SC-04

Course 4

Numerical Integration

Learn to approximate definite integrals using quadrature formulas. From basic trapezoidal and Simpson rules to advanced Gaussian quadrature, master the tools for numerical calculus.

Learning Objectives

Derive and apply Newton-Cotes quadrature formulas
Understand algebraic precision and error analysis
Construct composite quadrature rules for improved accuracy
Apply Richardson extrapolation and Romberg integration
Master Gaussian quadrature for optimal accuracy
Choose appropriate methods for different integration problems

1. Newton-Cotes Formulas

Newton-Cotes formulas approximate integrals by integrating an interpolating polynomial through equally-spaced nodes.

Definition 4.1: Trapezoidal Rule

Using linear interpolation through $(a, f(a))$ and $(b, f(b))$ :

\int_a^b f(x) \, dx \approx \frac{b-a}{2} \left( f(a) + f(b) \right)

Theorem 4.1: Trapezoidal Rule Error

If $f \in C^2[a, b]$ , the error is:

R_T(f) = -\frac{(b-a)^3}{12} f''(\eta), \quad \text{for some } \eta \in (a, b)

The trapezoidal rule has algebraic precision 1.

Definition 4.2: Simpson's Rule

Using quadratic interpolation through three points:

\int_a^b f(x) \, dx \approx \frac{b-a}{6} \left( f(a) + 4f\left(\frac{a+b}{2}\right) + f(b) \right)

Theorem 4.2: Simpson's Rule Error

If $f \in C^4[a, b]$ , the error is:

R_S(f) = -\frac{(b-a)^5}{2880} f^{(4)}(\eta), \quad \text{for some } \eta \in (a, b)

Simpson's rule has algebraic precision 3 (not 2, due to symmetry).

Definition 4.3: General Newton-Cotes Formula

Divide $[a, b]$ into $n$ equal parts with nodes $x_i = a + ih$ , $h = (b-a)/n$ :

\int_a^b f(x) \, dx \approx (b-a) \sum_{i=0}^{n} c_i^{(n)} f(x_i)

where $c_i^{(n)}$ are the Newton-Cotes coefficients.

Example: Newton-Cotes Coefficients

$n$	Name	Coefficients	Precision
1	Trapezoidal	$\frac{1}{2}, \frac{1}{2}$	1
2	Simpson	$\frac{1}{6}, \frac{4}{6}, \frac{1}{6}$	3
3	Simpson's 3/8	$\frac{1}{8}, \frac{3}{8}, \frac{3}{8}, \frac{1}{8}$	3
4	Boole	$\frac{7}{90}, \frac{32}{90}, \frac{12}{90}, \frac{32}{90}, \frac{7}{90}$	5

Remark:

For even $n$ , Newton-Cotes formulas achieve precision $n+1$ instead of just $n$ due to symmetry of the nodes and weights.

2. Composite Quadrature Rules

High-degree Newton-Cotes formulas can have negative coefficients and instability. Instead, apply low-degree rules on subintervals.

Definition 4.4: Composite Trapezoidal Rule

Divide $[a, b]$ into $n$ equal parts with $h = (b-a)/n$ :

T_n = \frac{h}{2} \left( f(a) + 2\sum_{k=1}^{n-1} f(a + kh) + f(b) \right)

Theorem 4.3: Composite Trapezoidal Error

If $f \in C^2[a, b]$ :

R(f, T_n) = -\frac{b-a}{12} h^2 f''(\eta) = O(h^2)

Definition 4.5: Composite Simpson's Rule

Divide $[a, b]$ into $2n$ equal parts:

S_n = \frac{h}{3} \left( f(a) + 4\sum_{k=1}^{n} f(x_{2k-1}) + 2\sum_{k=1}^{n-1} f(x_{2k}) + f(b) \right)

Theorem 4.4: Composite Simpson Error

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

R(f, S_n) = -\frac{b-a}{180} h^4 f^{(4)}(\eta) = O(h^4)

Note:

Successive halving: From $T_n$ to $T_{2n}$ :

T_{2n} = \frac{1}{2} T_n + \frac{h}{2} \sum_{k=1}^{n} f\left(a + \frac{2k-1}{2}h\right)

This avoids recomputing function values at existing nodes.

3. Romberg Integration

Richardson extrapolation systematically eliminates error terms from the trapezoidal rule.

Definition 4.6: Richardson Extrapolation

If $F_1(h)$ approximates $F^*$ with error $O(h^{p_1})$ , then:

F_2(h) = \frac{F_1(qh) - q^{p_1} F_1(h)}{1 - q^{p_1}}

approximates $F^*$ with error $O(h^{p_2})$ where $p_2 > p_1$ .

Theorem 4.5: Romberg Formula

With $q = 1/2$ for successive halving:

T_m^{(k)} = \frac{4^m T_{m-1}^{(k+1)} - T_{m-1}^{(k)}}{4^m - 1}, \quad m = 1, 2, \ldots

where $T_0^{(k)}$ is the trapezoidal rule with $2^k$ subintervals.

Example: Romberg Table

The Romberg table structure (each column has higher accuracy):

$T_0$	$T_1$	$T_2$	$T_3$
$T_0^{(0)}$
$T_0^{(1)}$	$T_1^{(0)}$
$T_0^{(2)}$	$T_1^{(1)}$	$T_2^{(0)}$
$T_0^{(3)}$	$T_1^{(2)}$	$T_2^{(1)}$	$T_3^{(0)}$

Note: $T_1^{(k)} = S_n$ (Simpson), and $T_m^{(k)}$ has error $O(h^{2m+2})$ .

4. Gaussian Quadrature

By choosing both nodes and weights optimally, Gaussian quadrature achieves the maximum possible precision with $n$ function evaluations.

Definition 4.7: Gaussian Quadrature

A quadrature formula $\int_a^b w(x) f(x) \, dx \approx \sum_{i=1}^{n} A_i f(x_i)$ is Gaussian if it has algebraic precision $2n-1$ .

Theorem 4.6: Gauss-Legendre Nodes

For $\int_{-1}^{1} f(x) \, dx$ , the optimal nodes are the zeros of the Legendre polynomial $P_n(x)$ .

The weights are: $A_i = \frac{2}{(1 - x_i^2)[P_n'(x_i)]^2}$

Example: Gauss-Legendre Nodes and Weights

$n$	Nodes $x_i$	Weights $A_i$
1	$0$	$2$
2	$\pm 1/\sqrt{3}$	$1, 1$
3	$0, \pm\sqrt{3/5}$	$8/9, 5/9, 5/9$

Gauss-Laguerre

For $\int_0^{\infty} e^{-x} f(x) \, dx$ , nodes are zeros of Laguerre polynomials.

Gauss-Hermite

For $\int_{-\infty}^{\infty} e^{-x^2} f(x) \, dx$ , nodes are zeros of Hermite polynomials.

Gauss-Chebyshev

For $\int_{-1}^{1} \frac{f(x)}{\sqrt{1-x^2}} \, dx$ , nodes are zeros of Chebyshev polynomials.

Note:

To integrate over $[a, b]$ instead of $[-1, 1]$ , use the transformation $x = \frac{b-a}{2}t + \frac{a+b}{2}$ .

Practice Quiz

Numerical Integration Quiz

Questions

Correct

Accuracy

The trapezoidal rule approximates

\int_a^b f(x)dx

as:

Easy

Not attempted

Simpson's rule has algebraic precision of degree:

Easy

Not attempted

The error in the composite trapezoidal rule is

O(h^k)

where

k

equals:

Easy

Not attempted

Romberg integration uses which technique to improve accuracy?

Medium

Not attempted

A quadrature formula has algebraic precision

m

if it is exact for all polynomials of degree:

Medium

Not attempted

The composite Simpson's rule divides

[a,b]

into

2n

equal parts and has error:

Medium

Not attempted

Gauss-Legendre quadrature with

n

points has algebraic precision:

Hard

Not attempted

The nodes of Gauss-Legendre quadrature on

[-1,1]

are:

Hard

Not attempted

For Newton-Cotes formulas with even

n

, the algebraic precision is:

Hard

Not attempted

The relationship

S_n = \frac{1}{3}(4T_{2n} - T_n)

connects:

Medium

Not attempted

Frequently Asked Questions

When should I use Simpson's rule vs. Gaussian quadrature?

Simpson's rule: Good for smooth functions when you can choose equally-spaced evaluation points. Easy to implement with adaptive refinement.

Gaussian quadrature: Better accuracy with fewer function evaluations, but nodes are not equally spaced. Ideal when function evaluations are expensive.

Why does Simpson's rule have precision 3 instead of 2?

Due to symmetry! The error term for degree-3 polynomials vanishes because the nodes and weights are symmetric about the midpoint. This "bonus" precision occurs for all even-order Newton-Cotes formulas.

How do I handle improper integrals?

Options include:

Variable transformation to remove singularity
Gauss-Laguerre for $[0, \infty)$ with exponential decay
Gauss-Hermite for $(-\infty, \infty)$ with Gaussian weight
Truncation with error estimation

What's the advantage of Romberg integration?

Romberg integration reuses function values from coarser grids, achieving high accuracy efficiently. It automatically produces error estimates by comparing successive approximations, enabling adaptive termination.

Why not always use the highest-degree Newton-Cotes formula?

High-degree Newton-Cotes formulas (n ≥ 8) have negative coefficients, causing potential cancellation errors. They can also be unstable.

Composite low-degree rules or Gaussian quadrature are preferred for accuracy and stability.