LU decomposition algorithm, which is a powerful technique for solving systems of linear equations and other numerical problems. Here’s how it works:
What is LU Decomposition?
LU decomposition (or LU factorization) is a method to factorize a given square matrix A into two triangular matrices: an upper triangular matrix (U) and a lower triangular matrix (L).
The product of these two matrices, L and U, gives the original matrix A.
Why Use LU Decomposition?
LU decomposition has several advantages over other methods like Gaussian elimination:
Efficiency: Once we compute the LU decomposition, we can solve multiple systems of equations with the same left-hand side (matrix A) using the same L and U matrices. This is especially useful when solving equations with different right-hand sides (vectors b).
Numerical Stability: LU decomposition with partial pivoting provides better numerical stability than Gaussian elimination.
Matrix Inversion: LU decomposition can be used to find the inverse of a matrix.
The Algorithm:
Given a square matrix A, we want to find matrices L and U such that A = LU.
Start with the original matrix A.
Perform Gaussian elimination (row operations) to transform A into an upper triangular matrix U while keeping track of the row operations.
The lower triangular matrix L is constructed by storing the multipliers used during the elimination process.
The diagonal elements of L are set to 1.
The final result is A = LU.
4. Example: Let’s illustrate this with a simple 3x3 matrix:
A = | 2 1 3 |
| 4 3 7 |
| 6 4 8 |
Perform Gaussian elimination:
Divide row 2 by 2: Row2 = Row2 - 2 * Row1
Divide row 3 by 3: Row3 = Row3 - 3 * Row1
Divide row 3 by 2: Row3 = Row3 - 2 * Row2
The resulting matrices are:
L: L = | 1 0 0 | | 2 1 0 | | 3 2 1 |
U: U = | 2 1 3 | | 0 1 1 | | 0 0 1 |
Applications:
Solving systems of linear equations: Given Ax = b, we can solve for x using LU decomposition.
Matrix inversion: If A is invertible, we can find its inverse using LU decomposition.
In summary, LU decomposition provides a structured way to factorize a matrix, making it useful for solving linear systems and related problems. It’s a valuable tool in numerical analysis and scientific computing!
// LU decomposition using Crout's method
function luDecomposition(matrix) {
const n = matrix.length;
const L = new Array(n).fill(0).map(() => new Array(n).fill(0));
const U = new Array(n).fill(0).map(() => new Array(n).fill(0));
for (let i = 0; i < n; i++) {
// Calculate U matrix
for (let j = i; j < n; j++) {
let sum = 0;
for (let k = 0; k < i; k++) {
sum += L[i][k] * U[k][j];
}
U[i][j] = matrix[i][j] - sum;
}
// Calculate L matrix
for (let j = i; j < n; j++) {
if (i === j) {
L[i][j] = 1;
} else {
let sum = 0;
for (let k = 0; k < i; k++) {
sum += L[j][k] * U[k][i];
}
L[j][i] = (matrix[j][i] - sum) / U[i][i];
}
}
}
return { L, U };
}
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