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Outline

Title
Abstract
FAQs
Figures
Introduction
The Jacobi Iterative Method
The Modified Jacobi Method
Accuracy and Convergence of Iterative Methods
Advantages of the Jacobi Method
Disadvantages of the Jacobi Method
References
All Topics
Mathematics
Numerical Analysis

On Some Iterative Methods for Solving Systems of Linear Equations

Profile image of Fadugba SundayFadugba Sunday

Abstract

This paper presents some iterative methods for solving system of linear equations namely the Jacobi method and the modified Jacobi method. The Jacobi method is an algorithm for solving system of linear equations with largest absolute values in each row and column dominated by the diagonal elements. The modified Jacobi method also known as the Gauss Seidel method or the method of successive displacement is useful for the solution of system of linear equations. The comparative results analysis of the two methods was considered. We also discussed the rate of convergence of the Jacobi method and the modified Jacobi method. Finally, the results showed that the modified Jacobi method is more efficient, accurate and converges faster than its counterpart “the Jacobi Method”

FAQs

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What are the key assumptions for the Jacobi method's convergence?add

The Jacobi method converges if the coefficient matrix A has non-zero diagonal entries and is strictly diagonally dominant. Specifically, the diagonal entry must be larger than the sum of the absolute values of the other entries in the row.

How does the speed of convergence compare between Jacobi and modified Jacobi methods?add

The modified Jacobi method converges approximately twice as fast as the standard Jacobi method. This is due to its use of the most recent values of all variables during computations.

What numerical advantages does the modified Jacobi method provide over Jacobi?add

The modified Jacobi method requires less computer storage and yields more accurate results compared to the Jacobi method. Additionally, it achieves a faster rate of convergence while minimizing the number of iterations.

Why are iterative methods preferred for large, sparse systems of equations?add

Iterative methods are less susceptible to round-off errors and are more efficient for large systems since they do not rely on full matrix storage. For systems with n greater than 50, these methods effectively handle sparsity and convergence.

What challenges does the Jacobi method face in achieving convergence?add

The Jacobi method may diverge if the matrix is not strictly diagonally dominant, which can greatly impede its effectiveness. It also requires more time to converge compared to the modified version.

Figures (13)
then rows or columns must be interchanged to obtain a coefficient matrix that has non zero entries on the main diagonal. In Jacobi method, each equation of the system is solved for the component of the solution vector associated with the diagonal element, that is, x; as follows:
then rows or columns must be interchanged to obtain a coefficient matrix that has non zero entries on the main diagonal. In Jacobi method, each equation of the system is solved for the component of the solution vector associated with the diagonal element, that is, x; as follows:
Equation (2.9) is called the first approximation. In the same way, the second approximation is formed by substituting the first approximation (x-values) into the right hand side of (2.9), and then we have By repeated iterations, a sequence of approximations that often converges to the actual solution is formed. In general,
Equation (2.9) is called the first approximation. In the same way, the second approximation is formed by substituting the first approximation (x-values) into the right hand side of (2.9), and then we have By repeated iterations, a sequence of approximations that often converges to the actual solution is formed. In general,
In the Jacobi method, all values of x'” are based on x”. The modified Jacobi method is similar to the Jacobi method, except that the most recently computed values of all x, are used in all computations. Like the Jacobi method, the modified Jacobi method requires diagonal dominance to ensure convergence. Thus, in the modified Jacobi method, each equation of the system is solved for the component of the solution vector associated with the diagonal element, that is, x; as follows:
In the Jacobi method, all values of x'” are based on x”. The modified Jacobi method is similar to the Jacobi method, except that the most recently computed values of all x, are used in all computations. Like the Jacobi method, the modified Jacobi method requires diagonal dominance to ensure convergence. Thus, in the modified Jacobi method, each equation of the system is solved for the component of the solution vector associated with the diagonal element, that is, x; as follows:
Let us consider the systems of linear equations of the form: We consider the system of linear equations of the form
Let us consider the systems of linear equations of the form: We consider the system of linear equations of the form
the modified Jacobi algorithm is obtained from the Jacobi algorithm, (2.5), by using x? values in the summation from (j = 1 toi — 1), (assuming the sweeps through the equations proceed fromi = / to n) is given by The modified Jacobi method is sometimes called the method of successive iteration because the most recent values of all x; are used in all the calculations. The modified Jacobi method which is also called the Gauss-Seidel method is generally converges faster than the Jacobi method.
the modified Jacobi algorithm is obtained from the Jacobi algorithm, (2.5), by using x? values in the summation from (j = 1 toi — 1), (assuming the sweeps through the equations proceed fromi = / to n) is given by The modified Jacobi method is sometimes called the method of successive iteration because the most recent values of all x; are used in all the calculations. The modified Jacobi method which is also called the Gauss-Seidel method is generally converges faster than the Jacobi method.
Table 3. The Jacobi Method for Problem 2
Table 3. The Jacobi Method for Problem 2
3.3. Discussion of Results and Conclusion Table 4. The Modified Jacobi Method for Problem 2
3.3. Discussion of Results and Conclusion Table 4. The Modified Jacobi Method for Problem 2
Table 2. The Modified Jacobi Method for Problem 1
Table 2. The Modified Jacobi Method for Problem 1
Table 1. The Jacobi Method for Problem 1
Table 1. The Jacobi Method for Problem 1

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References (76)

  1. -6.2000] [-7.4000] [-7.6000] [-7.8000] Table 4. The Modified Jacobi Method for Problem 2 n x1 x2 x3 x4
  2. -% JACOBI ITERATIVE METHOD FOR 3 BY 3 MATRICES
  3. -X1{1} = input('ENTER THE INITIAL VALUES OF X1 = ');
  4. -X2{1} = input('ENTER THE INITIAL VALUES OF X2 = ');
  5. -X3{1}= input('ENTER THE INITIAL VALUES OF X3 = ');
  6. 9-X4{1}=input('ENTER THE INITIAL VALUES OF X4 = ');
  7. -% ENTER VALUES FOR THE FIRST EQUATION 11-% X11+X12+X13+X14=A1 12-disp('Enter in this form a11X1+a12X2+a13X3=A1');
  8. -a11=input('Enter a11=');
  9. -a12=input('enter a12=');
  10. -a13=input('enter a13=');
  11. -a14=input('enter a14=');
  12. -A1=input('enter A1=');
  13. -% ENTER VALUES FOR THE SECOND EQUATION 19-% X21+X22+X23+X24=A2 20-disp('Enter in this form a21X1+a22X2+a23X3=A2');
  14. -a21=input('Enter a21=');
  15. -a22=input('enter a22=');
  16. -a23=input('enter a23=');
  17. -a24=input('enter a24=');
  18. -A2=input('enter A2=');
  19. -% ENTER VALUES FOR THE THIRD EQUATION 27-% X31+X32+X33+X34=A3 28-disp('Enter in this form a31X1+a32X2+a33X3=A1');
  20. -a31=input('enter a31=');
  21. -a32=input('enter a32=');
  22. -a33=input('enter a33=');
  23. -a34=input('enter a34=');
  24. -A3=input('enter A3=');
  25. -% ENTER VALUES FOR THE FOURTH EQUATION 35-% X31+X32+X33+X34=A3
  26. 36-%disp('Enter in this form a31X1+a32X2+a33X3=A1');
  27. -a41=input('enter a41=');
  28. -a42=input('enter a42=');
  29. 39-a43=input('enter a43=');
  30. -a44=input('enter a44=');
  31. -A4=input('enter A4=');
  32. -%X1{0}=X1; 43-%X2{0}=X2; 44-%X3{0}=X3; 45-%X4{0}=X4; 46-for N= 1:N0 47-% R CALCULATION 48-R1{N}=A1+(a11*X1{N})+(a12*X2{N})+(a13*X3{N});
  33. %+(a14*X4{N});
  34. -R2{N}=A2+(a21*X1{N})+(a22*X2{N})+(a23*X3{N});
  35. %+(a24*X4{N});
  36. -R3{N}=A3+(a31*X1{N})+(a32*X2{N})+(a33*X3{N});
  37. %+(a34*X4{N});
  38. -R4{N}=A4+(a41*X1{N})+(a42*X2{N})+(a43*X3{N});
  39. %+(a44*X4{N});
  40. -% calculate values for x 53-X1{N+1}=X1{N}+(R1{N}/abs(a11));
  41. -X2{N+1}=X2{N}+(R2{N}/abs(a11));
  42. -RESULT={N, X1{N+1}, X2{N+1}, X3{N+1}}, X4{N+1}}; 58-disp(RESULT);
  43. -if((X1{N}==X1{N+1}) && (X2{N}==X2{N+1}) && (X3{N}==X3{N+1})&&(X4{N}==X4{N+1}));
  44. -% RESULT={N, X1{N+1}, X2{N+1}, X3{N+1}, X4{N+1}};
  45. -%end The Modified Jacobi Method for 4x4 Matrix Using Matlab 1. clc;
  46. N0=1000;
  47. % enter initial values of X 6. X1{1} = input('ENTER THE INITIAL VALUES OF X1 = ');
  48. X2{1} = input('ENTER THE INITIAL VALUES OF X2 = ');
  49. X3{1}= input('ENTER THE INITIAL VALUES OF X3 = ');
  50. X4{1}=input('ENTER THE INITIAL VALUES OF X4 = ');
  51. % ENTER VALUES FOR THE FIRST EQUATION 11. % X11+X12+X13+X14=A1 12. disp('Enter in this form a11X1+a12X2+a13X3+a14x4=A1');
  52. a11=input('Enter a11=');
  53. A1=input('enter A1=');
  54. % ENTER VALUES FOR THE SECOND EQUATION 19. % X21+X22+X23+X24=A2 20. disp('Enter in this form a21X1+a22X2+a23X3+a24x4=A2');
  55. A2=input('enter A2=');
  56. % ENTER VALUES FOR THE THIRD EQUATION 27. % X31+X32+X33+X34=A3 28. disp('Enter in this form a31X1+a32X2+a33X3+a34X4=A3');
  57. A3=input('enter A3=');
  58. % ENTER VALUES FOR THE FOURTH EQUATION 35. % X31+X32+X33+X34=A4 36. disp('Enter in this form a41X1+a42X2+a43X3+a44X4=A4');
  59. A4=input('enter A4='); 42. for N= 1:N0
  60. % R1 CALCULATION 44. R1{N}=A1+(a11*X1{N})+(a12*X2{N})+(a13*X3{N})+(a14*X4{N});
  61. % calculate values for x1 46. X1{N+1}=X1{N}+(R1{N}/-a11);
  62. % R2 CALCULATION 48. R2{N}=A2+(a21*X1{N+1})+(a22*X2{N})+(a23*X3{N})+(a24*X4{N});
  63. % calculate values for x2 50. X2{N+1}=X2{N}+(R2{N}/-a11);
  64. % R3 CALCULATION 52. R3{N}=A3+(a31*X1{N+1})+(a32*X2{N+1})+(a33*X3{N})+(a34*X4{N});
  65. % R4 CALCULATION 56. R4{N}=A4+(a41*X1{N+1})+(a42*X2{N+1})+(a43*X3{N+1})+(a44*X4{N});
  66. % calculate values for x4 58. X4{N+1}=X4{N}+(R4{N}/-a11);
  67. RESULT={{N, X1{N+1}, X2{N+1}, X3{N+1}}, X4{N+1}; 60. disp(RESULT);
  68. % if((X1{N}==X1{N+1}) && (X2{N}==X2{N+1}) && (X3{N}==X3{N+1}) && (X4{N}==X4{N+1});
  69. % goto 85;
  70. References
  71. Atkinson, Kendall A. (1989), An Introduction to numerical analysis (2nd ed.), New York: John Wiley & Sons..
  72. Bolch, G. et al (2006), Queuing networks and markov chains: Modeling and performance evaluation with computer science applications (2nd ed.), Wiley-Interscience.
  73. Farebrother, R.W. (1988), Linear least squares computations, statistics: Textbooks and monographs, Marcel Dekker.
  74. Grcar, J. F. (2011b),"Mathematicians of Gaussian elimination", Notices of the American Mathematical Society.
  75. Ibrahim B.K. (2010), A Survey of Three iterative methods for the solution of linear equations, IJNM. Vol. 5 Number 1, page 153-162
  76. Lipson, M. and Lipschutz, S. (2001), Schaum's outline of theory and problems of linear algebra, Schaum's outline series, Fourth edition.

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