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How much computational time does it take to find the inverse of a square matrix using Gauss Jordan method? Part 1 of 2.
To understand the solution, you should be familiar with the Gauss Jordan method of finding the inverse of a square matrix. Peter Young of UCSC describes it briefly in this pdf file while if you like watching an example via a video, you can see PatrickJMT doing so. You also need to read a previous blog where we calculated the computational time needed for the forward elimination steps on a square matrix in the Naïve Gauss elimination method. We are now ready to estimate the computational time required for Gauss Jordan method of finding the inverse of a square matrix.
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So why use and waste time talking about LU Decomposition?
Because, LU Decomposition is computationally more efficient than Gaussian elimination when we are solving several sets of equations with the same coefficient matrix but different right hand sides. Case in point is when you are finding the inverse of a matrix [A]. If one is trying to find the inverse of nxn matrix, then it implies that one needs to solve n sets of simultaneous linear equations of [A][X]=[C] form with the n right hand sides [C] being the n columns of the nxn identity matrix, while the coefficient matrix [A] stays the same.
The computational time taken for solving a single set of n simultaneous linear equations is as follows:
So for LU decomposition method used to find the inverse of a matrix, the computational time is proportional to . Remember that the forward elimination only needs to be done only once on [A] to generate the L and U matrices for the LU decomposition method. However the forward and back substitution need to be done n times.
Now for Gaussian Elimination used to find the inverse of a matrix, the computational time is proportional to . Remember that both the forward elimination and back substitution need to be done n times.
Hence for large n, for LU Decomposition, the computational time is proportional to , while for Gaussian Elimination, the computational time is proportional to . So for large n, the ratio of the computational time for Gaussian elimination to computational for LU Decomposition is .
As an example, to find the inverse of a 2000×2000 coefficient matrix by Gaussian Elimination would take n/4=2000/4=500 times the time it would take to find the inverse by LU Decomposition.
So are you convinced now why we use LU Decomposition in certain cases? For textbook notes on this issue, examples of LU Decomposition to solve a set of equations, and finding inverse of a matrix using LU Decomposition, click here.
Reference: Numerical Methods for the STEM Undergraduate, http://numericalmethods.eng.usf.edu/topics/lu_decomposition.html
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