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The formula is valid for all index values, and for any n (when n = 0 or n = 1, this is the empty product). However, computing the formula above naively has a time complexity of O( n 2 ) , whereas the sign can be computed from the parity of the permutation from its disjoint cycles in only O( n log( n )) cost.
Moore–Penrose inverse. In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. [1] It was independently described by E. H. Moore in 1920, [2] Arne Bjerhammar in 1951, [3] and Roger Penrose in 1955. [4]
Einstein notation. In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity.
Not to be confused with the Dirac delta function, nor with the Kronecker symbol. In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson brackets: For example, because , whereas ...
numpy.org. NumPy (pronounced / ˈnʌmpaɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3] The predecessor of NumPy, Numeric, was originally created by Jim Hugunin with ...
Antisymmetric tensor. In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. [1][2] The index subset must generally either be all covariant or all contravariant. For example, holds when the tensor is ...
Tensor contraction. In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor (s) caused by applying the summation convention to a pair of dummy indices that are bound to ...
Specify the function to be minimized, , the interval to be searched as {X 1,X 4}, and their functional values F 1 and F 4. Calculate an interior point and its functional value F 2. The two interval lengths are in the ratio c : r or r : c where r = φ − 1; and c = 1 − r, with φ being the golden ratio.