The resulting \(U\) matrix from part(1) above is a banded matrix. The algorithm is illustrated in the following diagramĭerive special backward-substitution algorithm to solve the resulting upper semi-penta diagonal The advantage of this algorithm is that it will work on any central banded Matrix \(A\), tri-diagonal, Notice that no data copying is involved, and the data is processed in-place. Step, we create a new separate \(A x=b\) with its own \(A\) and \(b\) variables extracted from the original \(A\) and The \(b\) vector is updated all the time. Processing small submatrices along the way. Hence we travel down the main matrix from the top left corner to the bottom right corner, New submatrix boundaries are located as described above, and Gaussian elimination is called to This process is repeated by shifting one row down and one column to the right, and each time a Thisĭetermines the boundaries of the submatrix. The lower bound, and then looking right from that location to locate the first zero entry. Starting at first pivot in \(A(1,1)\), looking down and locating the first zero entry to determine The algorithm locates these submatrices which are bounded below and to the right by the first So as to process those by applying the standard Gaussian elimination algorithm on The main idea of the algorithm is to locate submatrices within the main matrix \(A\), This algorithm works on Matrices which contain only one band of specific width such as thoseįound in tri-diagonal and penta-diagonal matrices. The following algorithm was designed and developed to handle a general banded \(A\) matrix to solve 2 Derive special backward-substitution algorithmĭerive special Gauss-elimination strategy to transfer the resulting penta-diagonal system into an
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