Description
Return a composite-midpoint quadrature error bound.
A truncation term and a rigorous error bound require different information. Taylor and quadrature bounds need derivative bounds over the entire relevant interval; a derivative sampled at one point or a leading asymptotic term does not automatically control the full approximation error.
Return a composite-midpoint quadrature error bound.
When to use this tool
Use Midpoint Rule Supplied Curvature Bound when checking the stated approximation term or derivative-bound calculation.
How to use the tool and how it works
- Enter lower, upper, curvatureBound, panels. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read errorBound. Use the output convention below.
Return a composite-midpoint quadrature error bound. Require upper >= lower, curvatureBound >= 0 and panels 1..1000000; errorBound = M*(upper-lower)³/(24*panels²).1
The composite midpoint error bound uses an interval-wide bound on the absolute second derivative, with cubic interval-length and inverse-square panel-count scaling.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| lower | Lower | Required. Finite number. |
| upper | Upper | Required. Finite number. |
| curvatureBound | Interval-wide bound on absolute second derivative, not geometric curvature. | Required. Finite number. |
| panels | Panels | Required. Safe integer; minimum 1; maximum 1000000. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| errorBound | Error Bound | Required. Finite number. |
{
"input": {
"lower": 0,
"upper": 2,
"curvatureBound": 3,
"panels": 1
},
"output": {
"errorBound": 1
}
}Limitations and assumptions
- M must bound abs(f'') over the interval; it is a second-derivative bound rather than geometric curvature, and no integrand is inspected.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Establish the required smoothness and interval-wide bounds for a rigorous estimate; analyze the complete numerical approximation when only local derivative data are available.
References
Similar or alternative tools
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