tutorials/README.md

@@ -4,7 +4,7 @@ This package comes with a set of Jupyter notebooks designed as a structured tuto
 
 ## Core Tutorials
 
-1. [Tutorial 1: Vectors and Matrices](tutorial1_vectors.ipynb)
+1. [Tutorial 1: Vectors and Matrices](./tutorial1_vectors_matrices.ipynb)
 
    - Definitions of vectors and matrices.
    - Vector operations: addition, scalar multiplication, dot product, norms.
@@ -12,7 +12,7 @@ This package comes with a set of Jupyter notebooks designed as a structured tuto
    - Matrix and vector norms.
    - Examples with `numethods.linalg`.
 
-2. [Tutorial 2: Linear Systems of Equations](tutorial2_linear_systems.ipynb)
+2. [Tutorial 2: Linear Systems of Equations](./tutorial2_linear_systems.ipynb)
 
    - Gaussian elimination and Gauss–Jordan.
    - LU decomposition.
@@ -20,7 +20,7 @@ This package comes with a set of Jupyter notebooks designed as a structured tuto
    - Iterative methods: Jacobi and Gauss-Seidel.
    - Examples with `numethods.solvers`.
 
-3. [Tutorial 3: Orthogonalization and QR Factorization](tutorial3_orthogonalization.ipynb)
+3. [Tutorial 3: Orthogonalization and QR Factorization](./tutorial3_orthogonalization.ipynb)
 
    - Inner products and orthogonality.
    - Gram–Schmidt process (classical and modified).
@@ -28,7 +28,7 @@ This package comes with a set of Jupyter notebooks designed as a structured tuto
    - QR decomposition and applications.
    - Examples with `numethods.orthogonal`.
 
-4. [Tutorial 4: Root-Finding Methods](tutorial4_root_finding.ipynb)
+4. [Tutorial 4: Root-Finding Methods](./tutorial4_root_finding.ipynb)
 
    - Bisection method.
    - Fixed-point iteration.

tutorials/tutorial2_linear_systems.ipynb

@@ -15,7 +15,7 @@
     "## Motivation\n",
     "Why we care about solving Ax=b? in numerical methods (e.g., arises in ODEs, PDEs, optimization, physics).\n",
     "\n",
-    "Exact solution: $ x=A^{-1}b $, but computing $ A^{-1} $ explicitly is costly/unstable.\n",
+    "Exact solution: $x = A^{-1}b$, but computing $A^{-1}$ explicitly is costly/unstable.\n",
     "\n",
     "Numerical algorithms instead use factorizations or iterative schemes.\n",
     "\n",

tutorials/tutorial3_orthogonalization.ipynb

@@ -94,7 +94,7 @@
     "q_1 = \\frac{a_1}{\\|a_1\\|}\n",
     "$$\n",
     "$$\n",
-    "q_k = \\frac{a_k - \\sum_{j=1}^{k-1} (q_j \\cdot a_k) q_j}{\\left\\|a_k - \\sum_{j=1}^{k-1} (q_j \\cdot a_k) q_j\\right\\|}\n",
+    "q_k = \\frac{a_k - \\sum_{j=1}^{k-1} (q_j \\cdot a_k) q_j}{\\left\\|a_k - \\sum_{j=1}^{k-1} (q_j \\cdot a_k) q_j\\right\\|}, \\qquad k = 2, \\ldots, n\n",
     "$$\n",
     "\n",
     "Matrix form:\n",
@@ -236,17 +236,17 @@
     "\n",
     "We want to solve\n",
     "\n",
-    "$$ \\min_x \\|Ax - b\\|_2. $$\n",
+    "$$ \\min_x \\Vert Ax - b \\Vert_2^2. $$\n",
     "\n",
     "If $A = QR$, then\n",
     "\n",
-    "$$ \\min_x \\|Ax - b\\|_2 = \\min_x \\|QRx - b\\|_2. $$\n",
+    "$$ \\min_x \\Vert Ax - b \\Vert_2^2 = \\min_x \\Vert QRx - b \\Vert_2^2. $$\n",
     "\n",
-    "Since $Q$ has orthonormal columns:\n",
+    "Since $Q$ has orthonormal columns, and the normal equations boils down to\n",
     "\n",
-    "$$ R x = Q^T b. $$\n",
+    "$$ R x = Q^T b, $$\n",
     "\n",
-    "So we can solve using back-substitution.\n"
+    "we can therefore solve for $x$ by using back-substitution.\n"
    ]
   },
   {

tutorials/tutorial4_root_finding.ipynb

@@ -55,7 +55,7 @@
    "source": [
     "## 2. Bisection Method\n",
     "\n",
-    "**Assumption (Intermediate Value Theorem):** If f is continuous on ([a,b]) and (f(a),f(b) < 0),\n",
+    "**Assumption (Intermediate Value Theorem):** If f is continuous on $[a,b]$ and $f(a)f(b) < 0$,\n",
     "then there exists $x^\\star$ in (a,b) with $f(x^\\star)=0$.\n",
     "\n",
     "- Assumes $f$ is continuous on $[a,b]$ with $f(a)f(b)<0$.\n",