Triangularization method

Detailed information about Triangularization method or LU factorization or decomposition method.

1. Triangularization Method Kamran Ansari 6 th Semester

2. Contents Introduction Formula and method Limitations of Triangularizatoin method Advantages of Triangularization Method

3. Introduction Triangularization Method is also known as decomposition method or the factorization method. It is a type of direct method of solving linear simultaneous equations. It is also useful for determining the inverse of matrix.

4. Formula and method Consider the linear equations, can be written as AX = B ……(1) and B =  

5. The concept behind this method is that any matrix  A  can be expressed as the product of a lower triangular matrix L and an upper triangular matrix U provided all the principal minors of  A is non singular. A = L U To produce a unique solution, it is convenient to choose either, the method is called the Doolittle’s method, or the method is called the Crout’s method.  

6. By using Doolittle’s method, A = L U ……(2) where and We can calculate the elements of matrices L and U to use the equality of matrices . By equation (1) becomes, L U X = B write as the following two systems of equations, U X = V ……(3) L V = B ……(4) using equation (4), by substitution we can calculate the matrix V, and at last by back substitution, matrix X can be obtained.  

7. Once we know the matrices L and U than the inverse of matrix A can be determined from, A = L U  

8. Limitations of Triangularizatoin Method This method fails if any of the diagonal elements of matrices L and U (e.g. or ) is zero. All the principal minors of  A is non singular.  

9. Advantages of Triangularization Method This method is superior to Gauss elimination method and used for the solution of linear systems and finding the inverse of the matrix. The number of operations involved in terms of multiplication for a system of linear equations by triangularization method is less than Gauss method. Among the direct methods, factorization method is also preferred as the software for computers.

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