LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting...
Transcript of LU Decomposition - math.boisestate.edu · Partial pivoting and LU decomposition 21 Partial pivoting...
LU Decomposition
1
A L U=
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
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
⇤
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?
=
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)”
Elimination matrices : First step
How do we express ”(equation 2)�(2)(equation 1)”
How do we express ”(equation 3)�(�1)(equation 1)”
Elimination matrices : First step
How do we express ”(equation 2)�(2)(equation 1)”
How do we express ”(equation 3)�(�1)(equation 1)”
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
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
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)
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)
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)
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-
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
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=
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
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
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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:
LU Decomposition
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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
Solution procedure given LU=A
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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
Partial pivoting and LU decomposition
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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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Partial pivoting and LU decomposition
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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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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
6664
1 0 0 0
� 23 1 0 0
� 12 0 1 0
� 12 0 0 1
3
7775
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2
6664
�6 �2 9 �3
0 � 43 �2 4
0 1 � 112 � 21
2
0 �3 32 � 7
2
3
7775
<latexit sha1_base64="SpzJa14CZJOawid5mV15KpaecTU=">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</latexit><latexit sha1_base64="SpzJa14CZJOawid5mV15KpaecTU=">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</latexit><latexit sha1_base64="SpzJa14CZJOawid5mV15KpaecTU=">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</latexit><latexit sha1_base64="SpzJa14CZJOawid5mV15KpaecTU=">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</latexit>
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)
2
6664
�6 �2 9 �3
0 � 43 �2 4
0 1 � 112 � 21
2
0 �3 32 � 7
2
3
7775
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2
6664
�6 �2 9 �3
0 �3 32 � 7
2
0 1 � 112 � 21
2
0 � 43 �2 4
3
7775
<latexit sha1_base64="PWtB8eKO3rc8MkrXYB8ii5HrK3Y=">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</latexit><latexit sha1_base64="PWtB8eKO3rc8MkrXYB8ii5HrK3Y=">AAACqHicbVFbb9MwFHbCbYRbgUdeLCoqJLQqziq2vVXihQceNomsQ01VOe5Ja812ItuBVVb+237D3vg3OG0q6MaRbH3nO9/xuTivBDc2jn8H4YOHjx4/OXgaPXv+4uWr3us3F6asNYOUlaLUlzk1ILiC1HIr4LLSQGUuYJJffWnjk5+gDS/Vd7uuYCbpUvGCM2o9Ne/dZBoU/GKllFQtXEa1pmtjNVi2ahwZJk2U5bDkyuWSWs2vm+jwMx7gw8Rfpy04wlmGo3gLBzgrNGXuqHFJ01Jb93jj7nTkb4CQfWFC9pQdO2r8g7uioygD3+munXmvHw/jjeH7gHSgjzo7m/dus0XJagnKMkGNmZK4sjNHteVMgB+3NlBRdkWXMPVQUQlm5jabbvAHzyxwUWp/lMUb9t8MR6Uxa5l7pe9vZe7GWvJ/sWlti5OZ46qqLSi2LVTUAtsSt9+GF1wDs2LtAWWa+14xW1G/HOs/N/JLIHdHvg/SZHg6JOej/njcbeMAvUPv0UdE0DEao6/oDKWIBYPgW5AGF+Gn8DychD+20jDoct6iPQvzP4AVxcI=</latexit><latexit sha1_base64="PWtB8eKO3rc8MkrXYB8ii5HrK3Y=">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</latexit><latexit sha1_base64="PWtB8eKO3rc8MkrXYB8ii5HrK3Y=">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</latexit>
2
664
4132
3
775
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2
664
4231
3
775
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2
6664
1 0 0 0
� 12 1 0 0
� 12 � 1
3 1 0
� 23
49 0 1
3
7775
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2
6664
�6 �2 9 �3
0 �3 32 � 7
2
0 0 �5 � 353
0 0 � 83
509
3
7775
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Need to also permute L(1) before inserting elements for L(2)
permuted A(1)
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
2
664
4132
3
775
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3
7775
<latexit sha1_base64="O4kfqgHv0i4NlciCvAxz1WGAKx8=">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</latexit><latexit sha1_base64="O4kfqgHv0i4NlciCvAxz1WGAKx8=">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</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
<latexit sha1_base64="RzhjAFY+oYYihRh8QA3X4TDHf4U=">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</latexit><latexit sha1_base64="RzhjAFY+oYYihRh8QA3X4TDHf4U=">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</latexit><latexit sha1_base64="RzhjAFY+oYYihRh8QA3X4TDHf4U=">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</latexit><latexit sha1_base64="RzhjAFY+oYYihRh8QA3X4TDHf4U=">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</latexit>
permuted A(2)
26
LU decomposition with partial pivoting
p1 P12
664
4132
3
775
<latexit sha1_base64="Agbn64bfsgJphFC2aR0e2uOm2tU=">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</latexit><latexit sha1_base64="Agbn64bfsgJphFC2aR0e2uOm2tU=">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</latexit><latexit sha1_base64="Agbn64bfsgJphFC2aR0e2uOm2tU=">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</latexit><latexit sha1_base64="Agbn64bfsgJphFC2aR0e2uOm2tU=">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</latexit>
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>
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
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2
664
3 �2 �3 �24 0 �8 63 2 �10 �9�6 �2 9 �3
3
775
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P1 A L U2
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2
6664
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0 �3 32 � 7
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0 0 0 1069
3
7775
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=
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=">AAAClXicbVHLbhMxFHWGVzs8msKmEpsbEiQ2RDNlQUGiagWqWCARHqGV4ijyeO4kVv2Y2h7SaDRfx1fwCWzhB3DSsGjL2fjo3Ov7ODcrpXA+SX62ohs3b92+s7EZ3713/8FWe/vhN2cqy3HIjTT2JGMOpdA49MJLPCktMpVJPM5O3y7jx9/ROmH0V78ocazYVItCcOaDNGmPqTZC56h9XNOsgC8eS0jgNTRwxLg3FnqHPRDaG+gNJukhvIEPQ6BnZxXLgUosPLPWzIGqzJzXn838ucW84sviTQ8m7W7ST1aA6yRdky5ZYzDZbnVobnilwkBcMudGaVL6cc2sF1xiE9PKYcn4KZtivdq+gadByqEIoxZGe1ipl/KYcm6hspCpmJ+5q7Gl+L/YqPLF3rgWuqw8an7RqKgkBC+WVkIuLHIvF4EwbkWYEPiM2WBbMDymDsM19NTPaurx3M9FHvrUe/0XAVyFVTTOuVGK6bwOV7CqGaXjmpahgpQogXZoB7ppeJf0n9zEwdT0qoXXyXC3/6qfftrtHuyv3d0gj8kT8oyk5CU5IO/JgAwJJz/IL/Kb/Il2ov3oXXR0kRq11n8ekUuIPv4FYt3HXw==</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=">AAACnXicbVFNbxMxEHWWrxK+UjhymZBW4hRly4GCBKqEVHGoUFAJqVRHkdc7m1j1x9aebRqt9hfyC/gZXOGCk4ZDW0ay/Pxm7Hl+k5VaBRoMfraSO3fv3X+w9bD96PGTp88628+/B1d5iSPptPMnmQiolcURKdJ4UnoUJtM4zs4+rfLjC/RBOfuNliVOjJhZVSgpKFLTDnLrlM3RUrvmWQHHhCWk8B4aOHb6AmHnaM0vG/gAw2kK61PWAD8/r0QOXGNBwnu3AG4yd1kfOr8QPodQZSHqqcjZZmfa6Q36g3XAbZBuQI9tYjjdbnV57mRlojCpRQin6aCkSS08KamxafMqYCnkmZhhvXahgd1I5VA4H5clWLPX6oQJYWmi9l0jaB5u5lbk/3KnFRX7k1rZsiK08qpRUWkgBytLIVceJellBEJ6FRWCnAsvJEXj2zxgnIqd0bzmhJe0UHnsU+/338SQJn7F4kI6Y4TN6zgNb5rTdFLzMr6gNWrgXd6FXhr3FfxHN+1oanrTwttgtNd/10+/7vUOPm7c3WIv2Sv2mqXsLTtgn9mQjZhkP9gv9pv9SSA5TI6SL1elSWtz5wW7Fsn4L/+szCs=</latexit><latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit><latexit sha1_base64="XRunt1/gr4iXQ9IETerqEwroEAM=">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</latexit>
LU decomposition with complete pivoting
28
p1 p2
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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2
664
1234
3
775
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A(0)=A 2
664
�10 2 3 �9�8 0 4 6�3 �2 3 �29 �2 �6 �3
3
775
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p1 p22
664
3214
3
775
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2
664
3214
3
775
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permuted A(0)
2
6664
1 0 0 045 1 0 0310 0 1 0�910 0 0 1
3
7775
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2
6664
�10 2 3 �9
0 �85
85
665
0 �135
2110
710
0 �15
�3310
�11110
3
7775
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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
29
p1 p2 A(1) p1 p22
664
3214
3
775
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2
6664
�10 2 3 �9
0 �85
85
665
0 �135
2110
710
0 �15
�3310
�11110
3
7775
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2
664
3214
3
775
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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)
30
p1 p2 A(2) p1 p22
664
3214
3
775
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664
3214
3
775
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2
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2
6664
�10 �9 2 3
0 665
�85
85
0 0 �8333
13366
0 0 0 �26583
3
7775
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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)
LU decomposition with complete pivotingObtaining the decomposition:Expand the permutation vectors to matrices:
Then2
664
3 �2 �3 �24 0 �8 63 2 �10 �9�6 �2 9 �3
3
775
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P1 A
L U
=
p1 P1 p2 P22
664
3214
3
775
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2
6664
0 0 1 0
0 1 0 0
1 0 0 0
0 0 0 1
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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=">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</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=">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</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=">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</latexit><latexit sha1_base64="WCO+FJ4BrZHhlhcY+g9AeDXB37w=">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</latexit><latexit sha1_base64="WCO+FJ4BrZHhlhcY+g9AeDXB37w=">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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>
General LU Factorizations
33
From Demmel, pp. 39-40:
Top 10 Algorithms
Matrix decompositions, such as LU, are listed as one the top 10 algorithms of the 20th century.
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