What is lsolveAll?

The lsolveAll function is a computational method designed to find all possible solutions of a linear system of equations of the form L * x = b. In this context:

• L is an [n x n] lower triangular matrix.
• b is an [n] column vector.
• x is the solution vector or vectors satisfying the equation.

This function is particularly useful in scenarios where the system has infinite or non-unique solutions, providing a comprehensive set of all valid solutions.

Syntax of the lsolveAll Function

The syntax for using the lsolveAll function is as follows:

x = lsolveAll(L, b)

Here, L represents the lower triangular matrix, and b is the column vector. The result, x, is a set of all solution vectors that satisfy the equation L * x = b.

Examples of Using lsolveAll

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

• Example 1: Finding all solutions for a simple linear system.

  
  a = [-2, 3; 2, 1]
  b = [11, 9]
  x = lsolveAll(a, b)
  

  Result: x = [[-5.5], [20]]

The lsolveAll function ensures that all potential solutions are explored, especially in cases where the system has dependent equations.

Applications of lsolveAll

The lsolveAll function is widely used in various computational and mathematical applications, including:

• Handling underdetermined systems in numerical analysis.
• Exploring solution spaces for systems with infinite solutions.
• Analyzing dependencies in lower triangular matrices.
• Applications in engineering, data science, and physics.

Related Functions

For additional functionality and solutions to linear systems, explore these related functions:

• lsolve: Finds one solution of a linear system with a lower triangular matrix.
• lup: Performs LU decomposition with permutation.
• lusolve: Solves a linear system using LU decomposition.
• usolve: Solves systems involving upper triangular matrices.