Analyze iterative roots with practical controls here. View tolerance checks, tables, and approximate error instantly. Test custom functions and export organized iteration results easily.
Use explicit multiplication such as 2*x. Supported functions include sin, cos, tan, asin, acos, atan, sqrt, abs, exp, ln, log, log10, floor, ceil, round, pi, and e.
| Field | Example Value |
|---|---|
| f(x) | cos(x)-x |
| g(x) | cos(x) |
| Initial Guess | 0.5 |
| Tolerance | 0.000001 |
| Maximum Iterations | 50 |
| Relaxation Factor | 1 |
| Minimum Iterations | 1 |
| Expected Root | About 0.73908513 |
The fixed point iteration formula is:
xn+1 = g(xn)
This calculator also supports relaxation:
xn+1 = (1 - λ)xn + λg(xn)
Absolute error is computed as |xn+1 - xn|.
Relative error is computed as |xn+1 - xn| / max(|xn+1|, small number).
Residual is computed as |f(xn+1)|.
For reliable convergence, the transform should keep values near a fixed point and usually satisfy |g'(x)| < 1 around the root.
Fixed point iteration is a classic numerical method. It solves nonlinear equations through repeated substitution. You start with an initial guess. Then you apply a transformed function again and again. The process aims to reach a stable value. That stable value is the fixed point. If the transform is chosen well, that point is also the root of the original equation. This calculator helps you test that process quickly. It shows each step clearly. It also reports error, residual, and convergence behavior.
An iteration table gives more than a final answer. It shows how the approximation changes from one step to the next. That matters in root finding. A root may look stable at first, then drift away. Another transform may converge smoothly. By checking xn, g(xn), relative error, and residual, you can judge method quality. This is useful in maths, engineering, and computational science. It helps students verify manual work. It also helps analysts compare transforms before using a model in larger calculations.
The most important choice is g(x). A poor transform often causes divergence. A better transform usually keeps values near the target root. A good initial guess also helps. Smaller tolerance gives tighter accuracy. Higher maximum iterations gives the method more room. Relaxation can improve difficult cases. Values below one may damp oscillation. Values above one may speed progress in some problems. The derivative check is also useful. When |g'(x)| stays below one near the solution, local convergence is more likely. That simple test often explains why one form works and another fails.
Use this calculator when you need a practical fixed point iteration method calculator for nonlinear equations. It works well for educational exercises, convergence studies, and quick numerical experiments. It is also useful when comparing approximate error against residual error. The export tools make reporting easier. You can save the iteration history, attach the table to notes, or review it later. For best results, rewrite the equation carefully, use explicit multiplication, and start near the expected root. Good inputs make fixed point iteration simple, clear, and efficient.
It is a root finding method that rewrites an equation as x = g(x). The method starts from an initial guess and repeatedly applies g(x) until the values stabilize near a fixed point.
f(x) defines the original equation you want to solve. g(x) is the rearranged form used in the iteration step. Both are helpful because the calculator checks the root and also performs the repeated update.
Tolerance is the accuracy target. The iteration stops when the chosen error check becomes small enough. A lower tolerance usually means a more accurate answer but may require more iterations.
The transform may be unsuitable, the starting guess may be weak, or the derivative near the root may be too large. Try a different g(x), a closer initial guess, or a smaller relaxation factor.
The relaxation factor changes the update step size. A value of 1 gives the standard method. Lower values can reduce oscillation. Slightly higher values may accelerate some stable problems.
Use both checks for stricter control. Use approximate error for step size monitoring. Use residual when you care most about how closely the result satisfies the original equation.
Yes. You can use sin, cos, tan, asin, acos, atan, sqrt, abs, exp, ln, log, log10, floor, ceil, round, pi, and e. Write multiplication explicitly, such as 3*x.
Yes. After calculation, you can download the iteration table as CSV or PDF. That makes it easier to save, share, print, or attach the results to reports.
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.