Derivatives

FastInterpolations.jl provides analytical derivatives for all interpolation methods. No finite difference approximation needed—derivatives are computed directly from the spline coefficients.

Overview

One-Shot API

Use the deriv keyword argument:

using FastInterpolations

x = range(0.0, 2π, 50)
y = sin.(x)

# Value (default)
val = cubic_interp(x, y, 1.0)

# First derivative
d1 = cubic_interp(x, y, 1.0; deriv=1)

# Second derivative
d2 = cubic_interp(x, y, 1.0; deriv=2)

println("At x = 1.0:")
println("  Value:      ", round(val, digits=6), " (true: ", round(sin(1.0), digits=6), ")")
println("  1st deriv:  ", round(d1, digits=6), " (true: ", round(cos(1.0), digits=6), ")")
println("  2nd deriv:  ", round(d2, digits=6), " (true: ", round(-sin(1.0), digits=6), ")")

At x = 1.0:
  Value:      0.841471 (true: 0.841471)
  1st deriv:  0.540316 (true: 0.540302)
  2nd deriv:  -0.841529 (true: -0.841471)

Vector Evaluation

xq = range(0.0, 2π, 100)

# Evaluate derivatives at multiple points
values = cubic_interp(x, y, xq)
first_derivs = cubic_interp(x, y, xq; deriv=1)
second_derivs = cubic_interp(x, y, xq; deriv=2)

println("First 5 first derivatives: ", round.(first_derivs[1:5], digits=4))

First 5 first derivatives: [1.0, 0.998, 0.992, 0.9819, 0.9679]

In-Place Evaluation

out = zeros(5)
xq_small = [0.5, 1.0, 1.5, 2.0, 2.5]

cubic_interp!(out, x, y, xq_small; deriv=1)
println("In-place first derivatives: ", round.(out, digits=4))

In-place first derivatives: [0.8776, 0.5403, 0.0708, -0.4161, -0.8011]

Interpolant API

For repeated evaluation, use deriv1(), deriv2(), and deriv3() wrapper functions:

itp = cubic_interp(x, y)

# Create derivative views
d1 = deriv1(itp)  # First derivative
d2 = deriv2(itp)  # Second derivative
d3 = deriv3(itp)  # Third derivative (cubic only)

# Scalar evaluation
println("d1(1.0) = ", round(d1(1.0), digits=6))
println("d2(1.0) = ", round(d2(1.0), digits=6))

d1(1.0) = 0.540316
d2(1.0) = -0.841529

Broadcasting

Derivative views support broadcasting for efficient vector evaluation:

xq = range(0.0, 2π, 100)

# Broadcast over query points
first_derivs = d1.(xq)
second_derivs = d2.(xq)

println("Broadcast results (first 5): ", round.(first_derivs[1:5], digits=4))

Broadcast results (first 5): [1.0, 0.998, 0.992, 0.9819, 0.9679]

Fused Broadcast

Derivative views work seamlessly with Julia's fused broadcast:

# Fused broadcast example
result = @. 2.0 * d1(xq) + d2(xq)
println("Fused broadcast (first 5): ", round.(result[1:5], digits=4))

Fused broadcast (first 5): [2.0, 1.9326, 1.8571, 1.7747, 1.6844]

Derivatives with Boundary Conditions

Different boundary conditions affect derivative behavior at endpoints. See Cubic Splines for details on NaturalBC, ClampedBC, and PeriodicBC.

Derivatives with Extrapolation

Derivatives respect the extrap setting of the interpolant:

itp = cubic_interp(x, y; extrap=:extension)
d1 = deriv1(itp)
d1(-0.5)  # Uses :extension extrapolation for derivative too

See Extrapolation for available modes.

API Summary

One-Shot API

Interpolant API

See Also

• Visual Comparison: Side-by-side plots of all 4 methods
• Constant | Linear | Quadratic | Cubic