direct methods for solving systems of linear equations
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Presented by:
Acosta Creus Mileidy Lorena 2073699
Teacher:
Ph.D. Eduardo Carrillo Zambrano
Numerical Methods in Petroleum Engineering
INDUSTRIAL UNIVERSITY OF SANTANDER
2010
1. CONVENTIONAL METHODS OF SOLUTION
1.1. Mathematical background
1.2. Solving Small Numbers Of Equations
1.2.1. Graphical method
1.2.2. Cramer’s rule
1.2.3. Elimination of Unknowns
2. TECHIQUES FOR IMPROVING SOLUTIONS
2.1. Use of more significant figures
2. 2.Pivoting
2.3. Scaling
3. COMPLEMENTARY TECHNIQUES
3.1. Gauss-Jordan
3.2. LU Decomposition
4. BIBLIOGRAPHY
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mnmjmm
inijii
nj
nj
aaaa
aaaa
aaaa
aaaa
A
......
::::
......
::::
......
......
21
21
222221
111211
An matrix A of mxn is a
rectangular array of mn numbers
arranged in m rows and n
columns. The component or
element ij of A, denoted by aij is
the number that appears in the
row i and column j of A.
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TYPE DESCRIPTION EXAMPLE
SQUARE
Matrix A (mxn) where the
number of rows equals the
number of columns, ie m = n.
TRANSPOSE
To A = (aij) a matrix of m x n.
The transpose is obtained by
exchanging the rows by the
columns of A.
SYMMETRIC
It’s a square matrix that is
equal to its transpose
1 2 32 3
4 0 5 B1 5
3 1 2
A
1 2 4 61 2 4 6
2 7 3 5 2 7 3 5
4 3 8 0 4 3 8 06 5 0 4
6 5 0 4
tB B
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TYPE DESCRIPTION EXAMPLE
UPPER AND
LOWER
TRIANGULAR
Triangular matrix is a special
kind of square matrix whose
elements above or below its
main diagonal are zero
Uppper Lower
AUGMENTED
It’s obtained by combining two
matrices.
BANDED
It has all the elements equal to
zero, with the exception of a
band centered on the main
diagonal
11 12
21 22 23
32 33 34
43 44
a a
a a a
A
a a a
a a
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TYPE DESCRIPTION EXAMPLE
DIAGONAL
It’s a square matrix where all the
elements outside the main
diagonal are zero
IDENTITY
It’s a diagonal matrix where all
elements on the main diagonal
are equal to 1
11
22
33
44
0 0 0
0 0 0
0 0 0
0 0 0
a
a
A
a
a
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
A
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The following are several methods
that are appropriate for solving
small (n ≤3) sets of simultaneous
equations and that don’t require a
computer.
These are:
• Graphical method
• Determinants and Cramer’s rule
• Elimination of Unknowns
3
2
5
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A graphical solution is obtainable for
two equations by plotting them on
Cartesian coordinates with one axis
corresponding to x1 and the other to
x2, because each equation is a straight
line.
a11x
1 + a
12x
2 = b
1
a21
x1 + a
22x
2 = b
2
Despejando x2
3
2
5
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Beyond three equations, graphical methods break down and,
consequently, have little practical value for solving
simultaneous equations. There are three cases that can pose
problems when solving sets of linear equations:
x1
x2
x1
x2
x1
x2
a) No solution b) Infinite solutions c) ill-conditioned system
3
2
5
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EXAMPLE:
Let x1 be the abscissa. Solve both
equations for x2:
3x1 + 2x
2 = 18
- x1 + 2x
2 = 2
𝑥2 = −3
2𝑥1 + 9
𝑥2 =1
2𝑥1 + 1
Solve:
X1=4 ; X
2=3
With the graphical method
to solve:
3
2
5
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Each unknown in a system of linear
algebraic equations may be expressed as a
fraction of two determinants with
denominator D and with the numerator
obtained from D by replacing the column
of coefficients of the unknown in question
by the constants b1,
b2,…,
b
n.
For example,
FUENTE:
http://auladeblanca.blogspot.com
Gabriel Cramer
3
2
5
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With Cramer’s rule to solve:
0.3x1 + 0.52x
2 + x
3= -0.01
0.5x1 + x
2 + 1.9x
3= 0.67
0.1x1 + 0.3x
2 + 0.5x
3= -0.44
Solution
D=
0.3 0.52 10.5 1 1.90.1 0.3 0.5
EX
AM
PLE
To write the determinant D: 1
3
2
5
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A1=
1 1.90.3 0.5
=1(0.5)-1.9(0.3)=-0.07
A1=
0.5 1.90.1 0.5
=0.5(0.5)-1.9(0.1)=0.06
A1=
0.5 10.1 0.3
=0.5(0.3)-1(0.1)=0.05
To solve the minors: 2 Evaluating the determinant 3
D=0.3(-0.07)-0.52(0.06)+1(0.05)
D=-0.0022
These can be used to
evaluate the determinant,
so:
3
2
5
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x1=
−0.01 0.52 10.67 1 1.9
−0.44 0.3 0.5−0.0022
=
0.03278
−0.0022=-14.9
x2=
0.3 −0.01 10.5 0.67 1.90.1 −0.44 0.5
−0.0022=
0.0649
−0.0022=-29.5
x3=
0.3 0.52 −0.010.5 1 0.670.1 0.3 −0.445
−0.0022=
−0.04356
−0.0022=19.8
Employing Cramer’s rule: 4
«For more than three
equations, Cramer´s rule
becomes impractcal
because, as the number of
equatioms increases, the
determinants are time
consuming to evaluate by
hand»
3
2
5
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The basic strategy is to multiply the equations by constants so
that one of the unknowns will be eliminated when the two
equations are combined.
The result is a single equation that can be solved for the
remaining unknown.
a11x
1 + a
12x
2 = b
1 a
21a
11x
1 + a
21a
12x
2 = b
1a
21
a21
x1 + a
22x
2 = b
2 a
21a
11x
1 + a
22a
11x
2 = b
1a
11
3
2
5
SCHOOL OF PETROLEUM ENGINEERING
EXAMPLE:
3x1 + 2x
2 = 18
- x1 + 2x
2 = 2
𝑥1 =2 18 −2(2)
3 2 −2(−1)=4 𝑥2 =
3 2 −18(−1)
3 2 −2(−1)=3
With elimination of unknownse:
Using the last equations:
These values are equals to graphical methods.
ESCUELA DE INGENIERÍA DE PETRÓLEOS
2.1. Use of more
significant figures
• To use more significant
figures in the
computation.
2.2. Pivoting
• Problems occur
when a pivot
element is nearly or
is equal to zero.
• Partial pivoting
(Rows are switched)
• Complete partial
(Columns and rows
are exchanged)
2.3. Scaling
• It has utility in
minimizing round-off
those cases where
some of the equations
in a system have much
larger coefficients than
others.
ESCUELA DE INGENIERÍA DE PETRÓLEOS
ESCUELA DE INGENIERÍA DE PETRÓLEOS
Gaussian elimination is an algorithm of linear algebra to
determine the solutions of a system of linear equations, matrices
and inverse found.
To convert linear equation system to banded matrix 1
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
.....
.....
.....
2211
22222121
11213112
mmnmm
n
n
b
b
b
aaa
aaa
aaa
2
1
21
22221
11211
.......
........
........
........_ n21 xxx
ESCUELA DE INGENIERÍA DE PETRÓLEOS
Go to the leftmost nonzero column 2
mmnmm
n
n
b
b
b
aaa
aaa
aaa
2
1
21
22221
11211
.......
........
........
........_ n21 xxx
If the first line has a zero in this column, swap it with another
that does not have
3
0,,0 1211 aa
mmnmm
n
n
b
b
b
aaa
aaa
aaa
2
1
12
22122
11112
.......
........
........
ESCUELA DE INGENIERÍA DE PETRÓLEOS
Get zeros below the pivot, adding appropriate multiples of row
than the row below it
4
12 1 1
11 12 1 1
11 11 11
21 22 2 2
21 22 2 2
11
1 2
1 2
................
........ 1........( )
.......
.......
n
n
n
n
m m mn m
m m mn m
a a b
a a a b
a a a
a a a b Fa a a b
a
a a a b
a a a b
ESCUELA DE INGENIERÍA DE PETRÓLEOS
Repeat the operations with of the other rows to obtain
the higher triangular matrix
3
1.......0........0
...1....0....0
........1...0
........1
3
213
11312
n
n
n
a
aa
aaa
Starting with the last line is not zero, move up to get
the identity matrix.
3
1.......0........0
0...1....0....0
........1...0
.........1
)(*)1(
1.......0........0
...1....0....0
........1...0
........1
223
11312
33
213
11312
n
n
nn
n
n
aa
aaa
aFnnFa
aa
aaa
ESCUELA DE INGENIERÍA DE PETRÓLEOS
Solution of the system of equations 3
bn
b
b
b
xn
x
x
x
3
2
1
.......0........0
0...3....0....0
0....0....2...0
0...0....0...1 1 1
2 2
n n
x b
x b
x b
ESCUELA DE INGENIERÍA DE PETRÓLEOS
EX
AM
PLE
322
1123
82
zyx
zyx
zyx
To find the unknowns of the linear equation system
using the method Gauss-Jordan
3212
11213
8112
1
2
F
3212
11213
42/12/11
ESCUELA DE INGENIERÍA DE PETRÓLEOS
EX
AM
PLE
3212
11213
42/12/11
312
3 21
FF
FF
5120
12/12/10
42/12/11
2*2F
5120
2110
42/12/11
ESCUELA DE INGENIERÍA DE PETRÓLEOS
EX
AM
PLE
3212
11213
42/12/11
312
3 21
FF
FF
5120
12/12/10
42/12/11
2*2F
5120
2110
42/12/11
ESCUELA DE INGENIERÍA DE PETRÓLEOS
EX
AM
PLE
2 1
2 3
(1/ 2)
2
F F
F F
1100
2110
3101
5120
2110
42/12/11
1*3
13
23
F
FF
FF
1100
3010
2001
ESCUELA DE INGENIERÍA DE PETRÓLEOS
The primary appeal of LU decomposition is that the time-
consuming elimination step can be formulated so that it involves
only operations on the matrix of coefficients (A), Thus it is well
suited for those situations where many right-hand-side vectors
(B) must be evaluated for a single value of (A).
The linear equation system can be rearranged to give 1
[A][x]=[B]
ESCUELA DE INGENIERÍA DE PETRÓLEOS
To express the linear equation system as an upper
triangular system
3
[L][D]=[B]
Assume that there is a lower diagonal matrix with 1’s
on the diagonal
2
[U][x]=[D]
𝑈 =
𝑢11 𝑢12 𝑢13
0 𝑢22 𝑢23
0 0 𝑢∗
𝑥1
𝑥2
𝑥3
=
𝑑1
𝑑2
𝑑3
L=1 0 0
𝑙12 1 0𝑙13 𝑙23 1
ESCUELA DE INGENIERÍA DE PETRÓLEOS
a) Decomposition
[A] [X]= [B]
[U] [L]
[L] [D]= [B]
[U] [X]= [D]
[X]
b) Forwards
c) Backward
Substitutio
n
Sum
mary o
f steps
[ D ]
ESCUELA DE INGENIERÍA DE PETRÓLEOS
EX
AM
PLE
Consider solving the system of equations by LU decomposition
1 2 3
1 2 3
1 2 3
2 3 4 6
4 5 10 16
4 8 2 2
x x x
x x x
x x x
Solution
2 3 4
4 5 10
4 8 2
A
1 0 0 2 3 4
2 1 0 y U= 0 1 2
2 2 1 0 0 6
L
ESCUELA DE INGENIERÍA DE PETRÓLEOS
1
2
3
2 3 4 6
0 1 2 4
0 0 6 2
x
x
x
3
3
2
3 2
1
0.333
4 23.333
1
6 4 37.333
2
x
xx
x xx
3
3
2
3 2
1
0.333
4 23.333
1
6 4 37.333
2
x
xx
x xx
7.333
3.333
0.333
x
Ux=y
So,
Finally, the solution to the linear system is given:
ESCUELA DE INGENIERÍA DE PETRÓLEOS
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