There are several common algorithms used in numerical computations, depending on the specific problem and requirements. Here are a few examples:

  1. Newton-Raphson: This iterative method is used to find roots of a function. It starts with an initial guess and iteratively improves the estimate until a desired level of accuracy is achieved.

  2. Gaussian Elimination: This algorithm is used to solve systems of linear equations. It involves transforming the system into row-echelon form and then back-substituting to find the solution.

  3. QR Decomposition: This technique decomposes a matrix into the product of an orthogonal matrix and an upper triangular matrix. It is commonly used for solving least squares problems and eigenvalue calculations.

  4. Fast Fourier Transform (FFT): This algorithm efficiently computes the discrete Fourier transform of a sequence or signal. It has numerous applications in signal processing, image processing, and numerical simulations.

  5. Monte Carlo Methods: These algorithms use random sampling to numerically approximate solutions to problems involving probability and statistics. They are widely used in simulations and optimization problems.

  6. Runge-Kutta Methods: These numerical integration techniques approximate solutions to ordinary differential equations (ODEs). The most common variant is the fourth-order Runge-Kutta method, which provides accurate results with reasonable computational cost.

  7. Singular Value Decomposition (SVD): This algorithm decomposes a matrix into the product of three matrices to reveal its fundamental properties. It is used in data analysis, image compression, and collaborative filtering.

  8. Krylov Subspace Methods: These iterative algorithms solve linear systems and eigenvalue problems by iteratively constructing a basis for a subspace. Popular examples include the Conjugate Gradient and Arnoldi methods.

  9. Brent's Method: This root-finding algorithm combines bisection, secant, and inverse quadratic interpolation to efficiently find the roots of a function. It is a robust alternative to Newton-Raphson for functions with complex behavior.

  10. Clustering Algorithms: These algorithms group similar items together based on their characteristics. Examples include k-means, hierarchical clustering, and density-based clustering, which have applications in data mining and pattern recognition.

  These are just a few examples of the many algorithms used in numerical computations. The choice of algorithm depends on the specific problem and its requirements for accuracy, computational cost, and scalability.