Understanding Picard Iteration
Picard iteration is a fixed point method for initial value problems. It builds a solution curve by repeating an integral equation. The first curve is often a flat line through the starting value. Each new curve uses the previous curve inside the derivative function. This process can reveal how a solution grows, falls, or bends over an interval.
Why This Calculator Helps
Manual Picard work can become long. The derivative may contain x, y, powers, roots, or trigonometric terms. A table makes every step easier to inspect. This calculator uses a grid based numerical form of the Picard integral. It shows the final approximation, change between iterations, and optional exact error. You can test step size, integration style, tolerance, and iteration count.
Numerical Approach
The tool does not try to produce a symbolic polynomial. Instead, it evaluates the integral on small subintervals. The previous approximation supplies y values on the grid. The selected rule estimates each small area. Smaller steps usually improve the curve, but they also increase rows. More iterations may improve convergence until changes become small.
Good Input Choices
Start with a simple derivative. Use x and y as variables. Use expressions such as x+y, x^2-y, sin(x)+y, or exp(x)-y. Choose a step that reaches the target point with enough detail. Use a modest iteration count first. Then raise it when the maximum change stays large.
Reading the Output
The iteration summary shows the final value after each pass. The maximum change helps judge convergence. A small value means the new curve barely changed. The grid table shows the approximate solution at each x point. If you enter an exact solution, the table also displays absolute error.
Practical Uses
Picard iteration supports learning, checking, and reporting. It helps students see the link between differential equations and integral equations. It also helps teachers prepare examples. Engineers and analysts can use it for quick exploratory checks. Results should still be reviewed. Numerical settings always affect accuracy. Try several step sizes before trusting a final value.
Reporting
Use the exported files to compare runs. Keep the same equation. Change only one setting at a time. That habit makes numerical behavior easier to explain and repeat.