feat: add booklet generator

.claude/agents/algo-implementer.md

@@ -14,8 +14,10 @@ from its `roadmap/<domain>/<Name>/` spec into the codebase, following both exact
 2. Produce, per `roadmap/DEPLOYMENT.md`: the `.hpp`, coverage `.cpp`, `test/Test*.cpp`,
    `doc/<domain>/<Name>.md`, and the CMake edits.
 3. Build and test; fix until green (scope to the target/test where possible).
-4. Remove the algorithm's row from `ROADMAP.md` and add it to the matching category row
-   in `README.md`'s Documentation table.
+4. Remove the algorithm's row from `ROADMAP.md`; add it to the matching category row in
+   `README.md`'s Documentation table **and** to the algorithms table in `doc/<domain>/README.md`
+   (the booklet's ordering source). The booklet regenerates from `doc/` + these README tables in
+   CI — no manual booklet edit.
 5. Report file paths + test result. Nothing else.
 
 ## Hard rules

.github/agents/algo-implementer.agent.md

@@ -17,8 +17,10 @@ Authoritative rules: `AGENTS.md`. Recipe: `roadmap/DEPLOYMENT.md`. Follow both e
 2. Produce, per `roadmap/DEPLOYMENT.md`: the `.hpp`, coverage `.cpp`, `test/Test*.cpp`,
    `doc/<domain>/<Name>.md`, and the CMake edits.
 3. Build and test; fix until green (scope to the target/test where possible).
-4. Remove the algorithm's row from `ROADMAP.md` and add it to the matching category row
-   in `README.md`'s Documentation table.
+4. Remove the algorithm's row from `ROADMAP.md`; add it to the matching category row in
+   `README.md`'s Documentation table **and** to the algorithms table in `doc/<domain>/README.md`
+   (the booklet's ordering source). The booklet regenerates from `doc/` + these README tables in
+   CI — no manual booklet edit.
 5. Report file paths + test result. Nothing else.
 
 ## Hard rules

.github/workflows/build-booklet.yml

@@ -0,0 +1,79 @@
+name: Build Booklet
+
+on:
+  push:
+    branches: [main]
+    paths:
+      - "doc/**"
+      - "scripts/build-booklet.py"
+      - "scripts/booklet/**"
+      - ".github/workflows/build-booklet.yml"
+  pull_request:
+    paths:
+      - "doc/**"
+      - "scripts/build-booklet.py"
+      - "scripts/booklet/**"
+      - ".github/workflows/build-booklet.yml"
+  release:
+    types: [published]
+  workflow_dispatch:
+
+permissions:
+  contents: write
+  pages: write
+  id-token: write
+
+concurrency:
+  group: booklet-${{ github.ref }}
+  cancel-in-progress: true
+
+jobs:
+  build:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7.0.1
+        with:
+          fetch-depth: 0
+
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7.0.0
+        with:
+          python-version: "3.12"
+
+      - name: Install Pandoc and TeX toolchain
+        run: |
+          sudo apt-get update
+          sudo apt-get install -y --no-install-recommends \
+            pandoc texlive-xetex texlive-latex-extra texlive-fonts-recommended lmodern librsvg2-bin
+
+      - name: Build booklet (PDF + HTML)
+        run: python scripts/build-booklet.py --format all
+
+      - name: Upload PDF artifact
+        uses: actions/upload-artifact@ea165f8d65b6e75b540449e92b4886f43607fa02 # v4.6.2
+        with:
+          name: numerical-toolbox-booklet
+          path: build/booklet/NumericalToolbox.pdf
+
+      - name: Upload Pages artifact
+        if: github.event_name == 'push' || github.event_name == 'release'
+        uses: actions/upload-pages-artifact@56afc609e74202658d3ffba0e8f6dda462b719fa # v3.0.1
+        with:
+          path: build/booklet
+
+      - name: Attach PDF to release
+        if: github.event_name == 'release'
+        uses: softprops/action-gh-release@e7a8f85e1c67a31e6ed99a94b41bd0b71bbee6b8 # v2.0.9
+        with:
+          files: build/booklet/NumericalToolbox.pdf
+
+  deploy-pages:
+    if: github.event_name == 'push' || github.event_name == 'release'
+    needs: build
+    runs-on: ubuntu-latest
+    environment:
+      name: github-pages
+      url: ${{ steps.deployment.outputs.page_url }}
+    steps:
+      - name: Deploy to GitHub Pages
+        id: deployment
+        uses: actions/deploy-pages@d6db90164ac5ed86f2b6aed7e0febac5b3c0c03e # v4.0.5

.gitignore

@@ -5,3 +5,4 @@ build/
 out/
 megalinter-reports/
 install/
+*.pdf

README.md

@@ -33,6 +33,19 @@ Refer to the documentation to quickly integrate and utilize the library's signal
 
 Each category page lists its algorithms with a brief description and links to the detailed documentation.
 
+### Booklet
+
+The entire documentation set is also published as a single book — read it online as a
+[GitHub Pages site](https://embedded-pro.github.io/embedded-dsp-control/) or download the latest
+PDF from the [Releases page](../../releases/latest). Both are generated automatically from `doc/`
+(cover, Summary/table of contents, one chapter per category, consolidated references, back cover).
+
+Build it locally with [Pandoc](https://pandoc.org) + XeLaTeX installed:
+
+```bash
+python scripts/build-booklet.py --format all   # writes build/booklet/{NumericalToolbox.pdf,index.html}
+```
+
 ## Simulator
 
 The `simulator/` directory contains interactive Qt-based GUI applications for visualizing and experimenting with the library's algorithms. These are desktop tools intended for development and exploration, separate from the core embedded-targeted library.

ROADMAP.md

@@ -577,7 +577,9 @@ Every new component should follow the established repository conventions:
 - [ ] No heap, no recursion, bounded containers (`infra::BoundedVector`, `std::array`)
 - [ ] `static_assert` on supported types and dimensions
 - [ ] Typed tests (`TYPED_TEST`) for multi-type components; `TEST_F` for single-type; `StrictMock` only
-- [ ] Design-first `doc/<domain>/<Name>.md` following `doc/TEMPLATE.md`
+- [ ] Design-first `doc/<domain>/<Name>.md` following `doc/TEMPLATE.md`; add its row to
+      `doc/<domain>/README.md` and `README.md`'s Documentation table (the booklet regenerates from
+      these tables in CI — no manual booklet edit)
 - [ ] Explicit template instantiation `.cpp` guarded by `NUMERICAL_TOOLBOX_COVERAGE_BUILD` + `numerical_add_coverage_sources`
 - [ ] `CMakeLists.txt` via `numerical_add_header_library()` / `${NUMERICAL_VISIBILITY}`
 - [ ] Simulator + `.vscode/launch.json` entry where a visual demo adds value

doc/analysis/GoertzelAlgorithm.md

@@ -28,9 +28,9 @@ $$X[k] = s[N-1] - s[N-2]\,e^{-j2\pi k/N}$$
 
 Separating real and imaginary parts:
 
-$$\operatorname{Re}\{X[k]\} = s[N-1] - s[N-2]\cos\!\left(\tfrac{2\pi k}{N}\right)$$
+$$\mathrm{Re}\{X[k]\} = s[N-1] - s[N-2]\cos\!\left(\tfrac{2\pi k}{N}\right)$$
 
-$$\operatorname{Im}\{X[k]\} = s[N-2]\sin\!\left(\tfrac{2\pi k}{N}\right)$$
+$$\mathrm{Im}\{X[k]\} = s[N-2]\sin\!\left(\tfrac{2\pi k}{N}\right)$$
 
 The magnitude can be obtained without the final trigonometric products using the identity:
 
@@ -42,7 +42,7 @@ so $|X[k]| = \sqrt{s[N-1]^2 + s[N-2]^2 - c\,s[N-1]\,s[N-2]}$, saving two multipl
 
 A physical frequency $f_t$ sampled at $f_s$ maps to bin:
 
-$$k = \operatorname{round}\!\left(\frac{f_t}{f_s}\,N\right)$$
+$$k = \mathrm{round}\!\left(\frac{f_t}{f_s}\,N\right)$$
 
 Frequency resolution is $\Delta f = f_s / N$.
 
@@ -70,9 +70,9 @@ $c = 2\cos(\pi/2) = 0$
 
 $\cos(2\pi/4) = 0$, $\sin(2\pi/4) = 1$
 
-$\operatorname{Re}\{X[1]\} = 1 - (-1)\cdot 0 = 1$
+$\mathrm{Re}\{X[1]\} = 1 - (-1)\cdot 0 = 1$
 
-$\operatorname{Im}\{X[1]\} = (-1)\cdot 1 = -1$
+$\mathrm{Im}\{X[1]\} = (-1)\cdot 1 = -1$
 
 Direct DFT check: $X[1] = 1 - j$.  Matches.
 

doc/analysis/SignalDetectors.md

@@ -18,7 +18,7 @@ When $d = 1$ the peak is infinite-hold; when $d < 1$ it releases exponentially t
 
 A zero crossing occurs at sample $n$ when the sign of $x_n$ differs from the sign of $x_{n-1}$ and $|x_n|$ exceeds a hysteresis threshold $h \geq 0$:
 
-$$\text{crossed}_n = \bigl(\operatorname{sgn}(x_n) \neq \operatorname{sgn}(x_{n-1})\bigr) \land \bigl(|x_n| > h\bigr)$$
+$$\text{crossed}_n = \bigl(\mathrm{sgn}(x_n) \neq \mathrm{sgn}(x_{n-1})\bigr) \land \bigl(|x_n| > h\bigr)$$
 
 The instantaneous fundamental frequency of a periodic signal can be estimated from the cumulative count $C$ over a window of $N$ samples at sampling period $T_s$:
 

doc/control_analysis/ControllabilityObservability.md

@@ -22,15 +22,15 @@ The $n \times nm$ controllability matrix is
 
 $$\mathcal{C} = \begin{bmatrix} B & AB & A^2 B & \cdots & A^{n-1}B \end{bmatrix}$$
 
-The pair $(A, B)$ is controllable if and only if $\operatorname{rank}(\mathcal{C}) = n$.
+The pair $(A, B)$ is controllable if and only if $\mathrm{rank}(\mathcal{C}) = n$.
 
 ### Observability Matrix
 
 The $np \times n$ observability matrix is
 
 $$\mathcal{O} = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix}$$
 
-The pair $(A, C)$ is observable if and only if $\operatorname{rank}(\mathcal{O}) = n$.
+The pair $(A, C)$ is observable if and only if $\mathrm{rank}(\mathcal{O}) = n$.
 
 ### Duality
 
@@ -83,7 +83,7 @@ $\mathcal{O} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, rank $= 2$. Observab
 ## Variants & Generalizations
 
 - **Continuous-time Gramians**: satisfy $AW + WA^\top + BB^\top = 0$ and require a different solver (e.g., Bartels-Stewart or Hammarling).
-- **PBH test**: The eigenvector-based Popov-Belevitch-Hautus test checks $\operatorname{rank}[A - \lambda I \;|\; B] = n$ for each eigenvalue $\lambda$, revealing which modes are uncontrollable.
+- **PBH test**: The eigenvector-based Popov-Belevitch-Hautus test checks $\mathrm{rank}[A - \lambda I \;|\; B] = n$ for each eigenvalue $\lambda$, revealing which modes are uncontrollable.
 - **Balanced realization**: Simultaneously diagonalizes $W_c$ and $W_o$ via similarity transformation; the resulting Hankel singular values rank-order modes by joint excitability and observability.
 - **Stochastic observability**: Extends to systems driven by process noise; replaced by reachability Gramians.
 

doc/control_analysis/TransferFunctionStateSpace.md

@@ -38,7 +38,7 @@ $$H(s) = C\,(sI - A)^{-1} B + D = \frac{C\,\text{adj}(sI - A)\,B + D\,\det(sI -
 
 The Faddeev–Le Verrier algorithm computes the characteristic polynomial $\det(sI - A) = s^n + c_1 s^{n-1} + \cdots + c_n$ and the adjugate action $\text{adj}(sI - A)B$ in a single recursion of $n$ steps:
 
-$$M_0 = I, \quad c_k = -\frac{1}{k}\operatorname{tr}(A M_{k-1}), \quad M_k = A M_{k-1} + c_k I$$
+$$M_0 = I, \quad c_k = -\frac{1}{k}\mathrm{tr}(A M_{k-1}), \quad M_k = A M_{k-1} + c_k I$$
 
 The numerator coefficient for degree $n - k$ is $C M_k B$.
 
@@ -70,8 +70,8 @@ $$A_\text{ocf} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix}, \quad B_\text{o
 
 **Round-trip verification via Le Verrier ($n = 2$):**
 
-- $M_0 = I$; $c_1 = -\tfrac{1}{1}\operatorname{tr}(AM_0) = -\operatorname{tr}(A) = 3$; $M_1 = A + 3I = \begin{bmatrix}3 & 1 \\ -2 & 0\end{bmatrix}$.
-- $c_2 = -\tfrac{1}{2}\operatorname{tr}(AM_1) = -\tfrac{1}{2}\operatorname{tr}\!\begin{bmatrix}-2 & 0 \\ -6 & -2\end{bmatrix} = 2$.
+- $M_0 = I$; $c_1 = -\tfrac{1}{1}\mathrm{tr}(AM_0) = -\mathrm{tr}(A) = 3$; $M_1 = A + 3I = \begin{bmatrix}3 & 1 \\ -2 & 0\end{bmatrix}$.
+- $c_2 = -\tfrac{1}{2}\mathrm{tr}(AM_1) = -\tfrac{1}{2}\mathrm{tr}\!\begin{bmatrix}-2 & 0 \\ -6 & -2\end{bmatrix} = 2$.
 - Numerator coefficients: $CM_1B = [2,1]\begin{bmatrix}1\\0\end{bmatrix} = 2$; $CM_2B = CM_0B = 0$.
 
 Result: $H(s) = \dfrac{s + 2}{s^2 + 3s + 2}$. Identical to the original.

doc/math/HouseholderTransform.md

@@ -10,7 +10,7 @@ For a vector $x$, the Householder reflector is the orthogonal matrix
 
 $$H = I - \beta\, v v^\top$$
 
-chosen so that $Hx$ is zero below the pivot. With $\sigma = \sum_{i>\text{start}} x_i^2$ and $\|x\| = \sqrt{x_{\text{start}}^2 + \sigma}$, the reflector maps $x_{\text{start}} \mapsto \mp\|x\|$. The pivot sign is chosen as $-\operatorname{sign}(x_{\text{start}})\|x\|$ to avoid cancellation:
+chosen so that $Hx$ is zero below the pivot. With $\sigma = \sum_{i>\text{start}} x_i^2$ and $\|x\| = \sqrt{x_{\text{start}}^2 + \sigma}$, the reflector maps $x_{\text{start}} \mapsto \mp\|x\|$. The pivot sign is chosen as $-\mathrm{sign}(x_{\text{start}})\|x\|$ to avoid cancellation:
 
 $$v_{\text{start}} = 1, \quad v_i = x_i / v_0, \quad \beta = \frac{2 v_0^2}{\sigma + v_0^2}$$
 

doc/math/Quaternion.md

@@ -70,9 +70,9 @@ $$R(q) = \begin{pmatrix}
 
 Converting from unit quaternion to roll $\phi$, pitch $\theta$, yaw $\psi$:
 
-$$\phi = \operatorname{atan2}(2(wx+yz),\; 1-2(x^2+y^2))$$
+$$\phi = \mathrm{atan2}(2(wx+yz),\; 1-2(x^2+y^2))$$
 $$\theta = \arcsin(2(wy-zx))$$
-$$\psi = \operatorname{atan2}(2(wz+xy),\; 1-2(y^2+z^2))$$
+$$\psi = \mathrm{atan2}(2(wz+xy),\; 1-2(y^2+z^2))$$
 
 At $\theta = \pm 90°$ the $\phi$ and $\psi$ axes align (gimbal lock); the formula still
 returns a bounded value but the decomposition is no longer unique.
@@ -81,7 +81,7 @@ returns a bounded value but the decomposition is no longer unique.
 
 Spherical Linear Interpolation between unit quaternions $q_0$ and $q_1$ at fraction $t \in [0,1]$:
 
-$$\operatorname{Slerp}(q_0, q_1, t) = \frac{\sin((1-t)\Omega)}{\sin\Omega}\,q_0 + \frac{\sin(t\Omega)}{\sin\Omega}\,q_1,$$
+$$\mathrm{Slerp}(q_0, q_1, t) = \frac{\sin((1-t)\Omega)}{\sin\Omega}\,q_0 + \frac{\sin(t\Omega)}{\sin\Omega}\,q_1,$$
 
 where $\cos\Omega = q_0 \cdot q_1$. When $\Omega \approx 0$ (nearly parallel quaternions)
 the formula degenerates; a normalized linear interpolation (nlerp) is substituted.

doc/neural_network/losses/Loss.md

@@ -24,7 +24,7 @@ $$\frac{\partial \mathcal{L}}{\partial \hat{y}_i} = \frac{2}{m}(\hat{y}_i - y_i)
 
 $$\mathcal{L}_{\text{MAE}} = \frac{1}{m} \sum_{i=1}^{m} |\hat{y}_i - y_i|$$
 
-$$\frac{\partial \mathcal{L}}{\partial \hat{y}_i} = \frac{1}{m} \operatorname{sign}(\hat{y}_i - y_i)$$
+$$\frac{\partial \mathcal{L}}{\partial \hat{y}_i} = \frac{1}{m} \mathrm{sign}(\hat{y}_i - y_i)$$
 
 - Linear penalty → robust to outliers.
 - Non-differentiable at $\hat{y}_i = y_i$ (use sub-gradient in practice).

doc/regularization/Regularization.md

@@ -26,7 +26,7 @@ The factor $(1 - \eta\lambda)$ shrinks weights toward zero each step — hence t
 
 $$\Omega_{L1}(\theta) = \sum_{i=1}^{P} |\theta_i| = \|\theta\|_1$$
 
-$$\frac{\partial \Omega_{L1}}{\partial \theta_i} = \operatorname{sign}(\theta_i)$$
+$$\frac{\partial \Omega_{L1}}{\partial \theta_i} = \mathrm{sign}(\theta_i)$$
 
 **Effect:** L1 drives small weights exactly to zero, producing a **sparse** model. This is useful for feature selection — irrelevant connections are pruned automatically.
 
@@ -84,7 +84,7 @@ Regularization adds negligible computational cost — one pass over the paramete
 
 - **$\lambda$ too large.** The model underfits — weights are driven so close to zero that the network cannot represent the function. Cross-validate $\lambda$.
 - **$\lambda$ too small.** Negligible effect; overfitting persists.
-- **L1 non-differentiability.** At $\theta_i = 0$, the L1 gradient is undefined. Use sub-gradient $\operatorname{sign}(0) = 0$ or proximal operators for exact handling.
+- **L1 non-differentiability.** At $\theta_i = 0$, the L1 gradient is undefined. Use sub-gradient $\mathrm{sign}(0) = 0$ or proximal operators for exact handling.
 - **Regularizing biases.** Conventionally, bias parameters are excluded from regularization because they do not contribute to model complexity. This library regularizes all parameters in the flat vector — be aware of this if bias control matters.
 - **Fixed-point precision.** The regularization term can be much smaller than the main loss when $\lambda$ is small. In low-precision fixed-point, it may round to zero. Scale $\lambda$ or use a wider accumulator.
 

doc/robust_control/SlidingModeControl.md

@@ -34,17 +34,17 @@ This exists if and only if $S B$ is nonsingular, which is the relative-degree-on
 
 The **switching term** adds a robust push toward the surface:
 
-$$u_{sw} = (S B)^{-1} K \operatorname{sat}(s/\phi)$$
+$$u_{sw} = (S B)^{-1} K \mathrm{sat}(s/\phi)$$
 
 where $K \in \mathbb{R}^{m}$ is the per-channel switching gain and $\phi > 0$ is the **boundary-layer thickness**. The saturation function
 
-$$\operatorname{sat}(\sigma) = \begin{cases} \sigma & |\sigma| \le 1 \\ \operatorname{sign}(\sigma) & |\sigma| > 1 \end{cases}$$
+$$\mathrm{sat}(\sigma) = \begin{cases} \sigma & |\sigma| \le 1 \\ \mathrm{sign}(\sigma) & |\sigma| > 1 \end{cases}$$
 
-replaces the discontinuous $\operatorname{sign}(s)$ of ideal SMC with a continuous ramp inside $|s| \le \phi$, eliminating infinite-bandwidth chattering while maintaining the reaching property.
+replaces the discontinuous $\mathrm{sign}(s)$ of ideal SMC with a continuous ramp inside $|s| \le \phi$, eliminating infinite-bandwidth chattering while maintaining the reaching property.
 
 ### Total Control Law
 
-$$u = u_{eq} - u_{sw} = -(S B)^{-1}\bigl[S A x + K\,\operatorname{sat}(s/\phi)\bigr]$$
+$$u = u_{eq} - u_{sw} = -(S B)^{-1}\bigl[S A x + K\,\mathrm{sat}(s/\phi)\bigr]$$
 
 ### Reaching Condition
 
@@ -73,7 +73,7 @@ $$A = \begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}, \quad B = \begin{bmatrix}0\\1\en
 with surface $S = [1\; 1]$, gain $K = 2$, boundary layer $\phi = 0.05$.
 
 1. **Precompute** $SB = 1$, so $(SB)^{-1} = 1$.
-2. **At state** $x = [1, 0]^\top$: $s = 1 > \phi$, $\operatorname{sat}(s/\phi) = 1$.
+2. **At state** $x = [1, 0]^\top$: $s = 1 > \phi$, $\mathrm{sat}(s/\phi) = 1$.
 3. **Equivalent control**: $u_{eq} = -(SA)x = -[0\;1][1\;0]^\top = 0$.
 4. **Switching term**: $u_{sw} = 1 \cdot 2 \cdot 1 = 2$.
 5. **Total**: $u = 0 - 2 = -2$. The large negative input decelerates the state toward the surface.

doc/solvers/LuDecomposition.md

@@ -60,7 +60,7 @@ quantities), and `IsSingular()` reports whether the last `Decompose` aborted.
 The determinant of $A$ equals the product of $U$'s diagonal entries multiplied by the
 sign of $P$:
 
-$$\det(A) = \operatorname{sign}(P) \prod_{k=0}^{n-1} U_{kk}.$$
+$$\det(A) = \mathrm{sign}(P) \prod_{k=0}^{n-1} U_{kk}.$$
 
 The sign flips each time rows are swapped; it starts at $+1$ and is multiplied by $-1$ per swap.
 

doc/solvers/LyapunovSylvester.md

@@ -24,13 +24,13 @@ $i, j$.
 
 ### Kronecker Vectorisation
 
-Stacking $X$ column-by-column into $\operatorname{vec}(X) \in \mathbb{R}^{NM}$ and applying
-the identities $\operatorname{vec}(AXI) = (I_M \otimes A)\operatorname{vec}(X)$ and
-$\operatorname{vec}(IXB) = (B^\top \otimes I_N)\operatorname{vec}(X)$ converts the matrix
+Stacking $X$ column-by-column into $\mathrm{vec}(X) \in \mathbb{R}^{NM}$ and applying
+the identities $\mathrm{vec}(AXI) = (I_M \otimes A)\mathrm{vec}(X)$ and
+$\mathrm{vec}(IXB) = (B^\top \otimes I_N)\mathrm{vec}(X)$ converts the matrix
 equation into the standard linear system
 
 $$\underbrace{(I_M \otimes A + B^\top \otimes I_N)}_{K \in \mathbb{R}^{NM \times NM}}
-\operatorname{vec}(X) = \operatorname{vec}(C).$$
+\mathrm{vec}(X) = \mathrm{vec}(C).$$
 
 The $(iN+k, jN+l)$ entry of $K$ is $\delta_{ij} A_{kl} + B_{ji} \delta_{kl}$.  The system
 is solved by Gaussian elimination with partial pivoting; a near-zero pivot signals the
@@ -51,7 +51,7 @@ The discrete counterpart
 
 $$AXA^\top - X = -Q$$
 
-is solved by the Kronecker system $(A \otimes A - I_{N^2})\operatorname{vec}(X) = -\operatorname{vec}(Q)$,
+is solved by the Kronecker system $(A \otimes A - I_{N^2})\mathrm{vec}(X) = -\mathrm{vec}(Q)$,
 with the $(iN+k, jN+l)$ entry of $A \otimes A - I$ equal to $A_{ij}A_{kl} - \delta_{ij}\delta_{kl}$.
 When $A$ is Schur-stable ($\rho(A) < 1$) and $Q \succeq 0$, the unique solution equals the
 controllability gramian $X = \sum_{k=0}^{\infty} A^k Q (A^\top)^k$.
@@ -79,7 +79,7 @@ $C = I_2$.
 
 $$K = \begin{pmatrix}-2&1&0&0\\0&-1&0&0\\2&0&-2&1\\0&2&0&-1\end{pmatrix}.$$
 
-**Build $\operatorname{vec}(C) = (1, 0, 0, 1)^\top$.**
+**Build $\mathrm{vec}(C) = (1, 0, 0, 1)^\top$.**
 
 **Gaussian elimination** with partial pivoting produces $X_{00} = -0.5$, $X_{10} = 0$,
 $X_{01} = -1$, $X_{11} = -1$.
@@ -113,7 +113,7 @@ $X_{01} = -1$, $X_{11} = -1$.
   controllability Gramian used to evaluate reachability in balanced truncation.
 - **Pole-placement observers (Luenberger):** the Sylvester equation $AX - XF = LC$ arises when
   placing observer poles while decoupling the error dynamics.
-- **H₂ norm computation:** $\|G\|_2^2 = \operatorname{tr}(B^\top X B)$ where $X$ satisfies the
+- **H₂ norm computation:** $\|G\|_2^2 = \mathrm{tr}(B^\top X B)$ where $X$ satisfies the
   observability Gramian Lyapunov equation.
 
 ## Connections to Other Algorithms