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OSCAR EDUARDO MENDIVELSO OROZCO I study Petroleum Engineering Numerical Methods
INTRODUCTION The problem to be solved in this chapter is that of a system of  m  linear equations with  n  unknowns in a matrix defined by: Where  A ∈  IR (M, n)  and  b ∈  IR m  are data and  x ∈  IR n  is the vector incognita. Can be explicitly written as: (1) Usually assume that  m  =  n  and that the system has only solution, ie det  A  =0, or, rg  A  =  n.
LU FACTORIZATION METHOD The easiest way to explain the LU method is illustrating the basic Gauss method through an example, as is the case of the given matrix and then applying the procedure to a system of four equations with four unknowns:
In the first step, we multiply the first equation by  12 / 6 = 2  and subtract the second, then multiply the first equation by  3 / 6 = 1 / 2  and subtract the third and finally multiply the first equation by  -6 / 6 =- 1  and subtract the fourth. The numbers  2, ½  and  -1  are the multipliers of the first step in the process of elimination. Number  6  is the pivotal element of this first step and the first row, that remains unchanged is called the pivot row. The system now looks like this:
In the next step of the process, the second row is used as pivot row and  -4  as the new pivot element we apply the process: multiply the second row by -  12/-4 = 3  and the remainder of the third and then multiply the second row  2 / (-4) = - 1/2  and subtract the fourth.  The multipliers are in this case  3  and  -1/2   and the system of equations reduces to:
The last step is to multiply the third equation by  4 / 2 = 2  and subtract the fourth. The resulting system turns out to be: The resulting system is upper triangular and equivalent to the original system (the solutions of both systems overlap.) However, this system is easily solvable by applying the backward substitution algorithm. The solution of the system of equations turns out to be:
If we put the multiplier used to transform the system into a unit lower triangular matrix  (L)  occupying each position of helping to produce zero, we obtain the following matrix: Moreover, the upper triangular matrix  (U)  formed by the coefficients resulting after applying the Gauss algorithm (2) is:
These two matrices give us the  LU  factorization of the initial matrix of coefficients,  A,  expressed by equation (1):
 
 
REFERENCES http://www.mty.itesm.mx/etie/deptos/m/ma95-843/lecturas/l843-13.pdf
 

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Direct methods

  • 1. OSCAR EDUARDO MENDIVELSO OROZCO I study Petroleum Engineering Numerical Methods
  • 2. INTRODUCTION The problem to be solved in this chapter is that of a system of  m  linear equations with  n  unknowns in a matrix defined by: Where  A ∈  IR (M, n) and  b ∈  IR m are data and  x ∈  IR n is the vector incognita. Can be explicitly written as: (1) Usually assume that  m  =  n  and that the system has only solution, ie det  A  =0, or, rg  A  =  n.
  • 3. LU FACTORIZATION METHOD The easiest way to explain the LU method is illustrating the basic Gauss method through an example, as is the case of the given matrix and then applying the procedure to a system of four equations with four unknowns:
  • 4. In the first step, we multiply the first equation by  12 / 6 = 2  and subtract the second, then multiply the first equation by  3 / 6 = 1 / 2  and subtract the third and finally multiply the first equation by  -6 / 6 =- 1  and subtract the fourth. The numbers  2, ½  and  -1  are the multipliers of the first step in the process of elimination. Number  6  is the pivotal element of this first step and the first row, that remains unchanged is called the pivot row. The system now looks like this:
  • 5. In the next step of the process, the second row is used as pivot row and  -4  as the new pivot element we apply the process: multiply the second row by -  12/-4 = 3  and the remainder of the third and then multiply the second row  2 / (-4) = - 1/2  and subtract the fourth.  The multipliers are in this case  3  and  -1/2   and the system of equations reduces to:
  • 6. The last step is to multiply the third equation by  4 / 2 = 2  and subtract the fourth. The resulting system turns out to be: The resulting system is upper triangular and equivalent to the original system (the solutions of both systems overlap.) However, this system is easily solvable by applying the backward substitution algorithm. The solution of the system of equations turns out to be:
  • 7. If we put the multiplier used to transform the system into a unit lower triangular matrix  (L)  occupying each position of helping to produce zero, we obtain the following matrix: Moreover, the upper triangular matrix  (U)  formed by the coefficients resulting after applying the Gauss algorithm (2) is:
  • 8. These two matrices give us the  LU  factorization of the initial matrix of coefficients,  A,  expressed by equation (1):
  • 9.  
  • 10.  
  • 12.