Numerical Interpolation, Integration & Solution of ODEs Notes

Last reviewed 16 Sept 2026 · 7 min read

Finite differences

For equally spaced values $x_{0}, x_{1} = x_{0} + h, \dots$ with $y_{i} = f (x_{i})$ :

Operator	Definition
Forward difference $Δ$	$Δ y_{i} = y_{i + 1} - y_{i}$
Backward difference $\nabla$	$\nabla y_{i} = y_{i} - y_{i - 1}$
Central difference $δ$	$δ y_{i} = y_{i + 1/2} - y_{i - 1/2}$
Shift operator $E$	$E y_{i} = y_{i + 1}$
Averaging operator $μ$	$μ y_{i} = \frac{1}{2} (y_{i + 1/2} + y_{i - 1/2})$

Relations: $E = 1 + Δ$ ; $\nabla = 1 - E^{- 1}$ ; $δ = E^{1/2} - E^{- 1/2}$ ; $E = e^{h D}$ ( $D = d / d x$ ).

The nth differences of a polynomial of degree n are constant, and higher differences are zero.

Interpolation

Newton's forward difference formula

For $x$ near the beginning of the table, $p = \frac{x - x _{0}}{h}$ :

y = y_{0} + p Δ y_{0} + \frac{p ( p - 1 )}{2 !} Δ^{2} y_{0} + \frac{p ( p - 1 ) ( p - 2 )}{3 !} Δ^{3} y_{0} + \dots

Newton's backward difference formula

For $x$ near the end of the table, $p = \frac{x - x _{n}}{h}$ :

y = y_{n} + p \nabla y_{n} + \frac{p ( p + 1 )}{2 !} \nabla^{2} y_{n} + \dots

(Central difference formulas — Gauss, Stirling, Bessel, Everett — are used near the middle.)

Lagrange's interpolation formula

For unequally spaced points:

P (x) = i = 0 \sum n y_{i} L_{i} (x), L_{i} (x) = j \neq = i \prod \frac{x - x _{j}}{x _{i} - x _{j}}

Newton's divided difference formula

P (x) = f [x_{0}] + (x - x_{0}) f [x_{0}, x_{1}] + (x - x_{0}) (x - x_{1}) f [x_{0}, x_{1}, x_{2}] + \dots

with $f [x_{0}, x_{1}] = \frac{f ( x _{1} ) - f ( x _{0} )}{x _{1} - x _{0}}$ , etc. — convenient for adding points.

Interpolation error $\frac{f ^{(n + 1)} ( ξ )}{( n + 1 )!} \prod (x - x_{i})$ ; high-degree polynomials on equally spaced points may oscillate (Runge phenomenon) — spline interpolation is often preferred.
Extrapolation (outside the data range) is unreliable.

Numerical differentiation

From Newton's forward formula at $x = x_{0}$ :

(\frac{d y}{d x})_{x_{0}} = \frac{1}{h} (Δ y_{0} - \frac{1}{2} Δ^{2} y_{0} + \frac{1}{3} Δ^{3} y_{0} - \dots)

Simple approximations: forward $\frac{y _{1} - y _{0}}{h}$ (error $O (h)$ ); central $\frac{y _{1} - y _{- 1}}{2 h}$ (error $O (h^{2})$ ); second derivative $\frac{y _{1} - 2 y _{0} + y _{- 1}}{h ^{2}}$ .

Numerical integration

FormulaNewton–Cotes rules (h = step, n = number of intervals)

Trapezoidal rule:

\int_{x_{0}}^{x_{n}} y d x \approx \frac{h}{2} [(y_{0} + y_{n}) + 2 (y_{1} + y_{2} + \dots + y_{n - 1})]

Error $O (h^{2})$ (global); exact for linear functions.

Simpson's 1/3 rule ( $n$ even):

\int y d x \approx \frac{h}{3} [(y_{0} + y_{n}) + 4 (y_{1} + y_{3} + \dots) + 2 (y_{2} + y_{4} + \dots)]

Error $O (h^{4})$ ; exact for polynomials up to degree 3.

Simpson's 3/8 rule ( $n$ a multiple of 3):

\int y d x \approx \frac{3 h}{8} [(y_{0} + y_{n}) + 3 (y_{1} + y_{2} + y_{4} + y_{5} + \dots) + 2 (y_{3} + y_{6} + \dots)]

Error $O (h^{4})$ .

Weddle's rule ( $n$ multiple of 6): $\frac{3 h}{10} [y_{0} + 5 y_{1} + y_{2} + 6 y_{3} + y_{4} + 5 y_{5} + y_{6}]$ per 6 intervals.

Gaussian quadrature

Chooses optimal points and weights — an $n$ -point Gauss–Legendre rule is exact for polynomials of degree up to $2 n - 1$ .
Two-point rule on $[- 1, 1]$ : $\int_{- 1}^{1} f d x \approx f (- \frac{1}{3}) + f (\frac{1}{3})$ .
For $[a, b]$ , substitute $x = \frac{b - a}{2} t + \frac{b + a}{2}$ .
Widely used in finite element stiffness computations.

Romberg integration

Richardson extrapolation of trapezoidal results with successively halved step sizes: $I = I_{h /2} + \frac{I _{h /2} - I _{h}}{3}$ (gives Simpson accuracy), repeated for higher accuracy.

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Numerical Interpolation, Integration & Solution of ODEs