solving linear eqns - university of iowa
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© Dennis L. BrickerDept of Mechanical & Industrial EngineeringUniversity of Iowa
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Contents
• Preliminarieso Elementary Matrix Operationso Elementary Matriceso Echelon Matriceso Rank of Matrices
• Elimination Methodso Gauss Eliminationo Gauss-Jordan Elimination
• Factorization Methodso Product Form of Inverse (PFI)o LU Factorizationo LDLT Factorizationo Cholesky LLT Factorization
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CHOLESKY FACTORIZATION
Suppose that A is a symmetric & positive definite matrix.
Then the Cholesky factorization of A is
ˆ T̂A L L=where L̂ is a lower triangular matrix.
Computation:Suppose that we have the factorization
TA L D L=Then if 0i
iD ≥ , we can define a new diagonal matrix D̂ whereˆ i ii iD D≡
Then ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆTT T TA L D L L D D L LD LD L L= = = = where ˆ ˆL LD=
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Example:
We wish to find the Cholesky factorization of the matrix
2 0 10 1 11 1 2
A =
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Cholesky factorization…
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The lower triangular matrix L is found by performing (on the
identity matrix) the inverse of the row operations used to reduce
the A matrix:
3 3 1
3 3 2
1 0 012 0 1 0
1 1 12
R R RL
R R R
← + ⇒ = ← +
We now have the LU factorization of matrix A:
1 0 0 2 0 10 1 0 0 1 11 11 1 0 02 2
A LU
= =
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Define the diagonal matrix D:
1
1 0 02 0 0 20 1 0 0 1 0
1 0 0 20 0 2
D D−
= ⇒ =
Note that
1
1 0 0 2 0 12ˆ 0 1 0 0 1 1
0 0 2 10 0 211 0 2
0 1 10 0 1
U D U−
= =
=
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And so,
11 01 0 0 2 0 0 20 1 0 0 1 0 0 1 11 1 0 0 11 1 0 02 2
TA LDL
= =
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Define the diagonal matrix D̂ where ˆ i ii iD D≡ :
2 0 0ˆ 0 1 0
10 02
D
=
Then compute1 0 0 2 0 0 2 0 0
ˆ ˆ 0 1 0 0 1 0 0 1 01 1 1 11 1 0 0 12 2 2 2
L LD
= = =
So the Cholesky factorization is
12 02 0 0 2ˆ ˆ 0 1 0 0 1 1
1 1 11 0 02 2 2
TA LL
= =