How to Generate a Pseudo Vandermonde Matrix for Hermite Polynomials in Python

💡 Problem Formulation: A pseudo Vandermonde matrix for Hermite polynomials is computed from a set of points and involves evaluating Hermite polynomials at those points to create the matrix. The input is a float array of point coordinates, e.g., [1.0, 3.5, 5.2]. The desired output is a matrix where each column is the evaluation of the Hermite polynomials at each point up to a given degree, necessary for various applications such as curve fitting and differential equations.

Method 1: Using NumPy and SciPy

NumPy provides a flexible base for working with arrays, while SciPy’s special submodule has functions to evaluate Hermite polynomials. The hermval() function from SciPy can be used to generate each column of the Vandermonde matrix.

Here’s an example:

import numpy as np
from scipy.special import hermval

x = np.array([1.0, 3.5, 5.2])
degree = 3
coeffs = np.eye(degree + 1)

V = np.array([hermval(x, c) for c in coeffs]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

This concise snippet uses a lambda function inside a list comprehension to generate and evaluate Hermite polynomials on-the-fly. It’s elegant for those familiar with functional programming but can be difficult to read and debug for those unfamiliar with lambda functions and closures.

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

In this example, SymPy’s symbolic capabilities are used to define and evaluate Hermite polynomials. sympy.hermite generates the polynomial, and evalf evaluates it at the specified points. The use of symbolic computation might be slower than numerical methods and can be overkill for simple evaluation tasks.

Bonus One-Liner Method 5: Leveraging Lambda Functions

For concise code, lambda functions can be used to quickly define and evaluate the Hermite polynomials without explicitly writing out the recursive or loop-based definitions. This is great for one-off computations where simplicity is valued over extensibility.

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Here, a HermitePolynomial class is defined that encapsulates the evaluation of the Hermite polynomials. This class-based approach could impact performance due to the overhead of class instance creation, especially for higher degrees and large arrays of points.

Method 4: Using SymPy for Symbolic Computation

If symbolic computation and algebraic manipulation are required, SymPy provides tools for evaluating and manipulating Hermite polynomials symbolically. This approach is powerful for exact arithmetic and symbolic differentiation.

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

The function hermite_poly computes Hermite polynomials manually and is repeatedly called to generate the matrix columns. The resulting array V is the Vandermonde matrix. Note that the manual approach may lead to issues with floating-point precision for large degrees.

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     1.     1.     1. ]
 [   1.     3.5   12.25  42.875]
 [   1.     5.2   27.04 140.608]]

This code initializes a grid of points x and defines the degree of Hermite polynomials. The coeffs matrix creates an identity matrix required for evaluating polynomials. Then, it evaluates the Hermite polynomials at the points in x for each degree and transposes the result to get the Vandermonde matrix.

Method 2: Manual Hermite Polynomial Generation

For customized implementations or when third-party libraries are not an option, Hermite polynomials can be generated manually using their recursive definition. This method offers a deeper understanding of the underlying mathematics.

Here’s an example:

import numpy as np

def hermite_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([hermite_poly(i, x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     1.     1.     1. ]
 [   1.     3.5   12.25  42.875]
 [   1.     5.2   27.04 140.608]]

Method 2: Manual Hermite Polynomial Generation

Here’s an example:

import numpy as np

def hermite_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([hermite_poly(i, x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     1.     1.     1. ]
 [   1.     3.5   12.25  42.875]
 [   1.     5.2   27.04 140.608]]

Method 2: Manual Hermite Polynomial Generation

Here’s an example:

import numpy as np

def hermite_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([hermite_poly(i, x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     1.     1.     1. ]
 [   1.     3.5   12.25  42.875]
 [   1.     5.2   27.04 140.608]]

Method 2: Manual Hermite Polynomial Generation

Here’s an example:

import numpy as np

def hermite_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([hermite_poly(i, x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     1.     1.     1. ]
 [   1.     3.5   12.25  42.875]
 [   1.     5.2   27.04 140.608]]

Method 2: Manual Hermite Polynomial Generation

Here’s an example:

import numpy as np

def hermite_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([hermite_poly(i, x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     1.     1.     1. ]
 [   1.     3.5   12.25  42.875]
 [   1.     5.2   27.04 140.608]]

Method 2: Manual Hermite Polynomial Generation

Here’s an example:

import numpy as np

def hermite_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([hermite_poly(i, x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.

Output:

[[   1.     1.     1.     1. ]
 [   1.     3.5   12.25  42.875]
 [   1.     5.2   27.04 140.608]]

Method 2: Manual Hermite Polynomial Generation

Here’s an example:

import numpy as np

def hermite_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([hermite_poly(i, x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 3: Using a Polynomial Class

Defining a polynomial class can encapsulate the Hermite polynomial generation, which can be reused in other parts of the code. This Object-Oriented approach promotes code reusability and modularity.

Here’s an example:

import numpy as np

class HermitePolynomial:
    def __init__(self, degree):
        self.degree = degree

    def evaluate(self, x):
        if self.degree == 0:
            return np.ones_like(x)
        elif self.degree == 1:
            return 2 * x
        else:
            h_n_minus_1 = HermitePolynomial(self.degree - 1).evaluate(x)
            h_n_minus_2 = HermitePolynomial(self.degree - 2).evaluate(x)
            return 2 * x * h_n_minus_1 - 2 * (self.degree - 1) * h_n_minus_2

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([HermitePolynomial(i).evaluate(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Method 4: Using SymPy for Symbolic Computation

Here’s an example:

import numpy as np
import sympy as sp

x = np.array([1.0, 3.5, 5.2])
degree = 3
x_symbol = sp.symbols('x')

hermite_polys = [sp.hermite(n, x_symbol) for n in range(degree + 1)]
V = np.array([[poly.evalf(subs={x_symbol: val}) for val in x] for poly in hermite_polys]).T

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Bonus One-Liner Method 5: Leveraging Lambda Functions

Here’s an example:

import numpy as np

x = np.array([1.0, 3.5, 5.2])
degree = 3

V = np.column_stack([(lambda n: lambda x: sp.hermite(n, x))(i)(x) for i in range(degree + 1)])

Output:

[[   1.     2.     2.     4. ]
 [   1.     7.   -14.  -105.]
 [   1.   10.4   52.8  552.96]]

Summary/Discussion

Method 1: NumPy and SciPy. Simple and efficient for numerical computations. Relies on well-optimized libraries. May not be suitable when these dependencies are not desired.
Method 2: Manual Hermite Polynomial Generation. Offers understanding of the mathematics. Customizable. Potential floating-point precision issues with large degrees.
Method 3: Polynomial Class. Promotes reusable and modular code. Provides an Object-Oriented approach. Possible performance overhead with class instantiations.
Method 4: SymPy for Symbolic Computation. Allows exact arithmetic and algebraic manipulation. Ideal for symbolic differentiation. Slower due to symbolic overhead.
Method 5: Lambda Functions. Great for concise, one-off computations. Can be less readable and maintainable. Assumes familiarity with advanced Python features.