# LSMR

Least-squares minimal residual

## Usage

`IterativeSolvers.lsmr`

— Function`lsmr(A, b; kwrags...) -> x, [history]`

Same as `lsmr!`

, but allocates a solution vector `x`

initialized with zeros.

`IterativeSolvers.lsmr!`

— Function`lsmr!(x, A, b; kwargs...) -> x, [history]`

Minimizes $\|Ax - b\|^2 + \|λx\|^2$ in the Euclidean norm. If multiple solutions exists the minimum norm solution is returned.

The method is based on the Golub-Kahan bidiagonalization process. It is algebraically equivalent to applying MINRES to the normal equations $(A^*A + λ^2I)x = A^*b$, but has better numerical properties, especially if $A$ is ill-conditioned.

**Arguments**

`x`

: Initial guess, will be updated in-place;`A`

: linear operator;`b`

: right-hand side.

**Keywords**

`λ::Number = 0`

: lambda.`atol::Number = 1e-6`

,`btol::Number = 1e-6`

: stopping tolerances. If both are 1.0e-9 (say), the final residual norm should be accurate to about 9 digits. (The final`x`

will usually have fewer correct digits, depending on`cond(A)`

and the size of damp).`conlim::Number = 1e8`

: stopping tolerance.`lsmr`

terminates if an estimate of`cond(A)`

exceeds conlim. For compatible systems Ax = b, conlim could be as large as 1.0e+12 (say). For least-squares problems, conlim should be less than 1.0e+8. Maximum precision can be obtained by setting`atol`

=`btol`

=`conlim`

= zero, but the number of iterations may then be excessive.`maxiter::Int = maximum(size(A))`

: 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`

: approximated solution.

**if log is true**

`x`

: approximated solution.`ch`

: convergence history.

**ConvergenceHistory keys**

`:atol`

=>`::Real`

: atol stopping tolerance.`:btol`

=>`::Real`

: btol stopping tolerance.`:ctol`

=>`::Real`

: ctol stopping tolerance.`:anorm`

=>`::Real`

: anorm.`:rnorm`

=>`::Real`

: rnorm.`:cnorm`

=>`::Real`

: cnorm.`:resnom`

=>`::Vector`

: residual norm at each iteration.

## Implementation details

Adapted from: http://web.stanford.edu/group/SOL/software/lsmr/