Block Elimination and the Computation of Simple Turning Points

Abstract

We present the algorithm GMBE to solve a block linear system

$$\begin{array}{*{20}{c}} n \\ m \\ l \end{array}\,\left[ {\mathop {\begin{array}{*{20}{c}} {{A_{11}}} \\ 0 \\ {{A_{31}}} \end{array}}\limits_n \,\mathop {\begin{array}{*{20}{c}} {{A_{12}}} \\ {{A_{22}}} \\ {{A_{32}}} \end{array}}\limits_m \,\mathop {\begin{array}{*{20}{c}} {{A_{13}}} \\ {{A_{23}}} \\ {{A_{33}}} \end{array}}\limits_l } \right]\,\left[ {\mathop {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \end{array}}\limits_l } \right]\, = \,\left[ {\mathop {\begin{array}{*{20}{c}} {{f_1}} \\ {{f_2}} \\ {{f_3}} \end{array}}\limits_l } \right]\,\begin{array}{*{20}{c}} n \\ m \\ l \end{array}$$

or Mz = hwhich appears in numerical computation of turning points and symmetry-breaking bifurcation. GMBE principally uses 1 solve with A ^T₁₁ and A ^T₂₂ each and 2 solves with A₁₁and A₂₂each. M must be well—conditioned but A₁₁and A₂₂may be arbitrarily ill—conditioned, in fact singular to machine precision. The error analysis requires that the solvers for A₁₁,A₂₂,A ^T₁₁ and A ^T₂₂ are stable (in the sense of backward projection of the errors) and that the solvers for A₁₁,A₂₂are bounded (in a sense to be made precise). Both properties are typically possessed in practice by solvers based on either direct or iterative methods. GMBE is the first algorithm that solves Mz = hby using the solvers for A₁₁etc. as ‘black boxes’.