Residual Inverse Iteration for the Nonlinear Eigenvalue Problem

For the nonlinear eigenvalue problem A( λ ) x = 0, where A( · ) is a matrix-valued operator, residual inverse iteration with shift σ is defined by \[ a(l + 1) : = {const. }(x⁽ˡ⁾ - A(σ )- 1 A(λ l + 1 )x⁽ˡ⁾ ),\] where λ l + 1 is an appropriate approximation of λ. In the linear case, A(λ ) = A - λ I, this is theoretically equivalent to ordinary inverse iteration, but the residual formulation results in a considerably higher limit accuracy when the residual A(λ l + 1 )x⁽ˡ⁾ = Ax⁽ˡ⁾ - λ l + 1 x⁽ˡ⁾ is accumulated in double precision. In the nonlinear case, if σ is sufficiently close to λ, convergence is at least linear with convergence factor proportional to | σ - λ |. As with ordinary inverse iteration, the convergence can be accelerated by using variable shifts.