Systems of linear simultaneous equations can be solved by two methods:
- Direct Methods of Solution
- Iterative Methods of Solutions
First of all, we will discuss the direct methods of solution in which the answer is calculated directly without the iterative procedure. And then we will discuss the iterative procedures to solve the equations.
We have already discussed the solution of the system of non-linear algebraic equations, numerical solution of definite integrals, and the numerical solution of differential equations in our previous posts. If you want to learn these concepts then click on their respective links.
1. Direct Methods of Solution:
1.1 Gauss Elimination
In Gauss elimination, we do elementary row operations to convert the augmented matrix of the system of linear equations to the triangular form (upper or lower triangular form). After that, we do back substitution to get the unknowns. Gauss elimination has three cases: Unique Solution Case, Infinite Solution Case, and No Solution Case. One by one, we will discuss each case through three different videos. Watch all the three video lectures of Gauss elimination to grab the concepts thoroughly.
>>Video lecture for the case of unique solution:
>>Video lecture for the case of infinite solution:
>>Video lecture for the case of no solution:
1.2 Gauss Jordan Method
Gauss-Jordan method is the extension of the Gauss elimination method. In this method back substitution is not required because we convert the augmented matrix into diagonal form and get the solution easily. Hence we need a few extra elementary row operations than gauss elimination to convert the triangular matrix into diagonal form. Watch the video lecture to grasp the topic thoroughly.
1.3 LU Decomposition Method / Factorization Method / Doolittle’s Method
In LU decomposition method we decompose the coefficient matrix of the system of linear equations into the lower and upper triangular matrix. Then we solve the matrix equation to get the unknowns. Hence this method is called LU decomposition (L for Lower triangular matrix and U for upper triangular matrix). Watch the given video tutorial to grasp the topic.
2. Iterative Methods of Solution:
2.1 Jacobi’s Iterative Method
Jacobi’s method is the iterative method of solution for the systems of equations. In this method, we take initial values of unknowns as an approximation and proceed to find unknowns repeatedly until the desired accuracy is achieved. Video lecture given below will explain you this iterative procedure with the help of the example.
2.2 Gauss Siedel Iterative Method
Gauss Siedel iterative procedure is similar to Jacobi’s iterative procedure except in this method, we keep using the results of previous steps into next steps. Watch the video lecture to grab this topic and learn the difference between Jacob’s and Gauss Siedel method.
That is all for this post. Watch each video attentively and repeat the video if you miss the concept. Do check our other advanced engineering maths videos and basic maths videos. If you have any doubts please leave your comment below and we will try to resolve your query as soon as possible.