Advanced computer graphics spring 2009
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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

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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


Gaussian elimination5

Gaussian Elimination

  • Example


Geometry of linear systems

Geometry of Linear Systems

  • Consider


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) )


Advanced computer graphics spring 2009

Q & A?


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