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Unformatted text preview: Numerical Computing chapter 2 Direct methods for Linear Systems Dr. Nguyen V.M. Man Email: [email protected] Faculty of Computer Science and Engineering HCMUT Vietnam February 23, 2010 The structure of the chapter ——————————————————————————— Gauss Elimination Its Variations Cholesky decomposition of symmetric and positive definite matrices Matrix Norms ———————————– Interlude Most actual dependencies are nonlinear Linearity assumptions are widely spread, since usually the nonlinearity of the analyzed phenomenon does not have great impact on the results How to achieve solutions of linear system? ——————————————————————————— Design stage : a precise problem specification is formulated in the form A . x = b Realization stage : find numerical solution by hand or using packages (e.g., Matlab, Lapack) Verification stage : review the solution, maybe you have reformulate the model Gauss Elimination  Contents ——————————————————————————— I 2A: Three cases of A . x = b I 2B: Direct and indirect methods I 2C: Inconsistent systems and Consistent systems I 2D: LUfactorization: a Gauss Elimination’s Variation Gauss Elimination  Its Variations The object : a linear system of equations, written A . x = b in which A ∈ R m × n ; b ∈ R m , vector of unknowns x = ( x 1 , x 2 , ··· , x n ). This brief form stands for the below system b i = n X j =1 a ij x j , i = 1 .. m where the coefficients a i , j , b i are known real numbers, and the coefficient matrix A m × n = a 1 , 1 a 1 , 2 a 1 , 3 . . . a 1 , n 1 a 1 , n a 2 , 1 a 2 , 2 a 2 , 3 . . . a 2 , n 1 a 2 , n a 3 , 1 a 3 , 2 a 3 , 3 . . . a 3 , n 1 a 3 , n . . . . . . . . . . . . . . . . . . a m , 1 a m , 2 a m , 3 . . . a m , n 1 a m , n . (0.1) The aim: discuss numerical methods for finding unknowns x = ( x 1 , x 2 , ··· , x n ) Cases ——————————————————————————— We will deal with the three cases: ———————————– (i) The case where m = n : the number of equations and the number of unknowns are equal so that the coefcient matrix A m × n is square. (ii) The case where m < n so that we might have to find the minimum norm solution among the numerous solutions. (iii) The case where m > n (the number of equations is greater than the number of unknowns) so that there might exist no exact solution and we must find a solution based on global error minimization, like the LSE (Leastsquares error) solution....
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 Spring '08
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 Computer Science, Linear Systems, Data Structures, Upper Triangular Matrix

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