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9 - 9 LU Decomposition 9 Chapter Objectives understand an...

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9: LU Decomposition 9: LU Decomposition
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Oct 30, 2006 13:22 ECOR2606 -- Hassan & Holtz 2 Chapter Objectives understand an alternative way to solve [A]{x} = {b} . understand that LU decomposition involves factoring a matrix into the product of 2 triangular matrices. know that this can be a more efficient way to solve systems with multiple right-hand sides. know how to use Gauss Elimination to perform LU decomposition. given an LU decomposition, know how to efficiently solve using multiple right-hand side vectors. recognize that Cholesky's method is similar but even better for certain kinds of [A] matrices. understand in general terms how MATLAB's backslash operator works.
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Oct 30, 2006 13:22 ECOR2606 -- Hassan & Holtz 3 Motivation Many important engineering calculations involve solving [A]{x} = {b} . Sometimes these systems are very large (100's or 1000's of equations). Often there are multiple right-hand side vectors, {b} , and a solution vector, {x} , is required for each one. One example is Structural Analysis (determining internal forces and stresses within building or vehicle frames). [A] is a coefficient matrix that depends on geometry, material, connectivity and boundary conditions. {x} is a set of displacements and/or strains to be determined. {p} is several sets of applied forces. Each set are forces due to a specific effect: snow wind earthquake temperature/shrinkage live loads (use and occupancy) dead loads (self weight) FYI
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Oct 30, 2006 13:22 ECOR2606 -- Hassan & Holtz 4 Motivation (2) the analyst must look at each effect individually, but must also investigate combinations. For example: dead + live dead + wind dead + live + wind dead + wind + earthquake etc. thus, an analysis must solve for many right-hand sides. LU decomposition is a technique to make this more efficient: factoring is hard (expensive) backward (and forward) substitution is easy (cheap) so, do the hard thing once, and the easy things many times (once for each right hand side). FYI
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Oct 30, 2006 13:22 ECOR2606 -- Hassan & Holtz 5 9.1 Overview of LU Decomposition The equation we wish to solve: can be expressed: Suppose that Eq. (9.2) could be expressed as a upper triangular system: This is the state after f.e. of Gauss Elimination. {d} is the transformed right-hand side (the last column of the augmented matrix).
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