在Python中生成勒让德多项式和x、y、z复数点的伪范德蒙矩阵

要生成勒让德多项式的伪范德蒙矩阵，并使用x、y、z样本点，可以使用Python Numpy中的legendre.legvander3d()方法。该方法返回度数为deg且样本点为(x, y, z)的伪范德蒙矩阵。

参数x、y、z是点坐标数组，形状相同。根据元素是否为复数，其数据类型将转换为float64或complex128。标量将转换为1维数组。参数deg是形式为[x_deg, y_deg, z_deg]的最大度数列表。

步骤

首先，导入所需的库−

import numpy as np
from numpy.polynomial import legendre as L

使用 numpy.array() 方法创建形状相同的点坐标数组：

x = np.array([-2.+2.j, -1.+2.j])
y = np.array([0.+2.j, 1.+2.j])
z = np.array([2.+2.j, 3. + 3.j])

显示数组 –

print("Array1...\n",x)
print("\nArray2...\n",y)
print("\nArray3...\n",z)

显示数据类型 −

print("\nArray1 datatype...\n",x.dtype)
print("\nArray2 datatype...\n",y.dtype)
print("\nArray3 datatype...\n",z.dtype)

检查两个数组的尺寸 –

print("\nDimensions of Array1...\n",x.ndim)
print("\nDimensions of Array2...\n",y.ndim)
print("\nDimensions of Array3...\n",z.ndim)

检查两个数组的形状 –

print("\nShape of Array1...\n",x.shape)
print("\nShape of Array2...\n",y.shape)
print("\nShape of Array3...\n",z.shape)

使用Python Numpy中的legendre.legvander3d()方法，生成带有x、y、z样本点的Legendre多项式的伪Vandermonde矩阵。

x_deg, y_deg, z_deg = 2, 3, 4
print("\nResult...\n",L.legvander3d(x,y,z, [x_deg, y_deg, z_deg]))

示例

import numpy as np
from numpy.polynomial import legendre as L

# Create arrays of point coordinates, all of the same shape using the numpy.array() method
x = np.array([-2.+2.j, -1.+2.j])
y = np.array([0.+2.j, 1.+2.j])
z = np.array([2.+2.j, 3. + 3.j])

# Display the arrays
print("Array1...\n",x)
print("\nArray2...\n",y)
print("\nArray3...\n",z)

# Display the datatype
print("\nArray1 datatype...\n",x.dtype)
print("\nArray2 datatype...\n",y.dtype)
print("\nArray3 datatype...\n",z.dtype)

# Check the Dimensions of both the arrays
print("\nDimensions of Array1...\n",x.ndim)
print("\nDimensions of Array2...\n",y.ndim)
print("\nDimensions of Array3...\n",z.ndim)

# Check the Shape of both the arrays
print("\nShape of Array1...\n",x.shape)
print("\nShape of Array2...\n",y.shape)
print("\nShape of Array3...\n",z.shape)

# To generate a pseudo Vandermonde matrix of the Legendre polynomial with x, y, z sample points, use the legendre.legvander3d() method in Python Numpy
x_deg, y_deg, z_deg = 2, 3, 4
print("\nResult...\n",L.legvander3d(x,y,z, [x_deg, y_deg, z_deg]))

输出

Array1...
   [-2.+2.j -1.+2.j]

Array2...
   [0.+2.j 1.+2.j]

Array3...
   [2.+2.j 3.+3.j]

Array1 datatype...
complex128

Array2 datatype...
complex128

Array3 datatype...
complex128

Dimensions of Array1...
1

Dimensions of Array2...
1

Dimensions of Array3...
1

Shape of Array1...
(2,)

Shape of Array2...
(2,)

Shape of Array3...
(2,)

Result...
   [[ 1.00000000e+00 +0.0000000e+00j  2.00000000e+00 +2.0000000e+00j
     -5.00000000e-01 +1.2000000e+01j -4.30000000e+01 +3.7000000e+01j
     -2.79625000e+02 -3.0000000e+01j  0.00000000e+00 +2.0000000e+00j
     -4.00000000e+00 +4.0000000e+00j -2.40000000e+01 -1.0000000e+00j
     -7.40000000e+01 -8.6000000e+01j  6.00000000e+01 -5.5925000e+02j
     -6.50000000e+00 +0.0000000e+00j -1.30000000e+01 -1.3000000e+01j
      3.25000000e+00 -7.8000000e+01j  2.79500000e+02 -2.4050000e+02j
      1.81756250e+03 +1.9500000e+02j  0.00000000e+00 -2.3000000e+01j
      4.60000000e+01 -4.6000000e+01j  2.76000000e+02 +1.1500000e+01j
      8.51000000e+02 +9.8900000e+02j -6.90000000e+02 +6.4313750e+03j
     -2.00000000e+00 +2.0000000e+00j -8.00000000e+00 +0.0000000e+00j
     -2.30000000e+01 -2.5000000e+01j  1.20000000e+01 -1.6000000e+02j
      6.19250000e+02 -4.9925000e+02j -4.00000000e+00 -4.0000000e+00j
      0.00000000e+00 -1.6000000e+01j  5.00000000e+01 -4.6000000e+01j
      3.20000000e+02 +2.4000000e+01j  9.98500000e+02 +1.2385000e+03j
      1.30000000e+01 -1.3000000e+01j  5.20000000e+01 +0.0000000e+00j
      1.49500000e+02 +1.6250000e+02j -7.80000000e+01 +1.0400000e+03j
     -4.02512500e+03 +3.2451250e+03j  4.60000000e+01 +4.6000000e+01j
      0.00000000e+00 +1.8400000e+02j -5.75000000e+02 +5.2900000e+02j
     -3.68000000e+03 -2.7600000e+02j -1.14827500e+04 -1.4242750e+04j
     -5.00000000e-01 -1.2000000e+01j  2.30000000e+01 -2.5000000e+01j
      1.44250000e+02 +0.0000000e+00j  4.65500000e+02 +4.9750000e+02j
     -2.20187500e+02 +3.3705000e+03j  2.40000000e+01 -1.0000000e+00j
      5.00000000e+01 +4.6000000e+01j  0.00000000e+00 +2.8850000e+02j
     -9.95000000e+02 +9.3100000e+02j -6.74100000e+03 -4.4037500e+02j
      3.25000000e+00 +7.8000000e+01j -1.49500000e+02 +1.6250000e+02j
     -9.37625000e+02 +0.0000000e+00j -3.02575000e+03 -3.2337500e+03j
      1.43121875e+03 -2.1908250e+04j -2.76000000e+02 +1.1500000e+01j
     -5.75000000e+02 -5.2900000e+02j  0.00000000e+00 -3.3177500e+03j
      1.14425000e+04 -1.0706500e+04j  7.75215000e+04 +5.0643125e+03j]
    [ 1.00000000e+00 +0.0000000e+00j  3.00000000e+00 +3.0000000e+00j
     -5.00000000e-01 +2.7000000e+01j -1.39500000e+02 +1.3050000e+02j
     -1.41712500e+03 -6.7500000e+01j  1.00000000e+00 +2.0000000e+00j
     -3.00000000e+00 +9.0000000e+00j -5.45000000e+01 +2.6000000e+01j
     -4.00500000e+02 -1.4850000e+02j -1.28212500e+03 -2.9017500e+03j
     -5.00000000e+00 +6.0000000e+00j -3.30000000e+01 +3.0000000e+00j
     -1.59500000e+02 -1.3800000e+02j -8.55000000e+01 -1.4895000e+03j
      7.49062500e+03 -8.1652500e+03j -2.90000000e+01 -8.0000000e+00j
     -6.30000000e+01 -1.1100000e+02j  2.30500000e+02 -7.7900000e+02j
      5.08950000e+03 -2.6685000e+03j  4.05566250e+04 +1.3294500e+04j
     -1.00000000e+00 +2.0000000e+00j -9.00000000e+00 +3.0000000e+00j
     -5.35000000e+01 -2.8000000e+01j -1.21500000e+02 -4.0950000e+02j
      1.55212500e+03 -2.7667500e+03j -5.00000000e+00 +0.0000000e+00j
     -1.50000000e+01 -1.5000000e+01j  2.50000000e+00 -1.3500000e+02j
      6.97500000e+02 -6.5250000e+02j  7.08562500e+03 +3.3750000e+02j
     -7.00000000e+00 -1.6000000e+01j  2.70000000e+01 -6.9000000e+01j
      4.35500000e+02- 1.8100000e+02j  3.06450000e+03 +1.3185000e+03j
      8.83987500e+03 +2.3146500e+04j  4.50000000e+01 -5.0000000e+01j
      2.85000000e+02 -1.5000000e+01j  1.32750000e+03 +1.2400000e+03j
      2.47500000e+02 +1.2847500e+04j -6.71456250e+04 +6.7818750e+04j
     -5.00000000e+00 -6.0000000e+00j  3.00000000e+00 -3.3000000e+01j
      1.64500000e+02 -1.3200000e+02j  1.48050000e+03 +1.8450000e+02j
      6.68062500e+03 +8.8402500e+03j  7.00000000e+00 -1.6000000e+01j
      6.90000000e+01 -2.7000000e+01j  4.28500000e+02 +1.9700000e+02j
      1.11150000e+03 +3.1455000e+03j -1.09998750e+04 +2.2201500e+04j
      6.10000000e+01 +0.0000000e+00j  1.83000000e+02 +1.8300000e+02j
     -3.05000000e+01 +1.6470000e+03j -8.50950000e+03 +7.9605000e+03j
     -8.64446250e+04 -4.1175000e+03j  9.70000000e+01 +2.1400000e+02j
     -3.51000000e+02 +9.3300000e+02j -5.82650000e+03 +2.5120000e+03j
     -4.14585000e+04 -1.7194500e+04j -1.23016125e+05 -3.0981225e+05j]]