Enter the lower and upper bounds along with M = max |f⁽⁴⁾(x)| on [a,b] into the calculator to determine Simpson’s Rule error bound, or to find the minimum even number of subintervals n needed to achieve a target error tolerance.

Error Bound Calculator

Use this calculator to find common calculus and numerical analysis error bounds. It includes Simpson’s Rule, Trapezoidal Rule, Midpoint Rule, Taylor remainder bounds, and alternating series bounds.

Simpson Error Bound
Min n for Simpson
Trapezoidal Error
Midpoint Error
Taylor Remainder
Alternating Series
Composite Simpson’s Rule error bound:
|E| ≤ M(b − a)5 / (180n4)
Use M = max |f(4)(x)| on the interval.
Solve for the minimum even n so that the Simpson error bound stays below a target tolerance:
|E| ≤ M(b − a)5 / (180n4)
Composite Trapezoidal Rule error bound:
|E| ≤ M(b − a)3 / (12n2)
Use M = max |fʺ(x)| on the interval.
Composite Midpoint Rule error bound:
|E| ≤ M(b − a)3 / (24n2)
Use M = max |fʺ(x)| on the interval.
Taylor remainder bound after degree n:
|Rn(x)| ≤ M|x − a|n+1 / (n+1)!
Use M = max |f(n+1)(ξ)| on the interval between a and x.
For an alternating series, the truncation error satisfies:
|Rn| ≤ |an+1|
Enter the absolute value of the first omitted term.

Error Bound Formula

The following equations are used to calculate the Simpson’s Rule error bound and (optionally) the minimum even number of subintervals needed for a target tolerance.

\begin{aligned}
|E_S| &\le \frac{(b-a)^5}{180\,n^4}\,M \qquad (n\ \text{even})\\
n &\ge \left(\frac{(b-a)^5\,M}{180\,E}\right)^{1/4}
\end{aligned}
  • Where |ES| is the error bound (maximum possible magnitude of the Simpson’s Rule error),
  • a is the lower bound,
  • b is the upper bound,
  • n is the number of subintervals (must be an even integer),
  • M is the maximum value of |f⁽⁴⁾(x)| on the interval [a,b],
  • E is the target error tolerance (a maximum acceptable bound on the error).

Error Bound Definition

An error bound is an upper bound on the magnitude of the error between an approximation and the true value. In the context of the composite Simpson’s Rule, the error bound estimates how far the numerical integral may be from the exact integral based on the interval length, the number of subintervals, and an upper bound on the absolute value of the function’s 4th derivative.

Error Bound Example

Lets look at an example problem of how to use this formula.

  1. First, determine the upper bound. For this example we will assume it is 4.
  2. Next, determine the lower bound. For this example we will say its' 1.
  3. Next, determine M = max |f⁽⁴⁾(x)| on [1,4]. We will say this value is 3 for this problem.
  4. Choose a target error tolerance. For this example, let E = 0.001.
  5. Finally, solve for the minimum even n:
    n ≥ (( (b−a)5 M ) / (180 E))1/4 = ((35·3)/(180·0.001))1/4 ≈ 7.98, so the minimum even n is 8 (and the resulting bound is ≈ 9.89063×10−4).