A Shooting Algorithm for the Best Least Squares Solution of Two-Point Boundary Value Problems

Abstract

A shooting algorithm is presented for the best least squares solution (BLSS) of the linear two-point boundary value problem ${\bf y}' + A(t){\bf y} = {\bf g}(t)$, $M{\bf y}(a) + N{\bf y}(b) = {\bf d}$, in the noninvertible case. The initial vector of the BLSS is given by an explicit formula involving the generalized inverse of a characteristic matrix. The BLSS is obtained by solving $n + 1$ initial value problems, where n is the dimension of the system. The nth order scalar boundary value problem is also solved in general. In the invertible case, this method reduces to the Goodman–Lance method of adjoints, which gives the classical solution. The method is illustrated by an application to a noninvertible example.