and real root of nonlinear equations f(x)=0 depended on given initial point, we write the real iterative formula and the imaginary iterative formula of Newton -Raphson in the following procedure such that it is computing all roots of equations step by step and the number of iterations depended on the value of tolerance .Now Input informatios : The initial point z and tolerance  Output informations: The complex root 1 w or real root 2 w Procedure : Step1 : If z is real , Go to step9 .

4-A new modified Secant algorithm:
For compute the complex and the real roots of nonlinear equations f(x)=0 where f(x) has no derivative we write the following procedure step by step depended on two initial points and tolerance.Input informations: the initial points 0 1 , z z and tolerance  .Output informations: complex root 1 w or real root 2 w .procedures : , go to step14 .Step12: Replace

Numerical Computation to evaluate complex roots
Example 1 : Find the root of the following nonlinear equation by using the new modified Newton and Secant algorithm : where ( tol = 0.00001 ) .Solution : 1 -By the new modified Newton algorithm: We know that the equation f(x)=0 has no real roots if we use real Newton iterative formula ) ( x ) we get no solution .
If we use the new modified algorithm (imaginary iterative formula) starting by complex initial point ( o y ) then after some iteration we get a sequence of complex number    0 k k y converges to the complex root on the yaxis Now start with the complex initial point i y 5 .0 0  ,we get the following table table No.1 -0.0506 0+2.0500i 3 0+1.0003i-6.0985e-004 0+2.0006i 4 0+1.0000i-9.2922e-008 0+2.0012i after 5iteration we will be end with complex root y = 0 + 1.0000i.If we start with the initial point i y
when (tol = 0.00001) Solution: we know that the above function has 5roots and 1  x in this algorithm if we start by the real initial point 5 .0 0  x , we get a sequence of real numbers after some iterations using real Newton Iteration formula we get the real root 1  x .Now to find Complex roots let 0 y be any complex initial point we get a sequence of complex numbers tends to the complex root .As shown in table No

Conclusion
In this paper we modified Newton -Raphson and Secant algorithms to Compute the complex roots of nonlinear equations, then we wrote down their programs by Matlab applications version 7.8 by taking complex initial point to get the complex roots if it exists if not we real initial point to get real root .In this paper we study initial point have only real or imaginary part , as future work we modified these algorithm to compute the complex roots if the initial point is complex and real and imaginary part not equal to zero .