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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

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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 multiplications

• 3 divisions

• 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

• Example

• 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

• Splitting Method

• Issues

• Convergence

• Numerical Stability

### 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) )

Q & A?
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