Question

What are the different ways of solving linear systems of equations using Armadillo’s functions, such as the Moore-Penrose pseudoinverse and LU decomposition.

Answer 1

Armadillo provides various functions to solve linear systems of equations, including the Moore-Penrose pseudoinverse and LU decomposition.

1. Moore-Penrose Pseudoinverse: The 

pinv()

 function computes the pseudoinverse of a matrix, which can be used to solve overdetermined or underdetermined systems. Example:

#include <armadillo>

using namespace arma;

mat A = randu<mat>(4, 3);

vec b = randu<vec>(4);

mat A_pinv = pinv(A);

vec x = A_pinv * b;

2. LU Decomposition: The 

solve()

 function uses LU decomposition by default for square matrices. It factors the input matrix into lower (L) and upper (U) triangular matrices. Example:

#include <armadillo>

using namespace arma;

mat A = randu<mat>(3, 3);

vec b = randu<vec>(3);

vec x = solve(A, b);

3. Cholesky Decomposition: For positive definite matrices, use the 

chol()

 function to perform Cholesky decomposition. Example:

#include <armadillo>

using namespace arma;

mat A = randu<mat>(3, 3);

A = A.t() * A; // Make it positive definite

vec b = randu<vec>(3);

mat L = chol(A);

vec y = solve(trimatl(L), b);

vec x = solve(trimatu(L.t()), y);

4. QR Decomposition: Use the 

qr()

 function for least squares problems. Example:

#include <armadillo>

using namespace arma;

mat A = randu<mat>(4, 3);

vec b = randu<vec>(4);

mat Q, R;

qr(Q, R, A);

vec x = solve(R, Q.t() * b);