Simpson's Rule

Q: How accurate is Simpson's rule?

The error of the 1/3 rule is O(h^5) on a single panel and O(h^4) for the composite rule, two orders more accurate than the trapezoidal rule's O(h^2).

Q: What is the difference between the 1/3 and 3/8 rules?

The 1/3 rule is based on quadratic interpolation (3 points) and the 3/8 rule on cubic interpolation (4 points). Both are O(h^5), but the 3/8 rule is convenient when the number of subintervals is a multiple of 3.

Goal of this chapter

Understand the derivation of Simpson's rule (quadratic-polynomial approximation), the $O(h^4)$ error estimate, and the composite Simpson rule.

Prerequisites

Contents

1. Simpson's 1/3 Rule
2. Simpson's 3/8 Rule
3. Composite Simpson Rule
4. Error Estimate
5. Accuracy Comparison with the Trapezoidal Rule
6. FAQ
7. References

1. Simpson's 1/3 Rule

We approximate the integral by the area under the quadratic polynomial (a parabola) through the three points $(a, f(a))$, $(m, f(m))$, $(b, f(b))$ with $m = (a+b)/2$.

Simpson's 1/3 rule

$$\displaystyle\int_a^b f(x)\,dx \approx \dfrac{h}{3}\bigl[f(a) + 4f(m) + f(b)\bigr], \qquad h = \dfrac{b-a}{2}$$

The characteristic feature is the weight ratio $1:4:1$. This rule is exact not only for quadratics but also for cubic polynomials (the order of accuracy gains one extra degree).

Why it is exact up to cubics

Consider the difference between $f$ and its quadratic interpolant $p_2$. When $f$ is a cubic, $f - p_2$ is a cubic that vanishes at the three sample points $a,\,m,\,b$, so with leading coefficient $c$ it can be written

$$f(x) - p_2(x) = c\,(x-a)(x-m)(x-b).$$

Setting $m=\dfrac{a+b}{2}$, $h=\dfrac{b-a}{2}$, and $x = m + t$,

$$(x-a)(x-m)(x-b) = (t+h)\,t\,(t-h) = t^3 - h^2 t,$$

which is an odd function of $t$. The integral of an odd function over the symmetric interval $[-h,\,h]$ is $0$, so $\displaystyle\int_a^b \bigl(f - p_2\bigr)\,dx = 0$; that is, $\displaystyle\int_a^b f\,dx = \int_a^b p_2\,dx$ holds exactly. In other words, the fact that the three points are placed symmetrically (equally spaced) about the midpoint is what gives this "bonus" extra degree. For the same reason, the symmetric closed Newton-Cotes rules gain one extra degree of accuracy for even-degree interpolation.

0 / 0

Figure 1. Geometric meaning of Simpson's rule (animation). The true integral of the curve $y = f(x) = \operatorname{sinc} x = \dfrac{\sin \pi x}{\pi x}$ ($\displaystyle\int_{-4}^{4} \operatorname{sinc} x\,dx = \tfrac{2}{\pi}\operatorname{Si}(4\pi) \approx 0.9499$, which has no elementary closed form and is expressed by the sine integral $\operatorname{Si}$) is approximated by the area under a parabola (green) through three points over each adjacent pair of subintervals. This function has a sharp peak (value $1$) at $x=0$ with oscillating side lobes (ringing) that swing positive and negative, so at $n=2$ the wide parabola completely fails to capture the peak and the lobes and greatly overestimates it ($S_2 \approx 5.3$), making the effect of the approximation and refinement clearly visible. Use "▶ Auto-play" or "Next / Prev" to advance one step at a time; as the subdivision count $n$ doubles as $2 \to 4 \to 8 \to 16 \to 32$, $S_n$ converges to the true value far faster than the trapezoidal estimate $T_n$ at the same $n$ (the log shows both errors). Here $x=0$ is a removable singularity with $\operatorname{sinc} 0 = 1$.

2. Simpson's 3/8 Rule

This rule is based on cubic Lagrange interpolation using four points.

Simpson's 3/8 rule

$$\displaystyle\int_a^b f(x)\,dx \approx \dfrac{3h}{8}\bigl[f(x_0) + 3f(x_1) + 3f(x_2) + f(x_3)\bigr], \qquad h = \dfrac{b-a}{3}$$

The weight ratio is $1:3:3:1$, and it is exact for cubic polynomials. It is convenient when the number of subintervals is a multiple of 3.

3. Composite Simpson Rule

Divide $[a,b]$ into $n$ (even) equal parts and apply the 1/3 rule over each adjacent pair of subintervals. With $h = (b-a)/n$,

Composite Simpson rule

$$S_n = \dfrac{h}{3}\bigl[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n)\bigr]$$

The weights follow the pattern $1, 4, 2, 4, 2, \ldots, 4, 2, 4, 1$.

4. Error Estimate

Theorem (error of the composite Simpson rule)

If $f \in C^4[a,b]$, then

$$\displaystyle\int_a^b f(x)\,dx - S_n = -\dfrac{(b-a)h^4}{180}\,f^{(4)}(\xi), \qquad \xi \in (a,b)$$

that is, the error is $O(h^4)$.

This is two orders more accurate than the trapezoidal rule's $O(h^2)$. Halving the step size $h$ reduces the error by roughly $1/16$.

5. Accuracy Comparison with the Trapezoidal Rule

Figure 2. Convergence of the error for $\displaystyle\int_{-4}^{4} \operatorname{sinc} x\,dx$ with the trapezoidal and Simpson rules (log-log). The measured errors (the difference of $T_n$ and $S_n$ from the true value $\approx 0.9499$) at $n=64,128,256,512,1024$ are plotted. The trapezoidal rule has slope about $2$ ($O(h^2)$) and Simpson rule slope about $4$ ($O(h^4)$); once the mesh resolves the peak Simpson is orders of magnitude smaller at the same $n$ and decreases more steeply. The integrand is the same $\operatorname{sinc} x$ as in Figure 1.

Rule	Interpolation degree	Degree of accuracy	Error order
Trapezoidal	1	1	$O(h^2)$
Simpson 1/3	2	3 (+ bonus)	$O(h^4)$
Simpson 3/8	3	3	$O(h^4)$

6. FAQ

Q1. What is Simpson's rule?

It is a method of numerical integration that approximates a definite integral by the area under the parabola through three points. The 1/3 rule is $\dfrac{h}{3}[f(a)+4f(m)+f(b)]$ and returns the exact value for polynomials of degree up to 3.

Q2. How accurate is Simpson's rule?

The error of the composite Simpson rule is $O(h^4)$, two orders more accurate than the trapezoidal rule's $O(h^2)$.

Q3. What is the difference between the 1/3 and 3/8 rules?

The 1/3 rule uses quadratic interpolation (3 points) and the 3/8 rule cubic interpolation (4 points). Both are $O(h^5)$ on a single panel, but the 3/8 rule is convenient when the number of subintervals is a multiple of 3.

7. References

Wikipedia, "Simpson's rule"
R. L. Burden & J. D. Faires, Numerical Analysis, 10th ed., Cengage, 2016.
P. J. Davis & P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, 1984.