# Newton-Raphson and Bisection Methods

**Newton-Raphson Method***f*(*x*,*y*) = 0 and*g*(*x*,*y*) = 0*x*_{0},*y*_{0}) by graphical or trial and error method.*x*_{1},*y*_{1}) such that*x*_{1}=*x*_{0}+*h*and*y*_{1}=*y*_{0}+*k**âˆ´**f*(*x*_{0}+*h*,*y*_{0}+*k*) = 0 and*g*(*x*_{0}+*h*,*y*_{0}+*k*) = 0*From Taylorâ€™s series we get,**= 0 and = 0**Solving these equations we get**h*and*k*.*Thus we find**x*_{1}and*y*_{1}*From this value of**x*and_{1}*y*we find_{1}*x*and_{2}*y*in similar way_{2}**Bisection Method***f*(*x*) and for any two number*a*and*b*if*f*(*a*),*f*(*b*) > 0, then there is at least one root for the equation*f*(*x*) in between*a*and*b*.*x*_{1}=*a*and*x*_{2}=*b**We find the mid-point of**x*_{1}and*x*_{2}i.e.*x*_{0}*x*_{0}_{= }*If**f*(*x*) = 0,_{0}*x*is the root_{0}*If**f*(*x*)_{0}*f*(*x*) < 0, root is in between_{1}*x*and_{0}*x*_{1}*If**f*(*x*)_{0}*f*(*x*) < 0, root is in between_{2}*x*and_{0}*x*_{2}*and this process in continued with either (**x*,_{0}*x*) or with (_{1}*x*,_{0}*x*)._{2}