Least-squares problems

CGLS

Krylov.cgls — Function

(x, stats) = cgls(A, b::AbstractVector{FC};
                  M=I, ldiv::Bool=false, radius::T=zero(T),
                  λ::T=zero(T), 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}.

Solve the regularized linear least-squares problem

minimize ‖b - Ax‖₂² + λ‖x‖₂²

of size m × n using the Conjugate Gradient (CG) method, where λ ≥ 0 is a regularization parameter. This method is equivalent to applying CG to the normal equations

(AᴴA + λI) x = Aᴴb

but is more stable.

CGLS produces monotonic residuals ‖r‖₂ but not optimality residuals ‖Aᴴr‖₂. It is formally equivalent to LSQR, though can be slightly less accurate, but simpler to implement.

Input arguments

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

Keyword arguments

M: linear operator that models a Hermitian positive-definite matrix of size n used for preconditioning;
ldiv: define whether the preconditioner uses ldiv! or mul!;
radius: add the trust-region constraint ‖x‖ ≤ radius if radius > 0. Useful to compute a step in a trust-region method for optimization;
λ: regularization parameter;
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;
stats: statistics collected on the run in a SimpleStats structure.

References

M. R. Hestenes and E. Stiefel. Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49(6), pp. 409–436, 1952.
A. Björck, T. Elfving and Z. Strakos, Stability of Conjugate Gradient and Lanczos Methods for Linear Least Squares Problems, SIAM Journal on Matrix Analysis and Applications, 19(3), pp. 720–736, 1998.

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Krylov.cgls! — Function

solver = cgls!(solver::CglsSolver, A, b; kwargs...)

where kwargs are keyword arguments of cgls.

See CglsSolver for more details about the solver.

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CGLS-LANCZOS-SHIFT

Krylov.cgls_lanczos_shift — Function

(x, stats) = cgls_lanczos_shift(A, b::AbstractVector{FC};
                  M=I, λ::T=zero(T), atol::T=√eps(T), rtol::T=√eps(T),
                  radius::T=zero(T), itmax::Int=0, verbose::Int=0, history::Bool=false,
                  callback=solver->false)

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

Solve the regularized linear least-squares problem

minimize ‖b - Ax‖₂² + λ‖x‖₂²

using the Conjugate Gradient (CG) method, where λ ≥ 0 is a regularization parameter. This method is equivalent to applying CG to the normal equations

(AᴴA + λI) x = Aᴴb

but is more stable.

CGLS produces monotonic residuals ‖r‖₂ but not optimality residuals ‖Aᴴr‖₂. It is formally equivalent to LSQR, though can be slightly less accurate, but simpler to implement.

Input arguments

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

Keyword arguments

M: linear operator that models a Hermitian positive-definite matrix of size n used for preconditioning;
ldiv: define whether the preconditioner uses ldiv! or mul!;
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 vector of p dense vectors, each one of length n;
stats: statistics collected on the run in a LanczosShiftStats structure.

References

M. R. Hestenes and E. Stiefel. Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49(6), pp. 409–436, 1952.
A. Björck, T. Elfving and Z. Strakos, Stability of Conjugate Gradient and Lanczos Methods for Linear Least Squares Problems, SIAM Journal on Matrix Analysis and Applications, 19(3), pp. 720–736, 1998.

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Krylov.cgls_lanczos_shift! — Function

solver = cgls_lanczos_shift!(solver::CglsLanczosShiftSolver, A, b, shifts; kwargs...)

where kwargs are keyword arguments of cgls_lanczos_shift.

See CglsLanczosShiftSolver for more details about the solver.

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CRLS

Krylov.crls — Function

(x, stats) = crls(A, b::AbstractVector{FC};
                  M=I, ldiv::Bool=false, radius::T=zero(T),
                  λ::T=zero(T), 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}.

Solve the linear least-squares problem

minimize ‖b - Ax‖₂² + λ‖x‖₂²

of size m × n using the Conjugate Residuals (CR) method. This method is equivalent to applying MINRES to the normal equations

(AᴴA + λI) x = Aᴴb.

This implementation recurs the residual r := b - Ax.

CRLS produces monotonic residuals ‖r‖₂ and optimality residuals ‖Aᴴr‖₂. It is formally equivalent to LSMR, though can be substantially less accurate, but simpler to implement.

Input arguments

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

Keyword arguments

M: linear operator that models a Hermitian positive-definite matrix of size n used for preconditioning;
ldiv: define whether the preconditioner uses ldiv! or mul!;
radius: add the trust-region constraint ‖x‖ ≤ radius if radius > 0. Useful to compute a step in a trust-region method for optimization;
λ: regularization parameter;
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;
stats: statistics collected on the run in a SimpleStats structure.

Reference

D. C.-L. Fong, Minimum-Residual Methods for Sparse, Least-Squares using Golubg-Kahan Bidiagonalization, Ph.D. Thesis, Stanford University, 2011.

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Krylov.crls! — Function

solver = crls!(solver::CrlsSolver, A, b; kwargs...)

where kwargs are keyword arguments of crls.

See CrlsSolver for more details about the solver.

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LSLQ

Krylov.lslq — Function

(x, stats) = lslq(A, b::AbstractVector{FC};
                  M=I, N=I, ldiv::Bool=false,
                  window::Int=5, transfer_to_lsqr::Bool=false,
                  sqd::Bool=false, λ::T=zero(T),
                  σ::T=zero(T), etol::T=√eps(T),
                  utol::T=√eps(T), btol::T=√eps(T),
                  conlim::T=1/√eps(T), 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}.

Solve the regularized linear least-squares problem

minimize ‖b - Ax‖₂² + λ²‖x‖₂²

of size m × n using the LSLQ method, where λ ≥ 0 is a regularization parameter. LSLQ is formally equivalent to applying SYMMLQ to the normal equations

(AᴴA + λ²I) x = Aᴴb

but is more stable.

If λ > 0, we solve the symmetric and quasi-definite system

[ E      A ] [ r ]   [ b ]
[ Aᴴ  -λ²F ] [ x ] = [ 0 ],

where E and F are symmetric and positive definite. Preconditioners M = E⁻¹ ≻ 0 and N = F⁻¹ ≻ 0 may be provided in the form of linear operators. If sqd=true, λ is set to the common value 1.

The system above represents the optimality conditions of

minimize ‖b - Ax‖²_E⁻¹ + λ²‖x‖²_F.

For a symmetric and positive definite matrix K, the K-norm of a vector x is ‖x‖²_K = xᴴKx. LSLQ is then equivalent to applying SYMMLQ to (AᴴE⁻¹A + λ²F)x = AᴴE⁻¹b with r = E⁻¹(b - Ax).

If λ = 0, we solve the symmetric and indefinite system

[ E    A ] [ r ]   [ b ]
[ Aᴴ   0 ] [ x ] = [ 0 ].

The system above represents the optimality conditions of

minimize ‖b - Ax‖²_E⁻¹.

In this case, N can still be specified and indicates the weighted norm in which x and Aᴴr should be measured. r can be recovered by computing E⁻¹(b - Ax).

Main features

the solution estimate is updated along orthogonal directions
the norm of the solution estimate ‖xᴸₖ‖₂ is increasing
the error ‖eₖ‖₂ := ‖xᴸₖ - x*‖₂ is decreasing
it is possible to transition cheaply from the LSLQ iterate to the LSQR iterate if there is an advantage (there always is in terms of error)
if A is rank deficient, identify the minimum least-squares solution

Input arguments

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

Keyword arguments

M: linear operator that models a Hermitian positive-definite matrix of size m used for centered preconditioning of the augmented system;
N: linear operator that models a Hermitian positive-definite matrix of size n used for centered preconditioning of the augmented system;
ldiv: define whether the preconditioners use ldiv! or mul!;
window: number of iterations used to accumulate a lower bound on the error;
transfer_to_lsqr: transfer from the LSLQ point to the LSQR point, when it exists. The transfer is based on the residual norm;
sqd: if true, set λ=1 for Hermitian quasi-definite systems;
λ: regularization parameter;
σ: strict lower bound on the smallest positive singular value σₘᵢₙ such as σ = (1-10⁻⁷)σₘᵢₙ;
etol: stopping tolerance based on the lower bound on the error;
utol: stopping tolerance based on the upper bound on the error;
btol: stopping tolerance used to detect zero-residual problems;
conlim: limit on the estimated condition number of A beyond which the solution will be abandoned;
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;
stats: statistics collected on the run in a LSLQStats structure.
stats.err_lbnds is a vector of lower bounds on the LQ error–-the vector is empty if window is set to zero
stats.err_ubnds_lq is a vector of upper bounds on the LQ error–-the vector is empty if σ == 0 is left at zero
stats.err_ubnds_cg is a vector of upper bounds on the CG error–-the vector is empty if σ == 0 is left at zero
stats.error_with_bnd is a boolean indicating whether there was an error in the upper bounds computation (cancellation errors, too large σ ...)

Stopping conditions

The iterations stop as soon as one of the following conditions holds true:

the optimality residual is sufficiently small (stats.status = "found approximate minimum least-squares solution") in the sense that either
- ‖Aᴴr‖ / (‖A‖ ‖r‖) ≤ atol, or
- 1 + ‖Aᴴr‖ / (‖A‖ ‖r‖) ≤ 1
an approximate zero-residual solution has been found (stats.status = "found approximate zero-residual solution") in the sense that either
- ‖r‖ / ‖b‖ ≤ btol + atol ‖A‖ * ‖xᴸ‖ / ‖b‖, or
- 1 + ‖r‖ / ‖b‖ ≤ 1
the estimated condition number of A is too large in the sense that either
- 1/cond(A) ≤ 1/conlim (stats.status = "condition number exceeds tolerance"), or
- 1 + 1/cond(A) ≤ 1 (stats.status = "condition number seems too large for this machine")
the lower bound on the LQ forward error is less than etol * ‖xᴸ‖
the upper bound on the CG forward error is less than utol * ‖xᶜ‖

References

R. Estrin, D. Orban and M. A. Saunders, Euclidean-norm error bounds for SYMMLQ and CG, SIAM Journal on Matrix Analysis and Applications, 40(1), pp. 235–253, 2019.
R. Estrin, D. Orban and M. A. Saunders, LSLQ: An Iterative Method for Linear Least-Squares with an Error Minimization Property, SIAM Journal on Matrix Analysis and Applications, 40(1), pp. 254–275, 2019.

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Krylov.lslq! — Function

solver = lslq!(solver::LslqSolver, A, b; kwargs...)

where kwargs are keyword arguments of lslq.

See LslqSolver for more details about the solver.

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LSQR

Krylov.lsqr — Function

(x, stats) = lsqr(A, b::AbstractVector{FC};
                  M=I, N=I, ldiv::Bool=false,
                  window::Int=5, sqd::Bool=false, λ::T=zero(T),
                  radius::T=zero(T), etol::T=√eps(T),
                  axtol::T=√eps(T), btol::T=√eps(T),
                  conlim::T=1/√eps(T), atol::T=zero(T),
                  rtol::T=zero(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}.

Solve the regularized linear least-squares problem

minimize ‖b - Ax‖₂² + λ²‖x‖₂²

of size m × n using the LSQR method, where λ ≥ 0 is a regularization parameter. LSQR is formally equivalent to applying CG to the normal equations

(AᴴA + λ²I) x = Aᴴb

(and therefore to CGLS) but is more stable.

LSQR produces monotonic residuals ‖r‖₂ but not optimality residuals ‖Aᴴr‖₂. It is formally equivalent to CGLS, though can be slightly more accurate.

If λ > 0, LSQR solves the symmetric and quasi-definite system

[ E      A ] [ r ]   [ b ]
[ Aᴴ  -λ²F ] [ x ] = [ 0 ],

The system above represents the optimality conditions of

minimize ‖b - Ax‖²_E⁻¹ + λ²‖x‖²_F.

For a symmetric and positive definite matrix K, the K-norm of a vector x is ‖x‖²_K = xᴴKx. LSQR is then equivalent to applying CG to (AᴴE⁻¹A + λ²F)x = AᴴE⁻¹b with r = E⁻¹(b - Ax).

If λ = 0, we solve the symmetric and indefinite system

[ E    A ] [ r ]   [ b ]
[ Aᴴ   0 ] [ x ] = [ 0 ].

The system above represents the optimality conditions of

minimize ‖b - Ax‖²_E⁻¹.

In this case, N can still be specified and indicates the weighted norm in which x and Aᴴr should be measured. r can be recovered by computing E⁻¹(b - Ax).

Input arguments

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

Keyword arguments

M: linear operator that models a Hermitian positive-definite matrix of size m used for centered preconditioning of the augmented system;
N: linear operator that models a Hermitian positive-definite matrix of size n used for centered preconditioning of the augmented system;
ldiv: define whether the preconditioners use ldiv! or mul!;
window: number of iterations used to accumulate a lower bound on the error;
sqd: if true, set λ=1 for Hermitian quasi-definite systems;
λ: regularization parameter;
radius: add the trust-region constraint ‖x‖ ≤ radius if radius > 0. Useful to compute a step in a trust-region method for optimization;
etol: stopping tolerance based on the lower bound on the error;
axtol: tolerance on the backward error;
btol: stopping tolerance used to detect zero-residual problems;
conlim: limit on the estimated condition number of A beyond which the solution will be abandoned;
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;
stats: statistics collected on the run in a SimpleStats structure.

Reference

C. C. Paige and M. A. Saunders, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares, ACM Transactions on Mathematical Software, 8(1), pp. 43–71, 1982.

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Krylov.lsqr! — Function

solver = lsqr!(solver::LsqrSolver, A, b; kwargs...)

where kwargs are keyword arguments of lsqr.

See LsqrSolver for more details about the solver.

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LSMR

Krylov.lsmr — Function

(x, stats) = lsmr(A, b::AbstractVector{FC};
                  M=I, N=I, ldiv::Bool=false,
                  window::Int=5, sqd::Bool=false, λ::T=zero(T),
                  radius::T=zero(T), etol::T=√eps(T),
                  axtol::T=√eps(T), btol::T=√eps(T),
                  conlim::T=1/√eps(T), atol::T=zero(T),
                  rtol::T=zero(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}.

Solve the regularized linear least-squares problem

minimize ‖b - Ax‖₂² + λ²‖x‖₂²

of size m × n using the LSMR method, where λ ≥ 0 is a regularization parameter. LSMR is formally equivalent to applying MINRES to the normal equations

(AᴴA + λ²I) x = Aᴴb

(and therefore to CRLS) but is more stable.

LSMR produces monotonic residuals ‖r‖₂ and optimality residuals ‖Aᴴr‖₂. It is formally equivalent to CRLS, though can be substantially more accurate.

LSMR can be also used to find a null vector of a singular matrix A by solving the problem min ‖Aᴴx - b‖ with any nonzero vector b. At a minimizer, the residual vector r = b - Aᴴx will satisfy Ar = 0.

If λ > 0, we solve the symmetric and quasi-definite system

[ E      A ] [ r ]   [ b ]
[ Aᴴ  -λ²F ] [ x ] = [ 0 ],

The system above represents the optimality conditions of

minimize ‖b - Ax‖²_E⁻¹ + λ²‖x‖²_F.

For a symmetric and positive definite matrix K, the K-norm of a vector x is ‖x‖²_K = xᴴKx. LSMR is then equivalent to applying MINRES to (AᴴE⁻¹A + λ²F)x = AᴴE⁻¹b with r = E⁻¹(b - Ax).

If λ = 0, we solve the symmetric and indefinite system

[ E    A ] [ r ]   [ b ]
[ Aᴴ   0 ] [ x ] = [ 0 ].

The system above represents the optimality conditions of

minimize ‖b - Ax‖²_E⁻¹.

In this case, N can still be specified and indicates the weighted norm in which x and Aᴴr should be measured. r can be recovered by computing E⁻¹(b - Ax).

Input arguments

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

Keyword arguments

M: linear operator that models a Hermitian positive-definite matrix of size m used for centered preconditioning of the augmented system;
N: linear operator that models a Hermitian positive-definite matrix of size n used for centered preconditioning of the augmented system;
ldiv: define whether the preconditioners use ldiv! or mul!;
window: number of iterations used to accumulate a lower bound on the error;
sqd: if true, set λ=1 for Hermitian quasi-definite systems;
λ: regularization parameter;
radius: add the trust-region constraint ‖x‖ ≤ radius if radius > 0. Useful to compute a step in a trust-region method for optimization;
etol: stopping tolerance based on the lower bound on the error;
axtol: tolerance on the backward error;
btol: stopping tolerance used to detect zero-residual problems;
conlim: limit on the estimated condition number of A beyond which the solution will be abandoned;
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;
stats: statistics collected on the run in a LsmrStats structure.

Reference

D. C.-L. Fong and M. A. Saunders, LSMR: An Iterative Algorithm for Sparse Least Squares Problems, SIAM Journal on Scientific Computing, 33(5), pp. 2950–2971, 2011.

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Krylov.lsmr! — Function

solver = lsmr!(solver::LsmrSolver, A, b; kwargs...)

where kwargs are keyword arguments of lsmr.

See LsmrSolver for more details about the solver.

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USYMQR

Krylov.usymqr — Function

(x, stats) = usymqr(A, b::AbstractVector{FC}, c::AbstractVector{FC};
                    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, stats) = usymqr(A, b, c, x0::AbstractVector; kwargs...)

USYMQR can be warm-started from an initial guess x0 where kwargs are the same keyword arguments as above.

USYMQR solves the linear least-squares problem min ‖b - Ax‖² of size m × n. USYMQR solves Ax = b if it is consistent.

USYMQR is based on the orthogonal tridiagonalization process and requires two initial nonzero vectors b and c. The vector c is only used to initialize the process and a default value can be b or Aᴴb depending on the shape of A. The residual norm ‖b - Ax‖ monotonously decreases in USYMQR. When A is Hermitian and b = c, QMR is equivalent to MINRES. USYMQR is considered as a generalization of MINRES.

It can also be applied to under-determined and over-determined problems. USYMQR finds the minimum-norm solution if problems are inconsistent.

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 argument

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

Keyword arguments

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;
stats: statistics collected on the run in a SimpleStats structure.

References

M. A. Saunders, H. D. Simon, and E. L. Yip, Two Conjugate-Gradient-Type Methods for Unsymmetric Linear Equations, SIAM Journal on Numerical Analysis, 25(4), pp. 927–940, 1988.
L. Reichel and Q. Ye, A generalized LSQR algorithm, Numerical Linear Algebra with Applications, 15(7), pp. 643–660, 2008.
A. Buttari, D. Orban, D. Ruiz and D. Titley-Peloquin, A tridiagonalization method for symmetric saddle-point and quasi-definite systems, SIAM Journal on Scientific Computing, 41(5), pp. 409–432, 2019.
A. Montoison and D. Orban, BiLQ: An Iterative Method for Nonsymmetric Linear Systems with a Quasi-Minimum Error Property, SIAM Journal on Matrix Analysis and Applications, 41(3), pp. 1145–1166, 2020.

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Krylov.usymqr! — Function

solver = usymqr!(solver::UsymqrSolver, A, b, c; kwargs...)
solver = usymqr!(solver::UsymqrSolver, A, b, c, x0; kwargs...)

where kwargs are keyword arguments of usymqr.

See UsymqrSolver for more details about the solver.

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