Special matrices


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Special matrices

  1. 1. Philipp Ludwig von Seidel Johann Carl Friedrich Gauss 23 October 1821, Zweibrücken, Germany 30 April 1777 Brunswick – 23 February – 13 August 1896, Munich 1855 Brunswick
  2. 2. SPECIAL MATRICES BANDED MATRIX Banded matrix is a square matrix that has all elements equal to zero, with the exception of a band centered on the main diagonal. Banded system is frequently encountered in engineering and scientific practice because of they typically occurred in the solution of differential equation. The dimensions of a banded system can be quantified by two parameters: • The banded width BW • The half bandwidth HBW These two values are related by BW=2HBW+1, = − HBW
  3. 3. SPECIAL MATRICES TRIDIAGONAL BANDED A tridiagonal matrix is a matrix that is almost a diagonal matrix. To be exact: a tridiagonal matrix has nonzero elements only in the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal
  4. 4. JACOBI METHOD The 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. The method is named after German mathematician Carl Gustav Jakob Jacobi
  5. 5. JACOBI METHOD This method makes two assumptions: 1. That the system given by the next system equation has a unique solution 2. That the coefficient matrix A has no zeros on its main diagonal. If any of the diagonal entries are zero, then rows or columns must be interchanged to obtain a coefficient matrix that has nonzero entries on the main diagonal.
  6. 6. JACOBI METHOD To begin the Jacobi method, solve the first equation for the second equation for and so on, as follows Then make an initial approximation of the solution 1, 2 , 3,… , initial aproximation, and substitute xi these values of into the right-hand side of the rewritten equations to obtain the first approximation. After this procedure has been completed, one iteration has been performed. In the same way, the second approximation is formed by substituting the first approximation’s x-values into the right-hand side of the rewritten equations. By repeated iterations, you will form a sequence of approximations that often converges to the actual solution.
  7. 7. JACOBI METHOD Example Use the Jacobi method to approximate the solution of the following system of linear equations. Continue the iterations until two successive approximations are identical when rounded to three significant digits. Solution To begin, write the system in the form : Because you do not know the actual solution, choose Initial approximation
  8. 8. JACOBI METHOD As a convenient initial approximation. So, the first approximation is Continuing this procedure, you obtain the sequence of approximations shown in Table n 0 1 2 3 4 5 6 7 1 0,000 -0,200 0,146 0,192 0,181 0,185 0,186 0,186 2 0,000 0,222 0,203 0,328 0,332 0,329 0,331 0,331 3 0,000 -0,429 -0,517 -0,416 -0,421 -0,424 -0,423 0,423
  9. 9. JACOBI METHOD Because the last two columns in Table are identical, you can conclude that to three significant digits the solution is For the system of linear equations given in Example 1, the Jacobi method is said to converge. That is, repeated iterations succeed in producing an approximation that is correct to three significant digits. As is generally true for iterative methods, greater accuracy would require more iterations.
  10. 10. THE GAUSS-SEIDEL METHOD You will now look at a modification of the Jacobi method called the Gauss-Seidel method, named after Carl Friedrich Gauss (1777–1855) and Philipp L. Seidel (1821– 1896). This modification is no more difficult to use than the Jacobi method, and it often requires fewer iterations to produce the same degree of accuracy
  11. 11. THE GAUSS-SEIDEL METHOD With the Jacobi method, the values of obtained in the nth approximation remain unchanged until the entire( + 1) th approximation has been calculated. With the Gauss- Seidel method, on the other hand, you use the new values of each as soon as they are known. That is, once you have determined 1 from the first equation, its value is then used in the second equation to obtain the new 2 Similarly, the new 1 and 2 are used in the third equation to obtain the new 3 and so on. Example Use the Gauss-Seidel iteration method to approximate the solution to the system of equations given by
  12. 12. THE GAUSS-SEIDEL METHOD Solution To begin, write the system in the form : Using 1, 2 , 3 = 0,0,0 as the initial approximation, you obtain the following new value for 1 Now that you have a new value for 1 , however, use it to compute a new value for 2 That is,
  13. 13. THE GAUSS-SEIDEL METHOD Similarly, use 1 = −0,200 2 = 0,156 to compute a new value for 3 that is So the first approximation is 1 = −0,200 2 = 0,156 and 3 = −0,508 Continued iterations produce the sequence of approximations shown in Table n 0 1 2 3 4 5 1 0,000 -0,200 0,167 0,191 0,186 0,186 2 0,000 0,156 0,334 0,333 0,331 0,331 3 0,000 -0,508 -0,429 -0,422 -0,423 -0,423 Note that after only five iterations of the Gauss-Seidel method, you achieved the same accuracy as was obtained with seven iterations of the Jacobi method
  14. 14. THE GAUSS-SEIDEL METHOD An Example of Divergence Apply the Jacobi method to the system Repeated iterations produce the sequence of approximations shown in Table using the initial approximation 1, 2 = 0,0 and show that the method diverges. n 0 1 2 3 4 5 6 7 1, 0 -4 -34 -174 -1244 -6124 -42,874 -214,374 2 0 -6 -34 -244 -1244 -8574 -42874 -300,124 For this particular system of linear equations you can determine that the actual solution is x1=1 and x2=1. So you can see from Table, that the approximations given by the Jacobi method become progressively worse instead of better, and you can conclude that the method diverges.
  15. 15. THE GAUSS-SEIDEL METHOD The problem of divergence in Example 3 is not resolved by using the Gauss- Seidel method rather than the Jacobi method. In fact, for this particular system the Gauss-Seidel method diverges more rapidly, as shown in Table n 0 1 2 3 4 5 1 0 -4 -174 -6124 -214,374 -7,503,124 2 0 -34 -1224 -42,874 -1,500624 -52,521,874 With an initial approximation of 1, 2 = 0,0 neither the Jacobi method nor the Gauss-Seidel method converges to the solution of the system of linear equations given in Example
  16. 16. BIBLIOGRAPHY • http://college.cengage.com/mathematics/larson/elementary_linear/5e /students/ch08-10/chap_10_2.pdf • Numerical Methods for Engineers, Fifth Edition, Steven C. Chapra and Raymond P. Canale.