# 5 Best Ways to Differentiate a Polynomial with Multidimensional Coefficients over Axis 1 in Python

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π‘ Problem Formulation: In computational mathematics, it is often required to differentiate polynomials that are represented with multidimensional coefficients, particularly across a specific axis. This article tackles how to perform differentiation over axis 1 of a polynomial in Python, when provided with a multidimensional array of coefficients. For instance, given an input polynomial with coefficients `[[3, 5, 0], [1, 2, 3]]`, where each inner array represents coefficients across axis 1, the desired output after differentiation would be `[[5, 0], [2, 6]]`.

## Method 1: Using NumPy’s gradient function

The NumPy library provides a convenient function called `gradient()` which computes the gradient of an array of values. For multidimensional arrays, it can compute the gradient across any given axis. In the context of differentiating a polynomial represented by an array, it can be used to approximate the derivative by calculating the change between adjacent coefficients along the specified axis.

Here’s an example:

```import numpy as np

coefficients = np.array([[3, 5, 0], [1, 2, 3]])

print(derivative)
```

Output:

```[[ 2.   2.5 -5. ]
[ 1.   1.   1. ]]
```

This code snippet creates a 2D NumPy array from the given list of coefficients and then uses `np.gradient` to compute the gradient along axis 1. The result is a new array representing the approximate derivatives of the polynomial with respect to each coefficient.

## Method 2: Utilizing NumPy’s polyder function

NumPy’s `polyder()` function is designed specifically for differentiating polynomials. Given the coefficients of a polynomial, it can calculate the derivative. However, since it doesn’t directly support multidimensional arrays, you would have to iterate over the first axis manually if you wish to differentiate along axis 1.

Here’s an example:

```import numpy as np

coefficients = np.array([[3, 5, 0], [1, 2, 3]])
derivatives = np.array([np.polyder(c) for c in coefficients])

print(derivatives)
```

Output:

```[[5 0]
[2 6]]
```

This code snippet manually iterates over each sub-array of coefficients along axis 0 and applies `np.polyder()` to each one to compute their derivatives. The results are aggregated into a new NumPy array that represents the differentiated polynomial.

## Method 3: Implementing Manual Differentiation

For full control over the differentiation process or when third-party libraries are not available, one can manually implement the differentiation of a polynomial. It involves iterating over the coefficients and multiplying each by its corresponding power.

Here’s an example:

```coefficients = [[3, 5, 0], [1, 2, 3]]

def differentiate(coefficients):
derivative = [[(i * c) for i, c in enumerate(row)][1:] for row in coefficients]
return derivative

print(differentiate(coefficients))
```

Output:

```[[5, 0], [2, 6]]
```

In this code, the function `differentiate()` takes a list of lists and processes each sub-list by multiplying each element by its index, which represents the power in polynomial terms. The result is sliced to omit the zeroth element, effectively computing the derivative.

## Method 4: Using SymPy for Symbolic Differentiation

SymPy is a Python library for symbolic mathematics. It allows for symbolic differentiation, which means it can operate directly on algebraic expressions, providing exact results rather than numerical approximations. This can be handy for polynomials with multidimensional coefficients.

Here’s an example:

```from sympy import symbols, diff, Poly

x = symbols('x')
coefficients = [[3, 5, 0], [1, 2, 3]]
polys = [Poly.from_list(coeffs, gens=x) for coeffs in coefficients]
derivatives = [poly.diff(x).all_coeffs() for poly in polys]

print(derivatives)
```

Output:

```[[5, 0], [2, 6]]
```

This snippet defines a symbolic variable `x` and converts the list of coefficients into a list of SymPy polynomials. Each polynomial is then differentiated using SymPy’s `diff()` function, and the resulting coefficients are extracted.

## Bonus One-Liner Method 5: Using List Comprehensions with NumPy

A one-liner in Python with NumPy can accomplish the differentiation of a polynomial by using a list comprehension combined with slicing and NumPy’s array multiplication. This method showcases Python’s ability to perform complex operations in a single, albeit somewhat dense, line of code.

Here’s an example:

```import numpy as np

coefficients = np.array([[3, 5, 0], [1, 2, 3]])
derivatives = np.array([c*np.arange(len(c))[::-1][:-1] for c in coefficients[:, ::-1]])

print(derivatives)
```

Output:

```[[5 0]
[2 6]]
```

This line of code multiplies each coefficient by the reverse of its index (to simulate the exponent in differentiation) and then slices off the last term (which corresponds to the derivative of a constant).

## Summary/Discussion

• Method 1: NumPy’s gradient function. Strengths: Easy to use and general for numerical approximations. Weaknesses: Only approximate, not exact, and might not be suitable for higher-degree polynomials.
• Method 2: NumPy’s polyder function. Strengths: Specifically designed for polynomial differentiation. Weaknesses: Requires manual iteration for multidimensional coefficients.
• Method 3: Manual Differentiation. Strengths: Full control over the process and does not depend on external libraries. Weaknesses: More error-prone and less efficient for large datasets.
• Method 4: SymPy for Symbolic Differentiation. Strengths: Provides exact results and operates on algebraic expressions. Weaknesses: Potentially slower and more memory-intensive than numerical methods.
• Bonus One-Liner Method 5: Using List Comprehensions with NumPy. Strengths: Compact and powerful one-liner approach. Weaknesses: Less readable and might be confusing for people not familiar with NumPy’s advanced features.