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Lecture20110120_1

Lecture20110120_1 - Djoko Wirosoetisno CM322(Thurs 10-11:30...

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Djoko Wirosoetisno CM322 (Thurs 10-11:30) Homeworks by Friday Noon CM322 Outline Term 1. 1) Root-finding 2) Polynomial Interpolation 3) Numerical Differentiation Term 2. 4) Linear Algebra 5) l 2 approximations 6) Numerical Integration Linear Systems Problem: Solve, over = or , the system: a 11 x 1 C a 12 x 2 C ... C a 1 n x n = b 1 ... a n1 x 1 C ... C a nn x n = b n 5 Ax = b Issues i) Efficiency: minimise work for large n . (think n ~ 10 6 ) ii) Accuracy: Minimise round-off errors when using floating-point arithmetic. Linear Algebra Review A is an n # n matrix. Definition Transpose A u : a u ij = a ji A symmetric: A u = A A skew symmetric A u = K A Definition A non singular c b d x that solves Ax = b Facts A non-singular 0 d ! x that solves Ax = b A non-singular 5 d ! A K 1 st c b , x = A K 1 b A non-singular 5 det A 0 Definition A positive definite Ax $ x O 0 c x A positive semidefinite 5 Ax $ x 0 c x Fact A positive definite 0 A non-singular Gaussian Elimination Try to solve Ax = b given A , b Start with A of special forms. Definition
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L = l ij is lower triangular if l ij = 0 for j O i U = u ij is upper triangular if u ij = 0 for j ! i To solve
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