Solve the given system of equations by using Jacobi method correct up to 3 decimal places.
3x + 4y – 15z = 54.8
x + 12y + 3z = 39.66
10x + y – 2z = 7.74
Solution:
1st Iteration:
10x + y – 2z = 7.74
x1 = 7.74 – y0 + 2z0 / 10
x + 12y + 3z = 39.66
y1 = 39.66 – 3z0 – x0 / 12
3x + 4y – 15z = 54.8
z1 = 54.8 – 4y0 – 3x0 / 15
Suppose starting value:
x0 = y0 = z0 = 0
x1 = 7.74 – 0 + 2(0) / 10
x1 = 0.774
y1 = 39.66 – 3(0) – 0 / 12
y1 = 3.305
z1 = 54.8 – 4(0) – 3(0) / 15
z1 = 3.6333333333
2nd Iteration
x2 = 7.74 – y1 + 2z1 / 10
x2 = 1.170166667
y2 = 39.66 – 3z1 – x1 / 12
y2 = 2.3327166675
z2 = 54.8 – 4y1 – 3x1 / 15
z2 = 2.6172
3rd Iteration
x3 = 7.74 – y2 + 2z2 / 10
x3 = 1.064168333
y3 = 39.66 – 3z2 – x2 / 12
y3 = 2.553186111
z3 = 54.8 – 4y2 – 3x2 / 15
z3 = 2.797242222
4th Iteration
x4 = 7.74 – y3 + 2z3 / 10
x4 = 1.07163167
y4 = 39.66 – 3z3 – x3 / 12
y4 = 2.5169625
z4 = 54.8 – 4y3 – 3x3 / 15
z4 = 2.759627926
5th Iteration
x5 = 7.74 – y4 + 2z4 / 10
x5 = 1.074229335
y5 = 39.66 – 3z4 – x4 / 12
y5 = 2.525791129
z5 = 54.8 – 4y4 – 3x4 / 15
z5 = 2.767816999
6th Iteration
x6 = 7.74 – y5 + 2z5 / 10
x6 = 1.074984287
y6 = 39.66 – 3z5 – x5 / 12
y6 = 2.523526639
z6 = 54.8 – 4y5 – 3x5 / 15
z6 = 2.764904446
7th Iteration
x7 = 7.74 – y6 + 2z6 / 10
x7 = 1.074628225
y7 = 39.66 – 3z6 – x6 / 12
y7 = 2.524191865
z7 = 54.8 – 4y6 – 3x6 / 15
z7 = 2.765396039
8th Iteration
x8 = 7.74 – y7 + 2z7 / 10
x8 = 1.074660021
y8 = 39.66 – 3z7 – x7 / 12
y8 = 2.52409888
z8 = 54.8 – 4y7 – 3x7 / 15
z8 = 2.765289858
x = 1.074
y = 2.524
z = 2.765
Correct up to 3 decimal places.
No. of iterations |
x |
y |
z |
|
1 |
0.774 |
3.305 |
3.6333333333 |
|
2 |
1.17016667 |
2.3327166675 |
2.6172 |
|
3 |
1.064168333 |
2.553186111 |
2.797242222 |
|
4 |
1.07163167 |
2.5169625 |
2.759627926 |
|
5 |
1.074229335 |
2.525791129 |
2.767816999 |
|
6 |
1.074984287 |
2.523526639 |
2.764904446 |
|
7 |
1.074628225 |
2.524191865 |
2.765396039 |
|
8 |
1.074660021 |
2.52409888 |
2.765289858 |
x = 1.074
y = 2.524
z = 2.765
Correct up to 3 decimal places.
No. of iterations |
x |
y |
z |
|
1 |
0.774 |
3.305 |
3.6333333333 |
|
2 |
1.17016667 |
2.3327166675 |
2.6172 |
|
3 |
1.064168333 |
2.553186111 |
2.797242222 |
|
4 |
1.07163167 |
2.5169625 |
2.759627926 |
|
5 |
1.074229335 |
2.525791129 |
2.767816999 |
|
6 |
1.074984287 |
2.523526639 |
2.764904446 |
|
7 |
1.074628225 |
2.524191865 |
2.765396039 |
|
8 |
1.074660021 |
2.52409888 |
2.765289858 |
x = 1.074
y = 2.524
z = 2.765
Correct up to 3 decimal places.
Understanding the Jacobi Method of Iteration
When it comes to solving systems of linear equations, the Jacobi method is a mainstay of numerical and iterative techniques. This method, which bears the name of a well-known mathematician from the 19th century, Carl Gustav Jacob Jacobi, offers an organized way to iteratively approximate solutions. Its efficacy and simplicity combine to create elegance, making it a useful tool in a variety of professions, from computer science to engineering and beyond.
Historical Context and Development
There were major developments in mathematics during the time when the Jacobi method first appeared, especially in the area of linear algebra. It was created in the middle of the 1800s as a component of a larger initiative to organize methods for resolving linear equation systems. Carl Gustav Jacob Jacobi made a substantial contribution to this groundbreaking study, as did his colleagues Gauss and Gauss-Seidel.
Basic Concept and Mathematical Formulation
The fundamental goal of the Jacobi method is to iteratively solve the system of linear equations Ax = b. In this case, the square matrix A, the unknown vector x, and the constant vector b are all the same. Every component of x is updated by the procedure depending on the values from the preceding iteration, which begins with an initial guess for x.
The Jacobi method’s iterative formula is as follows:
𝑥𝑖(𝑘 + 1) = (1 / 𝐴𝑖𝑖 )(𝑏 𝑖−∑𝑗 ≠𝑖 𝐴𝑖𝑗 𝑥𝑗(𝑘 ))
where the vector x at the k-th iteration of the matrix A is denoted by x (k); the matrix Ai elements are 𝐴ij , and the vector b’s i-th member is bi . The key idea behind this formula is that as the number of iterations rises, convergence towards the solution of x is ensured because each component of y is updated using the values from the preceding iteration.
Convergence and Stability
The characteristics of the matrix A affect how well the Jacobi technique converges. In particular, the matrix A needs to be symmetric positive definite or diagonally dominant for the algorithm to converge. Each diagonal element Aii is guaranteed to be sufficiently greater than the sum of the absolute values of the other elements in the same row by diagonally dominant matrices. The Jacobi technique is also widely stable and easy to use, which makes it the method of choice for many practical applications where a certain amount of accuracy is acceptable.
Practical Applications
Applications of the Jacobi technique can be found in many different fields, such as:
Engineering: Applies to optimization issues, finite element calculations, and structural analysis.
Physics: Used in modeling and numerical simulations of physical processes.
Computer science: Used in algorithms with iterative solutions and large-scale matrix operations.
Because of its effectiveness and simplicity, it can be used in scenarios where other approaches could be too complex or costly to compute.
Comparison with Other Iterative Methods
Despite its effectiveness, the Jacobi method is frequently contrasted with other iterative methods, such successive over-relaxation (SOR) and the Gauss-Seidel method. For example, the Gauss-Seidel technique uses the most recent components to update each component of x, which may result in faster convergence than the Jacobi method. Conversely, SOR adds a relaxation factor to speed up convergence, but at the expense of more computational work.
Every approach has advantages, and the best one depends on the particulars of the task at hand, such as matrix properties, required accuracy, and available computing power.
Implementation Considerations
When applying the Jacobi technique, computational efficiency and matrix properties must be carefully taken into account. To ensure accuracy and performance, modern computational tools and libraries frequently offer optimized implementations of iterative techniques, such as the Jacobi method.
Conclusion
To sum up, the Jacobi technique of iteration is still a vital tool in numerical mathematics since it offers a dependable way for iteratively solving systems of linear equations. Its practical applications, historical significance, and simplicity highlight its ongoing relevance in a variety of sectors. Iterative techniques like the Jacobi method will remain essential for solving complicated problems quickly and effectively as computing power increases.
Comprehending the fundamental ideas underlying the Jacobi technique not only heightens our admiration for its mathematical sophistication but also provides us with necessary instruments to tackle practical issues in various fields. The power of iterative thinking in contemporary computational mathematics is exemplified by the Jacobi method, which has applications in computer science, physics, and engineering.
Frequently Asked Questions (FAQs):
Q) What is the Jacobi method of iteration?
One iterative method for solving systems of linear equations is the Jacobi method. With the goal of convergent toward the true solution, it iteratively updates each element of the solution vector based on the values of the preceding iteration.
Q) How does the Jacobi method work?
The solution vector x is first estimated using this method. Then, using a formula that depends on the components of the coefficient matrix A and the constant vector b, iteratively updates each component of x. Until the required degree of accuracy is attained, this iterative procedure is carried out.
Q) What are the requirements for convergence in the Jacobi method?
The matrix A needs to be symmetric positive definite or diagonally dominant in order for the Jacobi technique to converge. Each diagonal element must be greater than the total of the absolute values of the other components in the same row in order for there to be diagonal dominance.
Q) How do you determine if a matrix is diagonally dominant?
If the absolute value of the diagonal element (Aii) in each row i is greater than the total of the absolute values of the other elements in that row, then matrix A is diagonally dominating.
Q) What are the practical applications of the Jacobi method?
Numerous disciplines employ the Jacobi technique, including computer science (large-scale matrix operations algorithms), physics (numerical simulations), and engineering (structural analysis and finite element methods). When computing efficiency is crucial and iterative solutions are needed, it is especially helpful.
Q) How does the Jacobi method compare with the Gauss-Seidel method?
While the Gauss-Seidel approach updates each component sequentially as new values become available, the Jacobi method updates all components of the solution vector simultaneously depending on the previous iteration. As a result, the Gauss-Seidel approach may converge more quickly than the Jacobi approach, particularly for systems with significant diagonal dominance.
Q) Can the Jacobi method be used for non-linear systems?
No, the Jacobi technique is intended only to solve linear equation systems in which the coefficient matrix A is constant. Iterative techniques such as Newton-Raphson or fixed-point iteration are commonly employed for non-linear systems.
Q) Is the Jacobi method guaranteed to converge?
If matrix A is symmetric positive definite or diagonally dominant, the Jacobi method converges. Convergence isn’t assured for every matrix, though. Convergence may require the use of additional iterative procedures or preconditioning strategies if diagonal dominance is not satisfied.
Q) How do you implement the Jacobi method in practice?
Setting convergence criteria, such as the tolerance level, initializing an initial estimate for x, and iteratively applying the update formula until the solution converges are the steps involved in implementing the Jacobi method. Optimized implementations for accuracy and efficiency are frequently offered by contemporary computational tools and libraries.
Q) What are the advantages of using the Jacobi method?
Even individuals who are unfamiliar with numerical approaches can easily grasp and apply the Jacobi method. It may be used to solve a variety of issues in a variety of areas and offers a solid basis for learning more complex iterative techniques.
These frequently asked questions (FAQs) offer a thorough introduction to the Jacobi method and cover frequently asked queries on its use, applications, convergence characteristics, and implementation issues. For additional research, knowing the theoretical foundations and real-world applications of the Jacobi technique can improve how well it solves linear equations in a variety of contexts.