Slope Methods for Finding Roots83.22f(x)8.7977P=8g(x)= x-104.32f'(x)Table2.4FindingtheTimeWhentheHeightf()IsZerokTime, PkHeight, f(pk)Pk+1Pk08.000000000.7977310183.2209720018.797731010.055301606.6836970028.742429410.000254750.03050700Iteration4times38.742174670.000000010.0000010040.000000008.742174660.00000000Pyhas eightydecimalplaces(x, y) = (r(t),f(t)300/ofaccuracy, t ~ 8.7421x = r(8.742 1) = 1600(1- e-0.8742)200=932.49810009/22/20265:58AM2006004008001000
9/22/2026 5:58 AM 6 Iteration 4 times 𝒑𝟒 has eight decimal places of accuracy, 𝒕 ≈ 𝟖. 𝟕𝟒𝟐𝟏 ( ) 0 8742 8 7421 1600 1 932 498 . ( . ) . x r e− = = − = 1 83 22 8 8 7977 104 32 ( ) . ( ) - . '( ) . f x g x x p f x = = − = −
Speedof ConvergenceDefinition 2.5 order of convergence, Assume that (pnJm=o converges to p andset En =p-pn,if two positive constant A± O and R> O existinn→aThesequence is said toconvergetop withorder of convergence R.ThenumberAiscalledasymptoticerrorconstant.if R-l,the convergence of p,Jn=o is called linearif R=2,the convergence of (pnin=o is called quadraticEx.2.14QuadraticconvergenceataSimpleRootatp=2Ex. f(x) = x3 - 3x + 2.Pe= g(Pr-)= Pr- - i,-3Pr-1 +2 _ 2pi-23 pz-, -33pa-1-3R= 29/22/20265:58AM
9/22/2026 5:58 AM 7 Definition 2.5 order of convergence, Assume that 𝒑𝒏 𝒏=𝟎 ∞ converges to 𝒑 and set 𝑬𝒏 = 𝒑 − 𝒑𝒏 , if two positive constant 𝑨 ≠ 𝟎 𝒂𝒏𝒅 𝑹 > 𝟎 exist The sequence is said to converge to 𝒑 with order of convergence R. The number A is called asymptotic error constant. Ex.2.14 Quadratic convergence at a Simple Root at 𝒑 = 𝟐 1 1 lim lim n n R R n n n n p p E A p p E + + → → − = = − 0 1 ¥ , { } n n if R the convergence of p is called linear = = 3 Ex f x x . (x) . = − + 3 2 0 2 ¥ , { } n n if R the convergence of p is called quadratic = = 3 3 1 1 1 1 1 2 2 1 1 3 2 2 2 3 3 3 3 ( ) k k k k k k k k p p p p g p p p p − − − − − − − − + − = = − = − − R = 2