Numerical Analysis

Description

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.

Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.

The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.

Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.

Problems Within Domain

• 0-1 Linear Programming
• 2-Dimensional Poisson Problem
• 3-Dimensional Poisson Problem
• All Eigenpairs
• All Eigenvalues
• All Permutations
• Any Eigenpair
• Any Eigenvalue
• Cardinality Estimation
• Convex Optimization (Non-linear)
• Coset Enumeration
• Cyclic Permutations
• Determinant of Matrices with Integer Entries
• Eigenpair closest to mu
• Eigenpair with the Largest Eigenvalue
• Exact Laplacian Solver
• Factorization of Polynomials Over Finite Fields
• Fredholm Equations
• General Linear Programming
• General Linear System
• General Permutations
• General Root Computation
• Gröbner Bases
• Hyperbolic Spline Interpolation
• Integer Factoring
• Integer Linear Programming
• Integer Relation Among Integers
• Integer Relation Among Reals
• Linear Programming with Reals
• Maximum Likelihood Parameters
• Multiplication
• Non-Definite, Symmetric Matrix
• Polynomial Interpolation
• Positive Definite, Hermitian Matrix
• Rectangular Matrix LU Decomposition
• Root Computation with continuous first derivative
• Smallest Factor
• Solutions to Nonlinear Equations
• Sparse Linear System
• Square Matrix LU Decomposition
• Toeplitz Matrix
• Vandermonde Matrix
• Variance Calculations
• Volterra Equations