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ougur merged 3 commits from ougur/numethods:main into main 2025-09-17 13:31:15 +03:00
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tutorials/tutorial2_linear_systems.ipynb
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@@ -15,7 +15,7 @@
"## Motivation\n", "## Motivation\n",
"Why we care about solving Ax=b? in numerical methods (e.g., arises in ODEs, PDEs, optimization, physics).\n", "Why we care about solving Ax=b? in numerical methods (e.g., arises in ODEs, PDEs, optimization, physics).\n",
"\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", "\n",
"Numerical algorithms instead use factorizations or iterative schemes.\n", "Numerical algorithms instead use factorizations or iterative schemes.\n",
"\n", "\n",

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tutorials/tutorial3_orthogonalization.ipynb
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@@ -94,7 +94,7 @@
"q_1 = \\frac{a_1}{\\|a_1\\|}\n", "q_1 = \\frac{a_1}{\\|a_1\\|}\n",
"$$\n", "$$\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",
"\n", "\n",
"Matrix form:\n", "Matrix form:\n",
@@ -236,17 +236,17 @@
"\n", "\n",
"We want to solve\n", "We want to solve\n",
"\n", "\n",
"$$ \\min_x \\|Ax - b\\|_2. $$\n", "$$ \\min_x \\Vert Ax - b \\Vert_2^2. $$\n",
"\n", "\n",
"If $A = QR$, then\n", "If $A = QR$, then\n",
"\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", "\n",
"Since $Q$ has orthonormal columns:\n", "Since $Q$ has orthonormal columns, and the normal equations boils down to\n",
"\n", "\n",
"$$ R x = Q^T b. $$\n", "$$ R x = Q^T b, $$\n",
"\n", "\n",
"So we can solve using back-substitution.\n" "we can therefore solve for $x$ by using back-substitution.\n"
] ]
}, },
{ {

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tutorials/tutorial4_root_finding.ipynb
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@@ -55,7 +55,7 @@
"source": [ "source": [
"## 2. Bisection Method\n", "## 2. Bisection Method\n",
"\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", "then there exists $x^\\star$ in (a,b) with $f(x^\\star)=0$.\n",
"\n", "\n",
"- Assumes $f$ is continuous on $[a,b]$ with $f(a)f(b)<0$.\n", "- Assumes $f$ is continuous on $[a,b]$ with $f(a)f(b)<0$.\n",