Numerical Analysis By Shanker Rao

26 NUMERICAL ANALYSIS
Let x n – 1
and x n
denote two successive approximations to α, then we have
x
n
= φdxn−1i,
By mean value theorem we have
φdxn−1i − φbg
α
x − α
d n−1i bg.
∴ x − α = φ x − φ α
n−1
n
bg,
= φ′
β
where x n− 1 < β < α
d i bg d i bg. (4)
⇒ φ xn−1 − φ α = xn−1
− α φ′
β
Let λ be the maximum absolute value of φ′
bg x over I, then from (4) we have
x
n
− α ≤ λ x −1 − α
(5)
n
proceeding in this way, we get
⇒ x − α ≤ λ x − α
n−1 n−2
∴ x − α ≤ λ x −2 − α
(6)
n
2
n
x
n
n
− α ≤ λ x0 − α . (7)
If λ 1
⇒ x n − α will become indefinitely large, as n increases and the sequence approximations
does not converge.
Note
1. The smaller the value of φ′(x) the more rapid will be the convergence.
2. From (1) we have the relation