Spline Continuity and End Conditions

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Spline Continuity and End Conditions

This help page describes the interpolating, continuity, and end conditions used in CurveFitting[Spline].

The form of the resulting piecewise function returned depends on whether the degree d is odd or even, and whether or not the knots='data' option is specified in the even case.

Odd Degree

Even Degree

Even degree with knots='data'

Examples

Odd Degree

The resulting function created by CurveFitting[Spline] is of the form $piecewise (v < x_{1}, p_{1}, ..., p_{n})$ , where the n spline sections ${p_{1}, p_{2}, ..., p_{n}}$ are polynomials of degree at most d. These polynomials are given by the following $(d + 1) n$ conditions:

2n Interpolating Conditions

•	Force continuity at the knots.

$p_{i} (x_{i - 1}) = y_{i - 1}, and p_{i} (x_{i}) = y_{i}, for i = 1, 2, ..., n$

(d-1)(n-1) Continuity Conditions

•	Force continuity of the derivatives of order $1, 2, ..., d - 1$ at the knots.

$p_{i}^{(k)} (x_{i}) = p_{i + 1}^{(k)} (x_{i}), for i = 1, 2, ..., n - 1, and k = 1, 2, ..., d - 1$

d-1 End Conditions

•	Natural splines specified by endpoints='natural'.

Equate the derivates of order $\frac{(d + 1)}{2}, ..., d - 1$ at the end nodes to zero.

$p_{i}^{(k)} (x_{0}) = 0, and p_{n}^{(k)} (x_{n}) = 0, for k = \frac{d}{2} + \frac{1}{2}, ..., d - 1$

•	Not-a-knot splines specified by endpoints='notaknot'.

Force the continuity of the dth derivative at the knots $x_{i}$ , for $i = 1, 2, ..., \frac{(d - 1)}{2}$ and $i = n - \frac{(d - 1)}{2}, ..., n - 1$ .

$p_{i}^{(k)} (x_{i}) = p_{i + 1}^{(d)} (x_{i}), and p_{n - i}^{(d)} (x_{n - i}) = p_{n - i + 1}^{(d)} (x_{n - i}), for i = 1, 2, ..., \frac{d}{2} - \frac{1}{2}$

•	Periodic splines specified by endpoints='periodic'.

Match the derivatives of order $1, 2, ..., d - 1$ at the end nodes.

$p_{1}^{(k)} (x_{0}) = p_{n}^{(k)} (x_{n}), for k = 1, 2, ..., d - 1$

•	Clamped splines specified by endpoints=V.

Equate the derivates of order $1, 2, ..., \frac{(d - 1)}{2}$ at the end nodes to the specified values given in V, where V is either a list, Vector, or an Array, of dimension $d - 1$ containing the specified clamped conditions. Specifically,

$V = [p_{1}^{(1)} (x_{0}), ..., p_{1}^{[\frac{d}{2} - \frac{1}{2}]} (x_{0}), p_{n}^{(1)} (x_{n}), ..., p_{n}^{[\frac{d}{2} - \frac{1}{2}]} (x_{n})], with$

$p_{1}^{(k)} (x_{0}) = V_{k}, and p_{n}^{(k)} (x_{n}) = V_{k + \frac{d}{2} + \frac{1}{2}}, for k = 1, 2, ..., \frac{d}{2} - \frac{1}{2}$

•	Generalized splines given by endpoints=G.

Generalized end conditions can be specified involving any arbitrary linear combination of the values of the derivatives (of any order $1, 2, ..., d$ at the nodes $x_{0}$ , $x_{1}$ , $x_{n - 1}$ , and $x_{n}$ , where $1 < n$ ). Such end conditions can be represented by a linear system of the form $Ax = b$ , where $x$ is a vector of dimension $4 d$ , with

$x = [p_{1}^{(1)} (x_{0}), p_{2}^{(1)} (x_{1}), ..., p_{1}^{(d - 1)} (x_{0}), p_{2}^{(d - 1)} (x_{1}), p_{1}^{(d)}, p_{2}^{(d)}, p_{n - 1}^{(1)} (x_{n - 1}), p_{n}^{(1)} (x_{n}), ..., p_{n - 1}^{(d - 1)} (x_{n - 1}), p_{n}^{(d - 1)} (x_{n}), p_{n - 1}^{(d)}, p_{n}^{(d)}]$

$A$ is the corresponding coefficient matrix of dimension $d - 1$ by $4 d$ and $b$ , a vector of dimension $d - 1$ , represents the right-hand side of the linear system.

Generalized end conditions are specified with the optional parameter endpoints=G, where $G$ is a Matrix or an Array. Here, $G$ represents the augmented linear system $[A | b]$ , having dimensions $d - 1$ by $4 d + 1$ .

Even Degree

Without the knots='data' option, the resulting function created by CurveFitting[Spline] is of the form $piecewise (v < z_{1}, p_{1}, ..., v < z_{n}, p_{n}, p_{n + 1})$ , where $z_{i} = \frac{x_{i - 1}}{2} + \frac{x_{i}}{2}$ , for $i = 1, 2, ..., n$ (that is, the spline knots are defined at the midpoints of the nodes) and the $n + 1$ spline sections ${p_{1}, p_{2}, ..., p_{n + 1}}$ are polynomials of degree at most d. These polynomials are specified by the following $(d + 1) (n + 1)$ conditions.

n+1 Interpolating Conditions at the Nodes

•	Force continuity at the nodes.

$p_{i} (x_{i - 1}) = y_{i - 1}, for i = 1, 2, ..., n + 1$

n Interpolating Conditions at the Knots

•	Force continuity at the knots.

$p_{i} (z_{i}) = p_{i + 1} (z_{i}), for i = 1, 2, ..., n$

(d-1)n Continuity Conditions

•	Force continuity of the derivatives of order $1, 2, ..., d - 1$ at the knots.

$p_{i}^{(k)} (z_{i}) = p_{i + 1}^{(k)} (z_{i}), for i = 1, 2, ..., n and k = 1, 2, ..., d - 1$

d End Conditions

•	Natural splines specified by endpoints='natural'.

Equate the derivates of order $\frac{d}{2}, ..., d - 1$ at the end nodes to zero.

$p_{1}^{(k)} (x_{0}) = 0, and p_{n + 1}^{(k)} (x_{n}) = 0, for k = \frac{d}{2}, ..., d - 1$

•	Not-a-knot splines specified by endpoints='notaknot'.

Force the continuity of the dth derivative at the knots $z_{i}$ , for $i = 1, 2, ..., \frac{d}{2}$ and $i = n + 1 - \frac{d}{2}, ..., n - 1$ .

$p_{i}^{(d)} (z_{i}) = p_{i + 1}^{(d)} (z_{i}), and p_{n - i + 1}^{(d)} (z_{n - i + 1}) = p_{n + 2 - i}^{(d)} (z_{n - i + 1}), for i = 1, 2, ..., \frac{d}{2}$

•	Periodic splines specified by endpoints='periodic'.

Match the derivatives of order $1, 2, ..., d$ at the end nodes.

$p_{1}^{(k)} (x_{0}) = p_{n + 1}^{(k)} (x_{n}), for k = 1, 2, ..., d$

•	Clamped splines specified by endpoints=V.

Equate the derivates of order $1, 2, ..., \frac{d}{2}$ at the end nodes to the specified values given in V, where V is either a list, Vector, or an Array, of dimension $d$ containing the specified clamped conditions. Specifically,

$V = [p_{1}^{(1)} (x_{0}), ..., p_{1}^{(\frac{d}{2})} (x_{0}), p_{n + 1}^{(1)} (x_{n}), ..., p_{n + 1}^{(\frac{d}{2})} (x_{n})], with$

$p_{1}^{(k)} (x_{0}) = V_{k}, and p_{n + 1}^{(k)} (x_{n}) = V_{k + \frac{d}{2}}, for k = 1, 2, ..., \frac{d}{2}$

•	Generalized splines specified by endpoints=G.

Generalized end conditions can be specified involving any arbitrary linear combination of the values of the derivatives (of any order $1, 2, ..., d$ at the nodes $x_{0}$ and $x_{n}$ ). Such end conditions can be represented by a linear system of the form $Ax = b$ , where $x$ is a vector of dimension $2 d$ , with

$x = [p_{1}^{(1)} (x_{0}), ..., p_{1}^{(d - 1)} (x_{0}), p_{1}^{(d)}, p_{n + 1}^{(1)} (x_{n}), ..., p_{n + 1}^{(d - 1)} (x_{n}), p_{n + 1}^{(d)}]$

$A$ is the corresponding coefficient matrix of dimension $d$ by $2 d$ and $b$ , a vector of dimension $d$ , represents the right-hand side of the linear system.

Even degree with knots='data'

With the knots='data' option included, CurveFitting[Spline] will avoid creating knots at the midpoints of the nodes, and instead use the nodes for the knots in the even case. The resulting function is of the form $piecewise (v < x_{1}, p_{1}, ..., p_{n})$ , where the n spline sections ${p_{1}, p_{2}, ..., p_{n}}$ are polynomials of degree at most d. These polynomials are given by the following $(d + 1) n$ conditions:

2n Interpolating Conditions

•	Force continuity at the knots.

$p_{i} (x_{i - 1}) = y_{i - 1}, and p_{i} (x_{i}) = y_{i}, for i = 1, 2, ..., n$

(d-1)(n-1) Continuity Conditions

•	Force continuity of the derivatives of order $1, 2, ..., d - 1$ at the knots.

$p_{i}^{(k)} (x_{i}) = p_{i + 1}^{(k)} (x_{i}), for i = 1, 2, ..., n - 1, and k = 1, 2, ..., d - 1$

d-1 End Conditions

•	Natural splines specified by endpoints='natural'.

Equate the derivates of order $\frac{d}{2}, ..., d - 1$ at the left end node and the derivatives of order $\frac{d}{2}, ..., d - 2$ at the right end node to zero.

$p_{1}^{(k)} (x_{0}) = 0, for k = \frac{d}{2}, ..., d - 1, and$

$p_{n + 1}^{(k)} (x_{n}) = 0, for k = \frac{d}{2}, ..., d - 2$

•	Not-a-knot splines specified by endpoints='notaknot'.

Force the continuity of the dth derivative at the knots $z_{i}$ , for $i = 1, 2, ..., \frac{d}{2}$ and $i = n + 1 - \frac{d}{2}, ..., n - 1$ .

$p_{i}^{(d)} (z_{i}) = p_{i + 1}^{(d)} (z_{i}), for i = 1, 2, ..., \frac{d}{2}, and$

$p_{n - i}^{(d)} (z_{n - i}) = p_{n - i + 1}^{(d)} (z_{n - i}), for i = 1, 2, ..., \frac{d}{2} - 1$

Examples

>	$with (CurveFitting) &colon;$

>	$data ≔ [[0, 0], [1, 5], [2, - 1], [3, 0]]$

$data ≔ [[0, 0], [1, 5], [2, −1], [3, 0]]$

(1)

A quintic spline using the 'natural' end condition.

>	$Spline (data, v, degree = 5, endpoints ='natural')$

$\{\begin{array}{c} \frac{3}{11} v^{5} - \frac{101}{11} v^{2} + \frac{153}{11} v & v < 1 \\ - \frac{6}{11} v^{5} + \frac{45}{11} v^{4} - \frac{90}{11} v^{3} - v^{2} + \frac{108}{11} v + \frac{9}{11} & v < 2 \\ \frac{3}{11} v^{5} - \frac{45}{11} v^{4} + \frac{270}{11} v^{3} - \frac{731}{11} v^{2} + \frac{828}{11} v - \frac{279}{11} & otherwise \end{array}$

(2)

A cubic spline using the 'periodic' end condition.

>	$Spline (data, v, degree = 3, endpoints ='periodic')$

$\{\begin{array}{c} - 5 v^{3} + 4 v^{2} + 6 v & v < 1 \\ 6 v^{3} - 29 v^{2} + 39 v - 11 & v < 2 \\ - v^{3} + 13 v^{2} - 45 v + 45 & otherwise \end{array}$

(3)

A quadratic spline using the 'notaknot' end condition.

>	$Spline (data, v, degree = 2, endpoints ='notaknot')$

$\{\begin{array}{c} - 7 v^{2} + 12 v & v < \frac{3}{2} \\ 5 v^{2} - 24 v + 27 & otherwise \end{array}$

(4)

A clamped cubic spline with slope A and B at the two end nodes.

>	$Spline (data, v, endpoints = [A, B])$

$\{\begin{array}{c} (\frac{11 A}{15} - \frac{49}{5} + \frac{B}{15}) v^{3} + (\frac{74}{5} - \frac{26 A}{15} - \frac{B}{15}) v^{2} + A v & v < 1 \\ (- \frac{A}{5} + \frac{42}{5} - \frac{B}{5}) v^{3} + (- \frac{199}{5} + \frac{16 A}{15} + \frac{11 B}{15}) v^{2} + (- \frac{9 A}{5} + \frac{273}{5} - \frac{4 B}{5}) v + \frac{14 A}{15} - \frac{91}{5} + \frac{4 B}{15} & v < 2 \\ (- \frac{29}{5} + \frac{A}{15} + \frac{11 B}{15}) v^{3} + (\frac{227}{5} - \frac{8 A}{15} - \frac{73 B}{15}) v^{2} + (- \frac{579}{5} + \frac{7 A}{5} + \frac{52 B}{5}) v + \frac{477}{5} - \frac{6 A}{5} - \frac{36 B}{5} & otherwise \end{array}$

(5)

A cubic spline using the generalized end conditions with second derivative equal to 5 at the end nodes.

>	$G ≔ Matrix (2, 13, \{(1, 3) = 1, (1, 13) = 5, (2, 10) = 1, (2, 13) = 5\}) &colon;$

>	$Spline (data, v, endpoints = G)$

$\{\begin{array}{c} - \frac{22}{5} v^{3} + \frac{5}{2} v^{2} + \frac{69}{10} v & v < 1 \\ 6 v^{3} - \frac{287}{10} v^{2} + \frac{381}{10} v - \frac{52}{5} & v < 2 \\ - \frac{8}{5} v^{3} + \frac{169}{10} v^{2} - \frac{531}{10} v + \frac{252}{5} & otherwise \end{array}$

(6)

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