# Gaussian elimination

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## Gaussian elimination

[¦gau̇·sē·ən ə‚lim·ə′nā·shən]
(mathematics)
A method of solving a system of n linear equations in n unknowns, in which there are first n- 1 steps, the m th step of which consists of subtracting a multiple of the m th equation from each of the following ones so as to eliminate one variable, resulting in a triangular set of equations which can be solved by back substitution, computing the n th variable from the n th equation, the (n- 1)st variable from the (n- 1)st equation, and so forth.
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Gaussian elimination and its applications are covered next, followed by eigenvalues, eigenvectors, and determinants.
Hart does offer much to think about, however--among other things, that Gaussian elimination and general solutions of systems of linear equations are ultimately of non-literate origin and attested in China some seventeen centuries before Gauss.
includes chapters on Gaussian elimination and its variants, sensitivity of linear systems, the least squares problem, the singular value decomposition, eigenvalues and eigenvectors, and iterative methods for linear systems.
14 for one tensor at a time, attempting to add it to an existing energy vector set via Gaussian elimination to ensure that the rows in the state space matrix at any time are always linearly independent.
The Gaussian elimination method comprises of two steps, namely the forward elimination phase and the backward substitution phase.
For this purpose an algorithm is developed based on triangular finite element and Gaussian elimination method, incorporated into a computer application with some user friendly features in problem definition and visualisation of results.
Appendices offer primers on computer arithmetic, Newton's method, Gaussian elimination, B-splines, and iterative matrix methods.
The authors, an international trio of computer scientists and researchers in applied mathematics, provide case studies for each new concept as they introduce semiseparable and related matrices, focusing on definitions and properties, representation, and historical applications; linear systems with semiseparable and related matrices, including Gaussian elimination, the QR factorization, a Levinson-like and Schur-like solver, and inverting semiseparable and related matrices, including known factorizations, direct inversion methods, general formulas for inversions, scaling of symmetric positive definite semiseparable matrices and decay rates for the inverses of tridiagonal matrices.
In subsequent work, Alan George, Esmond Ng and Joseph Liu described data structures for sparse Gaussian elimination with partial pivoting and orthogonal factorization with Householder transformations.
Last three of the four mentioned methods are numerically too expensive whether because parallelization is too fine grained thus yielding too much communication or because parallel algorithms are much more computationaly expensive than simple serial Gaussian elimination and therefore impractical.
k], we perform a step of Gaussian elimination to generate the kth column of L, the kth row of U, and the update matrix [U.
Written by a professor at MIT, this textbook describes methods for solving linear equations using Gaussian elimination, vector spaces, orthogonality, and determinants, then addresses the challenge of finding eigenvalues and eigenvectors.

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