Secant Method to find the root of the non-linear equation

Secant Method [Super-linear convergence]

1.      Decide two initial points x₁ and x₂, and accuracy level E.

2.      Compute f(x₁) and f(x₂)

3.      Compute improved value x₃ with the help of x₁ an x₂,

i.e. x₃ = x₂ - ( ( f(x₂) (x₂-x₁) ) / ( f(x₂) - f(x₁) ) )

4.      Checking the accuracy level of improved estimation i.e. x₃.

If absolute value of (x₃ – x₂) / x₃ < E

            Write improved estimation or x₃ as the root of the equation and go to 5.

Otherwise

            Set x₁ = x₂ and f(x₁) = f(x₂)

            Set x₂ = x₃ and f(x₂) = f(x₃)

            Go to step 3

5.      Stop

Example: Use the secant method to estimate the root of the equation x²-4x-10 = 0 with the initial estimates of x₁ = 4 and x₂ = 2.

Solⁿ: Given f(x) = x2-4x-10 = 0

Step 1: Initial points x₁ = 4 and x₂ = 2, and Accuracy level E = 0.01

Step 2: Computing f(x₁) and f(x₂),

            f(x₁) = 4²- 4*4 – 10 = -10

            f(x₂) = 2² – 4*2 – 10 = -14

Step 3: Iteration I:

Computing improves root x3,

            x₃ = x₂ - ( ( f(x₂) (x₂-x₁) ) / ( f(x₂) - f(x₁) ) ) = 2 - ( ( ( -14 ) ( 2 – 4 ) ) / ( ( -14 ) - ( -10 ) ) )

                = 2-(28 / (-4)) = 9               

Step 4: Checking accuracy level,

Absolute value of (x₃ – x₂) / x₃ < E

(x₃ – x₂) / x₃ = (9-2) / 9 = 7/9 = 0.7778

Here 0.7778 < 0.01 is false

Error criteria are not satisfied.

So,       Set x₁ = x₂ = 2 and f(x₁) = f(x₂) = -14

Set x₂ = x₃ = 9 and f(x₂) = f(x₃) = f(9) = 9² - 4*9 – 10 = 35

            Go to step 3

Iteration II: Calculating improved estimation of x_3, using

x₃ = x₂ - ( ( f(x₂) (x₂-x₁) ) / ( f(x₂) - f(x₁) ) ) = …

Repeat the iterations until error criteria satisfied.

If error criteria is satisfied then,

Write improved estimation x₃ as the root of the equation and stop.