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import numpy as np from scipy import linalg def sor_solver (A, b, omega, initial_guess, convergence_criteria): """ This is an implementation of the pseudo-code provided in the Wikipedia article.
In Python, the function cholesky from the numpy.linalg module performs Cholesky decomposition. In Matlab, the chol function gives the Cholesky decomposition. Note that chol uses the upper triangular factor of the input matrix by default, i.e. it computes = where is upper triangular. A flag can be passed to use the lower triangular factor instead.
NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
High-performance multi-threaded primitives for large sparse matrices. Support operations for iterative solvers: multiplication, triangular solve, scaling, matrix I/O, matrix rendering. Many variants: e.g.: symmetric, hermitian, complex, quadruple precision. oneMKL: Intel C, C++, Fortran 2003 2023.1 / 03.2023 Non-free Intel Simplified Software ...
#!/usr/bin/env python3 import numpy as np def power_iteration (A, num_iterations: int): # Ideally choose a random vector # To decrease the chance that our vector # Is orthogonal to the eigenvector b_k = np. random. rand (A. shape [1]) for _ in range (num_iterations): # calculate the matrix-by-vector product Ab b_k1 = np. dot (A, b_k) # calculate the norm b_k1_norm = np. linalg. norm (b_k1 ...
Similarly, in Python, the SciPy package has scipy.sparse.linalg.eigsh which is also a wrapper for the SSEUPD and DSEUPD functions functions from ARPACK which use the Implicitly Restarted Lanczos Method. A Matlab implementation of the Lanczos algorithm (note precision issues) is available as a part of the Gaussian Belief Propagation Matlab Package.
The Python package NumPy provides a pseudoinverse calculation through its functions matrix.I and linalg.pinv; its pinv uses the SVD-based algorithm. SciPy adds a function scipy.linalg.pinv that uses a least-squares solver. The MASS package for R provides a calculation of the Moore–Penrose inverse through the ginv function. [24]
To solve a linear system Ax = b with a preconditioner K = K 1 K 2 ≈ A, preconditioned BiCGSTAB starts with an initial guess x 0 and proceeds as follows: r 0 = b − Ax 0 Choose an arbitrary vector r̂ 0 such that ( r̂ 0 , r 0 ) ≠ 0 , e.g., r̂ 0 = r 0