1.4 Newton-Raphson and Secant Methods 1.4.1 Slope Methods for Finding Roots
1.4 Newton-Raphson and Secant Methods 1.4.1 Slope Methods for Finding Roots
Theorem 1.5(Newton-Raphson Theorem). Assume that f E C a, b and there exists a number p e Ja, bl, where f(p)=0. If f(p)#0, then there exists a 8>0 such that the sequence pk igso defined by the iteration Dk=9h-1)=D,f=1) f(pk-1) for k= 1.2 (1.40) will converge to p for any initial approximation po E, p+8 Remark. The function g(a) defined by formula 9(x)=x
Corollary 1. 2(Newtons Iteration for Finding Square Roots). Assume that A>O is a real number and let po>0 be an initial approximation to vA.Define the sequence pr g-o using the recursive rule Pk-1+ pk-1 for k=1.2 1.47) 2 Then the sequence (pk )k-o converges to VA; that is, limn-oo Pk =VA
Example 1.11. Use Newton's square-root algorithm to findv5 Starting with po=2 and using formula(1.47), we compute 2+5/ 2.25 2.25+5/2.25 p =2.236111111 2236111111+5/2.23611111 =2.236067978 236067978+5/223606797 p4 =2.236067978
0=ft)=(Cn2+3221-0)-32Ct r=r()=C2(1-e1)