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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.
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
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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