Generate a Laguerre series with given complex roots in Python

To generate a Laguerre series with given complex roots, use the laguerre.lagfromroots() method in Python NumPy. This method returns a 1-D array of coefficients representing the Laguerre polynomial. If all roots are real, the output is a real array. If some roots are complex, the output is complex even if all coefficients in the result are real.

Syntax

numpy.polynomial.laguerre.lagfromroots(roots)

Parameters

roots: A sequence containing the roots of the polynomial.

Example with Complex Roots

Let's generate a Laguerre series with complex conjugate roots ?

from numpy.polynomial import laguerre as L

# Define complex unit j
j = complex(0, 1)

# Generate Laguerre series with complex conjugate roots (-j, j)
coefficients = L.lagfromroots((-j, j))

print("Laguerre series coefficients:")
print(coefficients)

# Check the datatype
print("\nDatatype:", coefficients.dtype)

# Check the shape
print("Shape:", coefficients.shape)

Laguerre series coefficients:
[ 3.+0.j -4.+0.j  2.+0.j]

Datatype: complex128
Shape: (3,)

Example with Real Roots

When using only real roots, the output array remains real ?

from numpy.polynomial import laguerre as L

# Generate Laguerre series with real roots
real_coefficients = L.lagfromroots([1, 2, 3])

print("Laguerre series coefficients (real roots):")
print(real_coefficients)
print("\nDatatype:", real_coefficients.dtype)

Laguerre series coefficients (real roots):
[ 6. -18.  14.  -2.]

Datatype: float64

How It Works

The method constructs a Laguerre polynomial whose roots are the specified values. For roots r?, r?, ..., r?, it builds the polynomial (x - r?)(x - r?)...(x - r?) and converts it to Laguerre series representation.

Conclusion

The lagfromroots() method efficiently generates Laguerre series coefficients from given roots. Complex roots result in complex coefficient arrays, while real roots produce real coefficient arrays.