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### Advanced Computer Graphics Spring 2009

K. H. Ko

Department of Mechatronics

Gwangju Institute of Science and Technology

Today’s Topics

- Linear Algebra
- Systems of Linear Equations
- Matrices
- Vector Spaces

Systems of Linear Equations

- Linear Equation
- System of Linear Equations (n equations, m unknowns)

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

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.

Systems of Linear Equations

- Division is more expensive than multiplication and addition.

- 3 additions
- 3 multiplications
- 3 divisions

- 3 additions
- 5 multiplications
- 2 divisions

Gaussian Elimination

- Forward elimination + back substitution = Gaussian elimination

Gaussian Elimination

- Basic Operations for Forward Elimination

Gaussian Elimination

- Basic Operations for Forward Elimination

Gaussian Elimination

- Basic Operations for Forward Elimination

Gaussian Elimination

- Basic Operations for Back Substitution

Gaussian Elimination

- Example

Geometry of Linear Systems

- Consider

Geometry of Linear Systems

- Consider 3 equations and 3 unknowns

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.

Numerical Issues

- A better algorithm involves searching the entries with the pivot that is largest in absolute magnitude.

No division by ε. -> Numerically robust and stable.

Numerical Issues

- However, even the previous approach can be a problem.
- Swap columns to avoid such problem.
- Blackboard!!!

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.

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.

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

Iterative Methods for Solving Linear Systems

- Formulate the linear system Ax=b as a minimization problem

Matrices

- Square matrices
- Identity matrix
- Transpose of a matrix
- Symmetric matrix: A = AT
- Skew-symmetric: A = -AT

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

Matrices

- Elementary Row Matrices

Matrices

- Elementary Row Matrices

Matrices

- Elementary Row Matrices
- The final result of forward elimination can be stated in terms of elementary row matrices Ek, … E1 applied to the augmented matrix [A|b].
- [U|v] = Ek… E1[A|b]

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

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 echelon.

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.

Matrices

- LDU Decomposition of the matrix A

Matrices

- In general the factorization can be written as PA = LDU.

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

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???

Vector Spaces

- Definition of a Vector Space (the triple (V,+,ᆞ) )

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