Extrapolation

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

Overview

All extrapolation modes are concrete subtypes of AbstractExtrap. Pass them via the extrap keyword argument:

# Using Extrap() factory (recommended)
cubic_interp(x, y, xq; extrap=Extrap(:clamp))
linear_interp(x, y, xq; extrap=Extrap(:extend))

# Using direct types (also supported)
cubic_interp(x, y, xq; extrap=ClampExtrap())
linear_interp(x, y, xq; extrap=ExtendExtrap())

# Interpolant: extrap is fixed at creation
itp = cubic_interp(x, y; extrap=Extrap(:extend))
itp(xq)  # uses ExtendExtrap

All interpolation methods (linear_interp, quadratic_interp, cubic_interp, constant_interp) support the same extrapolation modes.

AbstractExtrap
├── NoExtrap         # DomainError on out-of-domain (default)
├── ClampExtrap      # clamp to y[1] / y[end]
├── ExtendExtrap     # continue boundary polynomial
├── WrapExtrap       # modular coordinate mapping
├── FillExtrap{T}    # return constant fill value
└── InBounds         # skip domain checks (advanced/internal)

Examples

using FastInterpolations
using Plots

# 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)

NoExtrap() (Default)

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

julia> cubic_interp(x, y, -1.0; extrap=NoExtrap())  # 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=NoExtrap())  # vector query (xq includes out-of-domain points)
ERROR: DomainError with -1.5:
query point outside interpolation domain [0.0, 6.0]

Create an interpolant and plot directly — the recipe renders only the interior domain:

itp = cubic_interp(x, y; extrap=NoExtrap())
plot(itp)

ClampExtrap()

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

# One-shot evaluation
yq = cubic_interp(x, y, xq; extrap=ClampExtrap())

Create an interpolant and plot directly:

itp = cubic_interp(x, y; extrap=ClampExtrap())
plot(itp)

ExtendExtrap()

Extends the boundary polynomial beyond the domain.

# One-shot evaluation
yq = cubic_interp(x, y, xq; extrap=ExtendExtrap())

Create an interpolant and plot directly:

itp = cubic_interp(x, y; extrap=ExtendExtrap())
plot(itp)

WrapExtrap()

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.

# One-shot evaluation
yq = cubic_interp(x, y, xq; extrap=WrapExtrap())

Create an interpolant and plot directly:

itp = cubic_interp(x, y; extrap=WrapExtrap())
plot(itp)

FillExtrap(value)

Returns a constant fill value for all out-of-domain queries. The fill value can be any numeric type — 0.0, 42.0, NaN, etc.

cubic_interp(x, y; extrap=FillExtrap(0.0))       # zero outside domain
cubic_interp(x, y; extrap=FillExtrap(NaN))        # NaN outside domain
cubic_interp(x, y; extrap=Extrap(:fill; fill_value=0)) # factory form

Derivatives in the out-of-domain region are exactly zero (flat constant), and integrate accumulates fill_value × interval_length over any OOB segment.

# One-shot evaluation
yq = cubic_interp(x, y, xq; extrap=FillExtrap(0.0))

Create an interpolant and plot directly:

itp = cubic_interp(x, y; extrap=FillExtrap(0.0))
plot(itp)

Comparison

# All modes on same plot
y_clamp = cubic_interp(x, y, xq; extrap=ClampExtrap())
y_ext   = cubic_interp(x, y, xq; extrap=ExtendExtrap())
y_wrap  = cubic_interp(x, y, xq; extrap=WrapExtrap())
y_fill  = cubic_interp(x, y, xq; extrap=FillExtrap(0.0))

plot(title="Extrapolation Comparison", xlabel="x", ylabel="y",
     legend=:topright, size=(700, 400))
vspan!([x[1] - 1.5, x[1]], alpha=0.1, color=:gray, label=nothing)
vspan!([x[end], x[end] + 1.5], alpha=0.1, color=:gray, label=nothing)
plot!(xq, y_clamp, label="ClampExtrap", linewidth=2)
plot!(xq, y_ext,   label="ExtendExtrap", linewidth=2, linestyle=:dash)
plot!(xq, y_wrap,  label="WrapExtrap", linewidth=2, linestyle=:dashdot)
plot!(xq, y_fill,  label="FillExtrap(0.0)", linewidth=2, linestyle=:dot)
scatter!(x, y, label="data", markersize=7, color=:black)
vline!([x[1], x[end]], color=:gray, linestyle=:dot, alpha=0.5, label=nothing)

Summary

See Also

• Derivatives: Analytical derivatives with extrapolation
• Cubic Splines: Boundary conditions and smooth periodicity