## Warm Start

Most Krylov methods in this module accept a starting point as argument. The starting point is used as initial approximation to a solution.

```
solver = CgSolver(n, n, S)
cg!(solver, A, b, itmax=100)
if !issolved(solver)
cg!(solver, A, b, solver.x, itmax=100) # cg! uses the approximate solution `solver.x` as starting point
end
```

If the user has an initial guess `x0`

, it can be provided directly.

`cg(A, b, x0)`

It is also possible to use the `warm_start!`

function to feed the starting point into the solver.

```
warm_start!(solver, x0)
cg!(solver, A, b)
# the previous two lines are equivalent to cg!(solver, A, b, x0)
```

If a Krylov method doesn't have the option to warm start, it can still be done explicitly. We provide an example with `cg_lanczos!`

.

```
solver = CgLanczosSolver(n, n, S)
cg_lanczos!(solver, A, b)
x₀ = solver.x # Ax₀ ≈ b
r = b - A * x₀ # r = b - Ax₀
cg_lanczos!(solver, A, r)
Δx = solver.x # AΔx = r
x = x₀ + Δx # Ax = b
```

Explicit restarts cannot be avoided in certain block methods, such as TriMR, due to the preconditioners.

```
# [E A] [x] = [b]
# [Aᴴ F] [y] [c]
M = inv(E)
N = inv(F)
x₀, y₀, stats = trimr(A, b, c, M=M, N=N)
# E and F are not available inside TriMR
b₀ = b - Ex₀ - Ay
c₀ = c - Aᴴx₀ - Fy
Δx, Δy, stats = trimr(A, b₀, c₀, M=M, N=N)
x = x₀ + Δx
y = y₀ + Δy
```

## Restarted methods

The storage requierements of Krylov methods based on the Arnoldi process, such as FOM and GMRES, increase as the iteration progresses. For very large problems, the storage costs become prohibitive after only few iterations and restarted variants FOM(k) and GMRES(k) are prefered. In this section, we show how to use warm starts to implement GMRES(k) and FOM(k).

```
k = 50
solver = GmresSolver(A, b, k) # FomSolver(A, b, k)
solver.x .= 0 # solver.x .= x₀
nrestart = 0
while !issolved(solver) || nrestart ≤ 10
solve!(solver, A, b, solver.x, itmax=k)
nrestart += 1
end
```