Numerical Methods using Linear Algebra

1
Numerical Methods using Linear Algebra

• By Andy Cool

2
Outline

• History
• Matrices
• Matrix operations and Properties
• Gauss-Jordan Elimination
• Determinants Cramers Rule
• Ill-Conditioned Systems
• Error Detection and Numerical Analysis
• Conclusion

3
Matrices and History

• Linear algebra has been around at least 2000
  years, perhaps longer.
• Solving linear systems of equations is an
  important part of mathematics from splitting
  slices of bread to making advanced scientific
  conclusions.
• When multiple linear equations share unknowns,
  they can be put into a matrix

4
Equation to Matrix

5
Matrix Layout
6
Matrix History

• There are hints of matrix work being done as far
  back as second century BC.
• The Babylonians and Chinese both did similar
  works with multiple linear equations around the
  same time.

7
Old Babylonian Problem

• There are three types of corn, of which three
  bundles of the first, two of the second, and one
  of the third make 39 measures. Two of the first,
  three of the second and one of the third make 34
  measures. And one of the first, two of the second
  and three of the third make 26 measures. How many
  measures of corn are contained of one bundle of
  each type?

8
Babylonian Work
?
Take 3 times the second column and subtract it by
the third column as many times as possible.
9
Types of Matrices
?
Sub Matrix of A
Matrix A
10
Types
Lower Triangular Matrix
Identity Matrix
11
Types
A symmetric matrix is one where elements

12
Operations
?
Transpose of a Matrix flip element positions
13
Operations

• Add
• Subtract
• Scalar Multiple

14
Properties of Matrix Addition

• Summation is associative (AB)C A(BC)
• Summation is commutative AB BA
• A0A
• A (-A) 0
• Matrix multiplication is distributive c(AB)
  cA cB
• A0 0

15
Matrix Multiplication

• (M x N) and (N x P)

16
BAM!
17
Properties of Matrix Multiplication

• (cA)B A(cB)
• A(BC) (AB)C
• (AB)C ACBC

18
Gaussian Elimination

• As mentioned earlier, there is evidence of
  Gaussian elimination over 2200 years ago.
  However, Carl Friedrich Gauss who lived into the
  mid 19th century is the first one to publicize
  the method. The method involves converting
  systems of equations into upper-triangular form
  and then solving one equation at a time.

19
Solving the Matrix

• 2x 1y 3z 4
• 3y 5z 2
• 1z 2

20
Gauss-Jordan and Row Operations

• If the matrix is not in upper-triangular form, we
  must do elementary row operations
• multiply a row by a non-zero number
• switch two rows
• add a multiple of one row to another one

21
Determinants

• The determinant of an n x n matrix is defined as
• To find a determinant of a 2 x 2 matrix
• Det(A) ad-cd

22
Properties of Determinants

• k(A) kdet(A)
• If 2 rows are interchanged, then the determinant
  is negated.
• If 2 rows of columns of A are the same, then
  det(A)0.
• For any upper or lower triangular matrix, the
  determinant is equal the product of the matrixs
  diagonal elements.

23
Cramers Rule

• If the det(A)0, then Cramers rule can be used to
  solve the system
• x
  5
• y
  2

24
Ill-Conditioned Systems

• Some systems of equations may need very precise
  measurments to find the correct answer. Some
  Ill-Conditioned systems may have a round off on
  one coefficient that can yeild a very big change
  in the answer
• The closer that k gets to 1 the equations get
  closer to representing parallel lines.
• As k goes toward 1, the denominator goes toward
  0. This can present large changes in the answers
  for and. If k .99, then -296 and 300.
  However, if k .99999, then -299,996 and
  300,000.

25
Errors and Numerical Analysis

• The type of error found in an ill-conditioned
  system is a large part of the study of numerical
  analysis. Numerical Analysis is a way to do
  highly complicated mathematics with a computer
  and understand the errors that could result.
  Results from numerical analysis are always
  numerical and approximate.
• The errors that need to be recognized are from
  problems involving quantities with infinite
  precision. A computer can only deal with finite
  numbers when computing. The types of errors that
  occur then are round-off errors. They can occur
  when any data is collected or when the final
  computations are done on a computer.

26
Conclusion

• Linear algebra and the concept of matrices have
  been around as almost as far back as we can find
  elementary mathematics.
• Systems of equations have been used for a long
  time solving simple proportion problems.
• Solving systems of linear equations is very
  useful.
• In fact, our technology has made linear algebra
  methods able to be done so quickly and
  efficiently that we tent to look further into its
  possible uses.
• The study of error helps us to find more
  precision in our studies and understand the ways
  to get more from linear algebra and its
  applications.