Python中的拉格朗日插值
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Lagrange interpolation in Python
提问by rubik
I want to interpolate a polynomial with the Lagrange method, but this code doesn't work:
我想用拉格朗日方法插入多项式，但此代码不起作用：
def interpolate(x_values, y_values):
def _basis(j):
p = [(x - x_values[m])/(x_values[j] - x_values[m]) for m in xrange(k + 1) if m != j]
return reduce(operator.mul, p)
assert len(x_values) != 0 and (len(x_values) == len(y_values)), 'x and y cannot be empty and must have the same length'
k = len(x_values)
return sum(_basis(j) for j in xrange(k))
I followed Wikipedia, but when I run it I receive an IndexError at line 3!
我关注了Wikipedia，但是当我运行它时，我在第 3 行收到一个 IndexError ！
Thanks
谢谢
采纳答案by AndiDog
Check the indices, Wikipedia says "k+1 data points", but you're setting k = len(x_values)where it should be k = len(x_values) - 1if you followed the formula exactly.
检查索引，维基百科说“k+1 个数据点”，但如果你完全遵循公式，你正在设置k = len(x_values)它应该在哪里k = len(x_values) - 1。
回答by smichr
Try
尝试
def interpolate(x, x_values, y_values):
def _basis(j):
p = [(x - x_values[m])/(x_values[j] - x_values[m]) for m in xrange(k) if m != j]
return reduce(operator.mul, p)
assert len(x_values) != 0 and (len(x_values) == len(y_values)), 'x and y cannot be empty and must have the same length'
k = len(x_values)
return sum(_basis(j)*y_values[j] for j in xrange(k))
You can confirm it as follows:
您可以通过以下方式确认：
>>> interpolate(1,[1,2,4],[1,0,2])
1.0
>>> interpolate(2,[1,2,4],[1,0,2])
0.0
>>> interpolate(4,[1,2,4],[1,0,2])
2.0
>>> interpolate(3,[1,2,4],[1,0,2])
0.33333333333333331
So the result is the interpolated value based on the polynomial that goes through the points given. In this case, the 3 points define a parabola and the first 3 tests show that the stated y_value is returned for the given x_value.
所以结果是基于经过给定点的多项式的插值。在这种情况下，3 个点定义了一个抛物线，前 3 个测试表明为给定的 x_value 返回了规定的 y_value。
回答by gwg
I'm almost a decade late to the party, but I found this searching for a simple implementation of Lagrange interpolation. @smichr's answer is great, but the Python is a little outdated, and I also wanted something that would work nicely with np.ndarraysso I could do easy plotting. Maybe others will find this useful:
我参加聚会晚了将近十年，但我发现这是在寻找拉格朗日插值的简单实现。@smichr 的回答很好，但是 Python 有点过时了，我还想要一些可以很好地使用的东西，np.ndarrays这样我就可以轻松地进行绘图。也许其他人会发现这很有用：
import numpy as np
import matplotlib.pyplot as plt
class LagrangePoly:
def __init__(self, X, Y):
self.n = len(X)
self.X = np.array(X)
self.Y = np.array(Y)
def basis(self, x, j):
b = [(x - self.X[m]) / (self.X[j] - self.X[m])
for m in range(self.n) if m != j]
return np.prod(b, axis=0) * self.Y[j]
def interpolate(self, x):
b = [self.basis(x, j) for j in range(self.n)]
return np.sum(b, axis=0)
X = [-9, -4, -1, 7]
Y = [5, 2, -2, 9]
plt.scatter(X, Y, c='k')
lp = LagrangePoly(X, Y)
xx = np.arange(-100, 100) / 10
plt.plot(xx, lp.basis(xx, 0))
plt.plot(xx, lp.basis(xx, 1))
plt.plot(xx, lp.basis(xx, 2))
plt.plot(xx, lp.basis(xx, 3))
plt.plot(xx, lp.interpolate(xx), linestyle=':')
plt.show()
