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# Advanced Computer Graphics Spring 2009 - PowerPoint PPT Presentation

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
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) )