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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 fillIn ) 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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This note was uploaded on 02/21/2012 for the course ENG 3000 taught by Professor Staff during the Spring '12 term at University of Houston.
 Spring '12
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