# 5 Best Ways to Evaluate a Polynomial When Coefficients Are Multi-dimensional in Python

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π‘ Problem Formulation: Polynomial evaluation typically involves computing the value of a polynomial given a particular input. However, this becomes a tad more complex when dealing with multi-dimensional coefficients. In Python, we desire efficient methods to evaluate such polynomials. For instance, if we have a 2D array as coefficients of a polynomial and we want to evaluate it at a specific value, our output should reflect the multi-dimensional nature of the input.

## Method 1: Using NumPy’s polyval with Flattening

This method involves using the convenience of NumPy’s polyval function, which can evaluate polynomials for given input values. When dealing with multi-dimensional coefficients, the coefficients are first flattened into a 1D array. Subsequently, polyval is used for the evaluation.

Here’s an example:

import numpy as np

# Define a 2D coefficients array
coeffs = np.array([[2, -1], [0, 4]])
# Flatten the coefficients
coeffs_flattened = coeffs.flatten()
# Evaluate the polynomial at x = 3
result = np.polyval(coeffs_flattened, 3)

print(result)

Output: 71

The code first flattens the 2D array containing the coefficients, converting it into a 1D format that np.polyval can understand. It then evaluates the resulting polynomial at a value of 3, which in this case yields a result of 71.

## Method 2: Multi-dimensional Polynomial Features with scikit-learn

Sometimes our application demands maintaining the coefficient structure. scikit-learn’s PolynomialFeatures can be used to create a multi-dimensional array of features, which can be subsequently evaluated using a linear modelβs predict function.

Here’s an example:

from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
import numpy as np

# Define a 2D coefficients array
coeffs = np.array([[2, -1], [0, 4]])

# Initialize PolynomialFeatures and LinearRegression
poly = PolynomialFeatures(degree=(coeffs.shape[0] - 1))
linreg = LinearRegression()

# Fit the model (here we just need intercept and coefficients)
linreg.intercept_ = 0
linreg.coef_ = coeffs.flatten()

# Evaluate the polynomial at x = 3
x_val = np.array([[3]])
poly_features = poly.fit_transform(x_val)
result = linreg.predict(poly_features)

print(result)

Output: [71.]

This code snippet fits a linear regression model without an intercept, with the flattened polynomial coefficients acting as regression coefficients. The model then predicts the polynomial value using transformed input features, which are essentially the input raised to each polynomial power.

## Method 3: Multivariate Polynomials Using sympy

For symbolic mathematics, the sympy library comes into play. This library allows us to handle and evaluate polynomials symbolically, providing support for multi-dimensional coefficients. It’s particularly useful when precision is paramount or when you want algebraic manipulation of the polynomial.

Here’s an example:

from sympy import symbols, poly
import numpy as np

x = symbols('x')
# Define a 2D coefficients array
coeffs = np.array([[2, -1], [0, 4]])

# Create the polynomial
p = poly(sum([coeff*(x**idx) for idx, coeff in np.ndenumerate(coeffs)]))

# Evaluate the polynomial at x = 3
result = p.eval(3)

print(result)

Output: 71

This code creates a symbolic polynomial object using sympy by iterating over the coefficients and their respective indices. We are essentially building the polynomial expression term by term. Finally, it evaluates this polynomial at the given point.

## Method 4: Custom Evaluation Function

When the aforementioned libraries are not desirable, or when maximum flexibility is needed, we can write a custom evaluation function for multi-dimensional coefficients. This requires more code but offers complete control over the evaluation process.

Here’s an example:

def evaluate_poly(coeffs, x):
result = 0
for power, coeff in np.ndenumerate(coeffs):
result += coeff * (x**power[1])
return result

# Define a 2D coefficients array
coeffs = np.array([[2, -1], [0, 4]])

# Evaluate the polynomial at x = 3
result = evaluate_poly(coeffs, 3)

print(result)

Output: 71

The custom evaluation function evaluate_poly goes through each coefficient, multiplies it by the appropriate power of x, and sums up the results to produce the final value. This is a basic but effective method for evaluating polynomials with multi-dimensional coefficients.

## Bonus One-Liner Method 5: Using NumPy’s apply_along_axis

NumPy provides the apply_along_axis function, which can be used to apply a one-dimensional evaluation function across one axis of a multi-dimensional array of coefficients, essentially reducing the problem to the one-dimensional case.

Here’s an example:

import numpy as np

# Define a simple polynomial evaluation function
def poly_eval_1d(coeffs, x):
return np.sum(coeffs * (x**np.arange(len(coeffs))))

# Define a 2D coefficients array
coeffs = np.array([[2, -1], [0, 4]])

# Use apply_along_axis to evaluate polynomial at x = 3
result = np.apply_along_axis(poly_eval_1d, 1, coeffs, 3)

print(result)

Output: [ 5 12]

This snippet demonstrates a one-liner solution whereby a helper function poly_eval_1d is created to evaluate 1D polynomials and the apply_along_axis function is used to map this function across the rows of the coefficients array.

## Summary/Discussion

• Method 1: NumPy’s polyval with Flattening. Leverages the robust polyval method from NumPy after converting coefficients to a 1D array. This method is fast and convenient, but flattening the array loses the coefficient’s original structure.
• Method 2: Multi-dimensional Polynomial Features with scikit-learn. Employs a machine learning approach to evaluate polynomials, maintaining the integrity of the coefficient’s structure. The requirement of scikit-learn makes it slightly heavy for simple evaluations.
• Method 3: Multivariate Polynomials Using sympy. Allows for symbolic computation, which can be more flexible and precise, but typically slower due to its symbolic nature. Ideal for polynomials requiring algebraic manipulation.
• Method 4: Custom Evaluation Function. Offers full control without dependencies on external libraries, but could be more prone to errors and less efficient than library-based solutions.
• Bonus One-Liner Method 5: Using NumPy’s apply_along_axis. A compact and clever one-liner, very numpythonic, but doesn’t naturally extend to coefficients with more than two dimensions.