Estimate roots using repeated substitution, stopping rules, and convergence diagnostics. Save detailed tables with ease. Test convergence behavior before trusting your final answer today.
Use explicit multiplication such as 2*x. Supported functions include sin, cos, tan, sqrt, abs, log, log10, exp, and pow.
| Iteration | xn | xn+1 = cos(xn) | Absolute Error |
|---|---|---|---|
| 1 | 0.500000 | 0.877583 | 0.377583 |
| 2 | 0.877583 | 0.639012 | 0.238570 |
| 3 | 0.639012 | 0.802685 | 0.163673 |
| 4 | 0.802685 | 0.694778 | 0.107907 |
| 5 | 0.694778 | 0.768196 | 0.073418 |
The standard fixed point update is:
xn+1 = g(xn)
This calculator also supports relaxation:
xn+1 = (1 - λ)xn + λg(xn)
Here, λ is the relaxation factor. When λ = 1, the method becomes the standard fixed point iteration. The calculator tracks absolute error, relative error, and residual at each step.
Fixed point iteration is a numerical method for solving nonlinear equations. You first rewrite the problem in the form x = g(x). The calculator then starts with an initial guess and keeps generating new approximations. This method is useful in algebra, calculus, optimization, and applied mathematics. It is also helpful when an exact algebraic solution is difficult or impossible to obtain.
This calculator does more than produce one final value. It builds an iteration table and shows how the sequence changes from one step to the next. It reports the absolute error, relative error, and residual. These metrics help you judge stability and convergence. They also make it easier to compare different starting points, tolerances, and relaxation settings.
Convergence depends on the choice of g(x), the initial guess, and the local behavior of the function. A good transformation can converge quickly. A poor transformation can oscillate or diverge. The relaxation factor adds extra control. You can reduce the step size when the iteration moves too aggressively. This makes the tool more practical for classroom examples and real numerical work.
The CSV export is useful for spreadsheets, reports, and audits. The PDF export is useful when you want a quick record of the computed result and the leading iteration rows. These downloads help students, teachers, analysts, and engineers keep clean documentation. They also support review, comparison, and validation across several test cases.
Always check whether your chosen g(x) is suitable for fixed point iteration. Start with a reasonable initial guess. Use a tolerance that matches the precision you need. Review the residual before accepting the answer. If convergence is slow, change the rearrangement or adjust the relaxation factor. A careful setup usually produces a more stable and reliable fixed point estimate.
It solves equations written as x = g(x). The tool generates repeated approximations until the error or residual reaches the selected tolerance.
A fixed point is a value of x that satisfies g(x) = x. At that value, the iteration no longer changes significantly.
The method can fail when g(x) is poorly chosen, the starting guess is unsuitable, or the function causes oscillation or divergence near the target point.
The relaxation factor scales the update step. It can slow the method for better stability or keep the standard method when set to 1.
Absolute error is simple and direct. Residual is also important because it shows how closely the latest approximation satisfies the fixed point condition.
Yes. The calculator supports common functions such as sin, cos, tan, sqrt, abs, log, log10, exp, and pow.
The table shows whether the sequence is stabilizing, oscillating, or diverging. It helps you judge whether the reported answer is trustworthy.
Use explicit multiplication, such as 2*x or x*(x+1). Avoid implicit multiplication because it is not parsed by this calculator.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.