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
y=ft)=(2+01-c0)-3t
Example 1. 12. A projectile is fired with an angle of elevation bo=45 160ft/sec, and C= 10. Find the elapsed time until impact and find the range USing formulas (1.51) and (1.52), the equations of motion are y=f(t)=4800(1 31.534367, we will use the initial guess Po=8. The derivative is f'(t)=480e-l W e-10)-320 t and a=r(t)=16001-c-10). Since f(8)=8.227andf(9) 320, and its value f(po)=f (8)=-104.3220972 is used in formula(1.40) to get 83.22097200 8.797731010 104.3220972 a summary of the calculation is given in Table 1.4 The value p4 has eight decimal places of accuracy, and the time until impact is tA8.74217466 seconds. The range can now be computed using r(t); and we get r(874217461=16001-c-0812176)=932.4602t
Table 1.4 Finding the Time When the Height f(t) Is Zero k Time, Pk PR+1-Pk Height, f(pk) 080000000797310183.22097200 1|879773101-0.0530160-668369700 2874242941-0.00025475-0.03050700 38.74217467-0.000000-0.00100 48.74217460.00000000
1.4.2 The division-by-Zero Error
1.4.2 The Division-by-Zero Error
Definition 1.4(Order of a root). Assume that f(a) and its derivatives f'(c) f(M(a)are defined and continuous on an interval about x=p. We say that f(ar)=0 has a root of order M at c =p if and only if f(p)=0,f(p),…,f-(p)=0,andf(p)≠0.(153)
