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# 数学代写|数值分析代写Numerical analysis代考|MTH405 Piecewise polynomial approximation

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## 数学代写|数值分析代写Numerical analysis代考|Piecewise polynomial approximation

The focus of our discussions in the previous chapters has been the question of approximation of a given rd-continuous function $f$, defined on an interval $[a, b]$, by a polynomial on that interval either through Lagrange, $\sigma$-Lagrange, Hermite, or $\sigma$-Hermite interpolation polynomials. Each of these constructions was global in nature, in the sense that the approximation was defined by the same analytical expression on the whole interval $[a, b]$. In this chapter, we will present an alternative and more flexible way for approximation of a function $f$, and this approach is based on dividing the interval $[a, b]$ into a number of subintervals and looking for a piecewise approximation by polynomials of low degree. Such piecewise-polynomial approximations are called splines, and the endpoints of the subintervals are known as the knots.

Let $\mathbb{T}$ be a time scale with forward jump operator $\sigma$ and delta differentiation operator $\Delta$.

## 数学代写|数值分析代写Numerical analysis代考|Linear interpolating splines

We first discuss the piecewise approximation by the lowest degree polynomials, that is, by linear functions, called linear splines. Let $a, b \in \mathbb{T}, a<b$, and $m \in \mathbb{N}, m \geq 2$.
Definition 3.1. Suppose that $f \in \mathcal{C}{r d}([a, b])$ and $K=\left{x_0, x_1, \ldots, x_m\right}$ is a subset of $[a, b]$ such that $$a=x_0{j-1}} f\left(x_{j-1}\right)+\frac{x-x_{j-1}}{x_j-x_{j-1}} f\left(x_j\right), \quad x \in\left[x_{j-1}, x_j\right], \quad j \in{1, \ldots, m} .$$
The points $x_j, j \in{0,1, \ldots, m}$, are the knots of the spline, and $K$ is said to be the set of knots.
By Definition 3.1, it follows that
$$s_L\left(x_{j-1}\right)=f\left(x_{j-1}\right), \quad s_L\left(x_j\right)=f\left(x_j\right), \quad j \in{1, \ldots, m} .$$
We will now give an error estimate for the maximal error for the linear spline interpolation of a given function. Below denote
$$r_{j+1}=x_{j+1}-x_j, \quad j \in{0,1, \ldots, m-1}, \quad r=\max _{j \in{1, \ldots, m}} r_j .$$

## 数学代写|数值分析代写数值分析代考|线性插值样条

$$s_L\left(x_{j-1}\right)=f\left(x_{j-1}\right), \quad s_L\left(x_j\right)=f\left(x_j\right), \quad j \in{1, \ldots, m} .$$

$$r_{j+1}=x_{j+1}-x_j, \quad j \in{0,1, \ldots, m-1}, \quad r=\max _{j \in{1, \ldots, m}} r_j .$$

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