Adaptive Quadrature Result
Calculate Adaptive Quadrature Points
Enter an integrand, interval, tolerance, and method. The calculator estimates integral value, local error, accepted panels, recursion depth, function calls, and unique quadrature points.
Generated Quadrature Points Preview
After calculation, the first selected points are shown here. CSV export includes the full generated list.
| # | x point | f(x) | Nearest accepted panel |
|---|---|---|---|
| Run the calculator to view adaptive quadrature points. | |||
Example Data Table
| Function | Interval | Tolerance | Method | Expected Behavior |
|---|---|---|---|---|
| sin(x) | 0 to pi | 1e-6 | Adaptive Simpson | Low point count because the curve is smooth. |
| sqrt(x) | 0 to 1 | 1e-7 | Gauss-Kronrod style | More points near zero due to changing slope. |
| 1 / (1 + 50*x^2) | -1 to 1 | 1e-6 | Adaptive Simpson | More points near the central peak. |
| abs(x) | -2 to 2 | 1e-5 | Adaptive Trapezoid | Splitting increases near the sharp corner. |
Formula Used
Adaptive quadrature divides an interval until the local error is smaller than the requested tolerance. Each accepted panel adds quadrature points to the final point plan.
How to Use This Calculator
- Enter the function using x as the variable.
- Set the lower and upper integration limits.
- Choose absolute and relative tolerance values.
- Select a quadrature method for your numerical task.
- Set the initial panels and maximum recursion depth.
- Press Calculate Points to view integral and point estimates.
- Download CSV or PDF when you need a saved report.
Adaptive Quadrature Points Guide
Why Adaptive Points Matter
Adaptive quadrature helps estimate an integral when the curve changes in uneven ways. A fixed rule uses equal panels everywhere. That can waste points on smooth sections. It can also miss sharp bends, peaks, or oscillations. This calculator uses the interval, tolerance, method, and depth limit to predict how many points are needed.
How The Method Works
The main idea is simple. The interval is first tested with a coarse rule. Then each section is checked for local error. Sections with high error are split again. Smooth sections are accepted early. Difficult sections receive more points. This makes the point plan more efficient than a plain uniform grid.
Settings That Change Point Count
Point counts depend on several settings. A tight tolerance asks for smaller error. That usually increases panels and evaluations. A large interval also needs more work. A high difficulty factor represents rapid changes in the function. A higher safety factor gives extra margin. Maximum depth prevents endless splitting when the target is too strict.
Choosing A Method
The Simpson option is useful for smooth functions. It uses parabolic fitting and has strong accuracy for gentle curves. The trapezoid option is simpler. It may need more points for the same tolerance. The Gauss style option represents higher order rules. It can be efficient when the function is well behaved.
Interpreting Results
Use the generated point count as a planning guide. It helps size arrays, estimate runtime, and compare method choices before running a large numerical job. The estimated integral and error are helpful checks. They should not replace domain review. Very rough, discontinuous, or singular functions may need special handling.
Export And Review
The export tools support documentation. CSV gives a clean table for spreadsheets. PDF creates a report for sharing. The example table shows common scenarios. It also explains how interval length and tolerance change the point need.
Best Practice
For best results, choose realistic tolerances. Start with moderate precision. Then reduce tolerance only when needed. Check whether the point count grows quickly. Rapid growth may mean the function has sharp behavior. In that case, split the interval around the difficult area before using adaptive quadrature. This method keeps reports transparent. Each result shows assumptions, accepted panels, unique points, estimated calls, and practical calculation notes too.
FAQs
What are adaptive quadrature points?
They are x values selected by an adaptive integration method. More points are placed where the function is harder to approximate.
Why does tolerance affect point count?
A smaller tolerance asks for lower error. The algorithm usually splits more panels, so the total number of points increases.
Which method should I choose first?
Adaptive Simpson is a good first choice for smooth functions. Use the Gauss-Kronrod style option for higher order estimates.
What does maximum depth mean?
Maximum depth limits how many times a panel can be split. It prevents excessive recursion when the function is difficult.
Why are function calls higher than unique points?
Some methods evaluate extra test points inside each panel. Duplicate x values can also be reused in the final unique point list.
Can this handle discontinuous functions?
It can estimate them, but results need caution. Split the interval around jumps or singular points for better reliability.
What should I do if depth is reached?
Increase maximum depth, loosen tolerance, or split the interval manually. Depth warnings often mean the curve has sharp behavior.
What is included in the CSV file?
The CSV includes summary values and generated quadrature points. It also includes function values and nearest panel information.