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Odd Degree
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The resulting function created by CurveFitting[Spline] is of the form , where the n spline sections are polynomials of degree at most d. These polynomials are given by the following conditions:
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2n Interpolating Conditions
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Force continuity at the knots.
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(d-1)(n-1) Continuity Conditions
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Force continuity of the derivatives of order at the knots.
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d-1 End Conditions
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Natural splines specified by endpoints='natural'.
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Equate the derivates of order at the end nodes to zero.
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Not-a-knot splines specified by endpoints='notaknot'.
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Force the continuity of the dth derivative at the knots , for and .
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Periodic splines specified by endpoints='periodic'.
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Match the derivatives of order at the end nodes.
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Clamped splines specified by endpoints=V.
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Equate the derivates of order at the end nodes to the specified values given in V, where V is either a list, Vector, or an Array, of dimension containing the specified clamped conditions. Specifically,
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Generalized splines given by endpoints=G.
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Generalized end conditions can be specified involving any arbitrary linear combination of the values of the derivatives (of any order at the nodes , , , and , where ). Such end conditions can be represented by a linear system of the form , where is a vector of dimension , with
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is the corresponding coefficient matrix of dimension by and , a vector of dimension , represents the right-hand side of the linear system.
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Generalized end conditions are specified with the optional parameter endpoints=G, where is a Matrix or an Array. Here, represents the augmented linear system , having dimensions by .
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Even Degree
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Without the knots='data' option, the resulting function created by CurveFitting[Spline] is of the form , where , for (that is, the spline knots are defined at the midpoints of the nodes) and the spline sections are polynomials of degree at most d. These polynomials are specified by the following conditions.
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n+1 Interpolating Conditions at the Nodes
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Force continuity at the nodes.
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n Interpolating Conditions at the Knots
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Force continuity at the knots.
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(d-1)n Continuity Conditions
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Force continuity of the derivatives of order at the knots.
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d End Conditions
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Natural splines specified by endpoints='natural'.
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Equate the derivates of order at the end nodes to zero.
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Not-a-knot splines specified by endpoints='notaknot'.
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Force the continuity of the dth derivative at the knots , for and .
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Periodic splines specified by endpoints='periodic'.
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Match the derivatives of order at the end nodes.
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Clamped splines specified by endpoints=V.
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Equate the derivates of order at the end nodes to the specified values given in V, where V is either a list, Vector, or an Array, of dimension containing the specified clamped conditions. Specifically,
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Generalized splines specified by endpoints=G.
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Generalized end conditions can be specified involving any arbitrary linear combination of the values of the derivatives (of any order at the nodes and ). Such end conditions can be represented by a linear system of the form , where is a vector of dimension , with
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is the corresponding coefficient matrix of dimension by and , a vector of dimension , represents the right-hand side of the linear system.
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Generalized end conditions are specified with the optional parameter endpoints=G, where is a Matrix or an Array. Here, represents the augmented linear system , having dimensions by .
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Even degree with knots='data'
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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 , where the n spline sections are polynomials of degree at most d. These polynomials are given by the following conditions:
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2n Interpolating Conditions
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Force continuity at the knots.
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(d-1)(n-1) Continuity Conditions
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Force continuity of the derivatives of order at the knots.
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d-1 End Conditions
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Natural splines specified by endpoints='natural'.
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Equate the derivates of order at the left end node and the derivatives of order at the right end node to zero.
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Not-a-knot splines specified by endpoints='notaknot'.
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Force the continuity of the dth derivative at the knots , for and .
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Examples
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A quintic spline using the 'natural' end condition.
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A cubic spline using the 'periodic' end condition.
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A quadratic spline using the 'notaknot' end condition.
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A clamped cubic spline with slope A and B at the two end nodes.
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A cubic spline using the generalized end conditions with second derivative equal to 5 at the end nodes.
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