Extrapolation

Extrapolation controls behavior when query points fall outside the data domain [x[1], x[end]].

Overview

Use the extrap keyword argument to specify extrapolation behavior:

# One-shot: specify extrap per call
cubic_interp(x, y, xq; extrap=:constant)
linear_interp(x, y, xq; extrap=:extension)

# Interpolant: extrap is fixed at creation
itp = cubic_interp(x, y; extrap=:extension)  # all future calls use :extension
itp(xq)  # uses :extension

Both linear_interp and cubic_interp support the same extrapolation modes.

Mode	Behavior
`:none`	Throws `DomainError` (default)
`:constant`	Returns boundary values
`:extension`	Extends boundary polynomial
`:wrap`	Wraps coordinates periodically (no smoothness enforced)

Examples

using FastInterpolations

# Sample data
x = [0.0, 0.7, 1.5, 2.3, 3.0, 4.2, 5.0, 6.0]
y = [0.2, 1.1, 0.6, 1.8, 1.2, 0.4, 1.5, 0.8]

# Query points (including extrapolation region)
xq = range(x[1] - 1.5, x[end] + 1.5, 300)

`extrap=:none` (Default)

Throws DomainError for out-of-domain queries. Use when extrapolation is unexpected.

julia> cubic_interp(x, y, -1.0; extrap=:none)  # scalar query outside domain
ERROR: DomainError with -1.0:
query point outside interpolation domain [0.0, 6.0]

julia> cubic_interp(x, y, xq; extrap=:none)  # vector query (xq includes out-of-domain points)
ERROR: DomainError with -1.5:
query point outside interpolation domain [0.0, 6.0]

Only interior queries succeed:

yq = cubic_interp(x, y, range(x[1], x[end], 200); extrap=:none)

`extrap=:constant`

Returns boundary values: y[1] for left, y[end] for right.

yq = cubic_interp(x, y, xq; extrap=:constant)

`extrap=:extension`

Extends the boundary polynomial beyond the domain.

yq = cubic_interp(x, y, xq; extrap=:extension)

`extrap=:wrap`

Wraps coordinates periodically:

\[S(x + \tau) = S(x), \quad \tau = x_{\text{end}} - x_1\]

This is purely coordinate mapping—it does not enforce any physical conditions at the boundary. The spline may have discontinuities in value, slope, or curvature at the wrap point.

For Smooth Periodicity

If you need C² continuity at the periodic boundary, use bc=PeriodicBC() with cubic_interp. This enforces $S(x_1) = S(x_{\text{end}})$, $S'(x_1) = S'(x_{\text{end}})$, and $S''(x_1) = S''(x_{\text{end}})$.

yq = cubic_interp(x, y, xq; extrap=:wrap)

Comparison

# All three modes on same plot
y_const = cubic_interp(x, y, xq; extrap=:constant)
y_ext   = cubic_interp(x, y, xq; extrap=:extension)
y_wrap  = cubic_interp(x, y, xq; extrap=:wrap)

Summary

Mode	Behavior	Use Case
`:none`	`DomainError`	Strict domain enforcement (default)
`:constant`	Returns boundary values	Physical constraints
`:extension`	Continues boundary polynomial	Smooth continuation
`:wrap`	Wraps coordinates (no smoothness)	Cyclic data (see `PeriodicBC` for C² continuity)