Chapter: 3b System of Linear Equations

1
Chapter 3bSystem of Linear Equations

• Dr. Asaf Varol
• asvarol_at_mail.wvu.edu

2
Gaussian Elimination

• In the Gaussian Elimination Method, Elementary
  Row Operations (E.R.O.'s) are applied in a
  specific order to transform an augmented matrix
  into triangular echelon form as efficiently as
  possible 6.This is the essence of the method
  Given a system of m equations in n variables or
  unknowns, pick the first equation and subtract
  suitable multiples of it from the remaining m-1
  equations. In each case choose the multiple so
  that the subtraction cancels or eliminates the
  same variable, say x1. The result is that the
  remaining m-1 equations contain only n-1 unknowns
  (x1 no longer appears) 6.
• Now set aside the first equation and repeat the
  above process with the remaining m-1 equations in
  n-1 unknowns 6.
• Continue repeating the process. Each cycle
  reduces the number of variables and the number of
  equations. The process stops when either

3
Gaussian Elimination (Contd)

• There remains one equation in one variable. In
  that case, there is a unique solution and
  back-substitution is used to find the values of
  the other variables 6.
• There remain variables but no equations. In that
  case there is no unique solution 6.
• There remain equations but no variables (ie. the
  lowest row(s) of the augmented matrix contain
  only zeros on the left side of the vertical
  line). This indicates that either the system of
  equations is inconsistent or redundant. In the
  case of inconsistency the information contained
  in the equations is contradictory. In the case of
  redundancy, there may still be a unique solution
  and back-substitution can be used to find the
  values of the other variables 6.

4
Algorithm for Gaussian Elimination

• Transform the columns of the augmented matrix,
  one at a time, into triangular echelon form. The
  column presently being transformed is called the
  pivot column. Proceed from left to right, letting
  the pivot column be the first column, then the
  second column, etc. and finally the last column
  before the vertical line. For each pivot column,
  do the following two steps before moving on to
  the next pivot column 6
• Locate the diagonal element in the pivot column.
  This element is called the pivot. The row
  containing the pivot is called the pivot row.
  Divide every element in the pivot row by the
  pivot (ie. use E.R.O. 1) to get a new pivot row
  with a 1 in the pivot position 6.
• Get a 0 in each position below the pivot position
  by subtracting a suitable multiple of the pivot
  row from each of the rows below it (ie. by using
  E.R.O. 2).
• Upon completion of this procedure the augmented
  matrix will be in triangular echelon form and may
  be solved by back-substitution 6.

5
Example

• Use Gaussian elimination to solve the system of
  equations6

6
Solution

• Perform this sequence of E.R.O.'s on the
  augmented matrix. Set the pivot column to column
  1. Get a 1 in the diagonal position (underlined)

7
Solution (Contd)
8
Solution (Contd)
9
Results

• It is solved by back-substitution. Substituting
  z 3 from the third equation into the second
  equation gives y 5, and substituting z 3 and
  y 5 into the first equation gives x 7. Thus
  the complete solution is 6x 7, y 5, z
  3.

10
Example

• Express the following system in augmented matrix
  form and find an equivalent upper- triangular
  system and the solution 4.

11
Example (Contd)

• The augmented matrix is

12
Example (Contd)

• The first row is used to eliminate elements in
  the first column below the diagonal. We refer to
  the first row as pivotal row and the element
  a111 is called the pivotal element. The values
  mk1 are the multiples of row 1 then are to be
  subtracted from row k for k2,3,4. The result
  after elimination is 4

13
Example (Contd)

• The second row is used to eliminate elements in
  the second column that lie below the diagonal.
  The second row is the pivotal row and the values
  mk2 are the multiples of row 2 that are to be
  subtracted from row k for k3,4. The result after
  elimination is 4

14
Example (Contd)

• Finally, the multiple m43-1.9 of the third row
  is subtracted from the fourth row, and the result
  is the upper- triangular system 4.

15
Example (Contd)

• The back-substitution algorithm can be used to
  solve the previous matrix, and we get
• X42
• X34
• X2-1
• X13

16
Example

• a) Use MATLAB to construct the augmented matrix
  for the linear system of the below given matrix.
• b) Use the max command to find the element of
  greatest magnitude in the first column of the
  coefficient matrix A.
• C) Break the augmented matrix into the
  coefficient matrix U and constant matrix Y of the
  upper-triangular system UXY 4.

17
Answers a)

• gtgtA1 2 1 42 0 4 34 2 2 1-3 1 3 2
• gtgtB13 28 20 6
• gtgtAugA B

18
Answer b)

• In the following MATLAB display, a is the element
  of greatest magnitude in the first column of A
  and j is the row number
• gtgta,jmax(abs(A(14,1)))

19
Answer c)

• Let AugupUY be the upper-triangular matrix.

20
References

• Celik, Ismail, B., Introductory Numerical
  Methods for Engineering Applications, Ararat
  Books Publishing, LCC., Morgantown, 2001
• Fausett, Laurene, V. Numerical Methods,
  Algorithms and Applications, Prentice Hall, 2003
  by Pearson Education, Inc., Upper Saddle River,
  NJ 07458
• Rao, Singiresu, S., Applied Numerical Methods
  for Engineers and Scientists, 2002 Prentice Hall,
  Upper Saddle River, NJ 07458
• Mathews, John, H. Fink, Kurtis, D., Numerical
  Methods Using MATLAB Fourth Edition, 2004
  Prentice Hall, Upper Saddle River, NJ 07458
• Varol, A., Sayisal Analiz (Numerical Analysis),
  in Turkish, Course notes, Firat University, 2001
• http//mathonweb.com/help/backgd3e.htm