![]() The LU solve algorithm for solving the linear system written as code is: These two linear systems can then be solved one after the other using forward and back substitution. We have thus replaced with two linear systems: and. Where we first evaluate using forward substitution and then evaluate using back substitution.Īn equivalent way to write this is to introduce a new vector defined by. Knowing the LU decomposition for a matrix allows us to solve the linear system using a combination of forward and back substitution. These equations have no solution, so does not have an LU decomposition. The top-left and bottom-left entries give the two equations: If we try and find the LU decomposition of this matrix then we getĮquating the individual entries gives us four equations to solve. We use the terms decomposition and factorization interchangeably to mean writing a matrix as a product of two or more other matrices, generally with some defined properties (such as lower/upper triangular).Įxample: matrix for which LU decomposition failsĪn example of a matrix which has no LU decomposition is.This choice is somewhat arbitrary (we could have decided that must have 1 on the diagonal) but it is the standard choice. The reason that has all diagonal entries set to 1 is that this means the LU decomposition is unique.The LU decomposition provides an efficient means of solving linear equations.If the LU decomposition exists then it is unique.The LU decomposition may not exist for a matrix.The properties of the LU decomposition are: is a lower-triangular matrix with all diagonal entries equal to 1.The LU decomposition of a matrix is the pair of matrices and such that: zeros ( n ) for i in range ( n - 1, - 1, - 1 ): tmp = b for j in range ( i + 1, n ): tmp -= U * x x = tmp / U return x LU decomposition Import numpy as np def back_sub ( U, b ): """x = back_sub(U, b) is the solution to U x = bī must be a vector of the same leading dimension as U The code for the back substitution algorithm to solve is: The number of operations for the back substitution algorithm is as.The properties of the back substitution algorithm are: The back substitution solution works from the bottom up to give: The upper-triangular system can be written as the set of linear equations: It is the backwards version of forward substitution. The back substitution algorithm solves the linear system where is an upper-triangular matrix. zeros ( n ) for i in range ( n ): tmp = b for j in range ( i - 1 ): tmp -= L * x x = tmp / L return x Back substitution algorithm Import numpy as np def forward_sub ( L, b ): """x = forward_sub(L, b) is the solution to L x = bī must be a vector of the same leading dimension as L ![]() The code for the forward substitution algorithm to solve is:
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