update eigen.py

fixed -> matrix and vector imported twice from linalg

.gitignore

@@ -1,38 +1,3 @@
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-.gitignore
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-2 days ago
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 # ---> JupyterNotebooks
 # gitignore template for Jupyter Notebooks
 # website: http://jupyter.org/

README.md

@@ -10,55 +10,9 @@ A lightweight, from-scratch, object-oriented Python package implementing classic
 - Lightweight, no dependencies.
 - Consistent object-oriented API (.solve() etc).
 
----
-
 ## Tutorial Series
 
-This package comes with a set of Jupyter notebooks designed as a structured tutorial series in **numerical methods**, both mathematically rigorous and hands-on with code.
+This package comes with a set of Jupyter notebooks designed as a structured tutorial series in **numerical methods**, both mathematically rigorous and hands-on with code. See [Tutorials](./tutorials/README.md).
-
-### Core Tutorials
-
-1. [Tutorial 1: Vectors and Matrices](tutorials/tutorial1_vectors.ipynb)
-
-   - Definitions of vectors and matrices.
-   - Vector operations: addition, scalar multiplication, dot product, norms.
-   - Matrix operations: addition, multiplication, transpose, inverse.
-   - Matrix and vector norms.
-   - Examples with `numethods.linalg`.
-
-2. [Tutorial 2: Linear Systems of Equations](tutorials/tutorial2_linear_systems.ipynb)
-
-   - Gaussian elimination and Gauss–Jordan.
-   - LU decomposition.
-   - Cholesky decomposition.
-   - Iterative methods: Jacobi and Gauss-Seidel.
-   - Examples with `numethods.solvers`.
-
-3. [Tutorial 3: Orthogonalization and QR Factorization](tutorials/tutorial3_orthogonalization.ipynb)
-
-   - Inner products and orthogonality.
-   - Gram–Schmidt process (classical and modified).
-   - Householder reflections.
-   - QR decomposition and applications.
-   - Examples with `numethods.orthogonal`.
-
-4. [Tutorial 4: Root-Finding Methods](tutorials/tutorial4_root_finding.ipynb)
-
-   - Bisection method.
-   - Fixed-point iteration.
-   - Newton’s method.
-   - Secant method.
-   - Convergence analysis and error behavior.
-   - Trace outputs for iteration history.
-   - Examples with `numethods.roots`.
-
-- [Polynomial Regression Demo](tutorials/polynomial_regression.ipynb)
-
-- Step-by-step example of polynomial regression.
-- Shows how to fit polynomials of different degrees to data.
-- Visualizes fitted curves against the original data.
-
----
 
 ## Features
 

numethods/eigen.py

@@ -3,7 +3,6 @@ from .linalg import Matrix, Vector
 from .orthogonal import QRHouseholder
 from .solvers import LUDecomposition
 from .exceptions import NonSquareMatrixError, ConvergenceError
-from .linalg import Matrix, Vector
 import math
 
 

tutorials/README.md

@@ -0,0 +1,47 @@
+# Tutorial Series
+
+This package comes with a set of Jupyter notebooks designed as a structured tutorial series in **numerical methods**, both mathematically rigorous and hands-on with code.
+
+## Core Tutorials
+
+1. [Tutorial 1: Vectors and Matrices](./tutorial1_vectors_matrices.ipynb)
+
+   - Definitions of vectors and matrices.
+   - Vector operations: addition, scalar multiplication, dot product, norms.
+   - Matrix operations: addition, multiplication, transpose, inverse.
+   - Matrix and vector norms.
+   - Examples with `numethods.linalg`.
+
+2. [Tutorial 2: Linear Systems of Equations](./tutorial2_linear_systems.ipynb)
+
+   - Gaussian elimination and Gauss–Jordan.
+   - LU decomposition.
+   - Cholesky decomposition.
+   - Iterative methods: Jacobi and Gauss-Seidel.
+   - Examples with `numethods.solvers`.
+
+3. [Tutorial 3: Orthogonalization and QR Factorization](./tutorial3_orthogonalization.ipynb)
+
+   - Inner products and orthogonality.
+   - Gram–Schmidt process (classical and modified).
+   - Householder reflections.
+   - QR decomposition and applications.
+   - Examples with `numethods.orthogonal`.
+
+4. [Tutorial 4: Root-Finding Methods](./tutorial4_root_finding.ipynb)
+
+   - Bisection method.
+   - Fixed-point iteration.
+   - Newton’s method.
+   - Secant method.
+   - Convergence analysis and error behavior.
+   - Trace outputs for iteration history.
+   - Examples with `numethods.roots`.
+
+- [Polynomial Regression Demo](./polynomial_regression.ipynb)
+
+- Step-by-step example of polynomial regression.
+- Shows how to fit polynomials of different degrees to data.
+- Visualizes fitted curves against the original data.
+
+---

tutorials/tutorial1_vectors_matrices.ipynb

@@ -142,7 +142,7 @@
     "## 5. Basic operations\n",
     "\n",
     "### 5.1 Vector addition and subtraction\n",
-    "For $ u, v \\in \\mathbb{R}^n $:\n",
+    "For $u, v \\in \\mathbb{R}^n$:\n",
     "\n",
     "$$\n",
     "u + v = \\begin{bmatrix} u_1 + v_1 \\\\ u_2 + v_2 \\\\ \\vdots \\\\ u_n + v_n \\end{bmatrix},\n",
@@ -180,7 +180,7 @@
    "metadata": {},
    "source": [
     "### 5.2 Scalar multiplication\n",
-    "For $ \\alpha \\in \\mathbb{R}, v \\in \\mathbb{R}^n $:\n",
+    "For $\\alpha \\in \\mathbb{R}, v \\in \\mathbb{R}^n$:\n",
     "\n",
     "$$\n",
     "\\alpha v = \\begin{bmatrix} \\alpha v_1 \\\\ \\alpha v_2 \\\\ \\vdots \\\\ \\alpha v_n \\end{bmatrix}.\n",
@@ -215,7 +215,7 @@
    "metadata": {},
    "source": [
     "### 5.3 Matrix addition and subtraction\n",
-    "For $ A, B \\in \\mathbb{R}^{m \\times n} $:\n",
+    "For $A, B \\in \\mathbb{R}^{m \\times n}$:\n",
     "\n",
     "$$\n",
     "A + B = [ a_{ij} + b_{ij} ], \\quad\n",
@@ -254,7 +254,7 @@
    "metadata": {},
    "source": [
     "### 5.4 Matrix-Vector multiplication\n",
-    "For $ A \\in \\mathbb{R}^{m \\times n}, v \\in \\mathbb{R}^n $:\n",
+    "For $A \\in \\mathbb{R}^{m \\times n}, v \\in \\mathbb{R}^n$:\n",
     "\n",
     "$$\n",
     "(Av)_i = \\sum_{j=1}^n a_{ij} v_j.\n",
@@ -290,7 +290,7 @@
    "metadata": {},
    "source": [
     "### 5.5 Matrix-Matrix multiplication\n",
-    "For $ A \\in \\mathbb{R}^{m \\times n}, B \\in \\mathbb{R}^{n \\times p} $:\n",
+    "For $A \\in \\mathbb{R}^{m \\times n}, B \\in \\mathbb{R}^{n \\times p}$:\n",
     "\n",
     "$$\n",
     "(AB)_{ij} = \\sum_{k=1}^n a_{ik} b_{kj}.\n",
@@ -327,7 +327,7 @@
    "metadata": {},
    "source": [
     "### 5.6 Transpose\n",
-    "For $ A \\in \\mathbb{R}^{m \\times n} $:\n",
+    "For $A \\in \\mathbb{R}^{m \\times n}$:\n",
     "\n",
     "$$\n",
     "A^T_{ij} = A_{ji}.\n",

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",