Least-Squares Solutions

Least-Squares solutions are usually computed for an over-determined system of linear equations \(A_{m \times n}x = b\), where m > n. A least-squares solution x minimizes the Euclidean length of the residual vector r = Ax-b.

Least-squares problems with linear constraints and one right-hand side are systems of least-squares equations of the form

\[A_{m \times n} x \cong b,\]

subject to constraints and simple bounds

\[\begin{split}b_l \le Cx \le b_u \\ x_l \le x \le x_u\end{split}\]

Here A is the coefficient matrix of the least-squares equations, b is the right-hand side, and C is the coefficient matrix of the constraints. The vectors \(b_l, b_u, x_l\) and \(x_u\) are the lower and upper bounds on the constraints and the variables. This general problem is solved with function imsl.linalg.lin_lsq_lin_constraints().