LU Decomposition

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04/22/23http://

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LU Decomposition

Major: All Engineering Majors

Authors: Autar Kaw

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates

LU Decomposition

http://numericalmethods.eng.usf.edu

LU DecompositionLU 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.

MethodFor most non-singular matrix [A] that one could conduct Naïve Gauss Elimination forward elimination steps, one can always write it as

[A] = [L][U]where

[L] = lower triangular matrix

[U] = upper triangular matrix

LU Decomposition

How does LU Decomposition work?

LU DecompositionHow can this be used?

Given [A][X] = [C]

1. Decompose [A] into [L] and [U]

2. Solve [L][Z] = [C] for [Z]

3. Solve [U][X] = [Z] for [X]

Is LU Decomposition better than Gaussian Elimination?

Solve [A][X] = [B]

3 nnnTCT DE

nnTCT FS 44| 2 nnTCT BS 124| 2

3 nnnTCT FE

Forward Elimination Decomposition to LU

Back Substitution

Forward Substitution

nnTCT BS 124| 2

Back Substitution

T = clock cycle time and nxn = size of the matrix

Is LU Decomposition better than Gaussian Elimination?

To solve [A][X] = [B]Time taken by methods

T = clock cycle time and nxn = size of the matrix

So both methods are equally efficient.

Gaussian Elimination LU Decomposition

3 nnnT

To find inverse of [A]Time taken by Gaussian Elimination Time taken by LU

Decomposition

34 nnnT

CTCTn BSFE

23 nnnT

CTnCTnCT BSFSDE

To find inverse of [A]Time taken by Gaussian Elimination Time taken by LU

Decomposition

3 nnnT

n 10 100 1000 10000CT|inverse GE / CT|inverse LU 3.288 25.84 250.8 2501

Table 1 Comparing computational times of finding inverse of a matrix using LU decomposition and Gaussian elimination.

34 nnnT

For large n, CT|inverse GE / CT|inverse LU ≈ n/4

Method: [A] Decomposes to [L] and [U]

131211

101001

uuuuuu

[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] matrixUsing the Forward Elimination Procedure of Gauss Elimination

11214418641525

11214456.18.40

152556.212;56.2

RowRow

76.48.16056.18.40

152576.513;76.5

RowRow

Step 1:

Finding the [U] Matrix

Step 2:

76.48.16056.18.40

7.00056.18.40

15255.323;5.3

8.48.16

RowRow

7.00056.18.40

Matrix after Step 1:

Finding the [L] matrix

Using the multipliers used during the Forward Elimination Procedure

101001

56.22564

76.525

From the first step of forward elimination

11214418641525

Finding the [L] Matrix

15.376.50156.2001

From the second step of forward elimination

76.48.16056.18.40

15255.3

8.48.16

Does [L][U] = [A]?

7.00056.18.40

15.376.50156.2001

Using LU Decomposition to solve SLEs

Solve the following set of linear equations using LU Decomposition

227921778106

11214418641525

Using the procedure for finding the [L] and [U] matrices

7.00056.18.40

15.376.50156.2001

ExampleSet [L][Z] = [C]

Solve for [Z]

2.2792.1778.106

15.376.50156.2001

2.2795.376.52.17756.2

ExampleComplete the forward substitution to solve for [Z]

21.965.38.10676.52.2795.376.52.279

2.968.10656.22.177

56.22.1778.106

735.021.968.106

ExampleSet [U][X] = [Z]

Solve for [X] The 3 equations become

735021968106

7.00056.18.40

735.07.021.9656.18.48.106525

aaaaaa

ExampleFrom the 3rd equation

050170

7350735070

Substituting in a3 and using the second equation

219656184 32 .a.a.

701984

0501561219684

5612196

ExampleSubstituting in a3 and a2 using the first equation

8106525 321 .aaa

Hence the Solution Vector is:

050.170.19

2900.0

2900025

05017019581062558106 32

Finding the inverse of a square matrix

The inverse [B] of a square matrix [A] is defined as

[A][B] = [I] = [B][A]

Finding the inverse of a square matrixHow can LU Decomposition be used to find the inverse?Assume the first column of [B] to be [b11 b12 … bn1]T

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 MatrixFind the inverse of a square matrix [A]

11214418641525

7000561840

15376501562001

Using the decomposition procedure, the [L] and [U] matrices are found to be

Example: Inverse of a MatrixSolving for the each column of [B] requires two steps

1)Solve [L] [Z] = [C] for [Z]

2)Solve [U] [X] = [Z] for [X]

Step 1:

15.376.50156.2001

This generates the equations:

05.376.5056.21

zzzzzz

Example: Inverse of a MatrixSolving for [Z]

5625317650537650

56215620

z.z.z.

Example: Inverse of a Matrix

Solving [U][X] = [Z] for [X]

3.22.561

7.00056.18.40

2.37.056.256.18.4

312111

bbbbbb

Example: Inverse of a MatrixUsing Backward Substitution

04762.025

571.49524.05125

9524.08.4

571.4560.156.28.4560.156.2

571.47.02.3

312111

b So the first column of the inverse of [A] is:

571.49524.0

04762.0

Example: Inverse of a MatrixRepeating for the second and third columns of the inverse

Second Column Third Column

11214418641525

000.5417.108333.0

11214418641525

429.14643.0

03571.0

Example: Inverse of a MatrixThe inverse of [A] is

429.1000.5571.44643.0417.19524.0

03571.008333.004762.01A

To check your work do the following operation

[A][A]-1 = [I] = [A]-1[A]

Additional ResourcesFor 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_decomposition.html

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LU Decomposition

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