Homework3 - 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 Analysis I (3rd Homework Assignment) Exercise 9 ( Convergence of the Jacobi iteration) ) A matrix A R n × n is called strongly diagonally dominant, if | a ii | > X j 6 = i | a ij | , 1 i n . Prove the convergence of the Jacobi iteration applied to the linear algebraic system Ax = b , b R n in case of strongly diagonally dominant matrices. Hint: Use Gershgorin’s theorem: For any matrix A C n × n ,n N there holds σ ( A ) n [ i =1 { λ C | | λ - a ii | ≤ X j 6 = i | a ij |} . Exercise 10 ( Scaling invariance of the Gauss Seidel iteration ) Scaling means the multiplication of quantities of interest by constant scalars. In physical and technical applications, this typically means the transformation of units. A numerical method for a given problem is said to be scaling invariant, if its application to the scaled problem gives rise to the same results as in the unscaled
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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.

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Homework3 - Department of Mathematics University of Houston...

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