From 4e7737fbd93719213548d73983a89a5eb97f447d Mon Sep 17 00:00:00 2001 From: Alexis Montoison Date: Tue, 12 Sep 2023 15:22:12 -0500 Subject: [PATCH] Update the pseudocode of the Krylov processes --- docs/src/graphics/arnoldi.png | Bin 118889 -> 180677 bytes docs/src/graphics/golub_kahan.png | Bin 134956 -> 200294 bytes docs/src/graphics/hermitian_lanczos.png | Bin 107418 -> 162473 bytes docs/src/graphics/montoison_orban.png | Bin 176065 -> 268356 bytes docs/src/graphics/nonhermitian_lanczos.png | Bin 136590 -> 214667 bytes docs/src/graphics/saunders_simon_yip.png | Bin 149026 -> 223991 bytes docs/src/processes.md | 16 ++++++++-------- 7 files changed, 8 insertions(+), 8 deletions(-) diff --git a/docs/src/graphics/arnoldi.png b/docs/src/graphics/arnoldi.png index 9ef8bd3a3a54cfe2456d0c566f95e67a4d000f8b..9644249d709789e6d5db5a44dd3b14e5c0f726d9 100644 GIT binary patch literal 180677 zcmeFaXH->Lw=IfVW|1-n6f70N0H`F%CR9LzM9HF(lVr(Qij*P-P>?JlIp+)}6ePz@ 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\beta_2 & \alpha_2 & \ddots & \\ - & \ddots & \ddots & \beta_k \\ - & & \beta_k & \alpha_k + \alpha_1 & \bar{\beta}_2 & & \\ + \beta_2 & \alpha_2 & \ddots & \\ + & \ddots & \ddots & \bar{\beta}_k \\ + & & \beta_k & \alpha_k \end{bmatrix} , \qquad T_{k+1,k} = @@ -87,7 +87,7 @@ T_{k+1,k} = \beta_{k+1} e_{k}^T \end{bmatrix}. ``` -Note that $T_{k+1,k}$ is a real tridiagonal matrix even if $A$ is a complex matrix. +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 $T_{k+1,k}$. @@ -190,7 +190,7 @@ After $k$ iterations of the Golub-Kahan bidiagonalization process, the situation ```math \begin{align*} A V_k &= U_{k+1} B_k, \\ - A^H U_{k+1} &= V_k B_k^H + \alpha_{k+1} v_{k+1} e_{k+1}^T = V_{k+1} L_{k+1}^H, \\ + 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*} ``` @@ -218,7 +218,7 @@ B_k = \beta_{k+1} e_{k}^T \end{bmatrix}. ``` -Note that $L_k$ is a real bidiagonal matrix even if $A$ is a complex matrix. +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 $L_{k+1}$. @@ -240,7 +240,7 @@ After $k$ iterations of the Saunders-Simon-Yip process (also named the orthogona ```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^H V_k &= U_k T_k^H + \gamma_{k+1} u_{k+1} e_k^T = U_{k+1} T_{k,k+1}^H, \\ + 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*} ```