多项式包提供的便利类有：
名字 提供 多项式的 幂级数 切比雪夫 切比雪夫系 勒让德 勒让德级数 拉盖尔 Laguerre级数 埃尔米特 厄米特统 厄米 赫米特系列
名字
提供
多项式的
幂级数
切比雪夫
切比雪夫系
勒让德
勒让德级数
拉盖尔
Laguerre级数
埃尔米特
厄米特统
厄米
赫米特系列
本文中的级数是相应多项式基函数乘以系数的有限和。例如，电源系列看起来像
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并且有系数 . 具有相同系数的切比雪夫级数看起来像
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更一般地说
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在这种情况下 切比雪夫函数是度的函数吗？ 但也可以很容易地成为任何其他类的基函数。所有类别的惯例是 与I度的基函数一致。
所有类都是不可变的，并且具有相同的方法，尤其是它们实现了python数字运算符+、-， * ，//，%，divmod，， * * ，=，和！=最后两个可能有点问题，因为浮点舍入错误。我们现在使用numpy版本1.7.0快速演示各种操作。
首先，我们需要一个多项式类和一个多项式实例。类可以直接从多项式包或相关类型的模块导入。这里我们从包中导入并使用传统的多项式类，因为它的熟悉性：
>>> from numpy.polynomial import Polynomial as P >>> p = P([1,2,3]) >>> p Polynomial([1., 2., 3.], domain=[-1, 1], window=[-1, 1])
请注意，打印输出的长版本有三个部分。第一个是系数，第二个是域，第三个是窗口：
>>> p.coef array([ 1., 2., 3.]) >>> p.domain array([-1., 1.]) >>> p.window array([-1., 1.])
打印多项式会以更熟悉的格式生成多项式表达式：
>>> print(p) 1.0 + 2.0·x¹ + 3.0·x²
请注意，多项式的字符串表示默认使用Unicode字符（Windows除外）来表示幂和下标。还提供了基于ASCII的表示（Windows上的默认值）。多项式字符串格式可以在包级别通过 set_default_printstyle 功能：
set_default_printstyle
>>> numpy.polynomial.set_default_printstyle('ascii') >>> print(p) 1.0 + 2.0 x**1 + 3.0 x**2
或对具有字符串格式的单个多项式实例进行控制：
>>> print(f"{p:unicode}") 1.0 + 2.0·x¹ + 3.0·x²
当我们开始拟合时，我们将处理域和窗口，此时我们忽略它们并运行基本的代数和算术运算。
加减法：
>>> p + p Polynomial([2., 4., 6.], domain=[-1., 1.], window=[-1., 1.]) >>> p - p Polynomial([0.], domain=[-1., 1.], window=[-1., 1.])
乘法：：
>>> p * p Polynomial([ 1., 4., 10., 12., 9.], domain=[-1., 1.], window=[-1., 1.])
权力：：
>>> p**2 Polynomial([ 1., 4., 10., 12., 9.], domain=[-1., 1.], window=[-1., 1.])
师：
楼层除法，“/”，是多项式类的除法运算符，在这方面多项式被视为整数。对于小于3.x的python版本，''operator映射到'/'，就像对python一样，对于更高版本，''在某种程度上，它将被弃用：
>>> p // P([-1, 1]) Polynomial([5., 3.], domain=[-1., 1.], window=[-1., 1.])
剩余部分：
>>> p % P([-1, 1]) Polynomial([6.], domain=[-1., 1.], window=[-1., 1.])
Divmod：
>>> quo, rem = divmod(p, P([-1, 1])) >>> quo Polynomial([5., 3.], domain=[-1., 1.], window=[-1., 1.]) >>> rem Polynomial([6.], domain=[-1., 1.], window=[-1., 1.])
评价：：
>>> x = np.arange(5) >>> p(x) array([ 1., 6., 17., 34., 57.]) >>> x = np.arange(6).reshape(3,2) >>> p(x) array([[ 1., 6.], [ 17., 34.], [ 57., 86.]])
替代：
用多项式代替X并展开结果。在这里，我们用p来代替它，在展开后得到一个新的四次多项式。如果多项式被视为函数，则这是函数的组成：
>>> p(p) Polynomial([ 6., 16., 36., 36., 27.], domain=[-1., 1.], window=[-1., 1.])
根：：
>>> p.roots() array([-0.33333333-0.47140452j, -0.33333333+0.47140452j])
显式使用多项式实例并不总是很方便，因此在算术运算中，元组、列表、数组和标量都会自动强制转换：
>>> p + [1, 2, 3] Polynomial([2., 4., 6.], domain=[-1., 1.], window=[-1., 1.]) >>> [1, 2, 3] * p Polynomial([ 1., 4., 10., 12., 9.], domain=[-1., 1.], window=[-1., 1.]) >>> p / 2 Polynomial([0.5, 1. , 1.5], domain=[-1., 1.], window=[-1., 1.])
域、窗口或类中不同的多项式不能在算术中混合：：
>>> from numpy.polynomial import Chebyshev as T >>> p + P([1], domain=[0,1]) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "<string>", line 213, in __add__ TypeError: Domains differ >>> p + P([1], window=[0,1]) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "<string>", line 215, in __add__ TypeError: Windows differ >>> p + T([1]) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "<string>", line 211, in __add__ TypeError: Polynomial types differ
但是可以使用不同的类型进行替换。实际上，对于类型、域和窗口转换，多项式类之间的转换是这样完成的：
>>> p(T([0, 1])) Chebyshev([2.5, 2. , 1.5], domain=[-1., 1.], window=[-1., 1.])
它给出了多项式 p 以切比雪夫的形式。这是因为 替换 对于 不会更改原始多项式。然而，所有的乘法和除法都将使用切比雪夫级数来完成，因此是结果的类型。
它的目的是所有多项式实例都是不可变的，因此增加了运算 (+= ， -= 等等）以及任何其他违反多项式实例不可变性的功能都是有意未实现的。
+=
-=
多项式实例可以集成和区分。：：
>>> from numpy.polynomial import Polynomial as P >>> p = P([2, 6]) >>> p.integ() Polynomial([0., 2., 3.], domain=[-1., 1.], window=[-1., 1.]) >>> p.integ(2) Polynomial([0., 0., 1., 1.], domain=[-1., 1.], window=[-1., 1.])
第一个示例集成了 p 第二个示例将它集成了两次。默认情况下，集成的下限和集成常量为0，但两者都可以指定。：：
>>> p.integ(lbnd=-1) Polynomial([-1., 2., 3.], domain=[-1., 1.], window=[-1., 1.]) >>> p.integ(lbnd=-1, k=1) Polynomial([0., 2., 3.], domain=[-1., 1.], window=[-1., 1.])
在第一种情况下，积分的下限设置为-1，积分常数为0。在第二个过程中，积分常数也被设置为1。微分更简单，因为唯一的选择是多项式的微分次数：
>>> p = P([1, 2, 3]) >>> p.deriv(1) Polynomial([2., 6.], domain=[-1., 1.], window=[-1., 1.]) >>> p.deriv(2) Polynomial([6.], domain=[-1., 1.], window=[-1., 1.])
通过指定系数构造多项式只是获得多项式实例的一种方法，也可以通过指定其根、从其他多项式类型转换以及通过最小二乘拟合来创建多项式实例。拟合在本节中进行了讨论，其他方法如下所示：
>>> from numpy.polynomial import Polynomial as P >>> from numpy.polynomial import Chebyshev as T >>> p = P.fromroots([1, 2, 3]) >>> p Polynomial([-6., 11., -6., 1.], domain=[-1., 1.], window=[-1., 1.]) >>> p.convert(kind=T) Chebyshev([-9. , 11.75, -3. , 0.25], domain=[-1., 1.], window=[-1., 1.])
convert方法还可以转换域和窗口：
>>> p.convert(kind=T, domain=[0, 1]) Chebyshev([-2.4375 , 2.96875, -0.5625 , 0.03125], domain=[0., 1.], window=[-1., 1.]) >>> p.convert(kind=P, domain=[0, 1]) Polynomial([-1.875, 2.875, -1.125, 0.125], domain=[0., 1.], window=[-1., 1.])
在numpy版本中>=1.7.0 basis 和 cast 类方法也可用。cast方法的工作方式与convert方法类似，而basis方法返回给定次数的基多项式：
>>> P.basis(3) Polynomial([0., 0., 0., 1.], domain=[-1., 1.], window=[-1., 1.]) >>> T.cast(p) Chebyshev([-9. , 11.75, -3. , 0.25], domain=[-1., 1.], window=[-1., 1.])
类型之间的转换可能很有用，但是 not 建议日常使用。从50度切比雪夫级数到同阶多项式级数的数值精度损失，可以使数值计算结果基本上是随机的。
试衣是因为 domain 和 window 属性是便利类的一部分。为了说明这个问题，下面绘制了切比雪夫多项式5阶的值。
>>> import matplotlib.pyplot as plt >>> from numpy.polynomial import Chebyshev as T >>> x = np.linspace(-1, 1, 100) >>> for i in range(6): ax = plt.plot(x, T.basis(i)(x), lw=2, label="$T_%d$"%i) ... >>> plt.legend(loc="upper left") <matplotlib.legend.Legend object at 0x3b3ee10> >>> plt.show()
在-1范围内<= x <=1它们是很好的，等波纹函数位于+/-1之间。相同的图在范围-2<= x <=2看起来非常不同：
>>> import matplotlib.pyplot as plt >>> from numpy.polynomial import Chebyshev as T >>> x = np.linspace(-2, 2, 100) >>> for i in range(6): ax = plt.plot(x, T.basis(i)(x), lw=2, label="$T_%d$"%i) ... >>> plt.legend(loc="lower right") <matplotlib.legend.Legend object at 0x3b3ee10> >>> plt.show()
可以看出，“好”的部分已经缩小到微不足道。在使用切比雪夫多项式进行拟合时，我们希望使用 x 介于-1和1之间，这就是 window 指定。但是，要拟合的数据不太可能在该时间间隔内具有所有数据点，因此我们使用 domain 指定数据点所在的间隔。拟合完成后，首先通过线性变换将域映射到窗口，然后使用映射的数据点进行常用的最小二乘拟合。拟合的窗口和域是返回序列的一部分，并且在计算值、导数等时自动使用。如果在调用中没有指定，拟合例程将使用默认窗口和保存所有数据点的最小域。下面说明了适合于噪声正弦曲线的情况。
>>> import numpy as np >>> import matplotlib.pyplot as plt >>> from numpy.polynomial import Chebyshev as T >>> np.random.seed(11) >>> x = np.linspace(0, 2*np.pi, 20) >>> y = np.sin(x) + np.random.normal(scale=.1, size=x.shape) >>> p = T.fit(x, y, 5) >>> plt.plot(x, y, 'o') [<matplotlib.lines.Line2D object at 0x2136c10>] >>> xx, yy = p.linspace() >>> plt.plot(xx, yy, lw=2) [<matplotlib.lines.Line2D object at 0x1cf2890>] >>> p.domain array([ 0. , 6.28318531]) >>> p.window array([-1., 1.]) >>> plt.show()
numpy.polynomial.polynomial