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Minor fixes to 00, 06, and 09_part1 #6

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6 changes: 3 additions & 3 deletions 00_intro_numerical_methods.ipynb
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Expand Up @@ -702,7 +702,7 @@
"\n",
"Topics Covered: (see Syllabus on [Courseworks](https://courseworks2.columbia.edu/courses/107701/assignments/syllabus))\n",
"\n",
"1. Quick <font color=red>*Review*</font> of computational Tools (Python,numpy,matplotlib,scipy,git)\n",
"- Quick <font color=red>*Review*</font> of Computational Tools (Python,numpy,matplotlib,scipy,git)\n",
"- Sources of Error and Error Analysis\n",
"- Rootfinding/Optimization of non-linear functions $f(x)$ \n",
"- Interpolation\n",
Expand All @@ -723,11 +723,11 @@
"source": [
"### Topics Not Covered: \n",
"\n",
"1. Optimization -- linear programming, constrained optimization\n",
"- Optimization -- linear programming, constrained optimization\n",
"- Numerical Solution of PDE's (APMA E4301)\n",
"- Mathematical Modeling (maybe a little)\n",
"- Machine Learning/Data Science (APMA 4990 Intro Math of Data Science)\n",
"- Lot's of things that would be fun...\n"
"- Lots of things that would be fun...\n"
]
},
{
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8 changes: 4 additions & 4 deletions 06_interpolation.ipynb
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Expand Up @@ -397,7 +397,7 @@
}
},
"source": [
"But more fundamentally. A set of functions is **linearly independent** if the only linear combination that add to form the zero function, e.g.\n",
"But more fundamentally. A set of functions is **linearly independent** if the only linear combination that adds to form the zero function, e.g.\n",
"\n",
"$$\n",
" P_N(x) = p_0 1 + p_1 x + p_2 x^2 + \\ldots + p_n x^n = 0\n",
Expand Down Expand Up @@ -1274,7 +1274,7 @@
"axes.plot(x, f(x), 'k', label=\"True $f(x)$\")\n",
"axes.plot(data[:, 0], data[:, 1], 'ro', label=\"data\")\n",
" \n",
"axes.set_title(\"Interpolation of Runge's function\")\n",
"axes.set_title(\"Interpolation of the Weierstrass function\")\n",
"axes.set_xlabel(\"x\")\n",
"axes.set_ylabel(\"y\")\n",
"axes.legend(loc=1)\n",
Expand Down Expand Up @@ -2013,7 +2013,7 @@
"source": [
"### Transformation of intervals\n",
"\n",
"The previous algorithms are quite direct but can look a bit messy because every interval (or set of points) appears different...An important trick that is used in finite-element analysis is to to transform each interval (or element) \n",
"The previous algorithms are quite direct but can look a bit messy because every interval (or set of points) appears different... An important trick that is used in finite-element analysis is to to transform each interval (or element) \n",
"\n",
"$$\n",
" x \\in[x_k, x_{k+1}]\n",
Expand Down Expand Up @@ -2762,5 +2762,5 @@
}
},
"nbformat": 4,
"nbformat_minor": 1
"nbformat_minor": 4
}
2 changes: 1 addition & 1 deletion 09_ODE_ivp_part1.ipynb
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Expand Up @@ -1604,7 +1604,7 @@
"\n",
"$$\\begin{aligned}\n",
" K_1 &= \\Delta t f(U_n, t_n) \\\\\n",
" K_2 &= \\Delta t f(U_n + K1, t_n + \\Delta t )\\\\\n",
" K_2 &= \\Delta t f(U_n + K_1, t_n + \\Delta t )\\\\\n",
" U_{n+1} &= U_n + \\frac{1}{2}\\left[K_1 +K_2\\right] \\\\ \n",
"\\end{aligned}$$\n",
"\n",
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