数学与应用数学专业论文英文文献翻译

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

Chapter 3 Interpolation

Interpolation is the process of defining a function that takes on specified values at specified points. This chapter concentrates on two closely related interpolants, the piecewise cubic spline and the shape-preserving piecewise cu bic named “pchip”.

3.1 The Interpolating Polynomial

We all know that two points determine a straight line. More precisely, any two points in the plane, ),(11y x and ),(11y x , with )(21x x ≠ , determine a unique first degree polynomial in x whose graph passes through the two points. There are many different formulas for the polynomial, but they all lead to the same straight line graph.

This generalizes to more than two points. Given n points in the plane, ),(k k y x ，n k ,,2,1 =, with distinct k x ’s, there is a unique polynomial in x of degree less than n whose graph passes through the points. It is easiest to remember that n , the number of data points, is also the number of coefficients, although some of the leading coefficients might be zero, so the degree might actually be less than 1-n . Again, there are many different formulas for the polynomial, but they all define the same function.

This polynomial is called the interpolating polynomial because it exactly

胡鹤：《MATLAB 数值计算》（美）Cleve B.Moler 第三章插值第1节插值多项式

re- produces the given data.

n k y x P k k ,,2,1,)( ==，

Later, we examine other polynomials, of lower degree, that only approximate the data. They are not interpolating polynomials.

The most compact representation of the interpolating polynomial is the La- grange form.

∑∏???

?

??--=≠k k k j j

k j

y x x x x x P )( There are n terms in the sum and 1-n terms in each product, so this expression defines a polynomial of degree at most 1-n . If )(x P is evaluated at

k x x = , all the products except the k th are zero. Furthermore, the k th

product is equal to one, so the sum is equal to k y and the interpolation conditions are satisfied.

For example, consider the following data set:

x=0:3; y=[-5 -6 -1 16];

The command

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

disp([x;y]) displays

0 1 2 3 -5 -6 -1 16

The Lagrangian form of the polynomial interpolating this data is

)

16()

6()2)(1()1()2()3)(1()

6()

2()

3)(2()5()6()3)(2)(1()(--+----+---+-----=

x x x x x x x x x x x x x P

We can see that each term is of degree three, so the entire sum has degree at most three. Because the leading term does not vanish, the degree is actually three. Moreover, if we plug in 2,1,0=x or 3, three of the terms vanish and the fourth produces the corresponding value from the data set.

Polynomials are usually not represented in their Lagrangian form. More fre- quently, they are written as something like

523--x x

The simple powers of x are called monomials and this form of a polynomial is said to be using the power form.

The coefficients of an interpolating polynomial using its power form,

n n n n c x c x c x c x P ++++=---12211)(

can, in principle, be computed by solving a system of simultaneous linear equations

胡鹤：《MATLAB 数值计算》（美）Cleve B.Moler 第三章插值第1节插值多项式

???

??

?

??????=

??????????????????????????------n n n n n n n n n n n y y y c c c x x x x x x x x x 21212

122

2121211

11111 The matrix V of this linear system is known as a Vandermonde matrix. Its elements are

j

n k

j k x v -=, The columns of a Vandermonde matrix are sometimes written in the opposite order, but polynomial coefficient vectors in Matlab always have the highest power first.

The Matlab function vander generates Vandermonde matrices. For our ex- ample data set,

V = vander(x) generates

V = 0 0 0 1 1 1 1 1 8 4 2 1 27 9 3 1 Then c = V\y’

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

computes the coefficients

c = 1.0000 0.0000 -2.0000 -5.0000

In fact, the example data was generated from the polynomial 523--x x .

One of the exercises asks you to show that Vandermonde matrices are nonsin- gular if the points k x are distinct. But another one of the exercises asks you to show that a Vandermonde matrix can be very badly conditioned. Consequently, using the power form and the Vandermonde matrix is a satisfactory technique for problems involving a few well-spaced and well-scaled data points. But as a general-purpose approach, it is dangerous.

In this chapter, we describe several Matlab functions that implement various interpolation algorithms. All of them have the calling sequence

v = interp(x,y,u)

The first two input arguments, x and y , are vectors of the same length that define the interpolating points. The third input argument, u , is a vector of

胡鹤：《MATLAB数值计算》（美）Cleve B.Moler 第三章插值第1节插值多项式

points where the function is to be evaluated. The output, v, is the same length as u and has elements ))

terp

x

k

v

y

in

)

,

(

,

(

(k

u

Our first such interpolation function, polyinterp, is based on the Lagrange form. The code uses Matlab array operations to evaluate the polynomial at all the components of u simultaneously.

function v = polyinterp(x,y,u)

n = length(x);

v = zeros(size(u));

for k = 1:n

w = ones(size(u));

for j = [1:k-1 k+1:n]

w = (u-x(j))./(x(k)-x(j)).*w;

end

end

v = v + w*y(k);

To illustrate polyinterp, create a vector of densely spaced evaluation points.

u = -.25:.01:3.25;

Then

v = polyinterp(x,y,u);

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

plot(x,y,’o’,u,v,’-’)

creates figure 3.1.

Figure 3.1. polyinterp

The polyinterp function also works correctly with symbolic variables. For example, create

symx = sym(’x’)

Then evaluate and display the symbolic form of the interpolating polynomial with

P = polyinterp(x,y,symx)

胡鹤：《MATLAB数值计算》（美）Cleve B.Moler 第三章插值第1节插值多项式

pretty(P)

produces

-5 (-1/3 x + 1)(-1/2 x + 1)(-x + 1) - 6 (-1/2 x + 3/2)(-x + 2)x

-1/2 (-x + 3)(x - 1)x + 16/3 (x - 2)(1/2 x - 1/2)x

This expression is a rearrangement of the Lagrange form of the interpolating poly- nomial. Simplifying the Lagrange form with

P = simplify(P)

changes P to the power form

P =

x^3-2*x-5

Here is another example, with a data set that is used by the other methods in this chapter.

x = 1:6;

y = [16 18 21 17 15 12];

disp([x; y])

u = .75:.05:6.25;

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

v = polyinterp(x,y,u);

plot(x,y,’o’,u,v,’-’);

produces

1 2 3 4 5 6

16 18 21 17 15 12

creates figure 3.2.

Figure 3.2. Full degree polynomial interpolation Already in this example, with only six nicely spaced points, we can begin to see the primary difficulty with full-degree polynomial interpolation. In between the data points, especially in the first and last subintervals, the function shows excessive variation. It overshoots the changes in the data values. As a result, full- degree polynomial interpolation is hardly ever used

胡鹤：《MATLAB 数值计算》（美）Cleve B.Moler 第三章插值第1节插值多项式

for data and curve fitting. Its primary application is in the derivation of other numerical methods.

第三章 插值多项式

插值就是定义一个在特定点取给定值得函数的过程。本章的重点是介绍两个紧密相关的插值函数：分段三次样条函数和保形分段三次插值函数（称为“pchip ”） 3.1插值多项式

人们知道两点确定一条直线，或者更确切地说，平面上任意两点

),(11y x 和),(11y x ，只要)(21x x ≠，就唯一确定一个关于x 的一次多项式，其

图形经过这两个点。对于这个多项式，有多种不同的公式表示，但是它们都对应同一个图形。

把上述讨论推广到多于两个点的情况。则对于平面上有着不同k x 值的

n 个点，),(k k y x ，n k ,,2,1 =，存在唯一一个关于x 的次数小于n 的多项式，

使其图形经过这些点。很容易可看出，数据点的数目n 也是多项式系数的个数。尽管，一些首项的系数可能是零，但多项式的次数实际上也小于

1-n 。同样，这个多项式可能有不同的公式表达式，但它们都定义着同一

个函数。

这样的多项式称为插值（interpolating ）多项式，它可以准确地重新计算出初始给定的数据：

k k y x P =)(，n k ,,2,1 =

后面，我们会考察另外一些较低次的多项式，这些多项式只能接近给定的数据，因此它们不是插值多项式。

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

表示插值多项式的最紧凑的方式是拉格朗日（Lagrange ）形式

∑∏???

?

??--=≠k k k j j

k j

y x x x x x P )( 在这个公式中，对n 项进行亲和，而每一个连乘符号中含有1-n 项，因此它定义的多项式最高次数为1-n 。当k x x =时计算)(x P ，除了第k 项外，其他的乘积都为零，同时，这第k 项乘积正好为1，所以求和结果为k y ，满足插值条件。

例如，考虑下面一组数据。 x=0:3; y=[-5 -6 -1 16]; 输入命令

disp([x;y]) 其输出为

0 1 2 3 -5 -6 -1 16 这些数据的拉格朗日形式的多项式为

)

16()

6()2)(1()1()2()3)(1()

6()

2()

3)(2()5()6()3)(2)(1()(--+----+---+-----=

x x x x x x x x x x x x x P

可以看出上式为四个三次多项式求和，因此最后结果最高为三。由于求和后最高次项系数不为零，所以此式就是一个三次多项式。而且，如果将

210、、=x 或者3代入上式，其中有三项都为零，而第四项结果正好符合给

定数据。

胡鹤：《MATLAB 数值计算》（美）Cleve B.Moler 第三章插值第1节插值多项式

一个多项式通常不用拉格朗日形式表示，它更常见地写成类似

523--x x

的形式。其中简单的x 的次方项称为单项式（momomial ），而多项式的这种形式称为使用幂形式（power form ）的多项式。

插值多项式使用幂形式表示为

n n n n c x c x c x c x P ++++=---12211)(

其中的系数，原则上可以通过求解下面的线性代数方程组得到。

??

?

??

?

??????=

?????????????????????????

?------n n n n n n n n n n n y y y c c c x x x x x x x x x 21212

122

2121211

11111 这个线性方程组的系数矩阵记为V ，也被称为范德尔蒙（Vandermonde ）矩阵，该矩阵的各个元素为

j

n k

j k x v -=, 上述范德尔蒙矩阵的各列，有时也按相反的顺序排列，但在MATLAB 中，多项式系数向量，通常按从高次幂到低次幂排列。

MATLAB 中的函数vander 可以生成范德尔蒙矩阵，例如对于前面的那组数据，

V=vander(x)

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

生成

V =

0 0 0 1 1 1 1 1 8 4 2 1 27 9 3 1 然后，输入命令

c=V\y' 计算出插值系数。

c = 1.0000 0.0000 -2.0000 -5.0000

事实上，这个例子的数据就是根据多项式523--x x 生成的。

在本章的习题3.6中，要证明当插值点的位置k x 互不相同时，范德尔蒙矩阵是非奇异的。而在练习3.19中，则请读者证明范德尔蒙矩阵的条件可能非常差。通过两个练习我们可以发现，对于一组间隔比较均匀、函数值变化不大的数据，适合采用幂形式的插值多项式和范德尔蒙矩阵进行求解。但对于一般的问题，这个方法有时是危险的。

在本章中，将介绍几个能实现各种插值算法的MATLAB 函数，它们都采用下面的调用格式

胡鹤：《MATLAB数值计算》（美）Cleve B.Moler 第三章插值第1节插值多项式

x

terp

in

y

v=

)

(u

,

,

前两个输入参数，x和y，是长度相同的向量，它们定义了插值点。第三个参数u，为要计算函数值的范围上的点组成的向量。输出向量v和u长度相等，其分量))

u

y

terp

in

k

v=。

x

,

(k

,

(

(

)

要介绍的第一个这样的插值函数是polyinterp，它基于拉格朗日形式。程序使用了MATLAB的数组操作，来同时计算出多项式在u向量各分量上的值。

function v=plyinterp(x,y,u)

n=length(x);

v=zeros(size(u));

for k=1:n

w=ones(size(u));

for j=[1:k-1 k+1:n]

w=(u-x(j))./(x(k)-x(j)).*w;

end

v=v+w*y(k);

end

为了解释polyinterp函数的功能，先构造一个间隔很密的求值点向量。

u=-0.25:0.01:3.25;

然后输入命令

v=polyinterp(x,y,u);

plot(x,y,'o',u,v,'-');

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

可生成图3_1。

函数polyinterp也可以处理符号变量，例如，创建符号变量symx=sym('x')

然后用下面的命令，计算并显示插值多项式的符号形式P=polyinterp(x,y,symx)

Pretty(P)

其输出结果为

-5(-1/3x+1)(-1/2x+1)(-x+1)-6(-1/2x+3/2)(-x+2)x

-1/2(-x+3)(x-1)x+16/3(x-2(1/2x-1/2)x

将其进行简化，从而得到P的幂形式

P=

x^3-2*x-5

下面是另一个例子，使用的是本章另一种方法所用数据。

x=1:6;

y=[16 18 21 17 15 12];

disp([x;y]);

u=.75:.05:6.25;

v=polyinterp(x,y,u);

plot(x,y,’o’,u,v,’-’);

运行后结果为

1 2 3 4 5 6

胡鹤：《MATLAB数值计算》（美）Cleve B.Moler 第三章插值第1节插值多项式

16 18 21 17 15 12

同时输出图3_2。

图3_1 polyinterp 图3_2完整次数(full_degree)多项式插值

在这个仅包含6个正常间距插值值点的例子里，我们已经可以看出，完整次数多项式插值的主要问题了。在数据点之间，特别是第一个和第二个点之间，函数值表现出很大的变化。它超出了给定数据值的变化，因此，完整次数多项式插值很少用于实际的数据或曲线拟合，它主要用于推导出其他的数值方法。

2013届数学与应用数学专业毕业设计（论文）英文文献翻译

指导教师：

翻译：胡鹤

完

成时间：2013.3.10