1. ESCUELA DE INGENIERÍA DE PETROLEOS
DIRECT METHODS FOR THE
SOLUTION OF SYSTEMS OF
LINEAR EQUATIONS
NORAIMA ZARATE GARCIA
COD: 2073173
ING. DE PETROLEOS
NUMERICAL METHODS IN ING. PETROLEUM

2. ESCUELA DE INGENIERÍA DE PETROLEOS
LU DECOMPOSITION
Its name is derived from the English words "Lower" and "Upper", which in Spanish
translates as "Lower" and "Superior." Studying the process followed in the LU
decomposition is possible to understand the reason for this name analyzing how an
original matrix is decomposed into two triangular matrices, an upper and lower.
LU Decomposition involves only operations on the coefficient matrix [A], providing an
efficient means to calculate the inverse matrix or solving systems of linear algebra. First
you must obtain the matrix [L] and matrix [U]
[L] is a diagonal matrix with numbers less than 1 on the diagonal. [U] is an upper diagonal
matrix on the diagonal which does not necessarily have to be number one.
The first step is to break down or transform [A] [L] and [U], ie to obtain the lower
triangular matrix [L] and the upper triangular matrix [U]
NUMERICAL METHODS IN ING. PETROLEUM

3. ESCUELA DE INGENIERÍA DE PETROLEOS
STEPS TO FIND THE UPPER TRIANGULAR MATRIX [U]
.Hacer zero all values below the pivot without turning this into a.
Achieve the above.
 .In order is required to obtain a factor which is necessary to convert values to zero
below the pivot.
.Dicho factor is equal to the number you want to make the number zero pivot.
This factor multiplied by -1 is then multiplied by the pivot and this result is added value
that is in the position to change (the value in the position to become zero). That is:
- factor * pivot + position changes
NUMERICAL METHODS IN ING. PETROLEUM

4. ESCUELA DE INGENIERÍA DE PETROLEOS
STEPS TO FIND THE LOWER TRIANGULAR MATRIX [L]
To find the lower triangular matrix seeks to zero values above each pivot, as well as become
an every pivot. It uses the same concept of "factor" described above and are located all the
"factors" below the diagonal as appropriate for each.
Esquematicamnete seeks the following:
NUMERICAL METHODS IN ING. PETROLEUM

5. ESCUELA DE INGENIERÍA DE PETROLEOS
STEPS TO SOLVE A SYSTEM OF EQUATIONS
BY THE LU DECOMPOSITION METHOD
1.Obtener lower triangular matrix L and upper triangular matrix U.
2.Resolver Ly = b (to find y).
3.The result of previous step is saved in a new array named "y".
4.Realizar Ux = y (to find x).
5.The result of previous step is stored in a new array called "x",
which provides the values for the unknowns of the equation
NUMERICAL METHODS IN ING. PETROLEUM

6. ESCUELA DE INGENIERÍA DE PETROLEOS
METHOD OF THOMAS
This method emerges as a simplification of an LU factorization of a tridiagonal
matrix.
It follows that:
NUMERICAL METHODS IN ING. PETROLEUM

7. ESCUELA DE INGENIERÍA DE PETROLEOS
CHOLESKY FACTORIZATION
The method of Cholesky factorization is applied to positive definite matrices where the
system Ax = b can be written as L (LTX) = b where L is a lower triangular matrix and its
transpose Lt; be replaced LTX = y and solve the lower triangular system Ly = b by forward
substitution obtained and then resolve the upper triangular system LTX = y by back
substitution process.
This method is unique in that each pivot element is obtained by calculating the square root of
lii, lii where is the diagonal element of L which is in row i, column i.
Ax  b
U L T
LL x  b
T
NUMERICAL METHODS IN ING. PETROLEUM

8. ESCUELA DE INGENIERÍA DE PETROLEOS
L
LT
 L11   L11 L21  Ln 2,1 Ln 1,1 Ln,1 
L L22   L22  Ln 2, 2 Ln 1, 2 Ln, 2 
 21   
          
   
 Ln 2,1 Ln 2, 2  Ln 2,n 2   Ln 2,n 2 Ln 1,n 2 Ln,n 2 
 Ln 1,1 Ln 1, 2  Ln 1,n 2 Ln 1,n 1   Ln 1,n 1 Ln,n 1 
   
 Ln,1 Ln, 2
  Ln,n 2 Ln,n 1 Ln,n  
  Ln,n  
 a11 a12  a1,n 2 a1,n1 a1,n 
a a22  a2,n 2 a2,n1 a2,n 
A =LLT
 21 
       
 
an 2,1 an 2, 2  an 2,n 2 an 2,n 1 an 2,n 
 an 1,1 an 1, 2  an 1,n 2 an 1,n 1 an 1,n 
 
 an,1 an, 2
  an,n 2 an,n 1 an,n  
A NUMERICAL METHODS IN ING. PETROLEUM

9. ESCUELA DE INGENIERÍA DE PETROLEOS
From the product of the n-th row of L by the n-th column of LT we have:
Ln ,12  Ln , 2 2    Ln , n  2 2  Ln , n 12  Ln n 2  a n n
Ln n 2  a n n  Ln ,12  Ln , 2 2    Ln , n  2 2  Ln , n 12
n 1
Ln n 2
 an n  L
j 1
n, j
2
n 1
Ln n  an n  L
j 1
n, j
2
Making the sweep from k = 1 to n we have:
k 1
Lkk  akk   Lk , j
2
j 1
NUMERICAL METHODS IN ING. PETROLEUM

10. ESCUELA DE INGENIERÍA DE PETROLEOS
GAUSS ELIMINATION METHOD
Is to transform the coefficient matrix of the linear system
in an upper triangular matrix by appropriate linear combinations of the equations of
the system.
To transform the matrix A into an upper triangular matrix is proceder'a step by
step, column by column. For the column i-'esima is realizar'an Equation linear
combinations between the i-'esima and each of the remaining j-'esimas equations
with j = i + 1 i + 2;:::: n , so that the elements become zeros aji, for column i-
'esima. These linear combinations of equations affects the matrix and the vector b.
NUMERICAL METHODS IN ING. PETROLEUM

11. ESCUELA DE INGENIERÍA DE PETROLEOS
BIBLIOGRAPHY
 es. Wikipedia.org.com
Presentación de sistemas especiales por el
Profesor Eduardo Carrillo Zambrano.
NUMERICAL METHODS IN ING. PETROLEUM