Gaussian Elimination: Solving Systems of Linear Equations

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Gaussian elimination is indeed used in computer software to solve systems of linear equations, often with pivoting for improved numerical stability. However, it is not the only method available, and alternative techniques may be more effective in specific scenarios. For instance, methods like conjugate gradients are advantageous for sparse matrices, while matrix decomposition approaches are beneficial for solving Ax = b with multiple b values. LU decomposition, which is derived from Gaussian elimination, is one of several matrix decomposition methods utilized in these solutions. Other methods, such as Cholesky, LDLT, QR, and SVD, also play significant roles in solving linear equations.
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Is it correct that the Gaussian elimination procedure is used in computer software to solve systems of linear equations?
 
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In short: yes (with pivoting for numerical stability), but it's far from the only way, and in certain cases other methods may be better (for example, some method based on conjugate gradients is good when the matrix is sparse, and matrix decomposition is useful for when you want to solve Ax = b for several different b).
 
There are a variety of matrix decompositions that are used in solving sets of linear equations. LU decomposition is based on Gauss elimination:

http://en.wikipedia.org/wiki/LU_decomposition

but other methods (e.g. Cholesky, LDLT, QR, SVD, etc.) may not.
 

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