LU Decomposition

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LU Decomposition

Major: All Engineering Majors

Authors: Autar Kaw

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LU Decomposition

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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]

3204

38| 2

3 nnnTCT DE

nnTCT FS 44| 2 nnTCT BS 124| 2

3328

38| 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

3412

38 2

3 nnnT

3412

38 2

3 nnnT


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

Decomposition

3412

38

||2

34 nnnT

CTCTn BSFE

32012

332

|||

23 nnnT

CTnCTnCT BSFSDE


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

Decomposition

32012

332 2

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.

3412

38 2

34 nnnT

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


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

33

2322

131211

3231

21

000

101001

uuuuuu

ULA

[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

2564

RowRow

76.48.16056.18.40

152576.513;76.5

25144

RowRow

Step 1:


Finding the [U] Matrix

Step 2:

76.48.16056.18.40

1525

7.00056.18.40

15255.323;5.3

8.48.16

RowRow

7.00056.18.40

1525U

Matrix after Step 1:


Finding the [L] matrix

Using the multipliers used during the Forward Elimination Procedure

101001

3231

21

56.22564

11

2121

aa

76.525

144

11

3131

aa

From the first step of forward elimination

11214418641525


Finding the [L] Matrix

15.376.50156.2001

L

From the second step of forward elimination

76.48.16056.18.40

15255.3

8.48.16

22

3232

aa


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

7.00056.18.40

1525

15.376.50156.2001

UL ?


Using LU Decomposition to solve SLEs

Solve the following set of linear equations using LU Decomposition

227921778106

11214418641525

3

2

1

.

.

.

xxx

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

7.00056.18.40

1525

15.376.50156.2001

ULA


ExampleSet [L][Z] = [C]

Solve for [Z]

2.2792.1778.106

15.376.50156.2001

3

2

1

zzz

2.2795.376.52.17756.2

10

321

21

1

zzzzz

z


ExampleComplete the forward substitution to solve for [Z]

735.0

21.965.38.10676.52.2795.376.52.279

2.968.10656.22.177

56.22.1778.106

213

12

1

zzz

zzz

735.021.968.106

3

2

1

zzz

Z


ExampleSet [U][X] = [Z]

Solve for [X] The 3 equations become

735021968106

7.00056.18.40

1525

3

2

1

...

xxx

735.07.021.9656.18.48.106525

3

32

321

aaaaaa


ExampleFrom the 3rd equation

050170

7350735070

3

3

3

.a.

.a

.a.

Substituting in a3 and using the second equation

219656184 32 .a.a.

701984

0501561219684

5612196

2

2

32

.a.

...a

.a..a


ExampleSubstituting in a3 and a2 using the first equation

8106525 321 .aaa

Hence the Solution Vector is:

050.170.19

2900.0

3

2

1

aaa

2900025

05017019581062558106 32

1

.

...

aa.a


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]

0

01

1

21

11

nb

bb

A

0

10

2

22

12

nb

bb

A

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

A

7000561840

1525

15376501562001

.

.. ..

.ULA

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:

001

15.376.50156.2001

3

2

1

zzz

CZL

This generates the equations:

05.376.5056.21

321

21

1

zzzzzz


Example: Inverse of a MatrixSolving for [Z]

23

5625317650537650

56215620

56201

213

12

1

....

z.z.z.

.z. z

z

23562

1

3

2

1

..

zzz

Z


Example: Inverse of a Matrix

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

3.22.561

7.00056.18.40

1525

31

21

11

bbb

2.37.056.256.18.4

1525

31

3121

312111

bbbbbb


Example: Inverse of a MatrixUsing Backward Substitution

04762.025

571.49524.05125

51

9524.08.4

571.4560.156.28.4560.156.2

571.47.02.3

312111

3121

31

bbb

bb

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

571.49524.0

04762.0

31

21

11

bbb


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

Second Column Third Column

010

11214418641525

32

22

12

bbb

000.5417.108333.0

32

22

12

bbb

100

11214418641525

33

23

13

bbb

429.14643.0

03571.0

33

23

13

bbb


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

http://numericalmethods.eng.usf.edu/topics/gaussian_elimination.html

http://numericalmethods.eng.usf.edu/topics/lu_decomposition.html

http://numericalmethods.eng.usf.edu/topics/lu_decomposition.html

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LU Decomposition

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