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Update Krylov processes to return more arguments
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@amontoison
amontoison committed Sep 24, 2023
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54 changes: 27 additions & 27 deletions docs/src/processes.md
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Original file line number Diff line number Diff line change
Expand Up @@ -60,18 +60,18 @@ For matrices $C \in \mathbb{C}^{n \times n} \enspace$ and $\enspace T \in \mathb
\left\{\sum_{i=0}^{k-1} C^i T \, \Omega_i \, \middle \vert \, \Omega_i \in \mathbb{C}^{p \times p},~0 \le i \le k-1 \right\}.
```

## Hermitian Lanczos
## [Hermitian Lanczos](@id hermitian-lanczos)

![hermitian_lanczos](./graphics/hermitian_lanczos.png)

After $k$ iterations of the Hermitian Lanczos process, the situation may be summarized as
```math
\begin{align*}
A V_k &= V_k T_k + \beta_{k+1,k} v_{k+1} e_k^T = V_{k+1} T_{k+1,k}, \\
A V_k &= V_k T_k + \beta_{k+1,k} v_{k+1} e_k^T = V_{k+1} T_{k+1,k}, \\
V_k^H V_k &= I_k,
\end{align*}
```
where $V_k$ is an orthonormal basis of the Krylov subspace $\mathcal{K}_k (A,b)$,
where $V_k$ is an orthonormal basis of the Krylov subspace $\mathcal{K}_k(A,b)$,
```math
T_k =
\begin{bmatrix}
Expand All @@ -89,22 +89,22 @@ T_{k+1,k} =
```
Note that depending on how we normalize the vectors that compose $V_k$, $T_{k+1,k}$ can be a real tridiagonal matrix even if $A$ is a complex matrix.

The function [`hermitian_lanczos`](@ref hermitian_lanczos) returns $V_{k+1}$ and $T_{k+1,k}$.
The function [`hermitian_lanczos`](@ref hermitian_lanczos(::Any, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\beta_1$ and $T_{k+1,k}$.

Related methods: [`SYMMLQ`](@ref symmlq), [`CG`](@ref cg), [`CR`](@ref cr), [`CAR`](@ref car), [`MINRES`](@ref minres), [`MINRES-QLP`](@ref minres_qlp), [`MINARES`](@ref minares), [`CGLS`](@ref cgls), [`CRLS`](@ref crls), [`CGNE`](@ref cgne), [`CRMR`](@ref crmr), [`CG-LANCZOS`](@ref cg_lanczos) and [`CG-LANCZOS-SHIFT`](@ref cg_lanczos_shift).

```@docs
hermitian_lanczos
hermitian_lanczos(::Any, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)
```

## Non-Hermitian Lanczos
## [Non-Hermitian Lanczos](@id nonhermitian-lanczos)

![nonhermitian_lanczos](./graphics/nonhermitian_lanczos.png)

After $k$ iterations of the non-Hermitian Lanczos process (also named the Lanczos biorthogonalization process), the situation may be summarized as
```math
\begin{align*}
A V_k &= V_k T_k + \beta_{k+1} v_{k+1} e_k^T = V_{k+1} T_{k+1,k}, \\
A V_k &= V_k T_k + \beta_{k+1} v_{k+1} e_k^T = V_{k+1} T_{k+1,k}, \\
A^H U_k &= U_k T_k^H + \bar{\gamma}_{k+1} u_{k+1} e_k^T = U_{k+1} T_{k,k+1}^H, \\
V_k^H U_k &= U_k^H V_k = I_k,
\end{align*}
Expand All @@ -131,7 +131,7 @@ T_{k,k+1} =
\end{bmatrix}.
```

The function [`nonhermitian_lanczos`](@ref nonhermitian_lanczos) returns $V_{k+1}$, $T_{k+1,k}$, $U_{k+1}$ and $T_{k,k+1}^H$.
The function [`nonhermitian_lanczos`](@ref nonhermitian_lanczos(::Any, ::AbstractVector{FC}, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\beta_1$, $T_{k+1,k}$, $U_{k+1}$, $\bar{\gamma}_1$ and $T_{k,k+1}^H$.

Related methods: [`BiLQ`](@ref bilq), [`QMR`](@ref qmr), [`BiLQR`](@ref bilqr), [`CGS`](@ref cgs) and [`BICGSTAB`](@ref bicgstab).

Expand All @@ -140,7 +140,7 @@ Related methods: [`BiLQ`](@ref bilq), [`QMR`](@ref qmr), [`BiLQR`](@ref bilqr),
With these scaling factors, the non-Hermitian Lanczos process coincides with the Hermitian Lanczos process when $A = A^H$ and $b = c$.

```@docs
nonhermitian_lanczos
nonhermitian_lanczos(::Any, ::AbstractVector{FC}, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)
```

The non-Hermitian Lanczos process can be also implemented without $A^H$ (transpose-free variant).
Expand All @@ -158,7 +158,7 @@ The polynomials are defined from the recursions:
\end{align*}
```

Because $\alpha_k = u_k^H A v_k$ and $(\bar{\gamma}_{k+1} u_{k+1})^H (\beta_{k+1} v_{k+1}) = \gamma_{k+1} \beta_{k+1}$, we can define the coefficients of $T_{k+1,k}$ and $T_{k,k+1}^H$ as follows:
Because $\alpha_k = u_k^H A v_k$ and $(\bar{\gamma}_{k+1} u_{k+1})^H (\beta_{k+1} v_{k+1}) = \gamma_{k+1} \beta_{k+1}$, we can determine the coefficients of $T_{k+1,k}$ and $T_{k,k+1}^H$ as follows:
```math
\begin{align*}
\alpha_k &= \dfrac{1}{\gamma_k \beta_k} \langle~Q_{k-1}(A^H) c \, , \, A P_{k-1}(A) b~\rangle \\
Expand All @@ -169,7 +169,7 @@ Because $\alpha_k = u_k^H A v_k$ and $(\bar{\gamma}_{k+1} u_{k+1})^H (\beta_{k+1
\end{align*}
```

## Arnoldi
## [Arnoldi](@id arnoldi)

![arnoldi](./graphics/arnoldi.png)

Expand Down Expand Up @@ -197,25 +197,25 @@ H_{k+1,k} =
\end{bmatrix}.
```

The function [`arnoldi`](@ref arnoldi) returns $V_{k+1}$ and $H_{k+1,k}$.
The function [`arnoldi`](@ref arnoldi(::Any, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\beta$ and $H_{k+1,k}$.

Related methods: [`DIOM`](@ref diom), [`FOM`](@ref fom), [`DQGMRES`](@ref dqgmres), [`GMRES`](@ref gmres) and [`FGMRES`](@ref fgmres).

!!! note
The Arnoldi process coincides with the Hermitian Lanczos process when $A$ is Hermitian.

```@docs
arnoldi
arnoldi(::Any, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)
```

## Golub-Kahan
## [Golub-Kahan](@id golub-kahan)

![golub_kahan](./graphics/golub_kahan.png)

After $k$ iterations of the Golub-Kahan bidiagonalization process, the situation may be summarized as
```math
\begin{align*}
A V_k &= U_{k+1} B_k, \\
A V_k &= U_{k+1} B_k, \\
A^H U_{k+1} &= V_k B_k^H + \bar{\alpha}_{k+1} v_{k+1} e_{k+1}^T = V_{k+1} L_{k+1}^H, \\
V_k^H V_k &= U_k^H U_k = I_k,
\end{align*}
Expand Down Expand Up @@ -246,7 +246,7 @@ B_k =
```
Note that depending on how we normalize the vectors that compose $V_k$ and $U_k$, $L_k$ can be a real bidiagonal matrix even if $A$ is a complex matrix.

The function [`golub_kahan`](@ref golub_kahan) returns $V_{k+1}$, $U_{k+1}$ and $L_{k+1}$.
The function [`golub_kahan`](@ref golub_kahan(::Any, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $U_{k+1}$, $\beta_1$ and $L_{k+1}$.

Related methods: [`LNLQ`](@ref lnlq), [`CRAIG`](@ref craig), [`CRAIGMR`](@ref craigmr), [`LSLQ`](@ref lslq), [`LSQR`](@ref lsqr) and [`LSMR`](@ref lsmr).

Expand All @@ -255,17 +255,17 @@ Related methods: [`LNLQ`](@ref lnlq), [`CRAIG`](@ref craig), [`CRAIGMR`](@ref cr
It is also related to the Hermitian Lanczos process applied to $\begin{bmatrix} 0 & A \\ A^H & 0 \end{bmatrix}$ with initial vector $\begin{bmatrix} b \\ 0 \end{bmatrix}$.

```@docs
golub_kahan
golub_kahan(::Any, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)
```

## Saunders-Simon-Yip
## [Saunders-Simon-Yip](@id saunders-simon-yip)

![saunders_simon_yip](./graphics/saunders_simon_yip.png)

After $k$ iterations of the Saunders-Simon-Yip process (also named the orthogonal tridiagonalization process), the situation may be summarized as
```math
\begin{align*}
A U_k &= V_k T_k + \beta_{k+1} v_{k+1} e_k^T = V_{k+1} T_{k+1,k}, \\
A U_k &= V_k T_k + \beta_{k+1} v_{k+1} e_k^T = V_{k+1} T_{k+1,k}, \\
A^H V_k &= U_k T_k^H + \bar{\gamma}_{k+1} u_{k+1} e_k^T = U_{k+1} T_{k,k+1}^H, \\
V_k^H V_k &= U_k^H U_k = I_k,
\end{align*}
Expand All @@ -292,18 +292,18 @@ T_{k,k+1} =
\end{bmatrix}.
```

The function [`saunders_simon_yip`](@ref saunders_simon_yip) returns $V_{k+1}$, $T_{k+1,k}$, $U_{k+1}$ and $T_{k,k+1}^H$.
The function [`saunders_simon_yip`](@ref saunders_simon_yip(::Any, ::AbstractVector{FC}, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\beta_1$, $T_{k+1,k}$, $U_{k+1}$, $\bar{\gamma}_1$ and $T_{k,k+1}^H$.

Related methods: [`USYMLQ`](@ref usymlq), [`USYMQR`](@ref usymqr), [`TriLQR`](@ref trilqr), [`TriCG`](@ref tricg) and [`TriMR`](@ref trimr).

```@docs
saunders_simon_yip
```

!!! note
The Saunders-Simon-Yip is equivalent to the block-Lanczos process applied to $\begin{bmatrix} 0 & A \\ A^H & 0 \end{bmatrix}$ with initial matrix $\begin{bmatrix} b & 0 \\ 0 & c \end{bmatrix}$.

## Montoison-Orban
```@docs
saunders_simon_yip(::Any, ::AbstractVector{FC}, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)
```

## [Montoison-Orban](@id montoison-orban)

![montoison_orban](./graphics/montoison_orban.png)

Expand Down Expand Up @@ -347,7 +347,7 @@ F_{k+1,k} =
\end{bmatrix}.
```

The function [`montoison_orban`](@ref montoison_orban) returns $V_{k+1}$, $H_{k+1,k}$, $U_{k+1}$ and $F_{k+1,k}$.
The function [`montoison_orban`](@ref montoison_orban(::Any, ::Any, ::AbstractVector{FC}, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\beta$, $H_{k+1,k}$, $U_{k+1}$, $\gamma$ and $F_{k+1,k}$.

Related methods: [`GPMR`](@ref gpmr).

Expand All @@ -356,5 +356,5 @@ Related methods: [`GPMR`](@ref gpmr).
It also coincides with the Saunders-Simon-Yip process when $B = A^H$.

```@docs
montoison_orban
montoison_orban(::Any, ::Any, ::AbstractVector{FC}, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)
```
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