1 - x) N^{d - 1}(x - 1)}{d}.
  \label{2.4}$$ Note that the elements of a collocation matrix are rational, because the interpolation nodes are rational, and the de Boor–Cox recurrence formula (\[2.4\]) involves only basic arithmetic operations on rational coefficients. These elements are therefore exactly computable in (arbitrary precise) rational arithmetic.

Low degree cardinal B-splines
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Let $t_i^d = N^d(x_i^{\ast})$ denote the values of cardinal B-splines at knot averages (see (\[2.3\])). Then, the cardinal B-spline collocation matrix $A$ with interpolation nodes $x_i^{\ast}$ is a banded Töplitz matrix of order $n - d$, to be denoted by $$T_n^d = \begin{bmatrix}
    t_0^d    & \cdots & t_r^d  &        & 0      \\
    \vdots   & \ddots &        & \ddots &        \\
    t_r^d    &        & \ddots &        & t_r^d  \\
             & \ddots &        & \ddots & \vdots \\
    0        &        & t_r^d  & \cdots & t_0^d,
  \end{bmatrix}, \quad
  r = \left\lfloor \frac{d}{2} \right\rfloor.
  \label{3.1}$$ The matrix $T_n^d$ is represented by its first row, usually called the symbol, $$t = (t_0^d, \ldots, t_r^d, 0, \ldots, 0), \quad t \in \mathbb{R}^{n-d}.$$

It is useful to note that each B-spline of degree $d > 0$ is a unimodal function, i.e., it has only one local maximum on the support. 