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The direct sum of modules A and B is the module A direct sum B={a direct sum b|a in A,b in B}, (1) where all algebraic operations are defined componentwise. In particular, ...

Consider a knot as being formed from two tangles. The following three operations are called mutations. 1. Cut the knot open along four points on each of the four strings ...

The Sherman-Morrison formula is a formula that allows a perturbed matrix to be computed for a change to a given matrix A. If the change can be written in the form u tensor v ...

The square root method is an algorithm which solves the matrix equation Au=g (1) for u, with A a p×p symmetric matrix and g a given vector. Convert A to a triangular matrix ...

Any square matrix A can be written as a sum A=A_S+A_A, (1) where A_S=1/2(A+A^(T)) (2) is a symmetric matrix known as the symmetric part of A and A_A=1/2(A-A^(T)) (3) is an ...

Symmetry operations include the improper rotation, inversion operation, mirror plane, and rotation. Together, these operations create 32 crystal classes corresponding to the ...

Gaussian elimination is a method for solving matrix equations of the form Ax=b. (1) To perform Gaussian elimination starting with the system of equations [a_(11) a_(12) ... ...

The usual number of scalar operations (i.e., the total number of additions and multiplications) required to perform n×n matrix multiplication is M(n)=2n^3-n^2 (1) (i.e., n^3 ...

Simply stated, floating-point arithmetic is arithmetic performed on floating-point representations by any number of automated devices. Traditionally, this definition is ...

Let a tree S be a subgraph of a cubic graph G. The graph excision G circleminus S is the graph resulting from removing the tree, then merging the edges. For example, if in ...