LU decomposition technique

This technique (also known as the Grout reduction or Cholesky factorization) is based on the transfonnation of the matrix of coefficients in a system of algebraic equations into the product of lower and upper triangular matrices as 

Therefore a = /nxl = hi, fl2i = hi, etc. (elements in the first column of a a,re the same as the elements in the first column of /) similarly multiplying rows of / by columns of u and equating the result with the corresponding element of a all of the elements of lower and upper triangular matrices are found. The general formula for obtaining elements of / and u can be expressed as 

After obtaining the described decomposition the set of equations can be readily solved. This is because all of the information required for transfonnation of the coefficient matrix to an upper triangular fonn is essentially recorded in the lower triangle. Therefore modification of the right-hand side is quite straightforward and can be achieved using the lower triangular matrix as 

Hence the solution is found by back substitution based on 

In some applications the diagonal elements of the upper triangular matrix are not predetermined to be unity. The formula used for the LU decomposition procedure in these applications is slightly different from those given in Equations (6.10) to (6.12), (Press et al., 1987). 

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The right-hand side in Equation (6.18) is known and hence its solution yields the error 5x in the original solution. The procedure can be iterated to improve the solution step-by-step. Note that implementation of this algorithm in the context of finite element computations may be very expensive. A significant advantage of the LU decomposition technique now becomes clear, because using this technique [A] can be decomposed only once and stored. Therefore in the solution of Equation (6.18) only the right-hand side needs to be calculated. 

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