The Induced Dimension Reduction method is a family of simple and fast Krylov subspace algorithms for solving large nonsymmetric linear systems. The idea behind the IDR(s) variant is to generate residuals that are in the nested subspaces of shrinking dimensions.


idrs!(x, A, b; s = 8, kwargs...) -> x, [history]

Solve the problem $Ax = b$ approximately with IDR(s), where s is the dimension of the shadow space.


  • x: Initial guess, will be updated in-place;
  • A: linear operator;
  • b: right-hand side.


  • s::Integer = 8: dimension of the shadow space;
  • Pl::precT: left preconditioner,
  • abstol::Real = zero(real(eltype(b))), reltol::Real = sqrt(eps(real(eltype(b)))): absolute and relative tolerance for the stopping condition |r_k| ≤ max(reltol * |r_0|, abstol), where r_k = A * x_k - b is the residual in the kth iteration;
  • maxiter::Int = size(A, 2): maximum number of iterations;
  • log::Bool: keep track of the residual norm in each iteration;
  • verbose::Bool: print convergence information during the iterations.

Return values

if log is false

  • x: approximate solution.

if log is true

  • x: approximate solution;
  • history: convergence history.

Implementation details

The current implementation is based on the MATLAB version by Van Gijzen and Sonneveld. For background see [Sonneveld2008], [VanGijzen2011], the IDR(s) webpage and the IDR chapter in [Meurant2020].


IDR(s) can be used as an iterator.

  • Sonneveld2008IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems. P. Sonneveld and M. B. van Gijzen SIAM J. Sci. Comput. Vol. 31, No. 2, pp. 1035–1062, 2008
  • VanGijzen2011Algorithm 913: An Elegant IDR(s) Variant that Efficiently Exploits Bi-orthogonality Properties. M. B. van Gijzen and P. Sonneveld ACM Trans. Math. Software, Vol. 38, No. 1, pp. 5:1-5:19, 2011
  • Meurant2020The IDR family. G. Meurant and J. Duintjer Tebbens. In: Krylov Methods for Nonsymmetric Linear Systems. Springer Series in Computational Mathematics, vol 57. Springer, 2020. doi:10.1007/978-3-030-55251-0_10