Newton-Raphson method or only Newton's method is one of the more popular methods used for solving nonlinear algebraic equations of the form . Nonlinear algebraic equations contain powers of variable(s) and/or transcendental functions. One of the major achievements in mathematics was the proof (by Niels Henrik Abel (1802-1829) in 1824 that polynomial equations of degree greater than 4 cannot be solved by means of an algebraic formula (that is, in terms of radicals). 

The basis for Newton's method is that the actual root is estimated and the zero of the tangent to the function at that point is determined. 

If a real root is to be assumed for the function , then one may easily compute If we draw a line tangent to the curve at then the tangent line intersects the -axis at a point which is expected to be closer to the actual root than the assumed root From the figure we see that  

solving for gives 

If we repeat the procedure at and let be the -intercept of the second tangent line, we find 

Repetitive use of the procedure yields a sequence of approximations that we expect converges to the root  

Newton-Raphson Method 

Suppose is differentiable and suppose represents the unknown root of Let denote a number that is chosen arbitrarily as a first guess to then repetitive use or iteration of  

yields a sequence of approximations that we expect converges to the root  

Newton-Raphson Method 

Note that Newton's method unfortunately does not always converge to a root if is not sufficiently close to the actual root. 

• Newton (f, , n, ) computes an approximation to a root of using Newton's method.  x(0) is the first approximation, nis the number of iterations, and the tolerances, (x(n)-x(n-1)

• NewtonPlot (f, , n, , animates Newton's iteration process of finding a root of .  x(0)  is the starting value, nis the number of iterations, the tolerances,(x(n)-x(n-1), and c is the background color

See also NewtonsMethod . 

The formula of a fourth-degree equation is quite complicated as Maple shows below. 

>	restart:MathMaple:-ini():

 

>	f:=x->x^4-x-1;

 

>	solve(f(x)=0,{x}):

 

>	evalf(%);   #evaluates to floating points

 

The roots of the equation above can of course be approximated numerically using Maple's   fsolve . 

>	fsolve(f(x)=0,{x});   #all real values

 

>	fsolve(f(x)=0,{x},complex); #all values

 

The solution of  

>	f:=x->x^5-3*x^3+x^2-23*x+19: f(x)=0;

 

can be found using fsolve . 

>	fsolve(f(x)=0,{x},complex);

 

>	fsolve(f(x)=0,{x}); #real solutions

 

>	plot(f(x),x=-3..3,`f(x)`);

 

The following shows an iteration process which does not converge. 

>	f:=x->piecewise(x>0,sqrt(x),-sqrt(-x)): 'f(x)'=f(x);

 

First using Newton  

>	Newton(f,1,10,0.0001);

 

In this case Newton's method fails. The animation below indicates the reason. 

>	NewtonPlot(f, 1, 10,0.0001,grey);

 

Example 1 

a) The volume of a body is . Find by solving the equation . 

b) Solve = 0. 

Solution 

>	restart:MathMaple:-ini():

 

a) 

> V:=x->3*x^3-16*x^2+20*x-40;     > plot(V(x)-60,x=2..6);     We use the formula where   > f:=x->V(x)-60:	We start start with and use 6 iterations.  > x(0),N:=5,6;     > for n from 0 to N do    x(n+1):=evalf(x(n)-f(x(n))/D(f)(x(n))); od;                 The approximate value is   > x:='x':
b)  > f:=x->18*sin(x) - 6*sqrt(x+1): f(x)=0;     > plot(f(x),x=0..15,tickmarks=[10,10]);	The figure shows that there is 3 or 4 roots.  Newton method gives    > Newton(f,1,6,0.00001);     The solution is     > Newton(f,2,6,0.00001);     The solution is
> Newton(f,7,6,0.00001);     The solution is	> Newton(f,8,6,0.00001);     The solution is
Using fsolve , we get  > x[1]=fsolve(f(x)=0,x=1);	> x[2]=fsolve(f(x)=0,x=2);     > x[3]=fsolve(f(x)=0,x=7);
> x[4]=fsolve(f(x)=0,x=8);     To go from one approximation to the next approximation can be visualized by animation.	> NewtonPlot(f,7,6,0.00001,yellow);        Click on the figure and start the animation.
> NewtonPlot(f,8,6,0.00001,grey);	Let us start with a larger initial value.  > NewtonPlot(f,12,10,0.00001,grey);     Click on the figure and select Next Frame repeatedly.