### Modified Methods

 A multiple root occurs when a certain function has two or more roots which they are in the same position. For example, assume the function:   This function has a multiple root in the position of x = 3. In this situation most of the numerical formula face major difficulties for finding roots. For example when even multiple root occurs the function is tangential to the x axis in the position of the root. Therefore, many bracketing methods don’t work because the sign of the function is not changed near the position of the root. The Newton –Raphson and secant method face difficulties, too. In these methods, the convergence speed decreases near a multiple root. To overcome these defect, Ralston and Rabinowitz (1978) suggested a modification for the Newton-Raphson method. They defined a new function as follows: It can be shown that the roots of this new function are as same as the old function. The only difference is that the multiple roots in old function have been changed to single roots in the new function. Using this function, the multiple roots can be found by employing conventional numerical method without any major difficulties. For example the Newton-Raphson formula is changed to a new formula as follows: or    (Source Code in C++)