Newton's method
![The function ƒ is shown in blue and the tangent line is in red. We see that xn+1 is a better approximation than xn for the root x of the function f.](/Images/godic/202501/29/NewtonIteration_Ani0520.gif")
![The tangent lines of x3 - 2x + 2 at 0 and 1 intersect the x-axis at 1 and 0 respectively, illustrating why Newton's method oscillates between these values for some starting points.](/Images/godic/202501/29/NewtonsMethodConvergenceFailure.svg0520.png")
![Basins of attraction for x5 - 1 = 0; darker means more iterations to converge.](/Images/godic/202501/29/Newtroot_1_0_0_0_0_m10520.png")
![The graph of a strictly concave quadratic function is shown in blue, with its unique maximum shown as a red dot. Below the graph appears the contours of the function: The level sets are nested ellipses.](/Images/godic/202501/29/Max_paraboloid.svg0520.png")
In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
The Newton–Raphson method in one variable is implemented as follows: