What is lup?

The lup function performs the LU decomposition of a given matrix with partial pivoting. It decomposes matrix A into three matrices:

• L: A lower triangular matrix.
• U: An upper triangular matrix.
• P: A permutation matrix.

The decomposition satisfies the equation: P * A = L * U, where P rearranges the rows of A to ensure numerical stability during the decomposition.

Syntax of the lup Function

The syntax for using the lup function is as follows:

lup(m)

Here, m is the matrix to be decomposed. The result is an object or dictionary containing the matrices L, U, and P.

Examples of Using lup

Below are some examples demonstrating the usage of the lup function:

• Example 1: Performing LU decomposition on a dense matrix.

  
  lup([[2, 1], [1, 4]])
  

  Result: {"L": [[1, 0], [0.5, 1]], "U": [[2, 1], [0, 3.5]], "p": [0, 1]}

• Example 2: Decomposing a matrix using a matrix object.

  
  lup(matrix([[2, 1], [1, 4]]))
  

  Result: L: [[1, 0], [0.5, 1]], U: [[2, 1], [0, 3.5]], P: [0, 1]

• Example 3: Decomposing a sparse matrix.

  
  lup(sparse([[2, 1], [1, 4]]))
  

  Result: L: [[1, 0], [0.5, 1]], U: [[2, 1], [0, 3.5]], P: [0, 1]

Applications of lup

The lup function is widely used in various mathematical and computational scenarios, including:

• Solving linear systems of equations efficiently using LU decomposition.
• Calculating matrix inverses and determinants.
• Analyzing numerical stability and dependencies in matrices.
• Preprocessing matrices for iterative methods in numerical analysis.

Related Functions

For additional functionality and related matrix operations, explore these related functions:

• lusolve: Solves a linear system using LU decomposition.
• lsolve: Finds one solution of a linear system with a lower triangular matrix.
• usolve: Solves systems involving upper triangular matrices.
• slu: Performs sparse LU decomposition.
• qr: Computes the QR decomposition of a matrix.