# Integrate a Chebyshev series and set the order of integration using NumPy in Python

• Last Updated : 01 May, 2022

In this article, we will discuss how to integrate the Chebyshev series and set the order of integration in Python and NumPy.

## chebyshev.chebint method

Chebyshev polynomials are significant in approximation theory because the Chebyshev nodes are used as matching points for optimizing polynomial interpolation. To perform Chebyshev Integration, NumPy provides a function called Chebyshev.chebint() method, which is used to integrate the Chebyshev series with a given order. It will take two parameters, the first is c which takes an array and, the second is m which is used to set the integration order.

Syntax: chebyshev.chebint(c,m)

Parameter:

• c: an array
• m: order of integration.

Return: Chebyshev series coefficients of the integral.

### Example 1:

In this example, we will create a one-dimensional NumPy coefficient array with 6 elements and integrate the series by setting orders with 3,4,1, and 6.

## Python3

 `# 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``))`

Output:

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]

### Example 2:

In this example, we will create a two-dimensional NumPy coefficient array with 6 elements with shape 2×2 and, integrate the series by setting orders with 3,4,1, and 6.

## Python3

 `# 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``)) `

Output:

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]]

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