# (Inverse) power method

Solves the eigenproblem $Ax = λx$ approximately where $A$ is a general linear map. By default converges towards the dominant eigenpair $(λ, x)$ such that $|λ|$ is largest. Shift-and-invert can be applied to target a specific eigenvalue near `shift`

in the complex plane.

## Usage

`IterativeSolvers.powm`

— Function`powm(B; kwargs...) -> λ, x, [history]`

See `powm!`

. Calls `powm!(B, x0; kwargs...)`

with `x0`

initialized as a random, complex unit vector.

`IterativeSolvers.powm!`

— Function`powm!(B, x; shift = zero(eltype(B)), inverse::Bool = false, kwargs...) -> λ, x, [history]`

By default finds the approximate eigenpair `(λ, x)`

of `B`

where `|λ|`

is largest.

**Arguments**

`B`

: linear map, see the note below.`x`

: normalized initial guess. Don't forget to use complex arithmetic when necessary.

**Keywords**

`tol::Real = eps(real(eltype(B))) * size(B, 2) ^ 3`

: stopping tolerance for the residual norm;`maxiter::Integer = size(B,2)`

: maximum number of iterations;`log::Bool`

: keep track of the residual norm in each iteration;`verbose::Bool`

: print convergence information during the iterations.

When applying shift-and-invert to $Ax = λx$ with `invert = true`

and `shift = ...`

, note that the role of `B * b`

becomes computing `inv(A - shift I) * b`

. So rather than passing the linear map $A$ itself, pass a linear map `B`

that has the action of shift-and-invert. The eigenvalue is transformed back to an eigenvalue of the actual matrix $A$.

**Return values**

**if log is false**

`λ::Number`

approximate eigenvalue computed as the Rayleigh quotient;`x::Vector`

approximate eigenvector.

**if log is true**

`λ::Number`

: approximate eigenvalue computed as the Rayleigh quotient;`x::Vector`

: approximate eigenvector;`history`

: convergence history.

**ConvergenceHistory keys**

`:tol`

=>`::Real`

: stopping tolerance;`:resnom`

=>`::Vector`

: residual norm at each iteration.

**Examples**

```
using LinearMaps
σ = 1.0 + 1.3im
A = rand(ComplexF64, 50, 50)
F = lu(A - σ * I)
Fmap = LinearMap{ComplexF64}((y, x) -> ldiv!(y, F, x), 50, ismutating = true)
λ, x = powm(Fmap, inverse = true, shift = σ, tol = 1e-4, maxiter = 200)
```

`IterativeSolvers.invpowm`

— Function`invpowm(B; shift = σ, kwargs...) -> λ, x, [history]`

Find the approximate eigenpair `(λ, x)`

of $A$ near `shift`

, where `B`

is a linear map that has the effect `B * v = inv(A - σI) * v`

.

The method calls `powm!(B, x0; inverse = true, shift = σ)`

with `x0`

a random, complex unit vector. See `powm!`

**Examples**

```
using LinearMaps
σ = 1.0 + 1.3im
A = rand(ComplexF64, 50, 50)
F = lu(A - σ * I)
Fmap = LinearMap{ComplexF64}((y, x) -> ldiv!(y, F, x), 50, ismutating = true)
λ, x = invpowm(Fmap, shift = σ, tol = 1e-4, maxiter = 200)
```

`IterativeSolvers.invpowm!`

— Function`invpowm!(B, x0; shift = σ, kwargs...) -> λ, x, [history]`

Find the approximate eigenpair `(λ, x)`

of $A$ near `shift`

, where `B`

is a linear map that has the effect `B * v = inv(A - σI) * v`

.

The method calls `powm!(B, x0; inverse = true, shift = σ)`

. See `powm!`

.

## Implementation details

Storage requirements are 3 vectors: the approximate eigenvector `x`

, the residual vector `r`

and a temporary. The residual norm lags behind one iteration, as it is computed when $Ax$ is performed. Therefore the final resdiual norm is even smaller.