One Dimension · BasicInterpolators.jl

Here is a table of the types/constructors available for one-dimensional interpolation.

Interpolators	Method
`LinearInterpolator`	piecewise linear
`CubicInterpolator`	piecewise cubic (no smoothness guarantee)
`CubicSplineInterpolator`	cubic spline, natural or clamped
`ChebyshevInterpolator`	Chebyshev expansion

For the derivative of a ChebyshevInterpolator, use chebyderiv.

Here is a table of the functions available for one-dimensional interpolation.

Functions	Use
`quadratic`	quadratic interpolation of any 3 points
`cubic`	cubic interpolation of any 4 points
`neville`	$(n-1)$th order polynomial interpolation of any $n$ points
`vandermonde`	coefficients of $(n-1)$th order polynomial passing through $n$ points
`cubichermite`	cubic interpolation using two points with first derivatives

BasicInterpolators.LinearInterpolator — Type

LinearInterpolator(x, y, boundaries=StrictBoundaries())

Construct a LinearInterpolator for the points defined by coordinates x and values y

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LinearInterpolator(f, xa, xb, n, boundaries=StrictBoundaries())

Construct a LinearInterpolator for the function f using n evenly spaced function evaluations in the range [xa,xb]

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BasicInterpolators.CubicInterpolator — Type

CubicInterpolator(x, y, boundaries=StrictBoundaries())

Construct a CubicInterpolator for the points defined by coordinates x and values y

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CubicInterpolator(f, xa, xb, n, boundaries=StrictBoundaries())

Construct a CubicInterpolator for the function f using n evenly spaced function evaluations in the range [xa,xb]

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BasicInterpolators.CubicSplineInterpolator — Type

CubicSplineInterpolator(x, y, boundaries=StrictBoundaries())

Construct a CubicSplineInterpolator for the points defined by coordinates x and values y. This constructor creates a natural spline, where the second derivative is set to zero at the boundaries.

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CubicSplineInterpolator(x, y, dy₁, dyₙ, boundaries=StrictBoundaries())

Construct a CubicSplineInterpolator for the points defined by coordinates x and values y. This constructor creates a clamped spline, where the first derivatives at the boundaries are set by dy₁ and dyₙ.

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CubicSplineInterpolator(f, xa, xb, n, boundaries=StrictBoundaries())

Construct a CubicSplineInterpolator for the function f using n evenly spaced function evaluations in the range [xa,xb]. A natural spline is created.

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BasicInterpolators.ChebyshevInterpolator — Type

ChebyshevInterpolator(x, y)

Construct a ChebyshevInterpolator for the points defined by coordinates x and values y. The x coordinates must be arranged on a chebyshev grid, which can be generated using the chebygrid function.

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ChebyshevInterpolator(f, xa, xb, n)

Construct a ChebyshevInterpolator for the function f using n function evaluations in the range [xa,xb]. The function evaluations will occur on the chebyshev nodes.

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BasicInterpolators.quadratic — Function

quadratic(x, xₚ, yₚ)

Perform quadratic polynomial interpolation of the points defined by coordinates xₚ and values yₚ, at the coordinate x, using Neville's algorithm. xₚ and yₚ must both contain three points.

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BasicInterpolators.cubic — Function

cubic(x, xₚ, yₚ)

Perform cubic polynomial interpolation of the points defined by coordinates xₚ and values yₚ, at the coordinate x, using Neville's algorithm. xₚ and yₚ must both contain four points.

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BasicInterpolators.neville — Function

neville(x, xₚ, yₚ)

Perform polynomial interpolation of the points defined by coordinates xₚ and values yₚ, at the coordinate x, using Neville's algorithm with as many points as are provided. xₚ and yₚ must have the same length. With only 3 or 4 points the quadratic and cubic functions will be considerably faster.

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BasicInterpolators.vandermonde — Function

vandermonde(x, y)

Generate the coefficients of an arbitrary order polynomial passing through the ponts defined by coordinates x and value y. For n points, n coefficients $[c_0, c_1, ..., c_{n-1}]$ are returned forming the polynomial $c_0 + c_1x + ... + c_{n-1}x^{n-1}$

Warning

Solving for the the coefficients of a high-order polynomial is a notoriously ill-conditioned problem. It is not recommended for orders greater than 5 or 6, although it depends on the application. If you must interpolate with a high-order polynomial, it's better to use the neville function instead of computing coefficients.

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BasicInterpolators.cubichermite — Function

cubichermite(x, x₁, x₂, y₁, y₂, y₁′, y₂′)

Interpolate a cubic polynomial between two points, given its values and first derivatives at the points.

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