The Gaussian exception is basically the same as LU factorization. The xGETRF routine computes LU factorization (for example, DGETRF for real matrices with double precision). The U-factor corresponds to the matrix after Gaussian exclusion. The U-factor is stored in the upper triangular part (including the diagonal) of the matrix A at the output.
LU factorization / Gaussian exception is usually used to solve linear systems of equations. You can use the xGETRS procedure to solve a linear system after calculating the LU factorization.
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