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BiLQR

Krylov.bilqr — Function
(x, y, stats) = bilqr(A, b::AbstractVector{FC}, c::AbstractVector{FC};
                      transfer_to_bicg::Bool=true, atol::T=√eps(T),
                      rtol::T=√eps(T), itmax::Int=0,
                      timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
                      callback=solver->false, iostream::IO=kstdout)

T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.

(x, y, stats) = bilqr(A, b, c, x0::AbstractVector, y0::AbstractVector; kwargs...)

BiLQR can be warm-started from initial guesses x0 and y0 where kwargs are the same keyword arguments as above.

Combine BiLQ and QMR to solve adjoint systems.

[0  A] [y] = [b]
[Aᴴ 0] [x]   [c]

The relation bᴴc ≠ 0 must be satisfied. BiLQ is used for solving primal system Ax = b of size n. QMR is used for solving dual system Aᴴy = c of size n.

Input arguments

  • A: a linear operator that models a matrix of dimension n;
  • b: a vector of length n;
  • c: a vector of length n.

Optional arguments

  • x0: a vector of length n that represents an initial guess of the solution x;
  • y0: a vector of length n that represents an initial guess of the solution y.

Keyword arguments

  • transfer_to_bicg: transfer from the BiLQ point to the BiCG point, when it exists. The transfer is based on the residual norm;
  • atol: absolute stopping tolerance based on the residual norm;
  • rtol: relative stopping tolerance based on the residual norm;
  • itmax: the maximum number of iterations. If itmax=0, the default number of iterations is set to 2n;
  • timemax: the time limit in seconds;
  • verbose: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations;
  • history: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;
  • callback: function or functor called as callback(solver) that returns true if the Krylov method should terminate, and false otherwise;
  • iostream: stream to which output is logged.

Output arguments

  • x: a dense vector of length n;
  • y: a dense vector of length n;
  • stats: statistics collected on the run in an AdjointStats structure.

Reference

source
Krylov.bilqr! — Function
solver = bilqr!(solver::BilqrSolver, A, b, c; kwargs...)
solver = bilqr!(solver::BilqrSolver, A, b, c, x0, y0; kwargs...)

where kwargs are keyword arguments of bilqr.

See BilqrSolver for more details about the solver.

source

TriLQR

Krylov.trilqr — Function
(x, y, stats) = trilqr(A, b::AbstractVector{FC}, c::AbstractVector{FC};
                       transfer_to_usymcg::Bool=true, atol::T=√eps(T),
                       rtol::T=√eps(T), itmax::Int=0,
                       timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
                       callback=solver->false, iostream::IO=kstdout)

T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.

(x, y, stats) = trilqr(A, b, c, x0::AbstractVector, y0::AbstractVector; kwargs...)

TriLQR can be warm-started from initial guesses x0 and y0 where kwargs are the same keyword arguments as above.

Combine USYMLQ and USYMQR to solve adjoint systems.

[0  A] [y] = [b]
[Aᴴ 0] [x]   [c]

USYMLQ is used for solving primal system Ax = b of size m × n. USYMQR is used for solving dual system Aᴴy = c of size n × m.

Input arguments

  • A: a linear operator that models a matrix of dimension m × n;
  • b: a vector of length m;
  • c: a vector of length n.

Optional arguments

  • x0: a vector of length n that represents an initial guess of the solution x;
  • y0: a vector of length m that represents an initial guess of the solution y.

Keyword arguments

  • transfer_to_usymcg: transfer from the USYMLQ point to the USYMCG point, when it exists. The transfer is based on the residual norm;
  • atol: absolute stopping tolerance based on the residual norm;
  • rtol: relative stopping tolerance based on the residual norm;
  • itmax: the maximum number of iterations. If itmax=0, the default number of iterations is set to m+n;
  • timemax: the time limit in seconds;
  • verbose: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations;
  • history: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;
  • callback: function or functor called as callback(solver) that returns true if the Krylov method should terminate, and false otherwise;
  • iostream: stream to which output is logged.

Output arguments

  • x: a dense vector of length n;
  • y: a dense vector of length m;
  • stats: statistics collected on the run in an AdjointStats structure.

Reference

source
Krylov.trilqr! — Function
solver = trilqr!(solver::TrilqrSolver, A, b, c; kwargs...)
solver = trilqr!(solver::TrilqrSolver, A, b, c, x0, y0; kwargs...)

where kwargs are keyword arguments of trilqr.

See TrilqrSolver for more details about the solver.

source