Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 10/31/2024
Public
Document Table of Contents

Mathematical Conventions for Data Fitting Functions

This section explains the notation used for Data Fitting function descriptions. Spline notations are based on the terminology and definitions of [deBoor2001]. The Subbotin quadratic spline definition follows the conventions of [StechSub76]. The quasi-uniform partition definition is based on [Schumaker2007].

Mathematical Notation in the Data Fitting Component
Concept Mathematical Notation

Partition of interpolation interval [a, b] , where

  • xi denotes breakpoints.
  • [xi, xi+1) denotes a sub-interval (cell) of size Δi=xi+1-xi .

{xi}i=1,...,n, where a = x1 < x2<... <xn = b

Quasi-uniform partition of interpolation interval [a, b]

Partition {xi}i=1,...,n which meets the constraint with a constant C defined as

1 M/ mC,

where

  • M = maxi=1,...,n-1 (Δi)

  • m = mini=1,...,n-1 (Δi)

  • Δi = xi+1 - xi

Vector-valued function of dimension p being fit

ƒ(x) = (ƒ1(x),..., ƒp(x))

Piecewise polynomial (PP) function ƒ of order k+1

ƒ(x) ≔ Pi (x), if x ∈ [ xi, xi+1), i = 1,..., n-1

where

  • {xi}i= 1,..., n is a strictly increasing sequence of breakpoints.
  • Pi(x) = ci,0 + ci,1(x - xi) + ... + ci,k(x - xi)k is a polynomial of degree k (order k+1) over the interval x ∈ [ xi, xi+1).

Function p agrees with function ƒ at the points {xi}i=1,...,n .

For every point ζ in sequence {xi}i=1,...,n that occurs m times, the equality p(i-1)(ζ) = ƒ(i-1)(ζ) holds for all i = 1,...,m, where p(i)(t) is the derivative of the i-th order.

Interpolants to the Function ƒ at x1,..., xn and Boundary Conditions
Concept Mathematical Notation

Linear interpolant

Pi(x) = c1, i + c2, i(x - xi),

where

  • x ∈ [ xi, xi+1)
  • c1, i = ƒ(xi)
  • c2, i = [xi, xi+1 ]ƒ
  • i = 1,..., n-1

Piecewise parabolic interpolant

Pi(x) = c1, i + c2, i(x - xi) + c3, i(x - xi)2, x ∈ [ xi, xi+1)

Coefficients c1, i, c2, i, and c3, i depend on the conditions:

  • Pi(xi) = ƒ(xi)
  • Pi(xi+1) = ƒ(xi+1)
  • Pi((xi+1 + xi) / 2) = vi+1

where parameter vi+1 depends on the interpolant being continuously differentiable:

Pi-1(1)(xi) = Pi(1)(xi)

Piecewise parabolic Subbotin interpolant

P(x) = Pi(x) = c1,i+c2,i(x-xi)+c3,i(x-xi)2+d3,i((x-ti)+)2,

where

  • x ∈ [ ti, ti+1)

  • {ti}i=1,...,n+1 is a sequence of knots such that

    • t1 = x1, tn+1 = xn

    • ti ∈ (xi-1, xi), i = 2,..., n

Coefficients c1,i, c2,i, c3,i, and d3,i depend on the following conditions:

  • Pi(xi) = ƒ(xi), Pi(xi+1) = ƒ(xi+1)
  • P(x) is a continuously differentiable polynomial of the second degree on [ ti, ti+1), i = 1,..., n.

Piecewise cubic Hermite interpolant

Pi(x) = c1,i + c2,i(x - xi) + c3,i(x - xi)2 + c4,i(x - xi)3,

where

  • x ∈ [ xi, xi+1)
  • c1,i = ƒ(xi)
  • c2,i = si
  • c3,i = ([xi, xi+1]ƒ - si ) / (Δxi) - c4,i(Δxi)
  • c4,i = (si + si+1 - 2[xi, xi+1]ƒ) / (Δxi)2
  • i = 1,..., n-1
  • si = ƒ(1)(xi)

Piecewise cubic Bessel interpolant

Pi(x) = c1,i + c2,i(x - xi) + c3,i(x - xi)2 + c4,i(x - xi)3,

where

  • x ∈ [ xi, xi+1)
  • c1,i = ƒ(xi)
  • c2,i = si
  • c3,i = ([xi, xi+1]ƒ - si ) / (Δxi) - c4,i(Δxi)
  • c4,i = (si + si+1 - 2[xi, xi+1]ƒ) / (Δxi)2
  • i = 1,..., n-1
  • si = (Δxi[xi-1, xi]ƒ + Δxi-1[xi, xi+1]ƒ) / (Δxi + Δxi+1)