Fixed Point Iteration Method

The Fixed Point Iteration method (or Iteration Method) rewrites an equation $f(x) = 0$ in the form $x = \phi(x)$ . Starting from an initial guess $x_0$ , successive approximations are generated using the recurrence relation $x_{n+1} = \phi(x_n)$ .

Formula for Fixed Point Iteration:

x_{n+1} = \phi(x_n)

|\phi'(x)| < 1 \quad (\text{Convergence Condition})

Where:

$\phi(x)$ is derived from $f(x) = 0$ by isolating $x$ .
The iteration guarantees convergence if $|\phi'(x)| < 1$ near the root.

Example

Given Function

f(x) = x^3 - x - 1 = 0

Find root near $x_0 = 1.5$

Step 1: Rewrite Equation

Express $x^3 - x - 1 = 0$ as $x = (x + 1)^{1/3}$ :

\phi(x) = (x + 1)^{1/3}

Step 2: Check Convergence Condition

\phi'(x) = \frac{1}{3(x+1)^{2/3}}

|\phi'(1.5)| = \frac{1}{3(2.5)^{2/3}} \approx 0.18 < 1 \quad (\text{Convergent!})

Step 3: Perform Iterations

x_1 = (1.5 + 1)^{1/3} \approx 1.3572

x_2 = (1.3572 + 1)^{1/3} \approx 1.3309

x_3 = (1.3309 + 1)^{1/3} \approx 1.3259

Conclusion

The iterations converge to the root x ≈ 1.3247.

Fixed-Point (Iteration) Method Solver

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Unit 1•3 of 28

NextNewton-Raphson Method

Fixed Point Iteration Method

Formula for Fixed Point Iteration:

x_{n+1} = \phi(x_n)

|\phi'(x)| < 1 \quad (\text{Convergence Condition})

Where:

$\phi(x)$ is derived from $f(x) = 0$ by isolating $x$ .
The iteration guarantees convergence if $|\phi'(x)| < 1$ near the root.

Example

Given Function

f(x) = x^3 - x - 1 = 0

Find root near $x_0 = 1.5$

Step 1: Rewrite Equation

Express $x^3 - x - 1 = 0$ as $x = (x + 1)^{1/3}$ :

\phi(x) = (x + 1)^{1/3}

Step 2: Check Convergence Condition

\phi'(x) = \frac{1}{3(x+1)^{2/3}}

|\phi'(1.5)| = \frac{1}{3(2.5)^{2/3}} \approx 0.18 < 1 \quad (\text{Convergent!})

Step 3: Perform Iterations

x_1 = (1.5 + 1)^{1/3} \approx 1.3572

x_2 = (1.3572 + 1)^{1/3} \approx 1.3309

x_3 = (1.3309 + 1)^{1/3} \approx 1.3259

Conclusion

The iterations converge to the root x ≈ 1.3247.

Fixed-Point (Iteration) Method Solver

PreviousFalse Position Method

Unit 1•3 of 28

NextNewton-Raphson Method

Fixed Point Iteration Method

Example

Step 1: Rewrite Equation

Step 2: Check Convergence Condition

Step 3: Perform Iterations

Conclusion

Fixed-Point (Iteration) Method Solver

NETZ Engine

Fixed Point Iteration Method

Example

Step 1: Rewrite Equation

Step 2: Check Convergence Condition

Step 3: Perform Iterations

Conclusion

Fixed-Point (Iteration) Method Solver