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Iteration method or successive approximation method.

 Iteration method

To find the real root of an equation f (x)=0 ...(1)
Which can be expressed in the form x=πœ™(x) ......(2)
First we find an initial approximate value x₀ for equation (1). The better approximation x₁ for the root is obtained by replacing x by x₀  in R.H.S of equation (2) i.e x₁ = πœ™(x₀)
A still better approximation x₂ for the root is obtained by putting x=x₁ in the R.H.S of equation (2). Thus x₂ = πœ™(x₁).
This procedure is continued and we get
x₃ =  πœ™(x₂)
x₄ =  πœ™(x₃)
.................
xβ‚™ =  πœ™(xβ‚™₋₁)
If the sequence x₀,x₁,x₂,.......xβ‚™ of approximate roots converges to a limit 𝛂, then 𝛂 is taken as the root of the equation f(x)=0.
Remarks
The sufficient condition for convergence of iterations:
1) If I is the interval in which the root  𝛂 of the equation x=πœ™(x) lies, then |πœ™'(x)|< 1 for all x in the interval I.
2) The initial approximation x₀ for the root lies in I.
Example: Find the real root of the equation x³+x²-100=0, by iteration method.
Solution. Given f(x) = x³+x²-100 = 0
f(1) = -98
f(2) = -88
f(3) = -64
f(4) = 64+16-100 = -20 (-ve)
f(5) = 125+25-100 = 50 (+ve)
f(4) and f(5) are of opposite sign.
So the root of equation lie between 4 and 5.
The given equation can be written as,
x²(x+1) = 100
or x = 10/√(x+1)
or x = πœ™(x), where πœ™(x) = 10/√(x+1)
Therefore πœ™'(x) = 10×(-1/2)× 1/(x+1)³⁄ ² = -5/(x+1)³⁄ ²
or  |πœ™'(x)| = |-5/(x+1)³⁄ ²| < 1 in the interval (4,5).
So, the iteration method can be applied.
Taking x₀ = 4.2,  then approximations are
x₁ = πœ™(x₀) = 10/√(4.2+1) = 4.3852

x₂ = πœ™(x₁) = 10/√ (4.3852+1) = 4.3092

x₃ = πœ™(x₂) = 10/√ (4.3092+1) = 4.33995

x₄ = πœ™(x₃) = 10/√ (4.33995+1) = 4.32744

x₅ = πœ™(x₄) = 10/√ (4.32744+1) = 4.33252

x₆= πœ™(x₅) = 10/√ (4.33252+1) = 4.33045

x₇ = πœ™(x₆) = 10/√ (4.33045+1) = 4.33129

x₈= πœ™(x₇) = 10/√ (4.33129+1) = 4.33095

x₉ = πœ™(x₈) = 10/√ (4.33095+1) = 4.33109

x₁₀ = πœ™(x₉) = 10/√ (4.33109+1) = 4.33103

Since x₉ = x₁₀ = 4.3310 correct up to four decimal places.
Therefore, root  is x = 4.3310 Ans.

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