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LU Decomposition 1 A L U =

LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

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Page 1: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

LU Decomposition

1

A L U=

Page 2: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Algebra?

2

Suppose we want to solve several linear systems with the same coefficient matrix but different right hand sides:

Ax = bi, i = 1, 2, . . . ,M

Can we reuse the work performed from solving one of these systems with Gaussian elimination to reduce the computational cost for solving the other systems?

Yes! By using the LU decomposition of A

Solving systems with many right hand sides

Page 3: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Gaussian elimination using matrices

Recall : Multiplication on the left by a row vector results in a row vector.

Multiplication on the left can be thought of as a linear combination of rows.

Result is a row vector

⇥a b c

⇤ 2

42 4 51 9 �3

�2 1 7

3

5 =

a⇥2 4 5

⇤+ b

⇥1 9 �3

⇤+ c

⇥�2 1 7

Page 4: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrices

2x+ 4y � 2z = 2

4x+ 9y � 3z = 8

�2x� 3y + 7z = 10

2x+ 4y � 2z = 2

y + z = 4

4z = 8

2

42 4 �24 9 �3

�2 �3 7

3

5

2

42 4 �20 1 10 0 4

3

5

elimination

A U

2

4 M

3

5

Can we find a matrix M such that MA = U?

=

Page 5: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrices : First step

How do we express ”(equation 2)�(2)(equation 1)”

2x+ 4y � 2z = 2

4x+ 9y � 3z = 8

�2x� 3y + 7z = 10

elimination2x+ 4y � 2z = 2

y + z = 4

y + 5z = 12

2

41 0 0? ? ?? ? ?

3

5

2

42 4 �24 9 �3

�2 �3 7

3

5 =

2

42 4 �20 1 10 1 5

3

5eliminate

How do we express ”(equation 3)�(�1)(equation 1)”

Page 6: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrices : First step

How do we express ”(equation 2)�(2)(equation 1)”

How do we express ”(equation 3)�(�1)(equation 1)”

Page 7: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrices : First step

How do we express ”(equation 2)�(2)(equation 1)”

How do we express ”(equation 3)�(�1)(equation 1)”

Page 8: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrices : First step

How can we express both operations at once?

Note that

2

41 0 0

�2 1 01 0 1

3

5

2

42 4 �24 9 �3

�2 �3 7

3

5 =

2

42 4 �20 1 10 1 5

3

5

So, we can express the elimination as:

M (1)

M (1) M (1)A

Page 9: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrix : Second step

2x+ 4y � 2z = 2

y + z = 4

y + 5z = 12

2x+ 4y � 2z = 2

y + z = 4

4z = 8

elimination

How do we express ”(equation 3)�(1)(equation 2)”

eliminate

2

41 0 00 1 0? ? ?

3

5

2

42 4 �20 1 10 1 5

3

5 =

2

42 4 �20 1 10 0 4

3

5

Page 10: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrix : Second step

How do we express ”(equation 3)�(1)(equation 2)”

2

41 0 00 1 00 �1 1

3

5

2

42 4 �20 1 10 1 5

3

5 =

2

42 4 �20 1 10 0 4

3

5

U

We found a sequence of matrices that will take A to U

M (1)AM (2)

Page 11: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrices

2

41 0 00 1 00 �1 1

3

5

2

41 0 0

�2 1 01 0 1

3

5

2

42 4 �24 9 �3

�2 �3 7

3

5 =

2

42 4 �20 1 10 0 4

3

5

We have constructed elimination matrices

A U

Note that the product of elimination matrices is still lower triangular and has a simple form:

M (2) M (1)

M (1)M (2) M (2)M (1)

Page 12: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Elimination matrices

We have constructed elimination matrices

So, we can express the whole elimination process as

A U

2

41 0 00 1 00 �1 1

3

5

2

41 0 0

�2 1 01 0 1

3

5

2

42 4 �24 9 �3

�2 �3 7

3

5 =

2

42 4 �20 1 10 0 4

3

5

A UM (2) M (1)

M (2)M (1)

Page 13: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

General elimination matrices

This idea can be generalized to an n-by-n matrix A*:

*Assuming A is non-singular and pivoting is not necessary.

M (n�1)M (n�2) · · ·M (2)M (1)A = Uwhere

and

M (n�1)M (n�2) · · ·M (2)M (1) =

2

666666664

1�`21 1�`31 �`32 1...

.... . .

. . ....

.... . .

. . . 1�`n1 �`n2 · · · · · · �`n,n�1 1

3

777777775

M (1) =

2

666666664

11

�`31 1...

. . ....

. . .�`n1 1

3

777777775

M (2) =

2

666666664

11

�`32 1...

. . ....

. . .�`n2 1

3

777777775

M (n�1) =

2

66666664

11

1. . .

1`n,n�1 1

3

77777775-

Page 14: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

General elimination matrices

All the elimination matrices are invertible so:

One can easily verify that

and that

A = (M (1))�1(M (2))�1 · · · (M (n�2))�1(M (n�1))�1U

(M (1))�1 =

2

666666664

11

`31 1...

. . ....

. . .`n1 1

3

777777775

(M (2))�1 =

2

666666664

11`32 1...

. . ....

. . .`n2 1

3

777777775

etc.

(M (1))�1(M (2))�1 · · · (M (n�2))�1(M (n�1))�1 =

2

666666664

1`21 1`31 `32 1...

.... . .

. . ....

.... . .

. . . 1`n1 `n2 · · · · · · `n,n�1 1

3

777777775

= L

Page 15: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

General elimination matrices

All the elimination matrices are invertible so:

So we have found the factorization:

A = (M (1))�1(M (2))�1 · · · (M (n�2))�1(M (n�1))�1U

A L U=

Page 16: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Recall Gaussian Elimination2

45 �1 2 7

�2 6 9 0�7 5 �3 5

3

5Apply elementary row operations to the augmented matrix

2

64

5 �1 2 7

0 285

495

145

0 185 � 1

5745

3

75 (eqn 2)�✓�25

◆(eqn 1)

(eqn 3)�✓�75

◆(eqn 1)

2

64

5 �1 2 7

0 285

495

145

0 0 � 6510

655

3

75 (eqn 3)�

✓9

14

◆(eqn 2)

Solution : x1 = 3, x2 = 4, x3 = �2 16

Page 17: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Recall Gaussian Elimination2

45 �1 2 7

�2 6 9 0�7 5 �3 5

3

5Apply elementary row operations to the augmented matrix

2

64

5 �1 2 7

0 285

495

145

0 185 � 1

5745

3

75 (eqn 2)�✓�25

◆(eqn 1)

(eqn 3)�✓�75

◆(eqn 1)

2

64

5 �1 2 7

0 285

495

145

0 0 � 6510

655

3

75 (eqn 3)�

✓9

14

◆(eqn 2)

Solution : x1 = 3, x2 = 4, x3 = �2

PivotsMultipliers

17

Page 18: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

LU Decomposition

18

U =

2

64

5 �1 2

0 285

495

0 0 � 6510

3

75

Entries of L are simply given by the multipliers:

L =

2

64

1 0 0

� 25 1 0

� 75

914 1

3

75

Resulting upper triangular coefficient matrix U:

Page 19: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

LU Decomposition

19

The product LU is equal to A :

L U A

It does not cost us anything to store the multipliers. But by doing so, we can now solve many systems involving the matrix A.

2

64

1 0 0

� 25 1 0

� 75

914 1

3

75

2

64

5 �1 2

0 285

495

0 0 � 6510

3

75 =

2

64

5 �1 2

�2 6 9

�7 5 �3

3

75

Page 20: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Solution procedure given LU=A

20

How can we solve a system using the LU factorization?

Ax = b

Step 1 : Solve Ly = b Forward substitution

Step 2 : Solve Ux = y Back substitution

Step 0 : Factor A into LU Row-reduction

Page 21: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Partial pivoting and LU decomposition

21

Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how the rows get swapped with permutation matrix (or vector).

A permutation matrix has one 1 per row and column:

P =

2

40 0 11 0 00 1 0

3

5 .

Any n-by-n permutation matrix can be represented as a vector of length n:

P =

2

40 0 11 0 00 1 0

3

5 ()

2

4312

3

5 = p

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Page 22: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Partial pivoting and LU decomposition

22

Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how the rows get swapped with permutation matrix (or vector).

• Multiplying a matrix A on the left by P permutes the rows of A:

2

40 0 11 0 00 1 0

3

5

2

41 �2 2

�1 �1 33 1 �2

3

5 =

2

43 1 �21 �2 2

�1 �1 3

3

5

• Multiplying a matrix A on the right by P permutes the columns of A:

2

41 �2 2

�1 �1 33 1 �2

3

5

2

40 0 11 0 00 1 0

3

5 =

2

4�2 2 1�1 3 �1�1 �2 3

3

5

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Page 23: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

LU decomposition with partial pivoting

23

p1

L(1)

2

664

3 �2 �3 �24 0 �8 63 2 �10 �9�6 �2 9 �3

3

775

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2

664

1234

3

775

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A(0)=A p1 permuted A(0)Pivot: apply permutation on rows and columns

Eliminate: GE on first column of permuted matrix A(0)

A(1)

Inverse of the elimination matrix, or first column of the lower triangular matrix

2

664

�6 �2 9 �34 0 �8 63 2 �10 �93 �2 �3 �2

3

775

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2

664

4231

3

775

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2

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Page 24: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

24

p1 A(1) p1

LU decomposition with partial pivotingPivot: apply permutation on rows and columns

Eliminate: GE on second column of permuted matrix A(1)

L(2) A(2)

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Need to also permute L(1) before inserting elements for L(2)

permuted A(1)

Page 25: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

25

p1 A(2) p1

LU decomposition with partial pivotingPivot: apply permutation on rows and columns

Eliminate: GE on third column of permuted matrix A(2)

L(3)=L A(3)=U

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664

4132

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775

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0 0 �5 � 353

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7775

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0 0 �5 � 353

0 0 � 83

509

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permuted A(2)

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LU decomposition with partial pivoting

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3

7775

<latexit sha1_base64="AOPvfDfJSM2gEaDva8DNMUxJVKg=">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</latexit><latexit sha1_base64="AOPvfDfJSM2gEaDva8DNMUxJVKg=">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</latexit><latexit sha1_base64="AOPvfDfJSM2gEaDva8DNMUxJVKg=">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</latexit><latexit sha1_base64="AOPvfDfJSM2gEaDva8DNMUxJVKg=">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</latexit>

Obtaining the decomposition:

Expand the permutation vector to a matrix:

Then

2

6664

0 0 0 1

1 0 0 0

0 0 1 0

0 1 0 0

3

7775

<latexit sha1_base64="AOPvfDfJSM2gEaDva8DNMUxJVKg=">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</latexit><latexit sha1_base64="AOPvfDfJSM2gEaDva8DNMUxJVKg=">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</latexit><latexit sha1_base64="AOPvfDfJSM2gEaDva8DNMUxJVKg=">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</latexit><latexit sha1_base64="AOPvfDfJSM2gEaDva8DNMUxJVKg=">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</latexit>

2

664

3 �2 �3 �24 0 �8 63 2 �10 �9�6 �2 9 �3

3

775

<latexit sha1_base64="nucpFqxaPzVjUPqSzQfLO9GjWQk=">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</latexit><latexit sha1_base64="nucpFqxaPzVjUPqSzQfLO9GjWQk=">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</latexit><latexit sha1_base64="nucpFqxaPzVjUPqSzQfLO9GjWQk=">AAACVHicbVFdS8MwFE3r16xfVR99CQ7Fl412DnVvA198VHAqrGOk2d0WlqYlScVR9if3Iv4TXwTTropOLyScnHPvzc1JmHCmtOe9WfbK6tr6RmXT2dre2d1z9w8eVJxKCh0a81g+hUQBZwI6mmkOT4kEEoUcHsPJda4/PoNULBb3eppALyIjwYaMEm2ovsuDEEZMZGFEtGQvM+ccn+JaI99KFATYaRro5ccrs10UVK4WaX4htAqydvFV3Vp0cAIQg+/mfbfq1b0i8F/gl6CKyrjtu/NgENM0AqEpJ0p1fS/RvYxIzSiHmROkChJCJ2QEXQMFiUD1ssKVGT4xzAAPY2mW0Lhgf1ZkJFJqGoUm08w3VstaTv6ndVM9vOplTCSpBkEXFw1TjnWMc4vxgEmgmk8NIFQyMyumYyIJ1eYjHGOCv/zkv6DTqLfq/l2z2m6XblTQETpGZ8hHl6iNbtAt6iCK5ujdsizberU+7BV7bZFqW2XNIfoV9u4n9AOkkg==</latexit><latexit sha1_base64="nucpFqxaPzVjUPqSzQfLO9GjWQk=">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</latexit>

P1 A L U2

6664

1 0 0 0

� 12 1 0 0

� 12 � 1

3 1 0

� 23

49

815 1

3

7775

<latexit sha1_base64="O4kfqgHv0i4NlciCvAxz1WGAKx8=">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</latexit><latexit sha1_base64="O4kfqgHv0i4NlciCvAxz1WGAKx8=">AAACrXicfVFNbxMxEPUuX2X5aIAjF4sIhISI1qGI9haJC8dWIm1FHKJZZzaxantXtheIrP15/AFu/BucZCGlRYw00pv3ZuQ346JW0vk8/5mkN27eun1n72527/6Dh/u9R49PXdVYgWNRqcqeF+BQSYNjL73C89oi6ELhWXHxfq2ffUHrZGU++lWNUw0LI0spwEdq1vvOLRr8KiqtwcwDB2th5bxFL5ZtYINhm/ECF9KEQoO38lubMfqC5n+Sc5q95qUFEVgbhm0k2f/EXflm13upb9gJ2+qgDUe76jA6erudyjhGu789zXr9fJBvgl4HrAN90sXxrPeDzyvRaDReKHBuwvLaTwNYL4XCuHPjsAZxAQucRGhAo5uGzblb+jwyc1pWNqbxdMNengignVvpInZGf0t3VVuT/9ImjS8Pp0GauvFoxPahslHUV3T9d3QuLQqvVhGAsDJ6pWIJ8TI+/nAWj8CurnwdjIeDowE7OeiPRt019shT8oy8JIy8IyPygRyTMRHJq+Qk+ZRM0jw9TXn6eduaJt3ME/JXpItfZsLJDQ==</latexit><latexit sha1_base64="O4kfqgHv0i4NlciCvAxz1WGAKx8=">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</latexit><latexit sha1_base64="O4kfqgHv0i4NlciCvAxz1WGAKx8=">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</latexit>

2

6664

�6 �2 9 �3

0 �3 32 � 7

2

0 0 �5 � 353

0 0 0 1069

3

7775

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=

Page 27: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

LU decomposition with partial pivoting

27

How can we solve a system using this factorization?

Ax = b

Step 0 : FactorA into P1A = LU Row-reduction<latexit sha1_base64="2HMeYDwdZRXKjyas0+I8dkl05z0=">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</latexit><latexit sha1_base64="2HMeYDwdZRXKjyas0+I8dkl05z0=">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</latexit><latexit sha1_base64="2HMeYDwdZRXKjyas0+I8dkl05z0=">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</latexit><latexit sha1_base64="2HMeYDwdZRXKjyas0+I8dkl05z0=">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</latexit>

Step 1 : Solve Ly = P1b Forward substituton<latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit><latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit><latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit><latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit>

Page 28: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

LU decomposition with complete pivoting

28

p1 p2

L(1)

2

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3

775

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664

1234

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775

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p1 p22

664

3214

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775

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2

664

3214

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775

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Pivot: apply permutation on rows and columns

Eliminate: GE on first column of permuted matrix A(0)

A(1)

Inverse of the elimination matrix, or first column of the lower triangular matrix

Page 29: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

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LU decomposition with complete pivotingPivot: apply permutation on rows and columns

Eliminate: GE on second column of permuted matrix A(1)

L(2) A(2)

permuted A(1)

Page 30: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

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p1 p2 A(2) p1 p22

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LU decomposition with complete pivotingPivot: apply permutation on rows and columns

Eliminate: GE on third column of permuted matrix A(2)

L(3)=L A(3)=U

permuted A(2)

Page 31: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

LU decomposition with complete pivotingObtaining the decomposition:Expand the permutation vectors to matrices:

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Page 32: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

LU decomposition with complete pivoting

32

How can we solve a system using this factorization?

Ax = b

Step 0 : FactorA into P1AP2 = LU Row-reduction<latexit sha1_base64="e+B1Z/QZni9AX66yLlrFPJ63hL8=">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</latexit><latexit sha1_base64="e+B1Z/QZni9AX66yLlrFPJ63hL8=">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</latexit><latexit sha1_base64="e+B1Z/QZni9AX66yLlrFPJ63hL8=">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</latexit><latexit sha1_base64="e+B1Z/QZni9AX66yLlrFPJ63hL8=">AAACmHicbZFLbxMxEMed5VXCoykce5mQIHEh2g0HSqWiRkg8JA6hEFopjoLXO5tY9WNre0mj1X49vgPfgSuccdJwaMtc/Nd/xjPjn9NCCufj+GcjunHz1u07W3eb9+4/eLjd2nn01ZnSchxxI409SZlDKTSOvPASTwqLTKUSj9PTN6v88Xe0Thj9xS8LnCg20yIXnPlgTVvfqDZCZ6h9s6JpDp89FhDDPtTwlnFvLHQHXRDaG+gOp8lgOO3DAXwcAT07K1kGVGLumbVmAVSl5rw6MovnFrOSr/rXXZi2OnEvXgdcF8lGdMgmhtOdRptmhpcq7MQlc26cxIWfVMx6wSXWTVo6LBg/ZTOs1gBqeBqsDPKwbW60h7V7qY4p55YqDZWK+bm7mluZ/8uNS5/vTSqhi9Kj5heD8lJCwLGiCZmwyL1cBsG4FWFD4HNmA7nAvEkdhg/RMz+vqMdzvxBZmFPt9V6E4Co8ReOCG6WYzqrwEVbV42RS0SJ0kBIl0DZtQycJ50r+s+tmgJpcRXhdjPq9V73kU79z+HpDd4vskifkGUnIS3JI3pMhGRFOfpBf5Df5E+1Gg+hd9OGiNGps7jwmlyI6+gvFache</latexit>

Step 2 : Solve Uz = y Back substitution<latexit sha1_base64="yTN1hJ/Zdt5xAvgPfQHrOPjUHl0=">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</latexit><latexit sha1_base64="yTN1hJ/Zdt5xAvgPfQHrOPjUHl0=">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</latexit><latexit sha1_base64="yTN1hJ/Zdt5xAvgPfQHrOPjUHl0=">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</latexit><latexit sha1_base64="yTN1hJ/Zdt5xAvgPfQHrOPjUHl0=">AAACl3icbVHLbhMxFHWGVxleKexgc0OKxCrKhAVtJVAEEuqyVQmtFEeRx3MnserH1PY0DaP5PD6Cb2ALezzTsGjLkSwfnXt9H8dpIYXzw+HPTnTn7r37D7Yexo8eP3n6rLv9/JszpeU44UYae5oyh1JonHjhJZ4WFplKJZ6kZ5+b+MkFWieM/urXBc4UW2iRC858kObdOdVG6Ay1jyua5nDssYAR7EMNx0ZeIOxMWv17DR+gZesa6Pl5yTKgEnPPrDUroCo1l9Unxs/AlakLg5RN/Xpn3u0PB8MWcJskG9InGxzOtzs9mhleqjASl8y5aTIs/Kxi1gsusY5p6bAIjdgCq3b/Gt4EKYPc2HC0h1a9lseUc2uVhkzF/NLdjDXi/2LT0ue7s0roovSo+VWjvJTgDTRmQiYsci/XgTBuRZgQ+JJZxn2wPKYOw3/ohV9W1OOlX4ks9Kl2B+8CuAqraFxxoxTTWRX+wap6mswqWoQKUqIE2qM96Cfhbug/uY6DqclNC2+TyWiwN0iORv3xx427W+QVeU3ekoS8J2NyQA7JhHDyg/wiv8mf6GU0jr5EB1epUWfz5gW5hujoLwsIyhY=</latexit>

Step 3 : Compute x = P2z Permute z<latexit sha1_base64="WCO+FJ4BrZHhlhcY+g9AeDXB37w=">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</latexit><latexit sha1_base64="WCO+FJ4BrZHhlhcY+g9AeDXB37w=">AAACnnicbVFNbxMxEHWWrxI+msKRy4QEiVOUTQ8UJFClHuCCCILQSHEUeb2ziVV/bG0vSbraf8gf4G9whQPebTi05UmWn9/MeMbPSS6F88Phz1Z06/adu/f27rcfPHz0eL9z8OSbM4XlOOFGGjtNmEMpNE688BKnuUWmEomnydlJHT/9jtYJo7/6bY5zxZZaZIIzH6RFJ6PaCJ2i9u2SJhl88ZjDIbyBCk6MyguP0G8Cmwrewngxag4XFdDz84KlQCVmnllr1kBVYjblGK1qqqhifpVk5UXVr/qLTm84GDaAmyTekR7ZYbw4aHVpanihwmRcMudm8TD385JZL7jEqk0LhznjZ2yJZWNDBS+ClEJmbFjaQ6NeyWPKua1KQmY9m7seq8X/xWaFz47mpdC1HZpfNsoKCd5A7SmkwiL3chsI41aECYGvmGXcB+fb1GH4Fr30q5J63Pi1SEOf8mhwGMBVeIrGNTdKMZ2W4TusqmbxvKR5uEFKlEC7tAu9OOw1/SdX7WBqfN3Cm2QyGrwexJ9HveN3O3f3yDPynLwkMXlFjskHMiYTwskP8ov8Jn+ibvQ++hh9ukyNWruap+QKoulf3Y3MeA==</latexit><latexit sha1_base64="WCO+FJ4BrZHhlhcY+g9AeDXB37w=">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</latexit><latexit sha1_base64="WCO+FJ4BrZHhlhcY+g9AeDXB37w=">AAACnnicbVFNbxMxEHWWrxI+msKRy4QEiVOUTQ8UJFClHuCCCILQSHEUeb2ziVV/bG0vSbraf8gf4G9whQPebTi05UmWn9/MeMbPSS6F88Phz1Z06/adu/f27rcfPHz0eL9z8OSbM4XlOOFGGjtNmEMpNE688BKnuUWmEomnydlJHT/9jtYJo7/6bY5zxZZaZIIzH6RFJ6PaCJ2i9u2SJhl88ZjDIbyBCk6MyguP0G8Cmwrewngxag4XFdDz84KlQCVmnllr1kBVYjblGK1qqqhifpVk5UXVr/qLTm84GDaAmyTekR7ZYbw4aHVpanihwmRcMudm8TD385JZL7jEqk0LhznjZ2yJZWNDBS+ClEJmbFjaQ6NeyWPKua1KQmY9m7seq8X/xWaFz47mpdC1HZpfNsoKCd5A7SmkwiL3chsI41aECYGvmGXcB+fb1GH4Fr30q5J63Pi1SEOf8mhwGMBVeIrGNTdKMZ2W4TusqmbxvKR5uEFKlEC7tAu9OOw1/SdX7WBqfN3Cm2QyGrwexJ9HveN3O3f3yDPynLwkMXlFjskHMiYTwskP8ov8Jn+ibvQ++hh9ukyNWruap+QKoulf3Y3MeA==</latexit>

Step 1 : Solve Ly = P1b Forward substituton<latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit><latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit><latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit><latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit>

Page 33: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

General LU Factorizations

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From Demmel, pp. 39-40:

Page 34: LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting can be easily incorporated into the LU decomposition algorithm by tracking how

Top 10 Algorithms

Matrix decompositions, such as LU, are listed as one the top 10 algorithms of the 20th century.

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