newton' method why not use an array for the approximationsnewton' method is normally formulated with an iteration index nxnc xn xn xn seeing such an indexmany would implement this as [ + [nf( [ ])/dfdx( [ ]such an array is finebut requires storage of all the approximations in large industrial applicationswhere newton' method solves millions of equations at onceone cannot afford to store all the intermediate approximations in memoryso then it is important to understand that the algorithm in newton' method has no more need for xn when xnc is computed thereforewe can work with one variable and overwrite the previous valuex ( )/dfdx(xrunning naive_newton(fdfdx eps= results in the approximate solution smaller value of eps will produce more accurate solution unfortunatelythe plain naive_newton function does not return how many iterations it usednor does it print out all the approximations :which would indeed be nice feature if we insert such printouta rerun results in we clearly see that the iterations approach the solution quickly this speed of the search for the solution is the primary strength of newton' method compared to other methods making more efficient and robust implementation the naive_newton function works fine for the example we are considering here howeverfor more general usethere are some pitfalls that should be fixed in an improved version of the code an example may illustrate what the problem islet us solve tanh xd which has solution with jx : everything works fine for examplex leads to six iterations if :