Iteration Calculator Guide
What Iteration Means
Iteration is a simple idea with strong practical value. You start with one estimate. Then you apply a rule. The new value becomes the next estimate. This process repeats until the answer changes very little.
This calculator supports common numerical iteration work. Fixed point mode uses a direct rule, written as g(x). Newton mode uses a function, written as f(x). It then estimates the next value from the slope. You can enter a derivative, or let the tool estimate it numerically.
Why Controls Matter
Good iteration needs control. A starting value can change the final behavior. A relaxation factor can slow jumps and improve stability. A tolerance sets the required accuracy. A maximum iteration limit stops endless loops. These settings help you test convergence safely.
The result table shows every step. It lists the old estimate, raw next estimate, adjusted next estimate, error, relative error, and residual. This makes the process easy to audit. You can see whether values settle, oscillate, or grow without control.
Choosing Error Rules
Use absolute error when you want direct step size. Use relative error when the scale is large. Use residual when solving a function equation. Residual is useful in Newton mode because it measures how close f(x) is to zero.
The graph adds a visual check. The estimate curve should flatten near convergence. The error curve should usually drop. If the error rises, the method may be unstable. Try another starting value, smaller relaxation, or a different iteration rule.
Reporting Results
Exports help with reporting. CSV is useful for spreadsheets. PDF is useful for sharing a clear record. Both options support teaching, homework checks, engineering notes, and business estimates.
Always review the formula before trusting a result. Iteration is powerful, but it is not magic. Some formulas do not converge. Some starting values move toward the wrong point. Use the table, graph, and residual together. That gives a clearer decision. For best results, test simple examples first. Compare two tolerance levels. Save one report for each setting. This habit makes mistakes easier to find. It also shows how fast the chosen rule improves. In classrooms, the steps explain the method better than a final answer alone and builds stronger number sense.