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In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are ...
If A is invertible, then the factorization is unique if we require the diagonal elements of R to be positive. If instead A is a complex square matrix, then there is a decomposition A = QR where Q is a unitary matrix (so the conjugate transpose Q † = Q − 1 {\displaystyle Q^{\dagger }=Q^{-1}} ).
Notes and video on high-performance implementation of Cholesky factorization at The University of Texas at Austin. Cholesky : TBB + Threads + SSE is a book explaining the implementation of the CF with TBB, threads and SSE (in Spanish). library "Ceres Solver" by Google. LDL decomposition routines in Matlab. Armadillo is a C++ linear algebra package
An LU factorization refers to expression of A into product of two factors – a lower triangular matrix L and an upper triangular matrix U: =. Sometimes factorization is impossible without prior reordering of A to prevent division by zero or uncontrolled growth of rounding errors hence alternative expression becomes: P A Q = L U {\displaystyle ...
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
This factorization is also unique up to the choice of a sign. For example, + + + = + + + is a factorization into content and primitive part. Gauss proved that the product of two primitive polynomials is also primitive (Gauss's lemma). This implies that a primitive polynomial is irreducible over the rationals if and only if it is irreducible ...
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of .Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .