SciPy Extensions

def solve(a, b, assume_a='gen', lower=False, overwrite_a=False, overwrite_b=False, debug=None, check_finite=True):
    """
    Solve linear system ax = b for general matrices.
    
    Parameters:
    - a: array_like, coefficient matrix
    - b: array_like, dependent variable values  
    - assume_a: str, properties of matrix a
    - lower: bool, use lower triangular part only
    - overwrite_a: bool, discard data in a
    - overwrite_b: bool, discard data in b
    - check_finite: bool, check for infinite/NaN values
    
    Returns:
    cupy.ndarray: Solution to the system ax = b
    """

def solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False, overwrite_b=False, debug=None, check_finite=True):
    """Solve triangular system ax = b."""

def inv(a, overwrite_a=False, check_finite=True):
    """Compute matrix inverse."""

def pinv(a, cond=None, rcond=None, return_rank=False, check_finite=True):
    """Compute Moore-Penrose pseudoinverse."""

def det(a, overwrite_a=False, check_finite=True):
    """Compute matrix determinant."""

def lu_factor(a, permute_l=False, overwrite_a=False, check_finite=True):
    """Compute LU decomposition with partial pivoting."""

def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True):
    """Solve linear system using LU decomposition."""

def cholesky(a, lower=False, overwrite_a=False, check_finite=True):
    """Compute Cholesky decomposition."""

def cholesky_solve(cho_fac, b, overwrite_b=False, check_finite=True):
    """Solve linear system using Cholesky decomposition."""

def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False, check_finite=True, lapack_driver='gesdd'):
    """Compute singular value decomposition."""

def svdvals(a, overwrite_a=False, check_finite=True):
    """Compute singular values."""

def qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False, check_finite=True):
    """Compute QR decomposition."""

def eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False, check_finite=True, homogeneous_eigvals=False):
    """Solve general eigenvalue problem."""

def eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False, overwrite_b=False, type=1, check_finite=True):
    """Solve symmetric/Hermitian eigenvalue problem."""

def eigvals(a, b=None, overwrite_a=False, check_finite=True, homogeneous_eigvals=False):
    """Compute eigenvalues of general matrix."""

def eigvalsh(a, b=None, lower=True, overwrite_a=False, overwrite_b=False, type=1, check_finite=True):
    """Compute eigenvalues of symmetric/Hermitian matrix."""

class csr_matrix:
    """
    Compressed Sparse Row matrix.
    
    Parameters:
    - arg1: array_like/tuple, matrix data
    - shape: tuple, matrix dimensions  
    - dtype: data type
    - copy: bool, copy input data
    """
    def __init__(self, arg1, shape=None, dtype=None, copy=False): ...
    
    def dot(self, other): 
        """Matrix multiplication."""
    
    def multiply(self, other):
        """Element-wise multiplication."""
    
    def transpose(self, axes=None, copy=False):
        """Matrix transpose."""
    
    def tocsc(self, copy=False):
        """Convert to CSC format."""
    
    def tocoo(self, copy=False):
        """Convert to COO format."""
    
    def toarray(self):
        """Convert to dense array."""
    
    def todense(self):
        """Convert to dense matrix."""

class csc_matrix:
    """Compressed Sparse Column matrix."""
    def __init__(self, arg1, shape=None, dtype=None, copy=False): ...
    # Similar methods as csr_matrix

class coo_matrix:
    """Coordinate format sparse matrix."""  
    def __init__(self, arg1, shape=None, dtype=None, copy=False): ...
    # Similar methods as csr_matrix

class dia_matrix:
    """Sparse matrix with DIAgonal storage."""
    def __init__(self, arg1, shape=None, dtype=None, copy=False): ...

def identity(n, dtype='d', format=None):
    """Sparse identity matrix."""

def eye(m, n=None, k=0, dtype=float, format=None):
    """Sparse matrix with ones on diagonal."""

def diags(diagonals, offsets=0, shape=None, format=None, dtype=None):
    """Construct sparse matrix from diagonals."""

def spdiags(data, diags, m, n, format=None):
    """Return sparse matrix from diagonals."""

def random(m, n, density=0.01, format='coo', dtype=None, random_state=None):
    """Generate random sparse matrix."""

def spsolve(A, b, permc_spec=None, use_umfpack=True):
    """Solve sparse linear system Ax = b."""

def spsolve_triangular(A, b, lower=True, overwrite_A=False, overwrite_b=False):
    """Solve triangular sparse system."""

def fft(x, n=None, axis=-1, norm=None, overwrite_x=False):
    """
    Compute 1-D discrete Fourier Transform.
    
    Parameters:
    - x: array_like, input array
    - n: int, length of FFT
    - axis: int, axis along which FFT is computed
    - norm: str, normalization mode
    - overwrite_x: bool, allow overwriting input
    
    Returns:
    cupy.ndarray: FFT of input array
    """

def ifft(x, n=None, axis=-1, norm=None, overwrite_x=False):
    """Compute 1-D inverse discrete Fourier Transform."""

def fft2(x, s=None, axes=(0, 1), norm=None, overwrite_x=False):
    """Compute 2-D discrete Fourier Transform."""

def ifft2(x, s=None, axes=(0, 1), norm=None, overwrite_x=False):  
    """Compute 2-D inverse discrete Fourier Transform."""

def fftn(x, s=None, axes=None, norm=None, overwrite_x=False):
    """Compute N-D discrete Fourier Transform."""

def ifftn(x, s=None, axes=None, norm=None, overwrite_x=False):
    """Compute N-D inverse discrete Fourier Transform."""

def rfft(x, n=None, axis=-1, norm=None, overwrite_x=False):
    """Compute real 1-D discrete Fourier Transform."""

def irfft(x, n=None, axis=-1, norm=None, overwrite_x=False):
    """Compute inverse of rfft."""

def rfft2(x, s=None, axes=(0, 1), norm=None, overwrite_x=False):
    """Compute real 2-D discrete Fourier Transform."""

def irfft2(x, s=None, axes=(0, 1), norm=None, overwrite_x=False):
    """Compute inverse of rfft2."""

def rfftn(x, s=None, axes=None, norm=None, overwrite_x=False):
    """Compute real N-D discrete Fourier Transform."""

def irfftn(x, s=None, axes=None, norm=None, overwrite_x=False):
    """Compute inverse of rfftn."""

def hfft(x, n=None, axis=-1, norm=None, overwrite_x=False):
    """Compute FFT of Hermitian sequence."""

def ihfft(x, n=None, axis=-1, norm=None, overwrite_x=False):
    """Compute inverse of hfft."""

def fftfreq(n, d=1.0):
    """Return discrete Fourier Transform sample frequencies."""

def rfftfreq(n, d=1.0):
    """Return sample frequencies for rfft."""

def fftshift(x, axes=None):
    """Shift zero-frequency component to center."""

def ifftshift(x, axes=None):
    """Inverse of fftshift."""

def convolve(in1, in2, mode='full', method='auto'):
    """
    Convolve two N-dimensional arrays.
    
    Parameters:
    - in1: array_like, first input array
    - in2: array_like, second input array
    - mode: str, convolution mode ('full', 'valid', 'same')
    - method: str, computation method
    
    Returns:
    cupy.ndarray: Convolution of in1 and in2
    """

def correlate(in1, in2, mode='full', method='auto'):
    """Cross-correlate two N-dimensional arrays."""

def convolve2d(in1, in2, mode='full', boundary='fill', fillvalue=0):
    """Convolve two 2-dimensional arrays."""

def correlate2d(in1, in2, mode='full', boundary='fill', fillvalue=0):
    """Cross-correlate two 2-dimensional arrays."""

def sepfir2d(input, hrow, hcol):
    """Convolve with separable 2-D FIR filter."""

def fftconvolve(in1, in2, mode='full', axes=None):
    """Convolve using FFT."""

def oaconvolve(in1, in2, mode='full', axes=None):
    """Convolve using overlap-add method."""

def hilbert(x, N=None, axis=-1):
    """Compute analytic signal using Hilbert transform."""

def hilbert2(x, N=None):
    """Compute 2-D analytic signal using Hilbert transform."""

def decimate(x, q, n=None, ftype='iir', axis=-1, zero_phase=True):
    """Downsample signal after applying anti-aliasing filter."""

def detrend(data, axis=-1, type='linear', bp=0, overwrite_data=False):
    """Remove linear trend from data."""

def resample(x, num, t=None, axis=0, window=None, domain='time'):
    """Resample x to num samples using Fourier method."""

def resample_poly(x, up, down, axis=0, window='kaiser', padtype='constant', cval=None):
    """Resample using polyphase filtering."""

def gamma(z, out=None):
    """Gamma function."""

def gammaln(z, out=None):
    """Logarithm of gamma function."""

def digamma(z, out=None):
    """Digamma function (psi function)."""

def polygamma(n, z, out=None):
    """Polygamma function."""

def beta(a, b, out=None):
    """Beta function."""

def betaln(a, b, out=None):
    """Logarithm of beta function."""

def erf(z, out=None):
    """Error function."""

def erfc(z, out=None):
    """Complementary error function."""

def erfinv(z, out=None):
    """Inverse error function."""

def erfcinv(z, out=None):
    """Inverse complementary error function."""

def j0(z):
    """Bessel function of first kind of order 0."""

def j1(z):
    """Bessel function of first kind of order 1."""

def jv(v, z):
    """Bessel function of first kind of real order."""

def y0(z):
    """Bessel function of second kind of order 0."""

def y1(z):
    """Bessel function of second kind of order 1."""

def yv(v, z):
    """Bessel function of second kind of real order."""

def i0(z):
    """Modified Bessel function of first kind of order 0."""

def i1(z):
    """Modified Bessel function of first kind of order 1."""

def iv(v, z):
    """Modified Bessel function of first kind of real order."""

def k0(z):
    """Modified Bessel function of second kind of order 0."""

def k1(z):
    """Modified Bessel function of second kind of order 1."""

def kv(v, z):
    """Modified Bessel function of second kind of real order."""

def logsumexp(a, axis=None, b=None, keepdims=False, return_sign=False):
    """Compute log of sum of exponentials."""

def softmax(x, axis=None):
    """Compute softmax function."""

def log_softmax(x, axis=None):
    """Compute log-softmax function."""

def distance_matrix(x, y, p=2, threshold=1000000):
    """
    Compute distance matrix between arrays.
    
    Parameters:
    - x: array_like, input array of size m by k
    - y: array_like, input array of size n by k
    - p: float, p-norm for distance calculation
    - threshold: int, use dense computation if product m*n < threshold
    
    Returns:
    cupy.ndarray: Distance matrix of shape (m, n)
    """

def pdist(X, metric='euclidean', out=None):
    """Pairwise distances between observations."""

def cdist(XA, XB, metric='euclidean', out=None):
    """Distances between each pair of observations."""

def squareform(X, force='no', checks=True):
    """Convert between condensed and square distance matrices."""

import cupy as cp
import cupyx.scipy.linalg as linalg

# Solve linear system
A = cp.random.random((1000, 1000))
b = cp.random.random(1000)
x = linalg.solve(A, b)

# Matrix decompositions
L, P = linalg.lu_factor(A)
solution = linalg.lu_solve((L, P), b)

# Cholesky decomposition for positive definite matrices
symmetric_A = cp.dot(A, A.T)  # Make positive definite
chol = linalg.cholesky(symmetric_A, lower=True)

# SVD decomposition
U, s, Vh = linalg.svd(A, full_matrices=False)

# Eigenvalue problems
eigenvals, eigenvecs = linalg.eigh(symmetric_A)

import cupyx.scipy.sparse as sparse

# Create sparse matrices
data = cp.array([1, 2, 3, 4, 5])
row = cp.array([0, 0, 1, 2, 2])
col = cp.array([0, 2, 1, 0, 2])
coo = sparse.coo_matrix((data, (row, col)), shape=(3, 3))

# Convert between formats
csr = coo.tocsr()
csc = coo.tocsc()

# Sparse matrix operations
identity = sparse.identity(1000, format='csr')
random_sparse = sparse.random(5000, 5000, density=0.01, format='csr')

# Sparse-dense operations
dense_vector = cp.random.random(5000)
result = random_sparse.dot(dense_vector)

# Solve sparse linear system
b_sparse = cp.random.random(5000)
x_sparse = sparse.linalg.spsolve(random_sparse, b_sparse)

import cupyx.scipy.fft as fft

# 1D FFT
signal_1d = cp.sin(2 * cp.pi * cp.arange(1000) * 0.1)
fft_1d = fft.fft(signal_1d)
ifft_1d = fft.ifft(fft_1d)

# 2D FFT for images
image = cp.random.random((512, 512))
fft_2d = fft.fft2(image)
shifted_fft = fft.fftshift(fft_2d)

# Real FFT for real-valued signals
real_signal = cp.random.random(1000)
rfft_result = fft.rfft(real_signal)
irfft_result = fft.irfft(rfft_result)

# Frequency analysis
frequencies = fft.fftfreq(len(signal_1d))
power_spectrum = cp.abs(fft_1d) ** 2

import cupyx.scipy.signal as signal

# Convolution and correlation
sig1 = cp.random.random(1000)
sig2 = cp.random.random(100)
convolved = signal.convolve(sig1, sig2, mode='same')
correlated = signal.correlate(sig1, sig2, mode='same')

# 2D image filtering
image = cp.random.random((256, 256))
kernel = cp.array([[1, 2, 1], [2, 4, 2], [1, 2, 1]]) / 16  # Gaussian blur
filtered = signal.convolve2d(image, kernel, mode='same')

# Hilbert transform for analytic signals
analytic_signal = signal.hilbert(sig1)
amplitude = cp.abs(analytic_signal)
phase = cp.angle(analytic_signal)

# Signal resampling
upsampled = signal.resample(sig1, 2000)  # Upsample by factor of 2
decimated = signal.decimate(sig1, 2)      # Downsample by factor of 2

import cupyx.scipy.special as special

# Gamma and related functions
x = cp.linspace(0.1, 5, 1000)
gamma_vals = special.gamma(x)
gammaln_vals = special.gammaln(x)

# Error functions
z = cp.linspace(-3, 3, 1000)
erf_vals = special.erf(z)
erfc_vals = special.erfc(z)

# Bessel functions
bessel_j0 = special.j0(x)
bessel_j1 = special.j1(x)
modified_bessel_i0 = special.i0(x)

# Log-sum-exp for numerical stability
logits = cp.random.normal(0, 1, (100, 10))
logsumexp_result = special.logsumexp(logits, axis=1)
softmax_result = special.softmax(logits, axis=1)

import cupyx.scipy.spatial as spatial

# Distance matrix computation
points_a = cp.random.random((1000, 3))  # 1000 points in 3D
points_b = cp.random.random((500, 3))   # 500 points in 3D

# Pairwise distances
distances = spatial.distance_matrix(points_a, points_b)

# All pairwise distances within single set
pairwise_dist = spatial.pdist(points_a)
square_dist = spatial.squareform(pairwise_dist)

# Different distance metrics
manhattan_dist = spatial.cdist(points_a, points_b, metric='cityblock')

# Complete signal processing workflow
import cupy as cp
import cupyx.scipy as scipy

# Generate noisy signal
t = cp.linspace(0, 1, 1000)
clean_signal = cp.sin(2 * cp.pi * 10 * t) + 0.5 * cp.sin(2 * cp.pi * 20 * t)
noise = 0.1 * cp.random.normal(size=len(t))
noisy_signal = clean_signal + noise

# FFT-based filtering
fft_signal = scipy.fft.fft(noisy_signal)
frequencies = scipy.fft.fftfreq(len(t), t[1] - t[0])

# Apply low-pass filter
cutoff = 25  # Hz
fft_filtered = fft_signal.copy()
fft_filtered[cp.abs(frequencies) > cutoff] = 0

# Inverse FFT
filtered_signal = scipy.fft.ifft(fft_filtered).real

# Compute power spectral density
power_spectrum = cp.abs(fft_signal) ** 2

# Statistical analysis of results
snr_improvement = 20 * cp.log10(
    cp.std(clean_signal) / cp.std(filtered_signal - clean_signal)
)