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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MM512 LAGRANGE AND HERMITE INTERPOLATION

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|LAGRANGE AND HERMITE INTERPOLATION

Let $u_0, \ldots, u_n$ be given numbers. If each of them is different from the others, we can specify arbitrary numbers $f_0, \ldots, f_n$ and look for a function $h$ such that $h\left(u_i\right)=f_i$ for $i=0, \ldots, n$. This is called a Lagrange interpolation problem. The existence and uniqueness of its solution depends on additional conditions to be satisfied by the function $h$. In the linear space $\mathbb{R}[\cdot]_n$ the solution is unique; in other words, there exists a unique polynomial $h$ of degree at most $n$, satisfying the interpolation conditions considered above.

We obtain a more general problem by allowing some (or all) interpolation knots $u_0, \ldots, u_n$ to coincide. Obviously, we cannot specify two different function values for the same argument. But if a number $u_i$ appears $r$ times in the sequence $u_0, \ldots, u_n$ (we say that the knot $u_i$ has the multiplicity $r$ ), then we can demand that the function to be found at the knot $u_i$ and its derivatives up to the order $r-1$ take prescribed values. This interpolation problem, bearing the name of Charles Hermite, also has a unique solution in the space of real polynomials of degree at most $n$. Examples are shown in Figure A.1.

Functions of interpolation may be searched also in other function spaces, e.g. of splines or trigonometric polynomials; the existence of solutions depends on algebraic properties of those spaces. Having a Lagrange interpolation problem and a basis $\left{g_0, \ldots, g_n\right}$ of a function space, we can write the following system of linear equations:
$$\left[\begin{array}{ccc} g_0\left(u_0\right) & \ldots & g_n\left(u_0\right) \ \vdots & & \vdots \ g_0\left(u_n\right) & \ldots & g_n\left(u_n\right) \end{array}\right]\left[\begin{array}{c} a_0 \ \vdots \ a_n \end{array}\right]=\left[\begin{array}{c} f_0 \ \vdots \ f_n \end{array}\right]$$

## 数学代写|曲线和曲面代写Curves And Surfaces代考|THE DIVIDED DIFFERENCES ALGORITHM

The matrix of the system (A.1) written for the power basis $\left{1, x, \ldots, x^n\right}$ is full; it may be obtained with $O\left(n^2\right)$ operations and then the system may be solved with $O\left(n^3\right)$ operations, e.g. with the Gaussian elimination. This computational cost may be reduced by using a different basis. Let $p_0, \ldots, p_n$ be polynomials defined as follows:
\begin{aligned} p_0(x) &=1, \ p_1(x) &=x-u_0, \ p_2(x) &=\left(x-u_0\right)\left(x-u_1\right), \ & \vdots \ p_n(x) &=\left(x-u_0\right)\left(x-u_1\right) \ldots\left(x-u_{n-1}\right) . \end{aligned}

## 数学代写|曲线和曲面代写Curves And Surfaces代考|THE DNIDED DIFFERENCES ALGORITHM

$$p_0(x)=1, p_1(x) \quad=x-u_0, p_2(x)=\left(x-u_0\right)\left(x-u_1\right), \quad \vdots p_n(x)=\left(x-u_0\right)\left(x-u_1\right) \ldots\left(x-u_{n-1}\right) .$$

## MATLAB代写

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