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Homework1

Homework1 - Department of Mathematics University of Houston...

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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 ) , 0 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. (i) Show that in case of Gauss elimination without pivoting, the first step already generates nonzero entries at all places where the original entry was zero (”fill-in”).

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