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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. Linear Algebra Systems of Linear Equations Matrices Vector Spaces. Systems of Linear Equations. Linear Equation

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Advanced Computer Graphics Spring 2009

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  1. Advanced Computer Graphics Spring 2009 K. H. Ko Department of Mechatronics Gwangju Institute of Science and Technology

  2. Today’s Topics • Linear Algebra • Systems of Linear Equations • Matrices • Vector Spaces

  3. Systems of Linear Equations • Linear Equation • System of Linear Equations (n equations, m unknowns)

  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.

  6. Systems of Linear Equations • Division is more expensive than multiplication and addition. • 3 additions • 3 multiplications • 3 divisions • 3 additions • 5 multiplications • 2 divisions

  7. Gaussian Elimination • Forward elimination + back substitution = Gaussian elimination

  8. Gaussian Elimination • Basic Operations for Forward Elimination

  9. Gaussian Elimination • Basic Operations for Forward Elimination

  10. Gaussian Elimination • Basic Operations for Forward Elimination

  11. Gaussian Elimination • Basic Operations for Back Substitution

  12. Gaussian Elimination • Example

  13. Geometry of Linear Systems • Consider

  14. Geometry of Linear Systems • Consider 3 equations and 3 unknowns

  15. 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.

  16. 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.

  17. Numerical Issues • However, even the previous approach can be a problem. • Swap columns to avoid such problem. • Blackboard!!!

  18. 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.

  19. 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.

  20. 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

  21. Iterative Methods for Solving Linear Systems • Splitting Method • Issues • Convergence • Numerical Stability

  22. Iterative Methods for Solving Linear Systems • Formulate the linear system Ax=b as a minimization problem

  23. Matrices • Square matrices • Identity matrix • Transpose of a matrix • Symmetric matrix: A = AT • Skew-symmetric: A = -AT

  24. 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

  25. Matrices • Elementary Row Matrices

  26. Matrices • Elementary Row Matrices

  27. 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]

  28. 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 echelon.

  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.

  31. Matrices • LDU Decomposition of the matrix A

  32. 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

  34. 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?

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