MINRES is a short-recurrence version of GMRES for solving $Ax = b$ approximately for $x$ where $A$ is a symmetric, Hermitian, skew-symmetric or skew-Hermitian linear operator and $b$ the right-hand side vector.


minres!(x, A, b; kwargs...) -> x, [history]

Solve Ax = b for (skew-)Hermitian matrices A using MINRES.


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


  • initially_zero::Bool = false: if true assumes that iszero(x) so that one matrix-vector product can be saved when computing the initial residual vector;
  • skew_hermitian::Bool = false: if true assumes that A is skew-symmetric or skew-Hermitian;
  • 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

    The residual is computed only approximately.

  • maxiter::Int = size(A, 2): maximum number of iterations;
  • log::Bool = false: keep track of the residual norm in each iteration;
  • verbose::Bool = false: 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

MINRES exploits the tridiagonal structure of the Hessenberg matrix. Although MINRES is mathematically equivalent to GMRES, it might not be equivalent in finite precision. MINRES updates the solution as

\[x := x_0 + (V R^{-1}) (Q^*\|r_0\|e_1)\]

where $V$ is the orthonormal basis for the Krylov subspace and $QR$ is the QR-decomposition of the Hessenberg matrix. Note that the brackets are placed slightly differently from how GMRES would update the residual.

MINRES computes $V$ and $W = VR^{-1}$ via a three-term recurrence, using only the last column of $R.$ Therefore we pre-allocate only six vectors, save only the last two entries of $Q^*\|r_0\|e_1$ and part of the last column of the Hessenberg matrix.

Real and complex arithmetic

If $A$ is Hermitian, then the Hessenberg matrix will be real. This is exploited in the current implementation.

If $A$ is skew-Hermitian, the diagonal of the Hessenberg matrix will be imaginary, and hence we use complex arithmetic in that case.


MINRES can be used as an iterator.