SelisOnelLectureNotes_SolNonlinEq

SelisOnelLectureNotes_SolNonlinEq - Solving Nonlinear...

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Unformatted text preview: Solving Nonlinear Algebraic Equations: Functions of One Variable Selis Önel, PhD SelisÖnel© 2 Numerical Iterative Methods  Graphical method  Bisection method  Secant-type methods  Regula falsi  Secant method  Newton‟s method  Muller‟s Method  Minimization  Golden-section search  Brent‟s method SelisÖnel© 3 Exact Zeros of Nonlinear Algebraic Equations f(x)=x 2-3  Exact zeros (roots) of f(x) can be found by the quadratic formula  However, no such method exists for most nonlinear functions  Formula for finding the zeros of a cubic function is very complicated  Niels Henrik Abel (1802-1829): In 1824 Abel proved the impossibility of solving algebraically the general equation of the fifth and higher degrees SelisÖnel© 4 Abel, Niels Henrik (1802-1829) Norwegian mathematician: Born on August 5, 1802 in the small village of Findoe, Norway Abel's life was spent in poverty, caused by the large size of his family (he had six brothers and his father died when he was only eighteen) and the difficult economic situation in Norway at that time. Abel died of tuberculosis at the age of 26 after being forced to live in miserable conditions because of his inability to obtain a university post.  At the age of 16, Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had only held for rationals . At age 19, he showed there is no general algebraic solution for the roots of a quintic equation , or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of Galois ) an extremely important branch of mathematics known as group theory , which is invaluable not only in many areas of mathematics, but for much of physics as well. Among his other accomplishments, Abel wrote a monumental work on elliptic functions which, however, was not discovered until after his death. When asked how he developed his mathematical abilities so rapidly, he replied "by studying the masters, not their pupils."  Abel sent a paper on the unsolvability of the quintic equation to Gauss , who proceeded to discard without a glance what he believed to be the worthless work of a crank. In 1825, the Norwegian government funded Abel on a scholarly visit to France and Germany. Abel then traveled to Paris, where he gave an important paper revealing the double periodicity of the elliptic functions , which Legendre later described to Cauchy as "a monument more lasting than bronze" (borrowing a famous sentence by the Roman poet Horatius). However, Cauchy proceeded to misplace the manuscript. In Berlin, Abel met and was befriended by August Crelle , an amateur mathematician who had founded the famous Journal für die reine und angewandte Mathematik (Journal for pure and applied mathematics), which had published several papers by Abel....
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This note was uploaded on 03/14/2012 for the course CHEMICAL E kmu 206 taught by Professor Onel,selis during the Spring '08 term at Hacettepe Üniversitesi.

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SelisOnelLectureNotes_SolNonlinEq - Solving Nonlinear...

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