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Iterative Methods for Solution of Systems of Linear Equations By Erika Villarreal
1.  Jacobi Method Jacobi method  is an algorithm for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. Here comes your footer     Page  Given a square system of  n  linear equations: Where: Then  A  can be decomposed into a  diagonal  component  D , and the remainder  R : AX = b
1.  Jacobi Method Here comes your footer     Page  The system of linear equations may be rewritten as: and finally: The Jacobi method is an iterative technique that solves the left hand side of this expression for  x , using previous value for  x  on the right hand side. Analytically, this may be written as: The element-based formula is thus: Note that the computation of  x i ( k +1)  requires each element in  x ( k )  except itself.  Unlike the Gauss–Seidel method, we  can't overwrite  x i ( k )  with x i ( k +1) , as that value will be needed by the rest of the computation. This is the most meaningful difference between the Jacobi and Gauss–Seidel methods, and is the reason why the former can be implemented as a  parallel algorithm , unlike the latter. The minimum amount of storage is two vectors of size  n .
1.  Jacobi Method example Here comes your footer     Page  A linear system of the form  Ax  =  b  with initial estimate  x (0)  is given by We use the equation  x ( k  + 1)  =  D   − 1 ( b  −  Rx ( k ) ) , described above, to estimate  x . First,  we rewrite the equation in a more convenient form D   − 1 ( b  −  Rx ( k ) ) =  Tx ( k )  +  C , where   T  = −  D   − 1 R  and  C  =  D   − 1 b . Note that  R  =  L  +  U  where  L  and  U  are the strictly lower and upper parts of  A . From the known values we determine  T  = −  D   − 1 ( L  +  U )  as Further, C is found as
1.  Jacobi Method example Here comes your footer     Page  With T and C calculated, we estimate  x  as  x (1)  =  Tx (0)  +  C : The next iteration yields This process is repeated until convergence (i.e., until    is small). The solution after  25 iterations is
1.  Jacobi Method example Here comes your footer     Page
2.  Gauss–Seidel method Given a square system of  n  linear equations with unknown  x : where: Then  A  can be decomposed into a lower triangular component  L * , and a strictly upper triangular component  U : The system of linear equations may be rewritten as: Here comes your footer     Page
2.  Gauss–Seidel method Here comes your footer     Page  The Gauss–Seidel method is an iterative technique that solves the left hand side of this  expression for  x , using previous value for  x  on the right hand side. Analytically, this may be written as: However, by taking advantage of the triangular form of  L * , the elements of  x ( k +1)  can be  computed sequentially using forward substitution: Note that the sum inside this computation of  x i ( k +1)  requires each element in  x ( k )  except  x i ( k )  itself. The procedure is generally continued until the changes made by an iteration are below some tolerance .
2.  Gauss–Seidel method example Here comes your footer     Page  A linear system shown as   Ax=b  is given by   and   . We want to use the equation in the form where:   and   . We must decompose  A   into the sum of a lower triangular component L *  and a strict upper triangular component   U  :   and   The inverse of    is: .
2.  Gauss–Seidel method example Here comes your footer     Page  Now we can find: Now we have  T  and  C  and we can use them to obtain the vectors  X iteratively. First of all, we have to choose  X 0  : we can only guess. The better the guess, the quicker will perform the algorithm .We suppose: .
2.  Gauss–Seidel method example Here comes your footer     Page  We can then calculate: As expected, the algorithm converges to the exact solution: . In fact, the matrix A is diagonally dominant (but not positive definite).
Here comes your footer     Page  BIBLIOGRAPY This article incorporates text from the article  Jacobi_method  on  CFD-Wiki  that is under the  GFDL  license. Black, Noel; Moore, Shirley; and Weisstein, Eric W., " Jacobi method " from  MathWorld . Jacobi Method from  www.math-linux.com Module for Jacobi and Gauss–Seidel Iteration Numerical matrix inversion Gauss–Seidel from www.math-linux.com Module for Gauss–Seidel Iteration Gauss–Seidel  From Holistic Numerical Methods Institute Gauss Siedel Iteration from www.geocities.com The Gauss-Seidel Method

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Chapter 5

  • 1. Iterative Methods for Solution of Systems of Linear Equations By Erika Villarreal
  • 2. 1. Jacobi Method Jacobi method  is an algorithm for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. Here comes your footer  Page Given a square system of  n  linear equations: Where: Then  A  can be decomposed into a  diagonal  component  D , and the remainder  R : AX = b
  • 3. 1. Jacobi Method Here comes your footer  Page The system of linear equations may be rewritten as: and finally: The Jacobi method is an iterative technique that solves the left hand side of this expression for  x , using previous value for  x  on the right hand side. Analytically, this may be written as: The element-based formula is thus: Note that the computation of  x i ( k +1)  requires each element in  x ( k )  except itself. Unlike the Gauss–Seidel method, we can't overwrite  x i ( k )  with x i ( k +1) , as that value will be needed by the rest of the computation. This is the most meaningful difference between the Jacobi and Gauss–Seidel methods, and is the reason why the former can be implemented as a  parallel algorithm , unlike the latter. The minimum amount of storage is two vectors of size  n .
  • 4. 1. Jacobi Method example Here comes your footer  Page A linear system of the form  Ax  =  b  with initial estimate  x (0)  is given by We use the equation  x ( k  + 1)  =  D   − 1 ( b  −  Rx ( k ) ) , described above, to estimate  x . First, we rewrite the equation in a more convenient form D   − 1 ( b  −  Rx ( k ) ) =  Tx ( k )  +  C , where   T  = −  D   − 1 R  and  C  =  D   − 1 b . Note that  R  =  L  +  U  where  L  and  U  are the strictly lower and upper parts of  A . From the known values we determine  T  = −  D   − 1 ( L  +  U )  as Further, C is found as
  • 5. 1. Jacobi Method example Here comes your footer  Page With T and C calculated, we estimate  x  as  x (1)  =  Tx (0)  +  C : The next iteration yields This process is repeated until convergence (i.e., until   is small). The solution after 25 iterations is
  • 6. 1. Jacobi Method example Here comes your footer  Page
  • 7. 2. Gauss–Seidel method Given a square system of  n  linear equations with unknown  x : where: Then  A  can be decomposed into a lower triangular component  L * , and a strictly upper triangular component  U : The system of linear equations may be rewritten as: Here comes your footer  Page
  • 8. 2. Gauss–Seidel method Here comes your footer  Page The Gauss–Seidel method is an iterative technique that solves the left hand side of this expression for  x , using previous value for  x  on the right hand side. Analytically, this may be written as: However, by taking advantage of the triangular form of  L * , the elements of  x ( k +1)  can be computed sequentially using forward substitution: Note that the sum inside this computation of  x i ( k +1)  requires each element in  x ( k )  except  x i ( k )  itself. The procedure is generally continued until the changes made by an iteration are below some tolerance .
  • 9. 2. Gauss–Seidel method example Here comes your footer  Page A linear system shown as  Ax=b  is given by  and  . We want to use the equation in the form where:  and  . We must decompose A   into the sum of a lower triangular component L * and a strict upper triangular component  U :   and  The inverse of   is: .
  • 10. 2. Gauss–Seidel method example Here comes your footer  Page Now we can find: Now we have  T  and  C  and we can use them to obtain the vectors  X iteratively. First of all, we have to choose  X 0 : we can only guess. The better the guess, the quicker will perform the algorithm .We suppose: .
  • 11. 2. Gauss–Seidel method example Here comes your footer  Page We can then calculate: As expected, the algorithm converges to the exact solution: . In fact, the matrix A is diagonally dominant (but not positive definite).
  • 12. Here comes your footer  Page BIBLIOGRAPY This article incorporates text from the article  Jacobi_method  on  CFD-Wiki  that is under the  GFDL  license. Black, Noel; Moore, Shirley; and Weisstein, Eric W., " Jacobi method " from  MathWorld . Jacobi Method from www.math-linux.com Module for Jacobi and Gauss–Seidel Iteration Numerical matrix inversion Gauss–Seidel from www.math-linux.com Module for Gauss–Seidel Iteration Gauss–Seidel  From Holistic Numerical Methods Institute Gauss Siedel Iteration from www.geocities.com The Gauss-Seidel Method