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Polynomial factoring algorithms use basic polynomial operations such as products, divisions, gcd, powers of one polynomial modulo another, etc. A multiplication of two polynomials of degree at most n can be done in O(n 2) operations in F q using "classical" arithmetic, or in O(nlog(n) log(log(n)) ) operations in F q using "fast" arithmetic.
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. In R, the chol function gives the Cholesky decomposition.
If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several). For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field).
Macaulay's notation is commonly used in the static analysis of bending moments of a beam. This is useful because shear forces applied on a member render the shear and moment diagram discontinuous.
A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors. A 2 ...
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.
For a typical sparse matrix, the LU factors can be much less sparse than the original matrix — a phenomenon called fill-in. The memory requirements for using a direct solver can then become a bottleneck in solving linear systems.
The indicator function of A is the Iverson bracket of the property of belonging to A; that is, 1 A ( x ) = [ x ∈ A ] . {\displaystyle \mathbf {1} _{A}(x)=[x\in A].} For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers .