A New numerical method for multi-roots finding with the R software Kouider Mohammed Ridha Department of Mathematics, Applied Mathematics Laboratory , University of Mohamed Khider, Biskra, Algeria, mohakouider@gmail.com Abstract The most basic problem in numerical analysis (methods) is the root finding problem. In this paper we are interesting in new numerical method which based on the Modified Bisection Method(MBM) referred to by Tanakan superficially and he didn't know her as a numerical method for finding the roots of a function. Hence in this study we define a new numerical method base on MBM with error bound and the number iterations necessary. Finally we present our new MBM for multi-roots with the R software. Keywords: Nonlinear equation , Linear interpolation , Bisection Method , Modified Bisection Algorithm. 1. Introduction The process of finding the roots of function is involves finding the value of x for which   0 fx  . If the function equals zero, x is the root of the function. The simplest root-finding algorithm is the bisection method is also called the interval halving method. It works when   fx is a continuous function and it requires previous knowledge of two initial guesses  and  such that   f  and   f  have opposite signs, then find the midpoint of   ;  , and then decide whether the root lies on   ; /2       or   ; ;       repeat until the interval is sufficiently small. However, the bisection method is reliable, but it converges slowly, gaining one bit of accuracy with each iteration. In this paper we interesting in new numerical method based on MBM and like a linear interpolation method is also known as the method of regula falsi (false position) see [1], but with other way. Tanakan, (2013, [3]) in his paper proposed algorithm base in the bisection method and the value of s x as we defined in (1) and (2) for solve nonlinear equation, but his study did not define the value of 1 s x  in term of s x he defined in term of the midpoint of   ;  , and bending the bisection method and the value of s x , he named modified bisection algorithm [ for more detail see Tanakan, (2013, [3])].Firstly we present the value of s x and define a new MBM with her algorithm and the error bound with the iterations necessary for finding the root of function. In section (2) we programming the new MBM for multi-roots finding with the R software By the intermediate value theorem implies that a number * x exists in   ;  with   * 0 fx  if     0 f f    , mean that     0 f f    , which means that one of them is above the x -axis and the other one below the x-axis. Let   fx be a continuous function and defined on interval   ;  with     0 f f    Firstly, we set 1   and 1    . For an integer 1 s . Then, we can find the equation of straight line from the points     * * , s s f   and     * * , s s f   as y ax b   where     * * * * s s s s f f a        or     * * * * s s s s f f a        and   * * s s b f a     or   * * s s b f a     Hence, the x -intercept of the straight line is at a point s b x a   That is       * * * * * * s s s s s s s x f f f           (1) or       * * * * * * s s s s s s s x f f f           (2) It's clearly that       1 x f f f           (3) or