From 669fc3dd9d128723952f26c007ce343bfc1e246d Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Sat, 14 Mar 2026 15:12:58 +0100 Subject: [PATCH 1/4] fit gh rendering --- docs/1_fixed_effects_and_block_methods.md | 28 +++++++++++------------ docs/2_solver_architecture.md | 2 +- 2 files changed, 15 insertions(+), 15 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 6c31f0c..8950d4f 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -74,11 +74,11 @@ The $m \times m$ Gramian inherits a natural block structure from the factor part $$ G = \begin{pmatrix} -{\color{royalblue}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\ -C_{12}^\top & {\color{crimson}D_2} & C_{23} & \cdots & C_{2Q} \\ -C_{13}^\top & C_{23}^\top & {\color{forestgreen}D_3} & \cdots & C_{3Q} \\ +{\color{#4169E1}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\ +C_{12}^\top & {\color{#DC143C}D_2} & C_{23} & \cdots & C_{2Q} \\ +C_{13}^\top & C_{23}^\top & {\color{#228B22}D_3} & \cdots & C_{3Q} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{goldenrod}D_Q} +C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{#DAA520}D_Q} \end{pmatrix} $$ @@ -109,20 +109,20 @@ The Gramian has $Q = 3$ diagonal blocks and $\binom{3}{2} = 3$ cross-tabulation $$ G = \begin{pmatrix} -{\color{royalblue}D_W} & {\color{gray}C_{WF}} & {\color{gray}C_{WY}} \\ -{\color{gray}C_{WF}^\top} & {\color{crimson}D_F} & {\color{gray}C_{FY}} \\ -{\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{forestgreen}D_Y} +{\color{#4169E1}D_W} & {\color{#808080}C_{WF}} & {\color{#808080}C_{WY}} \\ +{\color{#808080}C_{WF}^\top} & {\color{#DC143C}D_F} & {\color{#808080}C_{FY}} \\ +{\color{#808080}C_{WY}^\top} & {\color{#808080}C_{FY}^\top} & {\color{#228B22}D_Y} \end{pmatrix} = \left(\begin{array}{ccc|cc|cc} -{\color{royalblue}2} & {\color{royalblue}0} & {\color{royalblue}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ -{\color{royalblue}0} & {\color{royalblue}2} & {\color{royalblue}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ -{\color{royalblue}0} & {\color{royalblue}0} & {\color{royalblue}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ +{\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} \\ +{\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#808080}2} & {\color{#808080}0} & {\color{#808080}1} & {\color{#808080}1} \\ +{\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#808080}0} & {\color{#808080}2} & {\color{#808080}1} & {\color{#808080}1} \\ \hline -{\color{gray}1} & {\color{gray}2} & {\color{gray}0} & {\color{crimson}3} & {\color{crimson}0} & {\color{gray}2} & {\color{gray}1} \\ -{\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{crimson}0} & {\color{crimson}3} & {\color{gray}1} & {\color{gray}2} \\ +{\color{#808080}1} & {\color{#808080}2} & {\color{#808080}0} & {\color{#DC143C}3} & {\color{#DC143C}0} & {\color{#808080}2} & {\color{#808080}1} \\ +{\color{#808080}1} & {\color{#808080}0} & {\color{#808080}2} & {\color{#DC143C}0} & {\color{#DC143C}3} & {\color{#808080}1} & {\color{#808080}2} \\ \hline -{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{forestgreen}3} & {\color{forestgreen}0} \\ -{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{forestgreen}0} & {\color{forestgreen}3} +{\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}2} & {\color{#808080}1} & {\color{#228B22}3} & {\color{#228B22}0} \\ +{\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}2} & {\color{#228B22}0} & {\color{#228B22}3} \end{array}\right) $$ diff --git a/docs/2_solver_architecture.md b/docs/2_solver_architecture.md index 619ebea..3b55b81 100644 --- a/docs/2_solver_architecture.md +++ b/docs/2_solver_architecture.md @@ -37,7 +37,7 @@ As discussed in [Part 1, Section 5.2](1_fixed_effects_and_block_methods.md#52-th ## 2. Graph Structure of the Gramian -Part 1 derived the block structure of $G = D^\top W D$, with diagonal blocks $D_q$ and cross-tabulation blocks $C_{qr}$. It is convenient to write $G = \mathcal{D} + \mathcal{C}$, where $\mathcal{D} = \operatorname{block-diag}(D_1, \ldots, D_Q)$ collects the diagonal blocks and $\mathcal{C}$ collects the off-diagonal cross-tabulation blocks. This section describes the graph-theoretic properties that drive the domain decomposition. +Part 1 derived the block structure of $G = D^\top W D$, with diagonal blocks $D_q$ and cross-tabulation blocks $C_{qr}$. It is convenient to write $G = \mathcal{D} + \mathcal{C}$, where $\mathcal{D} = \text{block-diag}(D_1, \ldots, D_Q)$ collects the diagonal blocks and $\mathcal{C}$ collects the off-diagonal cross-tabulation blocks. This section describes the graph-theoretic properties that drive the domain decomposition. ### 2.1 Factor-pair bipartite blocks From 80652c8780005c28a46416c00f7ea1a3e5910f48 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Sat, 14 Mar 2026 15:17:02 +0100 Subject: [PATCH 2/4] fix --- docs/1_fixed_effects_and_block_methods.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 8950d4f..e36e437 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -113,14 +113,12 @@ G = \begin{pmatrix} {\color{#808080}C_{WF}^\top} & {\color{#DC143C}D_F} & {\color{#808080}C_{FY}} \\ {\color{#808080}C_{WY}^\top} & {\color{#808080}C_{FY}^\top} & {\color{#228B22}D_Y} \end{pmatrix} -= \left(\begin{array}{ccc|cc|cc} += \left(\begin{array}{ccccccc} {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} \\ {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#808080}2} & {\color{#808080}0} & {\color{#808080}1} & {\color{#808080}1} \\ {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#808080}0} & {\color{#808080}2} & {\color{#808080}1} & {\color{#808080}1} \\ -\hline {\color{#808080}1} & {\color{#808080}2} & {\color{#808080}0} & {\color{#DC143C}3} & {\color{#DC143C}0} & {\color{#808080}2} & {\color{#808080}1} \\ {\color{#808080}1} & {\color{#808080}0} & {\color{#808080}2} & {\color{#DC143C}0} & {\color{#DC143C}3} & {\color{#808080}1} & {\color{#808080}2} \\ -\hline {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}2} & {\color{#808080}1} & {\color{#228B22}3} & {\color{#228B22}0} \\ {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}2} & {\color{#228B22}0} & {\color{#228B22}3} \end{array}\right) From 1edc4cd5980db73ca4f38901460252a5a0be00f2 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Sat, 14 Mar 2026 15:18:34 +0100 Subject: [PATCH 3/4] another attempt --- docs/1_fixed_effects_and_block_methods.md | 28 +++++++++++------------ 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index e36e437..034ffc9 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -74,11 +74,11 @@ The $m \times m$ Gramian inherits a natural block structure from the factor part $$ G = \begin{pmatrix} -{\color{#4169E1}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\ -C_{12}^\top & {\color{#DC143C}D_2} & C_{23} & \cdots & C_{2Q} \\ -C_{13}^\top & C_{23}^\top & {\color{#228B22}D_3} & \cdots & C_{3Q} \\ +{\color{blue}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\ +C_{12}^\top & {\color{red}D_2} & C_{23} & \cdots & C_{2Q} \\ +C_{13}^\top & C_{23}^\top & {\color{green}D_3} & \cdots & C_{3Q} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{#DAA520}D_Q} +C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{orange}D_Q} \end{pmatrix} $$ @@ -109,18 +109,18 @@ The Gramian has $Q = 3$ diagonal blocks and $\binom{3}{2} = 3$ cross-tabulation $$ G = \begin{pmatrix} -{\color{#4169E1}D_W} & {\color{#808080}C_{WF}} & {\color{#808080}C_{WY}} \\ -{\color{#808080}C_{WF}^\top} & {\color{#DC143C}D_F} & {\color{#808080}C_{FY}} \\ -{\color{#808080}C_{WY}^\top} & {\color{#808080}C_{FY}^\top} & {\color{#228B22}D_Y} +{\color{blue}D_W} & {\color{gray}C_{WF}} & {\color{gray}C_{WY}} \\ +{\color{gray}C_{WF}^\top} & {\color{red}D_F} & {\color{gray}C_{FY}} \\ +{\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{green}D_Y} \end{pmatrix} = \left(\begin{array}{ccccccc} -{\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} \\ -{\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#808080}2} & {\color{#808080}0} & {\color{#808080}1} & {\color{#808080}1} \\ -{\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#808080}0} & {\color{#808080}2} & {\color{#808080}1} & {\color{#808080}1} \\ -{\color{#808080}1} & {\color{#808080}2} & {\color{#808080}0} & {\color{#DC143C}3} & {\color{#DC143C}0} & {\color{#808080}2} & {\color{#808080}1} \\ -{\color{#808080}1} & {\color{#808080}0} & {\color{#808080}2} & {\color{#DC143C}0} & {\color{#DC143C}3} & {\color{#808080}1} & {\color{#808080}2} \\ -{\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}2} & {\color{#808080}1} & {\color{#228B22}3} & {\color{#228B22}0} \\ -{\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}1} & {\color{#808080}2} & {\color{#228B22}0} & {\color{#228B22}3} +{\color{blue}2} & {\color{blue}0} & {\color{blue}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ +{\color{blue}0} & {\color{blue}2} & {\color{blue}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ +{\color{blue}0} & {\color{blue}0} & {\color{blue}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ +{\color{gray}1} & {\color{gray}2} & {\color{gray}0} & {\color{red}3} & {\color{red}0} & {\color{gray}2} & {\color{gray}1} \\ +{\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{red}0} & {\color{red}3} & {\color{gray}1} & {\color{gray}2} \\ +{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{green}3} & {\color{green}0} \\ +{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{green}0} & {\color{green}3} \end{array}\right) $$ From 50dd839e933da478f688b535b3f331a00b1441f4 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Sat, 14 Mar 2026 15:52:13 +0100 Subject: [PATCH 4/4] another attempt --- docs/1_fixed_effects_and_block_methods.md | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 034ffc9..9ef7ad9 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -113,7 +113,10 @@ G = \begin{pmatrix} {\color{gray}C_{WF}^\top} & {\color{red}D_F} & {\color{gray}C_{FY}} \\ {\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{green}D_Y} \end{pmatrix} -= \left(\begin{array}{ccccccc} +$$ + +$$ += \begin{pmatrix} {\color{blue}2} & {\color{blue}0} & {\color{blue}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ {\color{blue}0} & {\color{blue}2} & {\color{blue}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ {\color{blue}0} & {\color{blue}0} & {\color{blue}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ @@ -121,7 +124,7 @@ G = \begin{pmatrix} {\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{red}0} & {\color{red}3} & {\color{gray}1} & {\color{gray}2} \\ {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{green}3} & {\color{green}0} \\ {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{green}0} & {\color{green}3} -\end{array}\right) +\end{pmatrix} $$ $D_W$ is $3 \times 3$ (one row/column per worker) with 2s on the diagonal because each worker appears in exactly 2 observations (e.g. W1 in obs 1, 2). Off-diagonals are zero because no observation belongs to two workers. $D_F$ is $2 \times 2$ with 3s on the diagonal because each firm appears in 3 observations (F1 in obs 1, 3, 4; F2 in obs 2, 5, 6). The cross-tabulation block $C_{WY}$ is $3 \times 2$ (3 workers $\times$ 2 years); entry $[j,k]$ counts observations where worker $j$ is observed in year $k$. Here every worker appears once per year, so $C_{WY}$ is all ones.