Directs Methods

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Directs Methods

  1. 1. DIRECTS METHODS<br />
  2. 2. GAUSSIAN ELIMINATION<br />A system of equations is solved by the Gauss when their solutions are obtained by reducing an equivalent system given in which each equation has one fewer variables than the last.<br />When applying this process, the resulting matrix is known as: “Form staggered”. <br />
  3. 3. EXAMPLE <br />Solve the following system of linear equations by Gaussian Elimination method.<br />
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  7. 7. The Gauss-Jordan is a variant of the method of Gauss. When you delete an unknown in an equation, Gauss-Jordan eliminates that unknown quantity in the rest of the equations, the basis for disposing the pivot equation. Also all the rows are normalized when taken as pivot equation. The end result of such disposal creates an identity matrix instead of a triangular Gauss as it does, so do not use the back substitution.<br />GAUSS-JORDAN<br />
  8. 8. Solve the following system of linear equations by Gauss-Jordan method.<br />EXAMPLE <br />
  9. 9. Then we write the augmented matrix of the system.<br />We take this matrix to reduced row-echelon form by elementary operations on the rows of the matrix, for this, write the matrix and then an arrow. Above this arrow will indicate the operations we're making so that the reader can follow the development.<br />
  10. 10. Development to obtain the reduced row echelon form.<br />
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  14. 14. The final reduced row echelon matrix indicates that:The solution of the system is x = 1, y =- 1 and z = 2.<br />
  15. 15. It is a kind of direct method for calculating matrices.<br />The system consists of taking a system of equations as an equation as a pivot in order to give a matrix identical to the system of equations. When you delete an unknown in an equation, Gauss-Jordan eliminates that unknown in the rest of the equations.<br />GAUSS-JORDANWITH PIVOTING<br />
  16. 16. The number arr in the coefficient matrix A that is used to eliminate akr, where k = r + 1, r + 2, . . . , N, is called the r thpivotal element, and the r throw is called the pivot row. <br />The following example illustrates how to use the operations to obtain an equivalent upper-triangular system UX = Y from a linear system AX = B, where A is an N × N matrix.<br />
  17. 17. EXAMPLE <br />Solve the following system of linear equations by Gauss-Jordan with pivoting method. <br />
  18. 18. Theaugmentedmatrixis:<br />
  19. 19. The first row is used to eliminate elements in the first column below the diagonal. We refer to the first row as the pivotal row and the element a11 = 1 is called the pivotal element. The values mk1 are the multiples of row 1 that are to be subtracted from row k for k = 2, 3, 4. The result after elimination is:<br />
  20. 20. The second row is used to eliminate elements in the second column that lie below the diagonal. The second row is the pivotal row and the values mk2 are the multiples of row 2 that are to be subtracted from row k for k = 3, 4. The result after elimination is: <br />
  21. 21. Finally, the multiple m43 = −1.9 of the third row is subtracted from the fourth row, and the result is the upper-triangular system. <br />
  22. 22. The back-substitution algorithm can be used to solve, and we get: x4= 2, x3 = 4, x2 = −1, x1 = 3.<br />
  23. 23. http://cbi.azc.uam.mx/archivos/varios/ProblemarioW.pdf<br />http://www.ecs.fullerton.edu/~mathews/n2003/gaussjordan/GaussianJordanProof.pdf<br />BIBLIOGRAPHY<br />

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