Great question! I’ve spent a fair amount of time working with Armadillo’s linear algebra tools, so let me walk you through exactly what happens here.

When you call solve(x, A, b) and matrix A is non-invertible (either singular, rank-deficient, or not square), Armadillo automatically switches to computing a least-squares approximate solution. Put simply, it finds the vector x that minimizes the Euclidean norm of the error: ||Ax - b||₂ (the square root of the sum of squared differences between Ax and b).

How Armadillo computes this under the hood #

Armadillo picks a numerical decomposition method based on the structure of A to ensure stability and efficiency:

• QR Decomposition with Column Pivoting (QRCP): This is the go-to for most cases, especially when A is "tall and skinny" (more rows than columns) or has linearly dependent columns. Column pivoting helps avoid numerical instability by reordering columns to prioritize those with the most information.
• Singular Value Decomposition (SVD): If A is a singular square matrix, or QRCP isn’t performing well numerically, Armadillo falls back to SVD. This method is more computationally expensive but extremely stable. It truncates very small singular values (to avoid division by zero) and produces the minimum-norm least-squares solution—meaning among all possible x that minimize the error, this is the one with the smallest Euclidean norm.

Example to illustrate #

Let’s take a quick code snippet with a singular matrix:

#include <armadillo>

int main() {
    arma::mat A = {{1, 2}, {2, 4}}; // Singular matrix (rank 1)
    arma::vec b = {3, 6};
    arma::vec x;

    solve(x, A, b);
    x.print("Least-squares solution:");
    // Output will be the minimum-norm solution: [0.6; 1.2]
    // Because A*x = [3; 6], which exactly matches b, and this x has the smallest possible norm
    return 0;
}

Optional: Force a specific method #

If you want explicit control over the solving method, you can use Armadillo’s solve options:

• solve(x, A, b, arma::solve_opts::least_squares): Forces least-squares mode regardless of matrix type
• solve(x, A, b, arma::solve_opts::svd): Explicitly uses SVD for the most stable result (great for ill-conditioned matrices)

内容的提问来源于stack exchange，提问作者Pasha Pashazade