Polynomial Functions

def poly(seq_of_zeros):
    """
    Find the coefficients of a polynomial with the given sequence of roots.
    
    Parameters:
    - seq_of_zeros: cupy.ndarray, sequence of polynomial roots
    
    Returns:
    cupy.ndarray: polynomial coefficients in descending powers
    """

def polyval(p, x):
    """
    Evaluate a polynomial at specific values.
    
    Parameters:
    - p: cupy.ndarray, polynomial coefficients in descending powers
    - x: cupy.ndarray, values at which to evaluate the polynomial
    
    Returns:
    cupy.ndarray: polynomial values at x
    """

def roots(p):
    """
    Return the roots of a polynomial with coefficients given in p.
    
    Parameters:
    - p: cupy.ndarray, polynomial coefficients in descending powers
    
    Returns:
    cupy.ndarray: roots of the polynomial
    """

def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False):
    """
    Least squares polynomial fit to data.
    
    Parameters:
    - x: cupy.ndarray, x-coordinates of data points  
    - y: cupy.ndarray, y-coordinates of data points
    - deg: int, degree of fitting polynomial
    - rcond: float, relative condition number for rank determination
    - full: bool, return additional information about the fit
    - w: cupy.ndarray, weights to apply to y-coordinates
    - cov: bool, return covariance matrix
    
    Returns:
    cupy.ndarray or tuple: polynomial coefficients, optionally with fit info
    """

def polyadd(a1, a2):
    """
    Find the sum of two polynomials.
    
    Parameters:
    - a1: cupy.ndarray, coefficients of first polynomial
    - a2: cupy.ndarray, coefficients of second polynomial
    
    Returns:
    cupy.ndarray: coefficients of sum polynomial
    """

def polysub(a1, a2):
    """
    Difference (subtraction) of two polynomials.
    
    Parameters:
    - a1: cupy.ndarray, coefficients of first polynomial
    - a2: cupy.ndarray, coefficients of second polynomial
    
    Returns:
    cupy.ndarray: coefficients of difference polynomial
    """

def polymul(a1, a2):
    """
    Find the product of two polynomials.
    
    Parameters:
    - a1: cupy.ndarray, coefficients of first polynomial
    - a2: cupy.ndarray, coefficients of second polynomial
    
    Returns:
    cupy.ndarray: coefficients of product polynomial
    """

def polydiv(u, v):
    """
    Returns the quotient and remainder of polynomial division.
    
    Parameters:
    - u: cupy.ndarray, dividend polynomial coefficients
    - v: cupy.ndarray, divisor polynomial coefficients
    
    Returns:
    tuple: (quotient coefficients, remainder coefficients)
    """

def polyder(p, m=1):
    """
    Return the derivative of the specified order of a polynomial.
    
    Parameters:
    - p: cupy.ndarray, polynomial coefficients
    - m: int, order of differentiation
    
    Returns:
    cupy.ndarray: coefficients of derivative polynomial
    """

def polyint(p, m=1, k=None):
    """
    Return an antiderivative (indefinite integral) of a polynomial.
    
    Parameters:
    - p: cupy.ndarray, polynomial coefficients  
    - m: int, order of integration
    - k: cupy.ndarray, integration constants
    
    Returns:
    cupy.ndarray: coefficients of antiderivative polynomial
    """

class poly1d:
    """
    A one-dimensional polynomial class for GPU arrays.
    
    Parameters:
    - c_or_r: cupy.ndarray, polynomial coefficients or roots
    - r: bool, if True, c_or_r specifies roots instead of coefficients  
    - variable: str, variable name for string representation
    """
    
    def __init__(self, c_or_r, r=False, variable=None): ...
    
    def __call__(self, val):
        """Evaluate polynomial at given values."""
    
    def __add__(self, other):
        """Add two polynomials."""
    
    def __sub__(self, other):
        """Subtract two polynomials."""
    
    def __mul__(self, other):
        """Multiply two polynomials."""
    
    def __div__(self, other):
        """Divide two polynomials."""
    
    def __pow__(self, val):
        """Raise polynomial to a power."""
    
    def deriv(self, m=1):
        """Return derivative of polynomial."""
    
    def integ(self, m=1, k=0):
        """Return antiderivative of polynomial."""
    
    @property
    def coeffs(self):
        """Polynomial coefficients."""
    
    @property  
    def order(self):
        """Order/degree of polynomial."""
    
    @property
    def roots(self):
        """Roots of polynomial."""

# Available through cupy.polynomial submodule
class Polynomial:
    """Modern polynomial class with enhanced functionality."""

class Chebyshev:
    """Chebyshev polynomial class."""

class Legendre:
    """Legendre polynomial class."""

class Laguerre:  
    """Laguerre polynomial class."""

class Hermite:
    """Hermite polynomial class."""

class HermiteE:
    """Hermite polynomial class (physicist's version)."""

import cupy as cp

# Basic polynomial evaluation
coeffs = cp.array([1, -2, 1])  # x^2 - 2x + 1
x = cp.linspace(-2, 2, 100)
y = cp.polyval(coeffs, x)

# Polynomial fitting
x_data = cp.linspace(0, 10, 50)
y_data = 2 * x_data**2 + 3 * x_data + 1 + cp.random.normal(0, 0.5, 50)
fit_coeffs = cp.polyfit(x_data, y_data, deg=2)

# Find roots
p = cp.array([1, -5, 6])  # x^2 - 5x + 6
polynomial_roots = cp.roots(p)  # Should be [2, 3]

# Polynomial arithmetic
p1 = cp.array([1, 2, 3])  # x^2 + 2x + 3
p2 = cp.array([1, 1])     # x + 1
sum_poly = cp.polyadd(p1, p2)
product_poly = cp.polymul(p1, p2)

# Using poly1d class
p = cp.poly1d([1, -2, 1])  # x^2 - 2x + 1
print(p(2))  # Evaluate at x=2
derivative = p.deriv()  # Get derivative
integral = p.integ()    # Get antiderivative

# Create polynomial from roots
roots_array = cp.array([1, 2, 3])
poly_from_roots = cp.poly(roots_array)

# Polynomial calculus
original = cp.array([1, 2, 3, 4])  # x^3 + 2x^2 + 3x + 4
derivative = cp.polyder(original, m=1)  # First derivative
second_derivative = cp.polyder(original, m=2)  # Second derivative
antiderivative = cp.polyint(derivative)  # Should recover original (up to constant)