极客教程在Python中使用NumPy对切比雪夫级数进行积分并设置积分顺序
在这篇文章中，我们将讨论如何在Python和NumPy中对切比雪夫级数进行积分并设置积分的顺序。
chebyshev.chebint 方法
切比雪夫多项式在近似理论中意义重大，因为切比雪夫节点被用作优化多项式插值的匹配点。为了进行切比雪夫积分，NumPy提供了一个名为Chebyshev.chebint()方法的函数，用于对给定阶数的切比雪夫数列进行积分。它需要两个参数，第一个是c，需要一个数组，第二个是m，用来设置积分顺序。
语法: chebyshev.chebint(c,m)
参数:
- c: 一个数组
- m：整合的顺序。
返回：积分的切比雪夫级数系数。
示例 1:
在这个例子中，我们将创建一个有6个元素的一维NumPy系数数组，并通过设置3,4,1,和6的阶数来进行积分。
# import the numpy module
import numpy
# import chebyshev
from numpy.polynomial import chebyshev
# Create an array of Chebyshev series
# coefficients with 6 elements
coefficient_array = numpy.array([1,2,3,4,3,5])
# display array
print("Coefficient array: ", coefficient_array)
# display the dimensions
print("Dimensions: ", coefficient_array.ndim)
# integrate chebyshev series with order 3
print("\nChebyshev series with order 3",
chebyshev.chebint(coefficient_array, m = 3))
# integrate chebyshev series with order 4
print("\nChebyshev series with order 4",
chebyshev.chebint(coefficient_array, m = 4))
# integrate chebyshev series with order 1
print("\nChebyshev series with order 1",
chebyshev.chebint(coefficient_array, m = 1))
# integrate chebyshev series with order 6
print("\nChebyshev series with order 6",
chebyshev.chebint(coefficient_array, m = 6))
输出 :
Coefficient array: [1 2 3 4 3 5]
Dimensions: 1
Chebyshev series with order 3 [ 0.08072917 0. 0.08854167 -0.01458333 -0.00104167 -0.00625
-0.00699405 0.00178571 0.00186012]
Chebyshev series with order 4 [ 0.00390625 0.03645833 0.00364583 0.01493056 -0.00104167 0.00059524
-0.00066964 -0.00063244 0.00011161 0.00010334]
Chebyshev series with order 1 [ 0.04166667 -0.5 -0.5 0. -0.125 0.3
0.41666667]
Chebyshev series with order 6 [ 2.28949653e-04 1.05251736e-03 3.25520833e-04 5.98338294e-04
1.02306548e-04 1.68960813e-04 1.55009921e-06 1.05923446e-05
-3.87524802e-06 -2.84184854e-06 3.10019841e-07 2.34863516e-07]
示例 2:
在这个例子中，我们将创建一个二维的NumPy系数数组，其中有6个元素，形状为2×2，并通过设置3、4、1和6的阶数来进行积分。
# import the numpy module
import numpy
# import chebyshev
from numpy.polynomial import chebyshev
# Create an 2 D array of Chebyshev series
# coefficients with 6 elements
coefficient_array = numpy.array([[1, 2, 3, 4, 3, 5], [5, 6, 8, 9, 0, 0]])
# display array
print("Coefficient array: ", coefficient_array)
# display the dimensions
print("Dimensions: ", coefficient_array.ndim)
# integrate chebyshev series with order 3
print("\nChebyshev series with order 3",
chebyshev.chebint(coefficient_array, m=3))
# integrate chebyshev series with order 4
print("\nChebyshev series with order 4",
chebyshev.chebint(coefficient_array, m=4))
# integrate chebyshev series with order 1
print("\nChebyshev series with order 1",
chebyshev.chebint(coefficient_array, m=1))
# integrate chebyshev series with order 6
print("\nChebyshev series with order 6",
chebyshev.chebint(coefficient_array, m=6))
输出:
Coefficient array: [[1 2 3 4 3 5]
[5 6 8 9 0 0]]
Dimensions: 2
Chebyshev series with order 3 [[0.078125 0.09375 0.125 0.140625 0. 0. ]
[0.125 0.25 0.375 0.5 0.375 0.625 ]
[0.10416667 0.125 0.16666667 0.1875 0. 0. ]
[0.04166667 0.08333333 0.125 0.16666667 0.125 0.20833333]
[0.02604167 0.03125 0.04166667 0.046875 0. 0. ]]
Chebyshev series with order 4 [[0.015625 0.03125 0.046875 0.0625 0.046875 0.078125 ]
[0.02604167 0.03125 0.04166667 0.046875 0. 0. ]
[0.02083333 0.04166667 0.0625 0.08333333 0.0625 0.10416667]
[0.01302083 0.015625 0.02083333 0.0234375 0. 0. ]
[0.00520833 0.01041667 0.015625 0.02083333 0.015625 0.02604167]
[0.00260417 0.003125 0.00416667 0.0046875 0. 0. ]]
Chebyshev series with order 1 [[1.25 1.5 2. 2.25 0. 0. ]
[1. 2. 3. 4. 3. 5. ]
[1.25 1.5 2. 2.25 0. 0. ]]
Chebyshev series with order 6 [[4.34027778e-04 8.68055556e-04 1.30208333e-03 1.73611111e-03
1.30208333e-03 2.17013889e-03]
[5.42534722e-04 6.51041667e-04 8.68055556e-04 9.76562500e-04
0.00000000e+00 0.00000000e+00]
[6.51041667e-04 1.30208333e-03 1.95312500e-03 2.60416667e-03
1.95312500e-03 3.25520833e-03]
[3.25520833e-04 3.90625000e-04 5.20833333e-04 5.85937500e-04
0.00000000e+00 0.00000000e+00]
[2.60416667e-04 5.20833333e-04 7.81250000e-04 1.04166667e-03
7.81250000e-04 1.30208333e-03]
[1.08506944e-04 1.30208333e-04 1.73611111e-04 1.95312500e-04
0.00000000e+00 0.00000000e+00]
[4.34027778e-05 8.68055556e-05 1.30208333e-04 1.73611111e-04
1.30208333e-04 2.17013889e-04]
[1.55009921e-05 1.86011905e-05 2.48015873e-05 2.79017857e-05
0.00000000e+00 0.00000000e+00]]