Patent Document ID: 20030212723
Application ID: 10140788
Patent Flag: 0

Claim One:
1. A computer method of solving a system of linear equations of the general form &lsqb;A&rsqb;&lsqb;X&rsqb;&equals;&lsqb;B&rsqb;, by means of a digital computer, comprising the steps of: (a) representing the linear equations system in said digital computer in the form of an extended matrix &lsqb;A&verbar;B&rsqb;, formed by N×N&plus;1 elements a ij, which are indexed as an arrangement of N rows and N&plus;1 columns; (b) selecting a first row of elements of said matrix having a first element different from zero; (c) normalizing the values of the elements of said first row by dividing all elements of said first row by the value of the first non-zero element; (d) setting the initial value of a first index “e”, associated with the sequential order of selection of the rows, to 2; (e) selecting a second row of said matrix; (f) calculating the values of the elements of a transformation vector &lsqb;K&rsqb; according to the formula: 19 K n = { &it; 1 &leq; n < e ; - a en - &Sum; i = 1 n - 1 &it; ( a in * K i ) &it; n = e ; 1 } where “n” is a second index associated with the sequential order of the n-th element of the vector &lsqb;K&rsqb; being calculated and “e” is said first index associated with the sequential index of the row being transformed; (g) transforming the values of each of the elements of the selected second row by performing the dot product of said vector &lsqb;K&rsqb; and a vector formed by those “e” elements of each column of the extended matrix &lsqb;A&verbar;B&rsqb; corresponding to the rows that have been transformed prior to the selection of said second row; (h) determining whether element a ee of matrix &lsqb;A&verbar;B&rsqb; is zero; (i) if element a ee is zero, then determining whether index “e” is equal to the number of columns “n”; If a ee is not zero, then modifying the value of said index “e” so that another row of said extended matrix is selected; (j) selecting a third row of said extended matrix and repeating steps (f) to (h) until a row is found with a nonzero element a ee ; (k) normalizing row “e” by dividing all elements of row “e” by the value of said element a ee ; (l) determining from the value of said first index “e” whether all rows of said matrix have been transformed; if all rows have not been transformed, then modifying the value of said index “e” so that another row of said extended matrix is selected; (m) repeating steps (e) to (j) until all rows of said matrix have been transformed; thereby obtaining an upper diagonal matrix &lsqb;U&rsqb; with all elements u ij located in the main diagonal having a value of 1, and all elements below the main diagonal having a value of zero; and (n) sequentially back-substituting the value of the nth element of vector &lsqb;x&rsqb; according to the following formula: 20 x i = ( b i - &Sum; j = i + 1 3 &it; u ij &it; x j ) u ii for i&equals;n−1, n−2,... 1; whereby the calculated vector &lsqb;x&rsqb; is the solution vector of the original system of linear equations.