BiLQR

Krylov.bilqrFunction
(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

TriLQR

Krylov.trilqrFunction
(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