| 释义 |
LU decomposition A square matrix A has an LU decomposition if A = LU, where L is a lower triangular matrix with all diagonal entries equalling 1 and U is an upper triangular matrix. The system of equations Ax = b can then be solved separately as Ly = b and then Ux = y, which is computationally more efficient. The system Ly = b efficiently yields y1,y2,…,yn in order and is known as forward substitution. The system Ux = y likewise yields xn,…,x2,x1 in that order and is known as backward substitution. Not all matrices A have LU decompositions, but there is always a permutation matrix P such that PA has an LU decomposition; P effectively just reorders the equations in the linear system.
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