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A ‘quasimatrix’ is, like a matrix, a rectangular scheme whose elements are indexed, but one discrete index is replaced by a continuous index. Likewise, a ‘cmatrix’, is continuous in both indices. As an example of a cmatrix, one can think of the kernel of an integral operator.
For example, we can conveniently require the lower triangular matrix L to be a unit triangular matrix, so that all the entries of its main diagonal are set to one. Then the system of equations has the following solution:
Decomposition method is a generic term for solutions of various problems and design of algorithms in which the basic idea is to decompose the problem into subproblems. The term may specifically refer to:
A decomposition paradigm in computer programming is a strategy for organizing a program as a number of parts, and usually implies a specific way to organize a program text. Typically the aim of using a decomposition paradigm is to optimize some metric related to program complexity, for example a program's modularity or its maintainability.
If one of the eigenvalues λ i has multiple linearly independent eigenvectors (that is, the geometric multiplicity of λ i is greater than 1), then these eigenvectors for this eigenvalue λ i can be chosen to be mutually orthogonal; however, if two eigenvectors belong to two different eigenvalues, it may be impossible for them to be orthogonal ...
For example, one can add N numbers either by a simple loop that adds each datum to a single variable, or by a D&C algorithm called pairwise summation that breaks the data set into two halves, recursively computes the sum of each half, and then adds the two sums. While the second method performs the same number of additions as the first and pays ...
A well-known example implicitly using additive state decomposition is the superposition principle, widely used in physics and engineering. The superposition principle states: For all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.
That is, in Benders decomposition, the variables of the original problem are divided into two subsets so that a first-stage master problem is solved over the first set of variables, and the values for the second set of variables are determined in a second-stage subproblem for a given first-stage solution.