Complex Number Support

All interpolation methods support complex-valued data natively. The type system automatically handles the distinction between the grid type (which must be real) and the value type (which can be real or complex).

Basic Usage

When interpolating complex data, the interpolant will return complex values. Analytic derivatives (deriv=1, deriv=2) also return complex values corresponding to the derivative of the real and imaginary parts respectively.

using FastInterpolations

x = 0.0:1.0:4.0
y = [1.0+0.5im, 0.5-1.0im, -2.0+1.5im, -1.0-0.5im, 2.0+2.0im]

# Create cubic spline with complex values
itp = cubic_interp(x, y)

# Evaluate at x = 1.5
val = itp(1.5)  # Returns ComplexF64

Type System

Internally, the interpolant separates the grid type (Tg) from the value type (Tv):

Grid (Tg): Always AbstractFloat (Float32 or Float64). This ensures that grid search and interval calculations remain performant and ordered.
Values (Tv): Can be Real or Complex.

If you provide inputs with different precisions (e.g., Float64 grid and ComplexF32 values), they will be promoted appropriately to ensure compatibility.

Boundary Conditions with Complex Values

Boundary conditions (BCPair) can specify complex values for derivatives. You can even mix real and complex boundary conditions if needed (though typically they match the data type).

# Left: First derivative = 1.0 + 2.0im (Complex)
# Right: Natural boundary (Second derivative = 0.0) (Real)
bc = BCPair(Deriv1(1.0 + 2.0im), Deriv2(0.0))

itp = cubic_interp(x, y; bc=bc)

Automatic Estimation: The CubicFit() boundary condition works seamlessly with complex data, estimating the complex derivatives at the endpoints using a 4-point polynomial fit.

# Estimates complex derivatives from the data points at boundaries
itp_auto = cubic_interp(x, y; bc=CubicFit())

Applications: Periodic Trajectories

A common use case in physics and engineering is interpolating a closed trajectory in the complex plane.

# Circle in complex plane
theta = range(0, 2π, 101)
trajectory = exp.(im .* theta)

# PeriodicBC ensures C² continuity at the wrap-around point (0 ↔ 2π)
itp_periodic = cubic_interp(theta, trajectory; bc=PeriodicBC())