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| # [Block Krylov processes](@id block-krylov-processes) | ||
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| ## [Block Hermitian Lanczos](@id block-hermitian-lanczos) | ||
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| If the vector $b$ in the [Hermitian Lanczos](@ref hermitian-lanczos) process is replaced by a matrix $B$ with $p$ columns, we can derive the block Hermitian Lanczos process. | ||
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| After $k$ iterations of the block Hermitian Lanczos process, the situation may be summarized as | ||
| ```math | ||
| \begin{align*} | ||
| A V_k &= V_{k+1} T_{k+1,k}, \\ | ||
| V_k^H V_k &= I_{pk}, | ||
| \end{align*} | ||
| ``` | ||
| where $V_k$ is an orthonormal basis of the block Krylov subspace $\mathcal{K}_k^{\square}(A,B)$, | ||
| ```math | ||
| T_{k+1,k} = | ||
| \begin{bmatrix} | ||
| \Omega_1 & \Psi_2^H & & \\ | ||
| \Psi_2 & \Omega_2 & \ddots & \\ | ||
| & \ddots & \ddots & \Psi_k^H \\ | ||
| & & \Psi_k & \Omega_k \\ | ||
| & & & \Psi_{k+1} | ||
| \end{bmatrix}. | ||
| ``` | ||
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| The function [`hermitian_lanczos`](@ref hermitian_lanczos(::Any, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\Psi_1$ and $T_{k+1,k}$. | ||
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| ```@docs | ||
| hermitian_lanczos(::Any, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat) | ||
| ``` | ||
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| ## [Block Non-Hermitian Lanczos](@id block-nonhermitian-lanczos) | ||
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| If the vectors $b$ and $c$ in the [non-Hermitian Lanczos](@ref nonhermitian-lanczos) process are replaced by matrices $B$ and $C$ with both $p$ columns, we can derive the block non-Hermitian Lanczos process. | ||
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|  | ||
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| After $k$ iterations of the block non-Hermitian Lanczos process, the situation may be summarized as | ||
| ```math | ||
| \begin{align*} | ||
| A V_k &= V_{k+1} T_{k+1,k}, \\ | ||
| A^H U_k &= U_{k+1} T_{k,k+1}^H, \\ | ||
| V_k^H U_k &= U_k^H V_k = I_{pk}, | ||
| \end{align*} | ||
| ``` | ||
| where $V_k$ and $U_k$ are bases of the block Krylov subspaces $\mathcal{K}^{\square}_k(A,B)$ and $\mathcal{K}^{\square}_k (A^H,C)$, respectively, | ||
| ```math | ||
| T_{k+1,k} = | ||
| \begin{bmatrix} | ||
| \Omega_1 & \Phi_2 & & \\ | ||
| \Psi_2 & \Omega_2 & \ddots & \\ | ||
| & \ddots & \ddots & \Phi_k \\ | ||
| & & \Psi_k & \Omega_k \\ | ||
| & & & \Psi_{k+1} | ||
| \end{bmatrix} | ||
| , \qquad | ||
| T_{k,k+1}^H = | ||
| \begin{bmatrix} | ||
| \Omega_1^H & \Psi_2^H & & \\ | ||
| \Phi_2^H & \Omega_2^H & \ddots & \\ | ||
| & \ddots & \ddots & \Psi_k^H \\ | ||
| & & \Phi_k^H & \Omega_k^H \\ | ||
| & & & \Phi_{k+1}^H | ||
| \end{bmatrix}. | ||
| ``` | ||
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| The function [`nonhermitian_lanczos`](@ref nonhermitian_lanczos(::Any, ::AbstractMatrix{FC}, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\Psi_1$, $T_{k+1,k}$, $U_{k+1}$ $\Phi_1^H$ and $T_{k,k+1}^H$. | ||
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| ```@docs | ||
| nonhermitian_lanczos(::Any, ::AbstractMatrix{FC}, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat) | ||
| ``` | ||
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| ## [Block Arnoldi](@id block-arnoldi) | ||
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| If the vector $b$ in the [Arnoldi](@ref arnoldi) process is replaced by a matrix $B$ with $p$ columns, we can derive the block Arnoldi process. | ||
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|  | ||
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| After $k$ iterations of the block Arnoldi process, the situation may be summarized as | ||
| ```math | ||
| \begin{align*} | ||
| A V_k &= V_{k+1} H_{k+1,k}, \\ | ||
| V_k^H V_k &= I_{pk}, | ||
| \end{align*} | ||
| ``` | ||
| where $V_k$ is an orthonormal basis of the block Krylov subspace $\mathcal{K}_k^{\square}(A,B)$, | ||
| ```math | ||
| H_{k+1,k} = | ||
| \begin{bmatrix} | ||
| \Psi_{1,1}~ & \Psi_{1,2}~ & \ldots & \Psi_{1,k} \\ | ||
| \Psi_{2,1}~ & \ddots~ & \ddots & \vdots \\ | ||
| & \ddots~ & \ddots & \Psi_{k-1,k} \\ | ||
| & & \Psi_{k,k-1} & \Psi_{k,k} \\ | ||
| & & & \Psi_{k+1,k} | ||
| \end{bmatrix}. | ||
| ``` | ||
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| The function [`arnoldi`](@ref arnoldi(::Any, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\Gamma$, and $H_{k+1,k}$. | ||
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| ```@docs | ||
| arnoldi(::Any, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat) | ||
| ``` | ||
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| ## [Block Golub-Kahan](@id block-golub-kahan) | ||
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| If the vector $b$ in the [Golub-Kahan](@ref golub-kahan) process is replaced by a matrix $B$ with $p$ columns, we can derive the block Golub-Kahan process. | ||
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|  | ||
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| After $k$ iterations of the block Golub-Kahan process, the situation may be summarized as | ||
| ```math | ||
| \begin{align*} | ||
| A V_k &= U_{k+1} B_k, \\ | ||
| A^H U_{k+1} &= V_{k+1} L_{k+1}^H, \\ | ||
| V_k^H V_k &= U_k^H U_k = I_{pk}, | ||
| \end{align*} | ||
| ``` | ||
| where $V_k$ and $U_k$ are bases of the block Krylov subspaces $\mathcal{K}_k^{\square}(A^HA,A^HB)$ and $\mathcal{K}_k^{\square}(AA^H,B)$, respectively, | ||
| ```math | ||
| B_k = | ||
| \begin{bmatrix} | ||
| \Omega_1 & & & \\ | ||
| \Psi_2 & \Omega_2 & & \\ | ||
| & \ddots & \ddots & \\ | ||
| & & \Psi_k & \Omega_k \\ | ||
| & & & \Psi_{k+1} \\ | ||
| \end{bmatrix} | ||
| , \qquad | ||
| L_{k+1}^H = | ||
| \begin{bmatrix} | ||
| \Omega_1^H & \Psi_2^H & & & \\ | ||
| & \Omega_2^H & \ddots & & \\ | ||
| & & \ddots & \Psi_k^H & \\ | ||
| & & & \Omega_k^H & \Psi_{k+1}^H \\ | ||
| & & & & \Omega_{k+1}^H \\ | ||
| \end{bmatrix}. | ||
| ``` | ||
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| The function [`golub_kahan`](@ref golub_kahan(::Any, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $U_{k+1}$, $\Psi_1$ and $L_{k+1}$. | ||
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| ```@docs | ||
| golub_kahan(::Any, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat) | ||
| ``` | ||
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| ## [Block Saunders-Simon-Yip](@id block-saunders-simon-yip) | ||
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| If the vectors $b$ and $c$ in the [Saunders-Simon-Yip](@ref saunders-simon-yip) process are replaced by matrices $B$ and $C$ with both $p$ columns, we can derive the block Saunders-Simon-Yip process. | ||
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|  | ||
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| After $k$ iterations of the block Saunders-Simon-Yip process, the situation may be summarized as | ||
| ```math | ||
| \begin{align*} | ||
| A U_k &= V_{k+1} T_{k+1,k}, \\ | ||
| A^H V_k &= U_{k+1} T_{k,k+1}^H, \\ | ||
| V_k^H V_k &= U_k^H U_k = I_{pk}, | ||
| \end{align*} | ||
| ``` | ||
| where $\begin{bmatrix} V_k & 0 \\ 0 & U_k \end{bmatrix}$ is an orthonormal basis of the block Krylov subspace $\mathcal{K}^{\square}_k \left(\begin{bmatrix} 0 & A \\ A^H & 0 \end{bmatrix}, \begin{bmatrix} B & 0 \\ 0 & C \end{bmatrix}\right)$, | ||
| ```math | ||
| T_{k+1,k} = | ||
| \begin{bmatrix} | ||
| \Omega_1 & \Phi_2 & & \\ | ||
| \Psi_2 & \Omega_2 & \ddots & \\ | ||
| & \ddots & \ddots & \Phi_k \\ | ||
| & & \Psi_k & \Omega_k \\ | ||
| & & & \Psi_{k+1} | ||
| \end{bmatrix} | ||
| , \qquad | ||
| T_{k,k+1}^H = | ||
| \begin{bmatrix} | ||
| \Omega_1^H & \Psi_2^H & & \\ | ||
| \Phi_2^H & \Omega_2^H & \ddots & \\ | ||
| & \ddots & \ddots & \Psi_k^H \\ | ||
| & & \Phi_k^H & \Omega_k^H \\ | ||
| & & & \Phi_{k+1}^H | ||
| \end{bmatrix}. | ||
| ``` | ||
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| The function [`saunders_simon_yip`](@ref saunders_simon_yip(::Any, ::AbstractMatrix{FC}, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\Psi_1$, $T_{k+1,k}$, $U_{k+1}$, $\Phi_1^H$ and $T_{k,k+1}^H$. | ||
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| ```@docs | ||
| saunders_simon_yip(::Any, ::AbstractMatrix{FC}, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat) | ||
| ``` | ||
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| ## [Block Montoison-Orban](@id block-montoison-orban) | ||
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| If the vectors $b$ and $c$ in the [Montoison-Orban](@ref montoison-orban) process are replaced by matrices $D$ and $C$ with both $p$ columns, we can derive the block Montoison-Orban process. | ||
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|  | ||
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| After $k$ iterations of the block Montoison-Orban process, the situation may be summarized as | ||
| ```math | ||
| \begin{align*} | ||
| A U_k &= V_{k+1} H_{k+1,k}, \\ | ||
| B V_k &= U_{k+1} F_{k+1,k}, \\ | ||
| V_k^H V_k &= U_k^H U_k = I_{pk}, | ||
| \end{align*} | ||
| ``` | ||
| where $\begin{bmatrix} V_k & 0 \\ 0 & U_k \end{bmatrix}$ is an orthonormal basis of the block Krylov subspace $\mathcal{K}^{\square}_k \left(\begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix}, \begin{bmatrix} D & 0 \\ 0 & C \end{bmatrix}\right)$, | ||
| ```math | ||
| H_{k+1,k} = | ||
| \begin{bmatrix} | ||
| \Psi_{1,1}~ & \Psi_{1,2}~ & \ldots & \Psi_{1,k} \\ | ||
| \Psi_{2,1}~ & \ddots~ & \ddots & \vdots \\ | ||
| & \ddots~ & \ddots & \Psi_{k-1,k} \\ | ||
| & & \Psi_{k,k-1} & \Psi_{k,k} \\ | ||
| & & & \Psi_{k+1,k} | ||
| \end{bmatrix} | ||
| , \qquad | ||
| F_{k+1,k} = | ||
| \begin{bmatrix} | ||
| \Phi_{1,1}~ & \Phi_{1,2}~ & \ldots & \Phi_{1,k} \\ | ||
| \Phi_{2,1}~ & \ddots~ & \ddots & \vdots \\ | ||
| & \ddots~ & \ddots & \Phi_{k-1,k} \\ | ||
| & & \Phi_{k,k-1} & \Phi_{k,k} \\ | ||
| & & & \Phi_{k+1,k} | ||
| \end{bmatrix}. | ||
| ``` | ||
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| The function [`montoison_orban`](@ref montoison_orban(::Any, ::Any, ::AbstractMatrix{FC}, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\Gamma$, $H_{k+1,k}$, $U_{k+1}$, $\Lambda$, and $F_{k+1,k}$. | ||
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| ```@docs | ||
| montoison_orban(::Any, ::Any, ::AbstractMatrix{FC}, ::AbstractMatrix{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat) | ||
| ``` |
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