Concepts inUpdating an LU Factorization with Pivoting
Pivot element
The pivot or pivot element is the element of a matrix, an array, or some other kind of finite set, which is selected first by an algorithm, to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called pivoting.
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LU decomposition
In linear algebra, LU decomposition (also called LU factorization) factorizes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. LU decomposition is a key step in several fundamental numerical algorithms in linear algebra such as solving a system of linear equations, inverting a matrix, or computing the determinant of a matrix. It can be viewed as the matrix form of Gaussian elimination.
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Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is derived from the accuracy of the algorithm. An opposite phenomenon is instability.
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Factorization
In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.
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Matrix (mathematics)
In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is Matrices of the same size can be added or subtracted element by element. The rule for matrix multiplication is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.
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Linear algebra
Linear algebra is the branch of mathematics concerning finite or countably infinite dimensional vector spaces, as well as linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations in several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics.
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