A brief history of Newton’s method CASA seminar 8 February 2006 Luiza Bondar

What is Newton’s method? where is a real valued function. Suppose is a function on a given interval and x is a guess for the solution. We want to find a better approximation x + h

What is Newton’s method? Newton iteration Start with initial guess Find the limit of the recurrence Nowadays questions what the initial condition needs to be ? speed of convergence ? conditions for the convergence of the recurrence process ?

Before Newton Babylon, BC initial guess with close to then and are approximations to take as a better approximation for the number Heron of Alexandria, 1 st century AD ( “professor” at Museum of Alexandria) included the method in Metrica I (on measurements) Francoise Viete, (lawyer, counsellor of King Henri IV ) “De numerosa potestatum”, approximation method to find the positive roots of polynomial equations from the 2 nd to the 6 th degree (extension of the work of Sharaf al-Din al-Muzaffar al-Tusi, , Iran)

Newton’s life and contribution to the method Sir Isaac Newton ( ) born in Woolsthorpe, parents were farmers Trinity College Cambridge 1661 for a law degree read Euclid’s Elements, Viete’s collected works, Geometria a Renato Des Cartes founded the modern calculus “De Methodis et Fluxionum Serierum and Serierum Infinitorum”, 1671 was not accepted “Principia Mathematica” 1687

Newton’s life and contribution to the method Approximate the solution of start with Solution “De Methodis et Fluxionum Serierum and Serierum Infinitorum”, written , published 1736 after his death

Newton’s life and contribution to the method Geometrical interpretation curve tangent curve tangent Newton does not refer to tangent nor the derivative does not give a geometrical interpretation of the method does not use or refer to a recurrence formula uses the method only for algebraic equations with rational coefficients describes a way to find bounds for the roots

Other contributions Joseph Raphson (Master of Arts degree at Cambridge) Analysis aequationum universalis 1690, contained the Newton method notices that the successive approximation can be written in an recurrence formula of the type but he did not refer to derivative Thomas Simpson (head of mathematics at the Royal Military Academy at Woolwich ) Essays on Mathematics, 1740 used the derivative in the recurrence formula used the method for equations with irrational and transcendental coefficients

Other contributions Jean Raymond Mourraille ( ) ( mathematician, astronomer, mayor of Marseille ) Traite de la resolution des equations numeriques 1768 geometric aspect of the method question of the choice of the starting point Augustin Louis Cauchy ( ) speed of convergence error estimation Newton method do determine complex roots Lecons sur le Calcul differentiel, 1829

Complex roots Arthur Cayley( ) (lawyer and mathematician) 1879 Given an initial input, to which root will Newton's method converge? take a piece of the complex plane each pixel represents a point in the complex plane used by the Newton method as an initial approximation the colour of the pixel indicates which of the roots the point converged to the intensity is proportional to the number of iterations required to approximate that root brighter pixels converged to the root more quickly than darker pixels white pixels did not converge at all after a specified number of iterations Graphic illustration

Complex roots

Mandelbrot sets for each complex c start a Newton iteration process with initial guess z=0

Future History continues… 4000 years ago nowadays

Any question?