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lecture30 - Solving Linear Systems IE170 Algorithms in...

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IE170: Algorithms in Systems Engineering: Lecture 30 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University April 18, 2007 Jeff Linderoth (Lehigh University) IE170:Lecture 30 Lecture Notes 1 / 18 Solving Linear Systems Last time we learned about how to solve systems Ax = b , when A was symmetric and positive-definite. The key was to factor the matrix into two triangular matrices A = LU In the case that A is spd , then we can always do this, and in fact U = L T . What if A is not spd ? The workhorse in this case is the LU-decomposition LU-decomposition is very related to (the well-known) Gaussian elimination, a fact we will try to make clear today... Jeff Linderoth (Lehigh University) IE170:Lecture 30 Lecture Notes 2 / 18 Gaussian Elimination An example for today. Let’s solve it... x 1 + x 2 + 2 x 3 = 3 2 x 1 + 3 x 2 + x 3 = 2 3 x 1 - x 2 - x 3 = 6 Subtract twice first equation from the second Subtract 3 times the first equation from the third Then add 4 times second equation to the third You’ve made a triangular system! What were the matrices that produce this? Jeff Linderoth (Lehigh University) IE170:Lecture 30 Lecture Notes 3 / 18 Elementary Dear Watson! We reduced the columns by taking linear combinations of the rows of the matrix. This implies that the reduction process can be thought of as a multiplication of A on the left by some matrix What does the matrix look like?
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