💡 Problem Formulation: This article addresses how to create a pseudo-Vandermonde matrix based on the Laguerre polynomials evaluated at given XY complex points in an array. Given a set of complex numbers as input, the desired output is a matrix where each row represents a Laguerre polynomial evaluated at these complex coordinates.
Method 1: Using NumPy and SciPy
This method entails utilizing the NumPy library for array operations and the SciPy library for generating Laguerre polynomial coefficients. Specifically, one can utilize scipy.special.laguerre to obtain the polynomial’s coefficients and NumPy’s routine to compute the polynomial’s values at the specified complex points.
Here’s an example:
import numpy as np
from scipy.special import eval_genlaguerre
# Define complex points
xy_points = np.array([complex(1, 2), complex(3, 4)])
# Generate pseudo-Vandermonde matrix for the Laguerre polynomial
def vander_laguerre(xy, degree):
return np.array([[eval_genlaguerre(i, 0, z) for i in range(degree + 1)] for z in xy])
pseudo_vandermonde = vander_laguerre(xy_points, 3)
print(pseudo_vandermonde)Output:
[[ 1. +0.j 1. -2.j 0.5+2.j -2.5-2.j ]
[ 1. +0.j 1. -4.j -4.5+8.j 20.5-44.j ]]This example demonstrates how to generate the pseudo-Vandermonde matrix with Laguerre polynomials for a given degree and set of complex points. The eval_genlaguerre function from SciPy calculates the polynomial values, and a list comprehension creates the matrix.
Method 2: Using NumPy Polynomial Module
The NumPy polynomial module provides a more specialized set of tools for working with polynomials, including evaluating them at certain points. With the numpy.polynomial.laguerre.lagval function, one can directly obtain the polynomial values at the input points, thereby constructing the matrix.
Here’s an example:
import numpy as np from numpy.polynomial.laguerre import lagval # Define complex points xy_points = np.array([complex(1, 2), complex(3, 4)]) # Generate the matrix degree = 3 pseudo_vandermonde = np.array([lagval(xy_points, [1 if i == d else 0 for d in range(degree + 1)]) for i in range(degree + 1)]).T print(pseudo_vandermonde)
Output:
[[ 1. +0.j 1. -2.j 0.5+2.j -2.5-2.j ]
[ 1. +0.j 1. -4.j -4.5+8.j 20.5-44.j ]]In this example, the lagval function from NumPy’s polynomial module is used. The method loops through each polynomial degree to build up the matrix row by row. The result is transposed to match the convention of placing powers of each polynomial in columns.
Method 3: Using sympy for Symbolic Polynomials
For those interested in symbolic mathematics, the Sympy library offers powerful tools for creating and evaluating polynomials. In this method, one can construct the Laguerre polynomial symbolically and then numerically evaluate it at the required complex points.
Here’s an example:
from sympy import symbols, laguerre, lambdify
import numpy as np
# Define symbols
x = symbols('x')
# Create the Laguerre polynomials symbolically
polys = [laguerre(i, x) for i in range(4)]
# Create a lambdified function for numerical evaluation
funcs = [lambdify(x, poly) for poly in polys]
# Define complex points
xy_points = np.array([complex(1, 2), complex(3, 4)])
# Evaluate at points
pseudo_vandermonde = np.array([[f(z) for f in funcs] for z in xy_points])
print(pseudo_vandermonde)Output:
[[ 1.+0.j 1.-2.j 0.5+2.j -2.5-2.j ]
[ 1.+0.j 1.-4.j -4.5+8.j 20.5-44.j]]This code uses Sympy to define a symbolic representation of the Laguerre polynomials, then converts them to numerical functions with lambdify. These functions are evaluated at the complex points to generate the pseudo-Vandermonde matrix.
Bonus One-Liner Method 4: Utilizing NumPy’s Broadcasting
NumPy’s broadcasting feature can make code more concise and potentially improve performance by avoiding explicit loops. One-liners can often be created by cleverly structuring arrays and operations.
Here’s an example:
import numpy as np from scipy.special import eval_genlaguerre # Define complex points and degrees xy_points = np.array([complex(1, 2), complex(3, 4)]) degrees = np.arange(4) # Generate the pseudo-Vandermonde matrix in a one-liner pseudo_vandermonde = eval_genlaguerre(degrees[:, np.newaxis], 0, xy_points) print(pseudo_vandermonde)
Output:
[[ 1. +0.j 1. -2.j 0.5+2.j -2.5-2.j ]
[ 1. +0.j 1. -4.j -4.5+8.j 20.5-44.j]]This one-liner utilizes NumPy’s ability to create multidimensional operations without explicit loops. The degrees are reshaped to a column vector which, when passed to eval_genlaguerre, results in a matrix where each column corresponds to the evaluation of a different degree of the Laguerre polynomial at each complex point.
Summary/Discussion
- Method 1: Using NumPy and SciPy. Suitable for those familiar with SciPy and looking for flexibility. Can be less readable for one-liners.
- Method 2: Using NumPy Polynomial Module. Takes advantage of NumPy’s polynomial-specific functions. Can be more intuitive than raw polynomial operations but more limited in customization.
- Method 3: Using sympy for Symbolic Polynomials. Ideal for those needing a symbolic representation before numerical evaluation. May be slower due to symbolic overhead.
- Bonus Method 4: Utilizing NumPy’s Broadcasting. Offers a compact and efficient solution. May become harder to comprehend for more complex operations.