lec3 - Introduction to Simulation Lecture 3 Basics of...

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1 SMA-HPC ©2003 MIT Introduction to Simulation - Lecture 3 Thanks to Deepak Ramaswamy, Michal Rewienski, Karen Veroy and Jacob White Basics of Solving Linear Systems Jacob White
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2 Outline Solution Existence and Uniqueness Gaussian Elimination Basics LU factorization Pivoting and Growth Hard to solve problems Conditioning
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3 SMA-HPC ©2003 MIT Application Problems [] N N N ns GVI xb M = • No voltage sources or rigid struts • Symmetric and Diagonally Dominant • Matrix is n x n
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4 SMA-HPC ©2003 MIT Systems of Linear Equations 11 22 12 N NN xb MM M ⎤⎡⎤ ⎡⎤ ↑↑ ⎥⎢⎥ ⎢⎥ = ↓↓ ⎣⎦ ⎦⎣⎦ " GG G " ## " 2 2 xM x M b + ++ = G " Find a set of weights, x, so that the weighted sum of the columns of the matrix M is equal to the right hand side b
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5 SMA-HPC ©2003 MIT Systems of Linear Equations Key Questions • Given Mx = b – Is there a solution? – Is the solution Unique? • Is there a Solution? 12 N N M MM b xx x + ++ = GG G " 1 There exists weights, suc , h that , N " A solution exists when b is in the span of the columns of M
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6 SMA-HPC ©2003 MIT Systems of Linear Equations Key Questions Continued • Is the Solution Unique? 12 0 N N MM yy y M ++ + = GG G " 1 Suppose there exists weigh , , not all zero ts, N " ( ) Then if , therefore M xb M x y b =+ = A solution is unique only if the columns of M are linearly independent.
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7 SMA-HPC ©2003 MIT Systems of Linear Equations Key Questions • Given Mx = b, where M is square – If a solution exists for any b, then the solution for a specific b is unique. Square Matrices For a solution to exist for any b, the columns of M must span all N-length vectors. Since there are only N columns of the matrix M to span this space, these vectors must be linearly independent. A square matrix with linearly independent columns is said to be nonsingular .
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8 SMA-HPC ©2003 MIT Gaussian Elimination Method for Solving M x = b A “Direct” Method Finite Termination for exact result (ignoring roundoff) • Produces accurate results for a broad range of matrices • Computationally Expensive Important Properties Gaussian Elimination Basics
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