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Linear Systems
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
Gaussian elimination is referred to as a direct method for solving a linear system, meaning a method that terminates in finitely many iterations. These are very different from our methods of Ch. 1; in infinite precision arithmetic, Gaussian elimination would always give us the right answer in finitely many operations (like the quadratic formula does for quadratic equations), whereas Newton's method would always give an approximation (even in infinite precision arithmetic). There are also iterative methods for linear systems that, like Newton's method, do not reach their answer in finitely many steps, and that are useful for certain types of large systems (as we will discuss in Ch. 3). We'll continue to focus on direct methods for now.
Linear Algebra
Published in Brian Vick, Applied Engineering Mathematics, 2020
The Gaussian elimination algorithm consists of two basic steps: (1) eliminate the elements below the diagonal and (2) back substitute to get the solution. The technique will be demonstrated for the 3-by-3 matrix [a11a12a13a21a22a23a31a32a33][x1x2x3]=[b1b2b3]
Linear Algebra
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
Gaussian elimination is a strategy for solving a system of linear equations. To find all solutions to the linear system of equations () a11x1+a12x2+⋯+a1nxn=b1⋮am1x1+am2x2+⋯+amnxn=bm
Linear Algebra on Parallel Structures Using Wiedemann Algorithm to Solve Discrete Logarithm Problem
Published in IETE Journal of Research, 2022
K S Spoorthi, R. Padmavathy, S K Pal, S Ravi Chandra
Gaussian Elimination is the oldest available algorithm to solve the system of linear equations. Gaussian elimination reduces the matrices to row echelon form, then solves the system of linear equations. For a matrix M of size , the complexity of the method can be given by or . The complexity is too high for large matrices. It does not utilize the sparsity of the matrix. Hence FFS does not use Gaussian Elimination.
Mathematically rigorous global optimization in floating-point arithmetic
Published in Optimization Methods and Software, 2018
However, such a method is most certainly bound to fail due to data dependencies. For Gaussian elimination it can be shown [68, Subsection 10.1] that, for a general matrix, this method of replacing operations by their corresponding interval operation (IGA) must fail for small dimensions, even for orthogonal matrices.