Introduction to LU Factorization Calculator:
LU factorization calculator is an amazing tool that helps you to perform LU decomposition of a given matrix. It takes an input matrix to convert the given matrix A into L and U for decomposition and provides you a solution in a few seconds.
It is a very useful tool for everyone who wants to solve the Lu factorization of a matrix problem as it is used in various fields like numerical computations, engineering, and scientific applications where matrix operations are commonly used.
What is Lu Factorization?
Lu Factorization is a method that is used in numerical analysis for solving systems of linear equations, finding determinants, and inverting matrices decomposition in linear algebra. It is known as LU factorization or LU decomposition.
It decomposes a matrix into a lower triangular matrix (L) and an upper triangular matrix (U). This decomposition helps in solving linear systems more efficiently than direct methods such as the Gaussian elimination, especially when the same matrix needs to be solved in multiple ways.
Working Procedure Behind Lu Decomposition Calculator
The (LU decomposition) LU factorization Calculator gives you the solution to LU factorization problem in the easiest way because it has advanced features that allow it to solve complicated problems quickly without any trouble.
Let's see how our upper triangular matrix calculator works during the evaluation of lu factorization problems with the help of an example.
Step-by-Step LU Decomposition
Step 1: Let's assume we have an n×n matrix A.
$$ A \;=\; \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{matrix} \right] $$
$$ Here\; A \;=\; \left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & 3 \\ 2 & 4 & 5 \\ \end{matrix} \right] $$
Step 2: To achieve the echelon form, apply the elimination method of row operation,
$$ R_2 \leftarrow R_2 - (-0.75) \times R_1 $$
$$ =\; \left[ \begin{matrix} 8 & -6 & 2 \\ 0 & 2.5 & 4.5 \\ 2 & 4 & 5 \\ \end{matrix} \right] $$
$$ R_3 \leftarrow R_3 - (0.25) \times R_1 $$
$$ =\; \left[ \begin{matrix} 8 & -6 & 2 \\ 0 & 2.5 & 4.5 \\ 0 & 5.5 & 4.5 \\ \end{matrix} \right] $$
$$ R_3 \leftarrow R_3 - (2.2) \times R_2 $$
$$ =\; \left[ \begin{matrix} 8 & -6 & 2 \\ 0 & 2.5 & 4.5 \\ 0 & 0 & -5.4 \\ \end{matrix} \right] $$
Step 3: Suppose the above matrix result is U in lu factorization because it has a lower triangle matrix term that is a necessary condition in the U matrix.
$$ U \;=\; \left[ \begin{matrix} u_{11} & u_{12} & \cdots & u_{1n} \\ 0 & u_{22} & \cdots & u_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & u_{nn} \\ \end{matrix} \right] $$
$$ U \;=\; \left[ \begin{matrix} 8 & -6 & 2 \\ 0 & 2.5 & 4.5 \\ 0 & 0 & -5.4 \\ \end{matrix} \right] $$
Step 4: For the value of L, again use the elimination method to make the diagonal value of matrix A is 1 and the upper triangle value is zero as given in the below matrix.After doing the elimination method on matrix A we get the solution of L as per its condition.
$$ L \;=\; \left[ \begin{matrix} 1 & 0 & \cdots & 0 \\ l_{21} & 1 & \cdots & 0 \\ l_{31} & l_{32} & \ddots & 0 \\ \vdots & \vdots & \ddots & 1 \\ \end{matrix} \right] $$
$$ L \;=\; \left[ \begin{matrix} 1 & 0 & 0 \\ -0.75 & 1 & 0 \\ 0.25 & 2.2 & 1 \\ \end{matrix} \right] $$
Step 5: To check whether the matrix L and matrix U are correct or not, you must verify it with the lu decomposition method in which if A=LU then your required solution is right otherwise it is incorrect.
$$ \left[ \begin{matrix} 1 & 0 & 0 \\ -0.75 & 1 & 0 \\ 0.25 & 2.2 & 1 \\ \end{matrix} \right] \times \left[ \begin{matrix} 8 & -6 & 2 \\ 0 & 2.5 & 4.5 \\ 0 & 0 & -5.4 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & 3 \\ 2 & 4 & 5 \\ \end{matrix} \right] $$
As the given solution shows A=LU, that means our decomposition solution is correct.
How to Use the Upper Triangular Matrix Calculator?
LU decomposition calculator with steps has an easy-to-use interface so that you can use it to calculate the LU decomposition questions. Before adding the input to this calculator to get solutions, you must follow some simple steps to avoid trouble during the calculation process. These steps are:
- Add the size of the matrix in the given field.
- Enter the elements of the matrix in the next input box.
- Review your input matrix value before hitting the calculate button to start the calculation process in the calculator.
- Click on the “Calculate” button to get the desired result of your given lu decomposition problem.
- If you want to try out our calculator first then you can use the load example for a better understanding.
- Click on the “Recalculate” button to get a new page for solving more matrix problems of LU Factorization.
What Output Lu Factorization Calculator Gives?
The upper triangular matrix calculator gives you the solution to a given square matrix problem when you add the input to it. It provides you with solutions to lu fractionization problems that may contain as:
- Result Option:
You can click on the result option and it provides you with a solution for Lu factorization.
- Possible Step:
When you click on the possible steps option it provides you with the solution of lu factorization where all calculation steps are included in detail.
Advantages of the Lu Decomposition Calculator With Steps:
The LU factorization calculator gives you tons of advantages whenever you use it to calculate a system of linear equation problems using LU decomposition. These advantages are:
- Our tool saves your time and effort from doing complex calculations of a square matrix A lu factorization in a few seconds.
- It is a free-of-cost tool so you can use it to find the lu decomposition of different matrices in the calculator.
- Upper triangular matrix calculator is an adaptive tool that allows you to solve various types of systems of matrix or linear equations and provide solutions in lu decomposition.
- You can use this LU Fractorization Calculator for practice so that you get a strong hold on this concept.
- It is a reliable tool that provides exact solutions as per your input matrix value whenever you use it to calculate the LU factorization problem.
- LU decomposition calculator with steps provides you a solution with a complete process in a step-by-step method so that you get more clarity.
