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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

Advanced Computer Graphics Spring 2009

K. H. Ko

Department of Mechatronics

Gwangju Institute of Science and Technology

today s topics
Today’s Topics
  • Linear Algebra
    • Systems of Linear Equations
    • Matrices
    • Vector Spaces
systems of linear equations
Systems of Linear Equations
  • Linear Equation
  • System of Linear Equations (n equations, m unknowns)
systems of linear equations1
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 equations2
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 equations3
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
Gaussian Elimination
  • Forward elimination + back substitution = Gaussian elimination
gaussian elimination1
Gaussian Elimination
  • Basic Operations for Forward Elimination
gaussian elimination2
Gaussian Elimination
  • Basic Operations for Forward Elimination
gaussian elimination3
Gaussian Elimination
  • Basic Operations for Forward Elimination
gaussian elimination4
Gaussian Elimination
  • Basic Operations for Back Substitution
geometry of linear systems1
Geometry of Linear Systems
  • Consider 3 equations and 3 unknowns
numerical issues
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 issues1
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 issues2
Numerical Issues
  • However, even the previous approach can be a problem.
  • Swap columns to avoid such problem.
    • Blackboard!!!
numerical issues3
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 issues4
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
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 systems1
Iterative Methods for Solving Linear Systems
  • Splitting Method
  • Issues
  • Convergence
  • Numerical Stability
iterative methods for solving linear systems2
Iterative Methods for Solving Linear Systems
  • Formulate the linear system Ax=b as a minimization problem
matrices
Matrices
  • Square matrices
  • Identity matrix
  • Transpose of a matrix
  • Symmetric matrix: A = AT
  • Skew-symmetric: A = -AT
matrices1
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
matrices2
Matrices
  • Elementary Row Matrices
matrices3
Matrices
  • Elementary Row Matrices
matrices4
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]
matrices5
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
matrices6
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.
matrices7
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.
matrices8
Matrices
  • LDU Decomposition of the matrix A
matrices9
Matrices
  • In general the factorization can be written as PA = LDU.
matrices10
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
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 spaces1
Vector Spaces
  • Definition of a Vector Space (the triple (V,+,ᆞ) )
ad