6. System of Linear Equations
Systems of linear equations are crucial in various engineering fields, providing solutions to real-world problems. This chapter discusses both direct methods, such as Gaussian Elimination and LU Decomposition, and iterative methods like Gauss-Jacobi and Gauss-Seidel for solving these systems. Understanding the efficiency and application of these methods is essential for tackling larger datasets and complex computational problems.
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What we have learnt
- Systems of linear equations form the basis of many engineering applications.
- Direct and iterative methods each have their strengths and are suitable for different types of problems.
- Gaussian Elimination and LU Decomposition are effective for smaller systems, while iterative methods excel in handling large sparse systems.
Key Concepts
- -- System of Linear Equations
- A collection of two or more linear equations involving the same set of variables that can be represented in matrix form.
- -- Gaussian Elimination
- A direct method for solving systems of linear equations that transforms the system into an upper triangular form before applying back-substitution.
- -- LU Decomposition
- A method that breaks down a matrix into the product of a lower triangular matrix and an upper triangular matrix, facilitating the solving of multiple systems.
- -- Iterative Methods
- Techniques for solving linear systems by approximating the solution through successive iterations, suited for large, sparse problems.
- -- GaussJacobi Method
- An iterative approach where each variable is solved in parallel based on the previous estimates of all other variables.
- -- GaussSeidel Method
- An iterative method that updates the solution variables as soon as new values are available, typically leading to faster convergence.
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