# Evaluate a 3D Legendre series on the Cartesian product of x, y and z with 4d array of coefficient in Python

To evaluate a 3D Legendre series on the Cartesian product of x, y and z use the polynomial.legendre.leggrid3d() method in Python Numpy. The method returns the values of the three dimensional Chebyshev series at points in the Cartesian product of x, and z. If c has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape.

The 1st parameter is x, y, z. The three dimensional series is evaluated at the points in the Cartesian product of x,y and z. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn’t an ndarray, it is treated as a scalar.

The 2nd parameter is c. Array of coefficients ordered so that the coefficient of the term of multidegree i,j is contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.

## Steps

At first, import the required library −

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

Create a 4d array of coefficients −

c = np.arange(48).reshape(2,2,6,2)

Display the array −

print("Our Array...\n",c)

Check the Dimensions −

print("\nDimensions of our Array...\n",c.ndim)

Get the Datatype −

print("\nDatatype of our Array object...\n",c.dtype)

Get the Shape −

print("\nShape of our Array object...\n",c.shape)

To evaluate a 3D Legendre series on the Cartesian product of x, y and z use the polynomial.legendre.leggrid3d() method in Python −

print("\nResult...\n",L.leggrid3d([1,2],[1,2],[1,2],c))

## Example

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

# Create a 4d array of coefficients
c = np.arange(48).reshape(2,2,6,2)

# Display the array
print("Our Array...\n",c)

# Check the Dimensions
print("\nDimensions of our Array...\n",c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n",c.dtype)

# Get the Shape
print("\nShape of our Array object...\n",c.shape)

# To evaluate a 3D Legendre series on the Cartesian product of x, y and z use the polynomial.legendre.leggrid3d() method in Python Numpy
print("\nResult...\n",L.leggrid3d([1,2],[1,2],[1,2],c))

## Output

Our Array...
[[[[ 0 1]
[ 2 3]
[ 4 5]
[ 6 7]
[ 8 9]
[10 11]]

[[12 13]
[14 15]
[16 17]
[18 19]
[20 21]
[22 23]]]

[[[24 25]
[26 27]
[28 29]
[30 31]
[32 33]
[34 35]]

[[36 37]
[38 39]
[40 41]
[42 43]
[44 45]
[46 47]]]]

Dimensions of our Array...
4

Datatype of our Array object...
int64

Shape of our Array object...
(2, 2, 6, 2)

Result...
[[[[ 552. 28911. ]
[ 900. 46566. ]]

[[ 972. 49765.5 ]
[ 1566. 79447.5 ]]]

[[[ 576. 29977.5 ]
[ 936. 48165.75 ]]

[[ 1008. 51365.25 ]
[ 1620. 81847.125]]]]