LU Decomposition

Solve A . x = b (system of linear equations) Decompose A = L . U * L : Lower Triangular Matrix U : Upper Triangular Matrix

[L][U]=[A]  [L][U]{x}={b} To solve [A]{x}={b} [L][U]=[A]  [L][U]{x}={b} Consider [U]{x}={d} [L]{d}={b} Solve [L]{d}={b} using forward substitution to get {d} Use back substitution to solve [U]{x}={d} to get {x}

[ L ] [ U ]

[ U ] Gauss Elimination   Coefficients used during the elimination step

[ L . U ]

[ L ] [ U ] Example: A = L . U Gauss Elimination Coefficients

Using the Forward Elimination Procedure of Gauss Elimination Finding the [U] matrix Using the Forward Elimination Procedure of Gauss Elimination Step 1: http://numericalmethods.eng.usf.edu

Finding the [U] Matrix Matrix after Step 1: Step 2: http://numericalmethods.eng.usf.edu

Using the multipliers used during the Forward Elimination Procedure Finding the [L] matrix Using the multipliers used during the Forward Elimination Procedure From the first step of forward elimination http://numericalmethods.eng.usf.edu

Finding the [L] Matrix From the second step of forward elimination http://numericalmethods.eng.usf.edu

Does [L][U] = [A]? ? http://numericalmethods.eng.usf.edu

Example Set [L][Z] = [C] Solve for [Z] http://numericalmethods.eng.usf.edu

Example Complete the forward substitution to solve for [Z] http://numericalmethods.eng.usf.edu

Example Math 685/CSI 700 Spring 08 George Mason University, Department of Mathematical Sciences

LU decomposition Gauss Elimination can be used to decompose [A] into [L] and [U]. Therefore, it requires the same total FLOPs as for Gauss elimination: In the order of (proportional to) N3 where N is the # of unknowns. lij values (the factors generated during the elimination step) can be stored in the lower part of the matrix to save storage. This can be done because these are converted to zeros anyway and unnecessary for the future operations. Saves computing time by separating time-consuming elimination step from the manipulations of the right hand side. Provides efficient means to compute the matrix inverse