UNIT 4 • LINEAR ALGEBRA & MATRIX METHODS

LU Decomposition
Doolittle Method Guide

LU Decomposition factors any square matrix $A = LU$ where $L$ is lower triangular (with 1s on diagonal) and $U$ is upper triangular. Once factored, solving $Ax = b$ becomes two simple triangular solves — making it ideal for multiple right-hand side problems.

TWO-PHASE SOLVE STRATEGYA = LU

A = LU \implies Ax = b \implies LUx = b

Let $y = Ux$ , then solve in two steps:

\text{Phase 1: } Ly = b \quad \text{(forward substitution)}

\text{Phase 2: } Ux = y \quad \text{(back substitution)}

Doolittle vs CroutDoolittle: L has 1s on diagonal. Crout: U has 1s on diagonal. Both are equivalent.

Key AdvantageFactor A once, then solve for any number of different

b

vectors efficiently.

When to UseFEM/FEA, circuit analysis, structural mechanics — any repeated solve with fixed A.

LABORATORY

Interactive Live Calculator

MATRIX LABORATORY

LU Decomposition Engine

Matrix Size:

💡 Yellow column = constants vector b. Coefficient matrix A is on the left. Full system: Ax = b.

PreviousGauss-Jordan Elimination

Unit 4•22 of 34

NextJacobi Method

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 4 • LINEAR ALGEBRA & MATRIX METHODS

LU Decomposition
Doolittle Method Guide

TWO-PHASE SOLVE STRATEGYA = LU

A = LU \implies Ax = b \implies LUx = b

Let $y = Ux$ , then solve in two steps:

\text{Phase 1: } Ly = b \quad \text{(forward substitution)}

\text{Phase 2: } Ux = y \quad \text{(back substitution)}

Doolittle vs CroutDoolittle: L has 1s on diagonal. Crout: U has 1s on diagonal. Both are equivalent.

Key AdvantageFactor A once, then solve for any number of different

b

vectors efficiently.

When to UseFEM/FEA, circuit analysis, structural mechanics — any repeated solve with fixed A.

LABORATORY

Interactive Live Calculator

MATRIX LABORATORY

LU Decomposition Engine

Matrix Size:

💡 Yellow column = constants vector b. Coefficient matrix A is on the left. Full system: Ax = b.

PreviousGauss-Jordan Elimination

Unit 4•22 of 34

NextJacobi Method

LU Decomposition Doolittle Method Guide

Interactive Live Calculator

LU Decomposition Engine

NETZ Engine

LU Decomposition Doolittle Method Guide

Interactive Live Calculator

LU Decomposition Engine

LU Decomposition
Doolittle Method Guide

LU Decomposition
Doolittle Method Guide