Commons Math学习笔记——多项式函数
2.2 多项式函数

看其他篇章到目录选择。

在Commons Math中的analysis.polynomials包中有所有的与多项式函数相关的类和接口定义。这一篇主要从这个包分析，来研究一下多项式函数的应用。

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Polynomials包中没有interface的定义，下属含有5个类：PolynomialFunction、PolynomialFunctionLagrangeForm、PolynomialFunctionNewtonForm、PolynomialSplineFunction和PolynomialsUtils。其中主要的只有PolynomialFunction和PolynomialSplineFunction，正如api doc中的介绍，PolynomialFunction类是Immutable representation of a real polynomial function with real coefficients——实数多项式的表示；PolynomialSplineFunction类是Represents a polynomial spline function.——样条曲线多项式的表示。另外两个表示拉格朗日和牛顿形式的多项式函数。而PolynomialsUtils类中提供了几个构造个别（比如切比雪夫多项式）多项式的静态方法。

我觉得最常用的应该就是实数系数的多项式了，因此以PolynomialFunction类为例来进行分析。实数系数的多项式函数形如：f(x) = ax^2 + bx + c。PolynomialFunction类实现了DifferentiableUnivariateRealFunction接口，因此必须实现value()和derivative()方法，并且实现该接口也表明这是一元可微分的实数函数形式。PolynomialFunction类定义了一组final double coefficients[]作为多项式系数，其中coefficients[0]表示常数项的系数，coefficients[n]表示指数为n的x^n次项的系数。因此，这个类所表达的多项式函数是这样的：f(x)=coeff[0] + coeff[1]x + coeff[2]x^2 + … + coeff[n]x^n。它的构造方法是PolynomialFunction(double [])就是接受这样的coefficients数组作为系数输入参数来构造多项式的。这个是很好表达也很方便理解的。那么它的value(double x)方法是通过调用double evaluate(double[] coefficients, double argument)实现的，本质用Horner's Method求解多项式的值，没有什么技术难点，非常好理解的一个给定参数和函数求值过程。剩余定义的一些加减乘等操作，都是通过一个类似public PolynomialFunction add(final PolynomialFunction p)这样的结构实现的。求导数的方法derivative()是通过这样的一个微分操作实现的。见源码：

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??1protected?static?double[]?differentiate(double[]?coefficients)?{
?2????????int?n?=?coefficients.length;
?3????????if?(n?<?1)?{
?4????????????throw?MathRuntimeException.createIllegalArgumentException("empty?polynomials?coefficients?array");
?5????????}
?6????????if?(n?==?1)?{
?7????????????return?new?double[]{0};
?8????????}
?9????????double[]?result?=?new?double[n?-?1];
10????????for?(int?i?=?n?-?1;?i??>?0;?i--)?{
11????????????result[i?-?1]?=?i?*?coefficients[i];
12????????}
13????????return?result;
14????}
15?

/**?*//**
?2?*?
?3?*/
?4package?algorithm.math;
?5
?6import?org.apache.commons.math.ArgumentOutsideDomainException;
?7import?org.apache.commons.math.analysis.polynomials.PolynomialFunction;
?8import?org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction;
?9
10/**?*//**
11?*?@author?Jia?Yu
12?*?@date?2010-11-21
13?*/
14public?class?PolinomialsFunctionTest?{
15
16????/**?*//**
17?????*?@param?args
18?????*/
19????public?static?void?main(String[]?args)?{
20????????//?TODO?Auto-generated?method?stub
21????????polynomials();
22????????System.out.println("-----------------------------------------------");
23????????polynomialsSpline();
24????}
25
26????private?static?void?polynomialsSpline()?{
27????????//?TODO?Auto-generated?method?stub
28????????PolynomialFunction[]?polynomials?=?{
29????????????????new?PolynomialFunction(new?double[]?{?0d,?1d,?1d?}),
30????????????????new?PolynomialFunction(new?double[]?{?2d,?1d,?1d?}),
31????????????????new?PolynomialFunction(new?double[]?{?4d,?1d,?1d?})?};
32????????double[]?knots?=?{?-1,?0,?1,?2?};
33????????PolynomialSplineFunction?spline?=?new?PolynomialSplineFunction(knots,
34????????????????polynomials);
35????????//output?directly
36????????System.out.println("poly?spline?func?is?"+spline);
37????????//?get?the?value?when?x?=?0.5
38????????try?{
39????????????System.out.println("f(0.5)?=?"+spline.value(0.5));
40????????}?catch?(ArgumentOutsideDomainException?e)?{
41????????????//?TODO?Auto-generated?catch?block
42????????????e.printStackTrace();
43????????}
44????????//?the?number?of?spline?segments
45????????System.out.println("spline?segments?number?is?"+spline.getN());
46????????//?the?polynomials?functions
47????????for(int?i=0;i<spline.getN();i++){
48????????????System.out.println("spline:f"+i+"(x)?=?"+spline.getPolynomials()[i]);
49????????}
50????????//function?derivative
51????????System.out.println("spline?func?derivative?is?"+spline.derivative());
52????}
53
54????private?static?void?polynomials()?{
55????????//?TODO?Auto-generated?method?stub
56????????double[]?f1_coeff?=?{?3.0,?6.0,?-2.0,?1.0?};
57????????double[]?f2_coeff?=?{?1.0,?2.0,?-1.0,?-2.0?};
58????????PolynomialFunction?f1?=?new?PolynomialFunction(f1_coeff);
59????????PolynomialFunction?f2?=?new?PolynomialFunction(f2_coeff);
60????????//?output?directly
61????????System.out.println("f1(x)?is?:?"?+?f1);
62????????System.out.println("f2(x)?is?:?"?+?f2);
63????????//?polynomial?degree
64????????System.out.println("f1(x)'s?degree?is?"?+?f1.degree());
65????????//?get?the?value?when?x?=?2
66????????System.out.println("f1(2)?=?"?+?f1.value(2));
67????????//?function?add
68????????System.out.println("f1(x)+f2(x)?=?"?+?f1.add(f2));
69????????//?function?substract
70????????System.out.println("f1(x)-f2(x)?=?"?+?f1.subtract(f2));
71????????//?function?multiply
72????????System.out.println("f1(x)*f2(x)?=?"?+?f1.multiply(f2));
73????????//?function?derivative
74????????System.out.println("f1'(x)?=?"?+?f1.derivative());
75????????System.out.println("f2''(x)?=?"
76????????????????+?((PolynomialFunction)?f2.derivative()).derivative());
77
78????}
79
80}
81

输出如下：
f1(x) is : 3.0 + 6.0 x - 2.0 x^2 + x^3
f2(x) is : 1.0 + 2.0 x - x^2 - 2.0 x^3
f1(x)'s degree is 3
f1(2) = 15.0
f1(x)+f2(x) = 4.0 + 8.0 x - 3.0 x^2 - x^3
f1(x)-f2(x) = 2.0 + 4.0 x - x^2 + 3.0 x^3
f1(x)*f2(x) = 3.0 + 12.0 x + 7.0 x^2 - 15.0 x^3 - 8.0 x^4 + 3.0 x^5 - 2.0 x^6
f1'(x) = 6.0 - 4.0 x + 3.0 x^2
f2''(x) = -2.0 - 12.0 x
-----------------------------------------------
poly spline func is org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction@69b332
f(0.5) = 2.75
spline segments number is 3
spline:f0(x) = x + x^2
spline:f1(x) = 2.0 + x + x^2
spline:f2(x) = 4.0 + x + x^2
spline func derivative is org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction@173a10f

PolynomialFunction类也是重写了toString方法和hashCode和equals方法的。

PolynomialSplineFunction类是多项式样条函数，样条是一种特殊的函数，由多项式分段定义。表示了一个由多个多项式组成的样条曲线。它的实现主要是内部定义了一个多项式函数组PolynomialFunction polynomials[]和一个样条分界节点数组double knots[]。这两个内部成员分别表示什么呢？分界节点表示整条曲线对应在x等于knots[i]的时候开始使用其他多项式样条，其构造方法public PolynomialSplineFunction(double knots[], PolynomialFunction polynomials[])完成这样的功能。

举例来说，一个多项式样条函数就是一个分段函数：

????? X^2+x??? [-1,0)

F(x) = x^2+x+2?? [0,1)

????? X^2+x+4?[1,2)

当然，构造方法中的参数，knots[]数组必须是递增的。

可以看到，直接输出PolynomialSplineFunction是多么丑陋啊~~，因为它没有重写toString方法。同样，它的导数也是一样的丑陋。其中如果给定的值不在定义域内，value方法还抛出异常ArgumentOutsideDomainException。

最后PolynomialFunctionLagrangeForm和PolynomialFunctionNewtonForm类完成的其实是多项式插值的功能，放到下一节研究的。

相关资料：

多项式：http://zh.wikipedia.org/zh-cn/%E5%A4%9A%E9%A1%B9%E5%BC%8F%E5%87%BD%E6%95%B0#.E5.A4.9A.E9.A0.85.E5.BC.8F.E5.87.BD.E6.95.B8.E5.8F.8A.E5.A4.9A.E9.A0.85.E5.BC.8F.E7.9A.84.E6.A0.B9

样条函数：http://zh.wikipedia.org/zh-cn/%E6%A0%B7%E6%9D%A1%E5%87%BD%E6%95%B0

Horner Methods：http://mathworld.wolfram.com/HornersMethod.html

Commons math包：http://commons.apache.org/math/index.html