Test equations, inspect tables, and compare convergence behavior. Download results as CSV or PDF instantly. Build confidence solving roots with transparent numerical steps today.
Enter a function in x. Use explicit multiplication such as 2*x and x*(x-1).
| Example Function | Initial Guess | Tolerance | Maximum Iterations | Approximate Root | Typical Outcome |
|---|---|---|---|---|---|
| cos(x)-x | 0.5 | 1e-8 | 25 | 0.73908513 | Fast convergence |
| x^3-x-2 | 1.5 | 1e-8 | 25 | 1.52137971 | Stable root finding |
| exp(-x)-x | 0.5 | 1e-8 | 25 | 0.56714329 | Good convergence |
Steffensen method is a derivative-free root-finding technique. It accelerates fixed-point style updates without directly computing derivatives.
The update formula is:
x_(n+1) = x_n - (f(x_n)^2) / (f(x_n + f(x_n)) - f(x_n))
Where:
x_n is the current approximation.f(x_n) is the function value at the current approximation.f(x_n + f(x_n)) - f(x_n) is the denominator used for acceleration.|x_(n+1) - x_n| is the step error.|f(x_(n+1))| is the residual.The calculator stops when the selected tolerance rule is met or when a stability limit is reached.
x.3*x or x*(x+1).The Steffensen method is a practical numerical technique for solving nonlinear equations when derivative evaluation is inconvenient or costly. It improves an initial guess by using repeated function values and a clever acceleration step. Because it avoids direct derivatives, it is useful in many applied mathematics, engineering, and computational science problems.
This method starts from a user-selected approximation and evaluates the function at the current point and again at a shifted point. Those two values create a denominator that estimates local behavior. The next approximation is then produced with an accelerated update. When the initial guess is sensible and the function behaves smoothly near the root, convergence can be very fast.
A detailed iteration table gives more insight than a final root alone. You can inspect each approximation, the denominator term, the step error, and the residual. That makes it easier to diagnose slow convergence, unstable updates, or denominator values that become too small. The optional true root field also helps compare approximate error with actual error.
Numerical work often needs documentation. The CSV export lets you move iteration data into spreadsheets or reports. The PDF export is useful for presentations, assignments, and records. The graph shows how approximations and residuals evolve across iterations, making convergence behavior easier to understand. Together, these features turn the calculator into a stronger learning and analysis tool for root-finding workflows.
It solves nonlinear equations of the form f(x)=0. The method updates an initial guess until the root estimate becomes stable and the residual becomes small.
Steffensen does not require an explicit derivative. That makes it useful when derivatives are hard to derive, expensive to evaluate, or not available in closed form.
A good initial guess is usually near the expected root. Better starting values often improve stability and reduce the number of iterations needed.
The residual is |f(x)| at the latest approximation. A smaller residual means the computed value is closer to satisfying the target equation.
The method can stop after convergence, after hitting the iteration limit, or when the denominator becomes too small or unstable for safe updating.
It prevents runaway values from growing without control. This helps detect unstable sequences and stops the process before useless iterations continue.
Yes. Common functions like sin, cos, tan, sqrt, abs, exp, log, ln, and log10 are supported. Use explicit multiplication in every expression.
If you already know the exact root, this field shows true error. That helps compare the computed answer against the actual solution quality.
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.