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

Chapter 1The Newton-RaphsonMethod for a SingleEquation1.1 The Geometry of the Newton-RaphsonMethodIn studying astronomy, Sir Isaac Newton needed to solve an equation f(x) = 0involving trigonometric functions such as sine. He could not do it by anyalgebra he knew and all he really needed was just a very good approximationto the equation’s solution. He then discovered the basic algorithm called theNewton-Raphson Method althoughNewtonfounditinapurelyalgebraicformatwhich was very difficult to use and understand. The general method and itsgeometric basis was actually first seen by Joseph Raphson (1648 - 1715) uponreading Newton’s workalthough Raphson only used it on polynomial equationsto find their real roots.The Newton-Raphson method is the true bridge between algebra (solvingequations of the form f(x) = 0 and factoring) and geometry (finding tangentlines to the graph of y = f(x)). What follows will explore the idea of theNewton-Raphson Method and how tangent lines will help us solve equationsboth quickly and easily although not for exact solutions, only approximateones.The reason that we are studying the Newton-Raphson Method in this bookis that it can also solve square non-linear systems of equations using matricesand their inverses as we shall see later. It is part of the wonderful effectivenessof the Newton-Raphson Method in that it can solve either a single equationor a square system of equations for its real or complex solutions, but only1