gauss y gauss jordan
Post on 25-May-2015
2.847 Views
Preview:
TRANSCRIPT
CONSTRUIMOS FUTURO
GAUSS AND GAUSS-JORDAN ELIMINATION
JONATHAN ORLANDO CELIS ARIASCOD: 2061226
CONSTRUIMOS FUTURO
CONTENT
Gauss Elimination
Gauss Elimination example
Gauss-Jordan Elimination
Gauss-Jordan Elimination example
Problems with Gauss-Jordan elimination
CONSTRUIMOS FUTURO
GAUSS ELIMINATION We will now explore a more versatile way than
the method of determinants to determine if a
system of equations has a solution.
We will indeed be able to use the results of this
method to find the actual solution(s) of the
system (if any).
It should be noted that this method can be
applied to systems of equations with an
unequal number of equations and unknowns.
CONSTRUIMOS FUTURO
GAUSS ELIMINATION EXAMPLE
• Consider the system of equations:
• To solve for x, y, and z we must eliminate some of the unknowns from some of the equations. Consider adding -2 times the first equation to the second equation and also adding 6 times the first equation to the third equation. The result is:
CONSTRUIMOS FUTURO
GAUSS ELIMINATION EXAMPLE
• We have now eliminated the x term from the last two equations. Now simplify the last two equations by dividing by 2 and 3, respectively:
• To eliminate the y term in the last equation, multiply the second equation by -5 and add it to the third equation:
CONSTRUIMOS FUTURO
GAUSS ELIMINATION EXAMPLE
• The third equation says z=-2.
• Substituting this into the second equation yields y=-1.
• Using both of these results in the first equation gives x=3.
• The process of progressively solving for the unknowns is called back-substitution, this is the essence of Gaussian elimination.
CONSTRUIMOS FUTURO
GAUSS-JORDAN ELIMINATION• This is a variation of Gaussian elimination.• It is done by manipulating the given matrix using the
elementary row operations to put the matrix into row echelon form.
• To be in row echelon form, a matrix must conform to the following criteria:
1. If a row does not consist entirely of zeros, then the first non zero number in the row is a 1.(the leading 1)
2. If there are any rows entirely made up of zeros, then they are grouped at the bottom of the matrix.
3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right that the leading 1 in the higher row.
CONSTRUIMOS FUTURO
GAUSS-JORDAN ELIMINATION EXAMPLE
• Consider the system of equations:
R2 - (-1)R1 --> R2 R3 - ( 3)R1 --> R3
(-1)R2 --> R2 R3 - (-10)R2 --> R3
CONSTRUIMOS FUTURO
GAUSS-JORDAN ELIMINATION EXAMPLE
(-1/52)R3 --> R3
R2 - (-5)R3 --> R2 R1 - (2)R3 --> R1
CONSTRUIMOS FUTURO
GAUSS-JORDAN ELIMINATION EXAMPLE
• It is now obvious, by inspection, that the solution to this linear system is x=3, y=1, and z=2. Again, by solution, it is meant the x, y, and z required to satisfy all the equations simultaneously
R1 - (1)R2 --> R1
CONSTRUIMOS FUTURO
PROBLEMS WITH GAUSS-JORDAN ELIMINATION
• There are some problems that could arise while searching for these
solutions.
1. If the lines are parallel then they will not intersect and thus provide no
solution.
2. Another problem that may arise is a division by zero. If a zero is placed in
the main diagonal of the row being operated on, when you divide that row
by the diagonal number the division by zero error will occur.
3. If the matrix is small then the error won't have time to propagate; but if the
matrix is large, the round off error could deem the output solution unreliable.
CONSTRUIMOS FUTURO
THANK YOU FOR YOUR ATTENTION
top related