L03 MAE507 LU Cholesky Decomp

7/24/2019 L03 MAE507 LU Cholesky Decomp

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September 3, 2015

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September 3, 2015

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Forward and Back Substitutions


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Example: Forward and Back Substitutions

A=[1 0 2 3; -1 2 2 -3; 0 1 1 4; 6 2 2 4];b=[1 -1 2 1]';

[L,U] = LU_factor(A);L =

1.0000 0 0 0-1.0000 1.0000 0 0

0 0.5000 1.0000 06.0000 1.0000 14.0000 1.0000

U =1 0 2 30 2 4 00 0 -1 40 0 0 -70

x=LU_solve(L,U,b)x =

-0.18570.2286-0.11430.4714

Forward and back

substitution

(without pivoting) MATLABM-file

LU_factor


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No need to consider {b} in decomposition


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September 3, 2015

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September 3, 2015

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September 3, 2015

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28/47

September 3, 2015

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Back

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Important


29/47

LU Decomposition with Pivoting

partial pivoting


30/47

LU Decomposition with PivotingA=[1 0 2 3; -1 2 2 -3; 0 1 1 4; 6 2 2 4];b=[1 -1 2 1]';

[L,U,P]=LU_pivot(A);L =

1.0000 0 0 0-0.1667 1.0000 0 00.1667 -0.1429 1.0000 0

0 0.4286 0 1.0000U =

6.0000 2.0000 2.0000 4.00000 2.3333 2.3333 -2.33330 0 2.0000 2.00000 0 0 5.0000

T1 =6 2 2 4

-1 2 2 -31 0 2 30 1 1 4

T2 =6 2 2 4-1 2 2 -31 0 2 3

0 1 1 4

T1 = [L][U]

T2 = [P][A]

Verify

T1= T2


31/47

LU Decomposition with Pivoting A=[1 0 2 3; -1 2 2 -3; 0 1 1 4; 6 2 2 4]; b=[1; -1; 2; 1]; [L,U,P]=LU_pivot(A); Pb=P*bPb =

1-112

x=LU_Solve(L,U,Pb)

d =1.0000-0.8333

00

d =1.0000-0.83330.7143

0d =

1.0000-0.83330.71432.3571

x =00

-0.11430.4714

x =0

0.2286-0.11430.4714

x =-0.18570.2286-0.11430.4714

x =-0.18570.2286-0.11430.4714

Forward Substitution Back Substitution

LU Decomposition

[L][U]{x} = [P]{b}


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September 3, 2015

Slide 32

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33/47

.3'"3K 7,*?;#*lu

A=[1 0 2 3; -1 2 2 -3; 0 1 1 4; 6 2 2 4];

b=[1; -1; 2; 1];

[L,U]=lu(A)

L =

0.1667 -0.1429 1.0000 0

-0.1667 1.0000 0 0

0 0.4286 0 1.0000

1.0000 0 0 0

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6.0000 2.0000 2.0000 4.0000

0 2.3333 2.3333 -2.3333

0 0 2.0000 2.0000

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L*U

ans =

1 0 2 3

-1 2 2 -3

0 1 1 4

6 2 2 4

d=L\b

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1.0000-0.8333

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2.3571

x=U\d

x =

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-0.1143

0.4714

Forward and

Back

SubstitutionsLU Decomposition

without Pivoting


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September 3, 2015

Slide 34

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Symmetric MatrixL= U

Compute only the upper triangular elements


36/47

A=[9 -6 12 -3; -6 5 -9 2; 12 -9 21 0; -3 2 0 6]A =

9 -6 12 -3-6 5 -9 212 -9 21 0-3 2 0 6

[L,U] = Cholesky(A)

L = 3 0 0 0-2 1 0 04 -1 2 0-1 0 2 1

U =3 -2 4 -10 1 -1 00 0 2 20 0 0 1

Symmetric

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41/47

September 3, 2015

Slide 41

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42/47

September 3, 2015

Slide 42

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September 3, 2015

Slide 43

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September 3, 2015

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September 3, 2015

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September 3, 2015

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L03 MAE507 LU Cholesky Decomp

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