updates 10/9

5 files changed, 127 insertions, 4 deletions

diff --git a/notes/Oct-4.org b/notes/Oct-4.org
new file mode 100644
index 0000000..8b8466f
--- /dev/null
+++ b/notes/Oct-4.org
@@ -0,0 +1,22 @@
+[[ a_{11} a_{12} \cdots a_{1n} | b_1]
+ [ 0  (a_{22} - \frac{a_{}_{21}}{a_{22}}a_{11}) \cdots a_{2n} | b_2 - \frac{a_{21}}{a_{11}}b_1 ]]
+
+#+BEGIN_SRC c
+  for (int i = 1; i < n; i++) {
+    float factor = -a[i][0] / a[0][0];
+    for (int j = 1; j < n; j++) {
+      a[i][j] = a[i][j] + factor * a[0][j];
+    }
+    b[i] = b[i] + factor * b[0];
+  }
+
+  for (int k = 0; k < (n - 1); k++) {
+    for (int i = k+1; i < n; i++) {
+      float factor = -a[i][k] / a[k][k];
+      for (int j = k+1; j < n; j++) {
+        a[i][j] = a[i][j] + factor * a[j][k];
+      }
+      b[i] = b[i] + factor*b[k];
+    }
+  }
+#+END_SRC
diff --git a/notes/Oct-6.org b/notes/Oct-6.org
new file mode 100644
index 0000000..8cbff29
--- /dev/null
+++ b/notes/Oct-6.org
@@ -0,0 +1,13 @@
+#+BEGIN_SRC c
+  for (int k = 0; i < (n - 1); k++) {
+    for (int i = k+1; i< n; i++) {
+      float factor = a[i][k] / a[k][k];
+      for (int j = k+1; j < k; j++) {
+        a[i][j] = a[i][j] - factor * a[k][j];
+      }
+      b[i] = b[i] - factor * b[k];
+    }
+  }
+#+END_SRC
+
+
diff --git a/notes/Sep-15.org b/notes/Sep-15.org
index d5bf371..8a64089 100644
--- a/notes/Sep-15.org
+++ b/notes/Sep-15.org
@@ -36,11 +36,11 @@ Again, $f'(a) \approx \frac{f(a+h) - f(a)}{h}$,
 
 $e = |\frac{1}{2} f''(a) + \frac{1}{3!}h^2 f'''(a) + \cdots$
 
-$R_2 = \frac{h}{2} f''(u)$
+$R_2 = \frac{h}{2} f''(\xi)$
 
-$|\frac{h}{2} f''(u)| \leq M h^1$
+$|\frac{h}{2} f''(\xi)| \leq M h^1$
 
-$M = \frac{1}{2}|f'(u)|$
+$M = \frac{1}{2}|f'(\xi)|$
 
 *** Another approximation
 
@@ -48,5 +48,5 @@ $\text{err} = |f'(a) - \frac{f(a) - f(a - h)}{h}|$
 
 $= f'(a) - \frac{1}{h}(f(a) - (f(a) + f'(a)(a - (a - h)) + \frac{1}{2}f''(a)(a-(a-h))^2 + \cdots))$
 
-$= |f'(a) - \frac{1}{h}(f'(a) + \frac{1}{2}f''(a)h)|$
+$= |f'(a) - (f'(a) + \frac{1}{2}f''(a)h)|$
 
diff --git a/notes/Sep-15.pdf b/notes/Sep-15.pdf
new file mode 100644
index 0000000..43a4f34
--- /dev/null
+++ b/notes/Sep-15.pdf
diff --git a/notes/Sep-15.tex b/notes/Sep-15.tex
new file mode 100644
index 0000000..52610ed
--- /dev/null
+++ b/notes/Sep-15.tex
@@ -0,0 +1,88 @@
+% Created 2023-09-29 Fri 10:00
+% Intended LaTeX compiler: pdflatex
+\documentclass[11pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{graphicx}
+\usepackage{longtable}
+\usepackage{wrapfig}
+\usepackage{rotating}
+\usepackage[normalem]{ulem}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{capt-of}
+\usepackage{hyperref}
+\author{Elizabeth Hunt}
+\date{\today}
+\title{}
+\hypersetup{
+ pdfauthor={Elizabeth Hunt},
+ pdftitle={},
+ pdfkeywords={},
+ pdfsubject={},
+ pdfcreator={Emacs 28.2 (Org mode 9.7-pre)}, 
+ pdflang={English}}
+\begin{document}
+
+\tableofcontents
+
+\section{Taylor Series Approx.}
+\label{sec:orgcc72ed1}
+Suppose f has \(\infty\) many derivatives near a point a. Then the taylor series is given by
+
+\(f(x) = \Sigma_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\)
+
+For increment notation we can write
+
+\(f(a + h) = f(a) + f'(a)(a+h - a) + \dots\)
+
+\(= \Sigma_{n=0}^{\infty} \frac{f^{(n)}(a)}{h!} (h^n)\)
+
+Consider the approximation
+
+\(e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |f'(a) - \frac{1}{h}(f(a + h) - f(a))|\)
+
+Substituting\ldots{}
+
+\(= |f'(a) - \frac{1}{h}((f(a) + f'(a) h + \frac{f''(a)}{2} h^2 + \cdots) - f(a))|\)
+
+\(f(a) - f(a) = 0\)\ldots{} and \(distribute the h\)
+
+\(= |-1/2 f''(a) h + \frac{1}{6}f'''(a)h^2 \cdots|\)
+
+\subsection{With Remainder}
+\label{sec:org7dfd6c7}
+We can determine for some u \(f(a + h) = f(a) + f'(a)h + \frac{1}{2}f''(u)h^2\)
+
+and so the error is \(e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |\frac{h}{2}f''(u)|\)
+
+\begin{itemize}
+\item\relax [\url{https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series}]
+\begin{itemize}
+\item > Taylor's Theorem w/ Remainder
+\end{itemize}
+\end{itemize}
+
+
+\subsection{Of Deriviatives}
+\label{sec:org1ec7c9b}
+
+Again, \(f'(a) \approx \frac{f(a+h) - f(a)}{h}\),
+
+\(e = |\frac{1}{2} f''(a) + \frac{1}{3!}h^2 f'''(a) + \cdots\)
+
+\(R_2 = \frac{h}{2} f''(u)\)
+
+\(|\frac{h}{2} f''(u)| \leq M h^1\)
+
+\(M = \frac{1}{2}|f'(u)|\)
+
+\subsubsection{Another approximation}
+\label{sec:org16193b9}
+
+\(\text{err} = |f'(a) - \frac{f(a) - f(a - h)}{h}|\)
+
+\(= f'(a) - \frac{1}{h}(f(a) - (f(a) + f'(a)(a - (a - h)) + \frac{1}{2}f''(a)(a-(a-h))^2 + \cdots))\)
+
+\(= |f'(a) - \frac{1}{h}(f'(a) + \frac{1}{2}f''(a)h)|\)
+\end{document}
\ No newline at end of file