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Advanced Computer Graphics Spring 2009
K. H. Ko
Department of MechatronicsGwangju Institute of Science and Technology
2
Today’s Topics
Linear AlgebraSystems of Linear EquationsMatricesVector Spaces
3
Systems of Linear Equations
Linear Equation
System of Linear Equations (n equations, m unknowns)
bxaxa mn 11
nmn
nmn
mn
bxdxd
bxcxc
bxaxa
11
111
111
4
Systems of Linear Equations
Solve a system of n linear equations in m unknown variables
A common problem in applications In most cases m = n. The system has three cases
No solutions, one solution or infinitely many solutions
How to solve the system? Forward elimination followed by back
substitution
5
Systems of Linear Equations
A closer look at two equations in two unknowns
When the solution method needs to be implemented for a computer, a practical concern is the amount of time required to compute a solution.
2222121
1212111
bxaxa
bxaxa
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Systems of Linear Equations
Division is more expensive than multiplication and addition.
• 3 additions
• 3 multiplications
• 3 divisions
• 3 additions
• 5 multiplications
• 2 divisions
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Gaussian Elimination
Forward elimination + back substitution = Gaussian elimination
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Gaussian Elimination
Basic Operations for Forward Elimination
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Gaussian Elimination
Basic Operations for Forward Elimination
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Gaussian Elimination
Basic Operations for Forward Elimination
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Gaussian Elimination
Basic Operations for Back Substitution
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Gaussian Elimination
Example
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Geometry of Linear Systems
Consider 2222121
1212111
bxaxa
bxaxa
021122211 aaaa0
0
121211
21122211
baba
aaaa 0
0
121211
21122211
baba
aaaa
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Geometry of Linear Systems
Consider 3 equations and 3 unknowns
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Numerical Issues
If the pivot is nearly zero, the division can be a source of numerical errors.
Use of floating point arithmetic with limited precision is the main cause.
/11
/1
/120
/11
1
1
21
1
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Numerical Issues
A better algorithm involves searching the entries with the pivot that is largest in absolute magnitude.
1
1
210
21
1
1
1
21
No division by ε. -> Numerically robust and stable.
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Numerical Issues
However, even the previous approach can be a problem.
Swap columns to avoid such problem.
Blackboard!!!
0,12
1
21212
211
xx
xx
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Numerical Issues
Generally, for a system of n equations in n unknowns…
Full Pivoting: Search the entire matrix of coefficients looking for the entry of largest absolute magnitude to be used as the pivot.
If that entry occurs in row r and column c, then rows r and 1 are swapped followed by a swap of column c and column 1.
After both swaps, the entry in row 1 and column 1 is the largest absolute magnitude entry in the matrix.
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Numerical Issues
Generally, for a system of n equations in n unknowns…
If that entry is nearly zero, the linear system is ill-conditioned and notify the user.
If you choose to continue, the division is performed and forward elimination begins.
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Iterative Methods for Solving Linear Systems Look for a good numerical
approximation instead of the exact mathematical solution.
Useful in sparse linear systems Approaches
Splitting Method Minimization problem
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Iterative Methods for Solving Linear Systems Splitting
Method
2222121
1212111
bxaxa
bxaxa
22
12122
11
21211
a
xabx
a
xabx
22
)(1212)1(
2
11
)(2121)1(
1
a
xabx
a
xabx
ii
ii
Issues
• Convergence
• Numerical Stability
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Iterative Methods for Solving Linear Systems Formulate the linear system Ax=b
as a minimization problem
0)()(),( 22222121
2121211121 bxaxabxaxaxxf
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Matrices
Square matrices Identity matrix Transpose of a matrix Symmetric matrix: A = AT
Skew-symmetric: A = -AT
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Matrices
Upper echelon matrix U = [uij](nxm) if uij = 0 for i > j If m=n, upper triangular matrix
Lower echelon matrix L = [lij](nxm) if lij = 0 for i < j If m=n, lower triangular matrix
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Matrices
Elementary Row Matrices
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Matrices
Elementary Row Matrices
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Matrices
Elementary Row Matrices The final result of forward elimination can be state
d in terms of elementary row matrices Ek, … E1 applied to the augmented matrix [A|b].
[U|v] = Ek … E1[A|b]
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Matrices
Inverse Matrix PA = I: P is a left inverse A-1A = I, AA-1 = I. Inverses are unique If A and B are invertible, so is AB. Its inverse
is (AB)-1 = B-1A-1
29
Matrices
LU Decomposition of the matrix A The forward elimination of a matrix A produces an
upper echelon matrix U. The corresponding elementary row matrices are Ek…E1
U = Ek…E1A., L = (Ek…E1)-1. L is lower triangular. A = LU: L is lower triangular and U is upper echelo
n.
30
Matrices
LDU Decomposition of the matrix A L is lower triangular, D is a diagonal matrix,
and U is upper echelon with diagonal entries either 1 or 0.
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Matrices
LDU Decomposition of the matrix A
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Matrices
In general the factorization can be written as PA = LDU.
33
Matrices
If A is invertible, its LDU decomposition is unique
If A is symmetric, U in the LDU decomposition must be U = LT.
A = LDLT. If the diagonal entries of D are
nonnegative, A = (LD1/2) (LD1/2)T
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Vector Spaces
The central theme of linear algebra is the study of vectors and the sets in which they live, called vector spaces.
What is the vector???
35
Vector Spaces
Definition of a Vector Space (the triple (V,+,ᆞ ) )
36
Q & A?