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LU Decomposition

- Major All Engineering Majors
- Authors Autar Kaw
- http//numericalmethods.eng.usf.edu
- Transforming Numerical Methods Education for STEM

Undergraduates

LU Decomposition http//numericalmethods.e

ng.usf.edu

LU Decomposition

LU Decomposition is another method to solve a set

of simultaneous linear equations Which is

better, Gauss Elimination or LU

Decomposition? To answer this, a closer look at

LU decomposition is needed.

LU Decomposition

Method For most non-singular matrix A that one

could conduct Naïve Gauss Elimination forward

elimination steps, one can always write it as A

LU where L lower triangular

matrix U upper triangular matrix

How does LU Decomposition work?

If solving a set of linear equations If A

LU then Multiply by Which gives Remember

L-1L I which leads to Now, if IU

U then Now, let Which ends with and AX

C LUX C L-1 L-1LUX

L-1C IUX L-1C UX

L-1C L-1CZ LZ C (1) UX

Z (2)

LU Decomposition

How can this be used?

- Given AX C
- Decompose A into L and U
- Solve LZ C for Z
- Solve UX Z for X

When is LU Decomposition better than Gaussian

Elimination?

- To solve AX B
- Table. Time taken by methods
- where T clock cycle time and n size of the

matrix - So both methods are equally efficient.

To find inverse of A

Time taken by Gaussian Elimination Time taken by

LU Decomposition

Table 1 Comparing computational times of finding

inverse of a matrix using LU decomposition and

Gaussian elimination.

Method A Decompose to L and U

U is the same as the coefficient matrix at the

end of the forward elimination step. L is

obtained using the multipliers that were used in

the forward elimination process

Finding the U matrix

Using the Forward Elimination Procedure of Gauss

Elimination

Step 1

Finding the U Matrix

Matrix after Step 1

Step 2

Finding the L matrix

Using the multipliers used during the Forward

Elimination Procedure

From the first step of forward elimination

Finding the L Matrix

From the second step of forward elimination

Does LU A?

?

Using LU Decomposition to solve SLEs

Solve the following set of linear equations using

LU Decomposition

Using the procedure for finding the L and U

matrices

Example

Set LZ C Solve for Z

Example

Complete the forward substitution to solve for Z

Example

Set UX Z Solve for X The 3

equations become

Example

Substituting in a3 and using the second equation

From the 3rd equation

Example

Substituting in a3 and a2 using the first equation

Hence the Solution Vector is

Finding the inverse of a square matrix

The inverse B of a square matrix A is defined

as AB I BA

Finding the inverse of a square matrix

How can LU Decomposition be used to find the

inverse? Assume the first column of B to be

b11 b12 bn1T Using this and the definition

of matrix multiplication First column of

B Second column of B

The remaining columns in B can be found in the

same manner

Example Inverse of a Matrix

Find the inverse of a square matrix A

Using the decomposition procedure, the L and

U matrices are found to be

Example Inverse of a Matrix

- Solving for the each column of B requires two

steps - Solve L Z C for Z
- Solve U X Z for X

Step 1

This generates the equations

Example Inverse of a Matrix

Solving for Z

Example Inverse of a Matrix

Solving UX Z for X

Example Inverse of a Matrix

Using Backward Substitution

So the first column of the inverse of A is

Example Inverse of a Matrix

Repeating for the second and third columns of the

inverse Second Column Third Column

Example Inverse of a Matrix

The inverse of A is

To check your work do the following

operation AA-1 I A-1A

Additional Resources

- For all resources on this topic such as digital

audiovisual lectures, primers, textbook chapters,

multiple-choice tests, worksheets in MATLAB,

MATHEMATICA, MathCad and MAPLE, blogs, related

physical problems, please visit - http//numericalmethods.eng.usf.edu/topics/lu_deco

mposition.html

- THE END
- http//numericalmethods.eng.usf.edu

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