Mathematical Conventions for Data Fitting Functions

Developer Reference for Intel® oneAPI Math Kernel Library for C

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Date 4/28/2026

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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 `\(x_i\)` denotes breakpoints. `\([x_i, x_{i+1})\)` denotes a sub-interval (cell) of size `\(\Delta_i = x_{i+1} - x_i\)`.	`\(\{x_i\}_{i=1,\ldots,n}\)`, where `\(a = x_1 < x_2 < \ldots < x_n = b\)`
Quasi-uniform partition of interpolation interval `\([a, b]\)`	Partition `\(\{x_i\}_{i=1,\ldots,n}\)` which meets the constraint with a constant `\(C\)` defined as `\(1 \leq M/m \leq C\)`, where `\(M = \max_{i=1,\ldots,n-1}(\Delta_i)\)` `\(m = \min_{i=1,\ldots,n-1}(\Delta_i)\)` `\(\Delta_i = x_{i+1} - x_i\)`
Vector-valued function of dimension `\(p\)` being fit	`\(f(x) = (f_1(x), \ldots, f_p(x))\)`
Piecewise polynomial (PP) function `\(f\)` of order `\(k+1\)`	`\(f(x) := P_i(x)\)`, if `\(x \in [x_i, x_{i+1})\)`, `\(i = 1,\ldots,n-1\)` where `\(\{x_i\}_{i=1,\ldots,n}\)` is a strictly increasing sequence of breakpoints `\(P_i(x) = c_{i,0} + c_{i,1}(x - x_i) + \ldots + c_{i,k}(x - x_i)^k\)` is a polynomial of degree `\(k\)` (order `\(k+1\)`) over the interval `\(x \in [x_i, x_{i+1})\)`
Function `\(p\)` agrees with function `\(f\)` at the points `\(\{x_i\}_{i=1,\ldots,n}\)`.	For every point `\(\zeta\)` in sequence `\(\{x_i\}_{i=1,\ldots,n}\)` that occurs `\(m\)` times, the equality `\(p^{(i-1)}(\zeta) = f^{(i-1)}(\zeta)\)` holds for all `\(i = 1,\ldots,m\)`, where `\(p^{(i)}(t)\)` is the derivative of the `\(i\)`-th order.
The `\(k\)`-th divided difference of function `\(f\)` at points `\(x_i, \ldots, x_{i+k}\)`. This difference is the leading coefficient of the polynomial of order `\(k+1\)` that agrees with `\(f\)` at `\(x_i, \ldots, x_{i+k}\)`.	`\([x_i, \ldots, x_{i+k}]f\)` In particular, `\([x_1]f = f(x_1)\)` `\([x_1, x_2]f = (f(x_1) - f(x_2))/(x_1 - x_2)\)`
A `\(k\)`-order derivative of interpolant `\(f(x)\)` at interpolation site `\(\tau\)`.	`\(f^{(k)}(\tau)\)`

Interpolants to the Function\(f\)at\(x_1,\ldots,x_n\)and Boundary Conditions

Concept	Mathematical Notation
Linear interpolant	`\(P_i(x) = c_{1,i} + c_{2,i}(x - x_i)\)`, where `\(x \in [x_i, x_{i+1})\)` `\(c_{1,i} = f(x_i)\)` `\(c_{2,i} = [x_i, x_{i+1}]f\)` `\(i = 1,\ldots,n-1\)`
Piecewise parabolic interpolant	`\(P_i(x) = c_{1,i} + c_{2,i}(x - x_i) + c_{3,i}(x - x_i)^2\)`, `\(x \in [x_i, x_{i+1})\)` Coefficients `\(c_{1,i}\)`, `\(c_{2,i}\)`, and `\(c_{3,i}\)` depend on the conditions: `\(P_i(x_i) = f(x_i)\)` `\(P_i(x_{i+1}) = f(x_{i+1})\)` `\(P_i((x_{i+1} + x_i)/2) = v_{i+1}\)` where parameter `\(v_{i+1}\)` depends on the interpolant being continuously differentiable: `\(P_{i-1}^{(1)}(x_i) = P_i^{(1)}(x_i)\)`
Piecewise parabolic Subbotin interpolant	`\(P(x) = P_i(x) = c_{1,i} + c_{2,i}(x - x_i) + c_{3,i}(x - x_i)^2 + d_{3,i}((x - t_i)_+)^2\)`, where `\(x \in [t_i, t_{i+1})\)` `\(\{t_i\}_{i=1,\ldots,n+1}\)` is a sequence of knots such that `\(t_1 = x_1\)`, `\(t_{n+1} = x_n\)` `\(t_i \in (x_{i-1}, x_i)\)`, `\(i = 2,\ldots,n\)` `\(x_+ = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}\)` Coefficients `\(c_{1,i}\)`, `\(c_{2,i}\)`, `\(c_{3,i}\)`, and `\(d_{3,i}\)` depend on the following conditions: `\(P_i(x_i) = f(x_i)\)` `\(P_i(x_{i+1}) = f(x_{i+1})\)` `\(P(x)\)` is a continuously differentiable polynomial of the second degree on `\([t_i, t_{i+1})\)`, `\(i = 1,\ldots,n\)`
Piecewise cubic Hermite interpolant	`\(P_i(x) = c_{1,i} + c_{2,i}(x - x_i) + c_{3,i}(x - x_i)^2 + c_{4,i}(x - x_i)^3\)`, where `\(x \in [x_i, x_{i+1})\)` `\(c_{1,i} = f(x_i)\)` `\(c_{2,i} = s_i\)` `\(c_{3,i} = ([x_i, x_{i+1}]f - s_i)/(\Delta x_i) - c_{4,i}(\Delta x_i)\)` `\(c_{4,i} = (s_i + s_{i+1} - 2[x_i, x_{i+1}]f)/(\Delta x_i)^2\)` `\(i = 1,\ldots,n-1\)` `\(s_i = f^{(1)}(x_i)\)`
Piecewise cubic Bessel interpolant	`\(P_i(x) = c_{1,i} + c_{2,i}(x - x_i) + c_{3,i}(x - x_i)^2 + c_{4,i}(x - x_i)^3\)`, where `\(x \in [x_i, x_{i+1})\)` `\(c_{1,i} = f(x_i)\)` `\(c_{2,i} = s_i\)` `\(c_{3,i} = ([x_i, x_{i+1}]f - s_i)/(\Delta x_i) - c_{4,i}(\Delta x_i)\)` `\(c_{4,i} = (s_i + s_{i+1} - 2[x_i, x_{i+1}]f)/(\Delta x_i)^2\)` `\(i = 1,\ldots,n-1\)` `\(s_i = (\Delta x_i[x_{i-1}, x_i]f + \Delta x_{i-1}[x_i, x_{i+1}]f)/(\Delta x_i + \Delta x_{i+1})\)`
Piecewise cubic Akima interpolant	`\(P_i(x) = c_{1,i} + c_{2,i}(x - x_i) + c_{3,i}(x - x_i)^2 + c_{4,i}(x - x_i)^3\)`, where `\(x \in [x_i, x_{i+1})\)` `\(c_{1,i} = f(x_i)\)` `\(c_{2,i} = s_i\)` `\(c_{3,i} = ([x_i, x_{i+1}]f - s_i)/(\Delta x_i) - c_{4,i}(\Delta x_i)\)` `\(c_{4,i} = (s_i + s_{i+1} - 2[x_i, x_{i+1}]f)/(\Delta x_i)^2\)` `\(i = 1,\ldots,n-1\)` `\(s_i = (w_{i+1}[x_{i-1}, x_i]f + w_{i-1}[x_i, x_{i+1}]f)/(w_{i+1} + w_{i-1})\)` `\(w_i = \|[x_i, x_{i+1}]f - [x_{i-1}, x_i]f\|\)`
Piecewise natural cubic interpolant	`\(P_i(x) = c_{1,i} + c_{2,i}(x - x_i) + c_{3,i}(x - x_i)^2 + c_{4,i}(x - x_i)^3\)`, where `\(x \in [x_i, x_{i+1})\)` `\(c_{1,i} = f(x_i)\)` `\(c_{2,i} = s_i\)` `\(c_{3,i} = ([x_i, x_{i+1}]f - s_i)/(\Delta x_i) - c_{4,i}(\Delta x_i)\)` `\(c_{4,i} = (s_i + s_{i+1} - 2[x_i, x_{i+1}]f)/(\Delta x_i)^2\)` `\(i = 1,\ldots,n-1\)` Parameter `\(s_i\)` depends on the condition that the interpolant is twice continuously differentiable: `\(P_{i-1}^{(2)}(x_i) = P_i^{(2)}(x_i)\)`.
Not-a-knot boundary condition.	Parameters `\(s_1\)` and `\(s_n\)` provide `\(P_1 = P_2\)` and `\(P_{n-1} = P_n\)`, so that the first and the last interior breakpoints are inactive.
Free-end boundary condition.	`\(f''(x_1) = f''(x_n) = 0\)`
Look-up interpolator for discrete set of points `\((x_1, y_1), \ldots, (x_n, y_n)\)`.	`\(y(x) = \begin{cases} y_1 & \text{if } x = x_1 \\ y_2 & \text{if } x = x_2 \\ \vdots & \vdots \\ y_n & \text{if } x = x_n \\ \text{error} & \text{otherwise} \end{cases}\)`
Step-wise constant continuous right interpolator.	`\(y(x) = \begin{cases} y_1 & \text{if } x_1 \leq x < x_2 \\ y_2 & \text{if } x_2 \leq x < x_3 \\ \vdots & \vdots \\ y_{n-1} & \text{if } x_{n-1} \leq x < x_n \\ y_n & \text{if } x = x_n \end{cases}\)`
Step-wise constant continuous left interpolator.	`\(y(x) = \begin{cases} y_1 & \text{if } x = x_1 \\ y_2 & \text{if } x_1 < x \leq x_2 \\ y_3 & \text{if } x_2 < x \leq x_3 \\ \vdots & \vdots \\ y_n & \text{if } x_{n-1} < x \leq x_n \end{cases}\)`

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Developer Reference for Intel® oneAPI Math Kernel Library for C

Mathematical Conventions for Data Fitting Functions