Description
Return first-order output half-width from a supplied derivative.
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 first-order output half-width from a supplied derivative.
When to use this tool
Use Scalar Derivative Error Propagation Factor when checking the stated approximation term or derivative-bound calculation.
How to use the tool and how it works
- Enter derivative, inputHalfwidth. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read outputHalfwidth. Use the output convention below.
Return first-order output half-width from a supplied derivative. Require inputHalfwidth >= 0; outputHalfwidth = abs(derivative)*inputHalfwidth.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| derivative | Derivative | Required. Finite number. |
| inputHalfwidth | Input Halfwidth | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| outputHalfwidth | Output Halfwidth | Required. Finite number. |
{
"input": {
"derivative": -3,
"inputHalfwidth": 0.2
},
"output": {
"outputHalfwidth": 0.6000000000000001
}
}Limitations and assumptions
- This is local linear propagation, not a nonlinear interval enclosure or a statistical confidence interval.
- 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
- Calculus: Scalar Central Difference Error Term
Return the leading central-difference derivative error term.
- Scalar Second Order Taylor Remainder Bound
Return a quadratic Taylor remainder bound from a supplied third-derivative bound.