lecture29 - IE170 Algorithms in Systems Engineering Lecture...

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Unformatted text preview: IE170: Algorithms in Systems Engineering: Lecture 29 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University April 16, 2007 Jeff Linderoth (Lehigh University) IE170:Lecture 29 Lecture Notes 1 / 23 Taking Stock Last Time Matrix Review This Time Solving Triangular Systems Solving Symmetric Positive Definite Systems Least Squares Jeff Linderoth (Lehigh University) IE170:Lecture 29 Lecture Notes 2 / 23 Systems of Equations: Ax = b From our previous discussion, we know that the system of equations Ax = b has a unique solution if and only if the matrix A is square and invertible This is true if the columns A j are linearly independent From now on, we will consider only invertible systems. In fact, today we will consider special versions of A The $64 Question How do we solve a systems of equations? We factor the matrix A into a simpler form Jeff Linderoth (Lehigh University) IE170:Lecture 29 Lecture Notes 3 / 23 Triangular Systems Let’s suppose that we are able to find two n × n matrices L , U such that A = LU where L is upper triangular. U is lower triangular with 1’s on the diagonal. How could use such a decomposition to solve the system Ax = b ? Jeff Linderoth (Lehigh University) IE170:Lecture 29 Lecture Notes 4 / 23 Using a Triangular Decomposition Once we have an triangular decomposition, we can use it to easily solve the system Ax = b . Note that the system Ax = b is equivalent to the original system, which is then equivalent to LUx = b . We can solve the system in two steps: First solve the system Ly = b (forward substitution)....
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lecture29 - IE170 Algorithms in Systems Engineering Lecture...

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