Principles of Linear Algebra With Maple The Newton–Raphson ...

4 Chapter 1. Newton-Raphson Method for a Single Equation> display({plot guess, plotfwithtang, plot xint, plot soln});y2004 4.5 5 5.5x–20Figure 1.2: Tangent line to our polynomial at x = 5UponinspectionofFigure1.2, notethatthetangentlineatx = 5crossesthex-axis much closer to our solution near x = 4 than our very rough estimate ofx = 5forthe placementofthistangentline. The x-interceptofthis tangentlineis the solution for x to the tangent line’s equation y = f(5)+ dfdx (5)(x−5) = 0which isx = 5− f(5)dfdx (5)(1.2)= 307 ≈ 4.285714286So x-intercepts of tangent lines seem to move you closer to the x-interceptsof their function f(x). This is what Raphson saw which Newton did not seebecause Newton forgot to look at the geometry of the situation and instead heconcentrated on the algebra.Raphson’s next idea was to try to move even closer to the root of f(x) (thex-intercept of y = f(x) or solution to f(x) = 0) by repeating the tangent line,but now at the point given by this new value of x = 4.285714286 which is thex-intercept of the previous tangent line. Let’s do it repeating the above workand see if Raphson was correct to do this. Let’s now call our starting guessnear the root by x 0 = 5 and the x-intercept of the tangent line at x = 5 byx 1 = 4.285714286. The value of x 1 is definitely closer to the root than thestarting guess x 0 .> x0:= 5.:> x1:= x0 - f(x0)/df(x0);x1 := 4.285714286