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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).

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

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


到目前为止,我们已经讨论了数值分析中最常出现的三个问题中的两个:非线性方程的求根问题和线性系统的解。这些问题各自都很重要,但除此之外,其他类型的问题在每一步都需要这些类中的一个来解决问题也并不罕见;一个例子是系统的牛顿方法(第2.4节),它要求在每一步解一个线性系统


多项式插值本身就很有趣,但它作为设计和分析数值方法的理论工具也很有趣。我们已经看到了线性插值(例如,在第1.1节的假位置方法中)和二次插值(例如,在第1.6节的布伦特方法中)是如何对推导方法有用的工具,第7节将使用三次插值。插值在论证那些方法对于足够光滑的函数应该工作得很好时也很有用。我们将从更正式地开发这个强大的工具开始,然后看看寻找通过给定点集的光滑曲线方程的相关技术。这些将在以后的章节中应用


假设我们有一组$n+1$点$x_0, x_1, \ldots, x_n$,称为节点(或断点),其顺序为$x_0<x_1<\cdots<x_n$,以及一组相关的$y$ -值$\nu_0, y_1, \ldots, y_n$。可能$y$ -值是从$y_i=f\left(x_i\right)$在$\left[x_0, x_n\right]$上定义的函数$f(x)$中找到的,也可能它们只是测量数据。多项式插值问题是找到一个多项式$p(x)$的度不超过$n$来插值数据$\left(x_0, y_0\right),\left(x_1, y_1\right), \ldots,\left(x_n, y_n\right)$,我们的意思是
$$
\begin{aligned}
y_0=& p\left(x_0\right) \
y_1=& p\left(x_1\right) \
& \vdots \
y_n=& p\left(x_n\right)
\end{aligned}
$$
,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

.


有很多原因,我们可能需要近似一个函数。在大多数标准机器上,任何要计算的东西都必须在某种程度上分解为加、减、乘或除(加上簿记操作)${ }^4$。因此,当计算器返回$\sin (3)$的值时,这只是因为函数$y=\sin (x)$被一个有理函数(即多项式的比率)近似。在更高的层次上,我们经常需要近似在特定问题中出现的特殊函数。在这些情况下,我们还远远不清楚我们是否需要一个插值函数本身,它需要通过某些点;我们只需要控制近似的误差。稍后我们将更笼统地讨论近似


另一个需要函数逼近的常见例子是ODE(或函数的数值积分)的数值解,它产生一组离散的点,通过这些点我们可以画出一条光滑的曲线。我们也可能已经测量了我们想要通过曲线的数据(例如,从一个数字化的图形或跟踪一个移动的物体)。在这种情况下,通常需要让曲线通过已知的数据点,因此需要一个插值函数,而不仅仅是一个近似值


多项式插值有很多用途。然而,正如我们在前一节中指出的,多项式插值通常不是逼近函数的正确选择。高阶多项式插值的振荡性质,加上对大的$n$求$x^n$的误差,使它们不能作为通用工具使用。振荡插补是一个完美的插补$\left(p\left(x_i\right)=y_i\right)$但它是一个非常差的曲线近似值,我们总是想要得到底层函数的一个很好的近似值。泰勒多项式更糟糕;它们保证只在一个点上插值,也就是它们的中心,虽然它们在那里是一个很好的近似,但近似的质量很快就会下降。此外,它们需要关于函数的导数信息,而这些信息很少能得到。多项式插值和泰勒多项式在数值分析中有很大的用处,但它们主要用于推导和分析方法,而不是作为方法本身。我们将在接下来的两章中看到这方面的例子(数值积分和ode的数值解)

数学代写|数值分析代写numerical analysis代考

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