1. 4 Newton-Raphson and secant methods
1.4 Newton-Raphson and Secant Methods
1. 4. 1 Slope Methods for Finding Roots
1.4.1 Slope Methods for Finding Roots
Theorem 1.5(Newton-Raphson Theorem). Assume that f E Ca, b and there exists a number p e la, 6, where f(p)=0. If f(p)#0, then there exists a 8>0 such hat the sequence pklk-o defined by the iteration P=9(h-1)=pk-1 f(pk-1) for k= 1.2 will converge to p for any initial approximation Po E lp-8, p+8 Remark. The function g()defined by formula 9()=x
Corollary 1.2(Newtons Iteration for Finding Square Roots ). Assume that A>0 is a real number and let po>0 be an initial approximation to vA. Define the sequence ipr lgo using the recursive rule 1+ for k=1.2 Then the sequence pk ) o converges to VA; that is, limn-oo Pk =VA
Example 1.11. Use Newton's square-root algorithm to find v5 Starting with po=2 and using formula(1.47), we compute 2+5/2 =2.25 225+5/2.25 =2.236111111 223611111+5/223611111 P3 2.236067978 2 236067978+5/2236067978 p4 =2.236067978