# 数学代写|数值分析代写numerical analysis代考|MATH3820

## 数学代写|数值分析代写numerical analysis代考|Lagrange Interpolating Polynomials

So far we’ve discussed two of the three most commonly occurring problems in numerical analysis: Root-finding for nonlinear equations and the solution of linear systems. These problems are each important in their own right but in addition it is not uncommon for another type of problem to require the solution of a problem from one of these classes at each step; an example is Newton’s method for systems (Sec. 2.4), which requires the solution of a linear system at each step.

Polynomial interpolation is of interest in and of itself, but it is also of interest as a theoretical tool to devise and analyze numerical methods. We’ve already seen how linear interpolation (for example, in the method of false position, Sec. 1.1) and quadratic interpolation (for example, in Brent’s method, Sec. 1.6), were useful tools for deriving methods, and cubic interpolation will be used in Ch. 7. Interpolation has also been useful for arguing that those methods should work well for sufficiently smooth functions. We will start by developing this powerful tool somewhat more formally, and then look at related techniques for finding the equation of a smooth curve through a given set of points. These will be applied in later chapters.

Suppose we have a set of $n+1$ points $x_0, x_1, \ldots, x_n$, called nodes (or breakpoints), ordered so that $x_0<x_1<\cdots<x_n$, and a set of associated $y$-values $\nu_0, y_1, \ldots, y_n$. Possibly the $y$-values are found from a function $f(x)$ defined on $\left[x_0, x_n\right]$ by $y_i=f\left(x_i\right)$, or possibly they’re just measured data. The polynomial interpolation problem is to find a polynomial $p(x)$ of degree at most $n$ that interpolates the data $\left(x_0, y_0\right),\left(x_1, y_1\right), \ldots,\left(x_n, y_n\right)$, by which we mean that
\begin{aligned} y_0=& p\left(x_0\right) \ y_1=& p\left(x_1\right) \ & \vdots \ y_n=& p\left(x_n\right) \end{aligned}
and we say that $p$ interpolates $f$ (or the data $y_0, y_1, \ldots, y_n$ ) at $x_0, x_1, \ldots, x_n$, and that $p$ is an interpolant. We stress that an interpolant must agree with the function $f$ or the values $y_0, y_1, \ldots, y_n$ at the corresponding points $x_0, x_1, \ldots, x_n$ : An interpolant, unlike a least squares curve, must go through every data point.

## 数学代写|数值分析代写numerical analysis代考|Piecewise Linear Interpolation

There are many reasons that we might need to approximate a function. On most standard machines, anything that is to be computed must be broken down, at some level, to addition, subtraction, multiplication, or division (plus book-keeping operations) ${ }^4$. Hence when a calculator returns a value for $\sin (3)$ it is only because the function $y=\sin (x)$ has been approximated by a rational function (that is, a ratio of polynomials). At a higher level, we often need to approximate special functions that occur in particular problems. In these cases it is far from clear that we need an interpolant per se, which is required to pass through certain points; we really just need to control the error in our approximation. We’ll discuss approximation in more generality later.

Another common example of a situation that calls for approximation of a function is the numerical solution of an ODE (or the numerical integral of a function), which yields a discrete set of points through which we would like to draw a smooth curve. We might also have measured data through which we would like to pass a curve (say, from a digitized graph or from tracking a moving object). In these cases it is usually desirable to pass the curve through the known data points and so an interpolant, rather than just an approximant, is needed.

Polynomial interpolation has many uses. As we have indicated in the previous section, however, polynomial interpolants are usually not the right choice for approximating a function. The oscillatory nature of high-order polynomial interpolants, plus the errors in evaluating $x^n$ for large $n$, prevent them from being useful as a general-purpose tool. An oscillatory interpolant will be a perfect interpolant $\left(p\left(x_i\right)=y_i\right)$ but will be a very poor approximation of the curve between those points, and we always want to get a good approximation of the underlying function. Taylor polynomials are worse; they are guaranteed to interpolate only at a single point, their center, and while they are an excellent approximation there the quality of the approximation drops off rapidly. In addition, they require derivative information about the function that is rarely available. Polynomial interpolants and Taylor polynomials are of great use in numerical analysis but they are principally used to derive and analyze methods, not as methods in and of themselves. We’ll see examples of this in the next two chapters (numerical integration and the numerical solution of ODEs).

# 凸优化代考

## 数学代写|数值分析代写数值分析代考|拉格朗日插值多项式

\begin{aligned} y_0=& p\left(x_0\right) \ y_1=& p\left(x_1\right) \ & \vdots \ y_n=& p\left(x_n\right) \end{aligned}
，我们说$p$在$x_0, x_1, \ldots, x_n$插值$f$(或数据$y_0, y_1, \ldots, y_n$)，而$p$是一个插值函数。我们强调插值必须与函数$f$或$y_0, y_1, \ldots, y_n$在对应点$x_0, x_1, \ldots, x_n$一致:与最小二乘曲线不同，插值必须经过每一个数据点

## 数学代写|数值分析代写数值分析代考|分段线性插值

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