C++ Program to Perform LU Decomposition of Any MatrixA popular method in numerical analysis for resolving linear equation systems and computing matrix inverses is LU decomposition. The process entails breaking down a matrix into its product of an upper triangular matrix (U) and a lower triangular matrix (L). Engineering, physics, and computational mathematics are just a few of the disciplines that use the LU decomposition. The following explains the theory underlying the LU decomposition approach and how it is implemented in the given C++ program: 1. LU DecompositionThe LU decomposition method takes an n×n matrix A as input and finds two matrices, L and U, such that A = LU. With 1s on the major diagonal, L is a lower triangular matrix, and U is an upper triangular matrix. 2. Algorithmic StepsVarious strategies, like Doolittle's method or Crout's method, which entail methodically breaking down the original matrix into L and U, are frequently used to achieve the LU decomposition. The values of the original matrix are used to determine the elements of L and U throughout the decomposition process. 3. Implementation in the C++ ProgramThe supplied C++ program generates empty matrix L and U and initializes a sample matrix A. The input matrix A is divided into the upper triangular matrix U and the bottom triangular matrix L by the decomposeLU function. The generated L and U matrices are then printed by the program. 4. Application and UsageLU decomposition is a useful tool for efficiently computing matrix inverses, determinants, and systems of linear equations. It is used in many different numerical techniques, such as structural analysis, electrical circuit analysis, and differential equation solutions. Example:Let us take an example to illustrate the use of LU decomposition of matrix in C++. Output: Lower Triangular Matrix (L): 1 0 0 -2 1 0 -2 -1 1 Upper Triangular Matrix (U): 2 -1 -2 0 4 -1 0 0 3 Explanation: 1. File headers and global constants
2. Function for LU Decomposition decomposeLU
3. LU Decomposition
4. Main Function
5. LU Decomposition call The input matrix A and the empty matrices L and U are passed to the decomposeLU function. 6. Printing the Results The higher triangular matrix U and the lower triangular matrix L are printed to the console by the program. 7. Output The output displays the upper and lower triangular matrices that were produced by decomposing the sample matrix A using the LU method. |