Homework1 - Department of Mathematics University of Houston...

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Unformatted text preview: Department of Mathematics University of Houston Numerical Analysis I Dr. Ronald H.W. Hoppe Numerical Mathematics I (1. Homework Assignment) Exercise 1 ( Gauss elimination for diagonally dominant matrices ) A matrix A R n n is called diagonally dominant, if | a ii | X j 6 = i | a ij | , 1 i n . Let A ( k ) , k n- 1, with A (0) := A be the intermediate matrices that arise during Gauss elimination. Show by an induction argument that the matrices A ( k ) , 1 k n- 1, are also diagonally dominant and satisfy n X j = k +1 | a ( k ) ij | n X j =1 | a ij | . Show further that this implies | a ( k ) ij | 2 max i,j | a ij | . Exercise 2 ( Avoiding fill-In ) Let A R 5 5 be a regular matrix of the following form A = . Here, the symbol denotes a nonzero entry, whereas all other matrix entries are supposed to be zero....
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Homework1 - Department of Mathematics University of Houston...

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