Examples
Examples
Try the starting example
- Restricted real expression f(x)
- x^3
- Integration start
- 0
- Integration end
- 2
- Absolute error-estimate tolerance
- 1e-08
- Method: adaptive-simpson or composite-trapezoid
- adaptive-simpson
- Numerical integral estimate
- 4
- Method error diagnostic
- 0
- Function evaluations
- 5
Try another scenario
- Restricted real expression f(x)
- x
- Integration start
- 0
- Integration end
- 2
- Absolute error-estimate tolerance
- 1e-08
- Method: adaptive-simpson or composite-trapezoid
- composite-trapezoid
- Numerical integral estimate
- 2
- Method error diagnostic
- 0
- Function evaluations
- 3
Calculation method
Calculation method
Numerical integral estimate
Adaptive Simpson recursively refines halves and applies the delta/15 correction. Composite trapezoid doubles uniform panels until its step-difference/3 diagnostic meets tolerance. Reverse bounds reverse the integral sign.
Method error diagnostic
Sum accepted Simpson |delta|/15 estimates, or final trapezoid difference/3; this is a smooth-function diagnostic, not a certified error bound.
Function evaluations
Count finite function evaluations; a 20,000-evaluation budget and refinement limits prevent an unbounded run.
Symbols and units
- a
- Restricted real expression f(x)
- b
- Integration start
- c
- Integration end
- d
- Absolute error-estimate tolerance
- e
- Method: adaptive-simpson or composite-trapezoid
Use the units shown. A list keeps the order in which you enter its values. The symbol ^ means “raised to the power of”.
Assumptions & limits
Assumptions & limits
- Finite smooth real functions over a finite interval. Singularities and unresolved refinement produce errors; discontinuities, cancellation and unsampled features can defeat numerical error estimates. Restricted expressions allow x (and y for ODEs), pi, e, + - * / ^, and sin, cos, tan, sec, csc, cot, exp, log, sqrt, abs, sinh, cosh, tanh, sech, csch, coth. Trigonometric arguments are in radians. Double-precision results are approximate. Numeric tables use 12 significant digits and display components at roundoff scale (32 machine epsilons relative to their row or relevant calculation scale) as zero; scalar diagnostics retain full precision.
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Further reading
SciPy: numerical integration and ODE method references ↗
Background reading; this does not imply endorsement or independent review of SumFizz.