The two-point boundary value problem \((1)\quad y'(t)=A(t)y(t)+q(t),\quad 0<t<1,(2)\quad (B_{11},B_{12})y(0)=\beta_ 1,\quad (B_{21},B_{22})y(1)=\beta_ 2,\)where \(\beta_ 1\in {\mathbb{R}}^ p\), \(\beta_ 2\in {\mathbb{R}}^ q\), \(B_{12}\in R^{p\times p}\) (and is assumed nonsingular), \(B_{21}\in {\mathbb{R}}^{q\times q}\) and \(p+q=n\), the dimension of y, is considered. After partioning \(y(t)=\left( \begin{matrix} y_ 1(t)\\ y_ 2(t)\end{matrix} \right),\) \(A(t)=\left( \begin{matrix} A_{11}\\ A_{21}\end{matrix} \begin{matrix} A_{12}\\ A_{22}\end{matrix} \right),\) \(q(t)=\left( \begin{matrix} q_ 1(t)\\ q_ 2(t)\end{matrix} \right)\) in accordance with (2), the problem can be solved by means of a double sweep integration process which is called the Riccati method. The major motivation is the desire to solve singular perturbation problems.
Forward sweep: solve the initial value problems (3) \(R'=A_{21}+A_{22}R-RA_{11}-RA_{12}R\), \(0\leq t\leq 1\), \(R(0)=- B^{-1}_{12}B_{11}\) and \(V'=(A_{22}-RA_{12})V-Rq_ 1+q_ 2,\) \(0\leq t\leq 1\), \(V(0)=B^{-1}_{12}\beta_ 1\). Backward sweep: solve \(y_ 1'=(A_{11}+A_{12}R)y_ 1+A_{12}V+q_ 1,\) \(1\geq t\geq 0\), \(y_ 1(1)=[B_{21}+B_{22}R(1)]^{-1}(\beta_ 2-B_{22}V(1)).\) Then one recovers the full solution via \(y_ 2(t)=R(t)y_ 1(t)+V(t).\) This algorithm generally needs to be reformulated on subintervals \(I_ h\) of [0,1] in terms of the “local” variables \(y^ h(t)=P^ hy(t)\), where P is the cumulative permutation matrix defined such that solvability of (3) in the new variables is guaranteed.