6 files changed, 191 insertions, 8 deletions
diff --git a/notes/Sep-11.org b/notes/Sep-11.org
index 1568618..3d71f2f 100644
--- a/notes/Sep-11.org
+++ b/notes/Sep-11.org
@@ -31,17 +31,18 @@ Table of Errors
   (lizfcm.utils:table (:headers '("u" "v" "e_{abs}" "e_{rel}")
                        :domain-order (u v)
                        :domain-values *u-v*)
-    (fround (eabs u v) 4)
-    (fround (erel u v) 4))
+    (eabs u v)
+    (erel u v))
 #+END_SRC
 
 #+RESULTS:
-|    u |    v | e_{abs} | e_{rel} |
-|    1 | 0.99 |  0.0 |  0.0 |
-|    1 | 1.01 |  0.0 |  0.0 |
-| -1.5 | -1.2 |  0.0 |  0.0 |
-|  100 | 99.9 |  0.0 |  0.0 |
-|  100 |   99 |  0.0 |  0.0 |
+|    u |    v |        e_{abs} |         e_{rel} |
+|    1 | 0.99 |  0.00999999 |   0.00999999 |
+|    1 | 1.01 |  0.00999999 |   0.00999999 |
+| -1.5 | -1.2 |  0.29999995 |   0.19999997 |
+|  100 | 99.9 | 0.099998474 | 0.0009999848 |
+|  100 |   99 |           1 |        1/100 |
+
 
 Look at $u \approx 0$ then $v \approx 0$, $e_{abs}$ is better error since $e_{rel}$ is high.
 
diff --git a/notes/Sep-13.org b/notes/Sep-13.org
new file mode 100644
index 0000000..0ebff2b
--- /dev/null
+++ b/notes/Sep-13.org
@@ -0,0 +1,16 @@
+* Homework 2
+1. maceps - single precision
+
+2. maceps - double precision
+
+3. 2-norm of a vector
+
+4. 1-norm of a vector
+
+5. infinity-norm of a vector (max-norm)
+
+6. 2-norm distance between 2 vectors
+
+7. 1-norm distance between 2 vectors
+
+8. infinity-norm distance
diff --git a/notes/Sep-15.org b/notes/Sep-15.org
new file mode 100644
index 0000000..d5bf371
--- /dev/null
+++ b/notes/Sep-15.org
@@ -0,0 +1,52 @@
+* Taylor Series Approx.
+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...
+
+$= |f'(a) - \frac{1}{h}((f(a) + f'(a) h + \frac{f''(a)}{2} h^2 + \cdots) - f(a))|$
+
+$f(a) - f(a) = 0$... and $distribute the h$
+
+$= |-1/2 f''(a) h + \frac{1}{6}f'''(a)h^2 \cdots|$
+
+** With Remainder
+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 $e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |\frac{h}{2}f''(u)|$
+
+- [https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series]
+ + > Taylor's Theorem w/ Remainder
+
+
+** Of Deriviatives
+
+Again, $f'(a) \approx \frac{f(a+h) - f(a)}{h}$,
+
+$e = |\frac{1}{2} f''(a) + \frac{1}{3!}h^2 f'''(a) + \cdots$
+
+$R_2 = \frac{h}{2} f''(u)$
+
+$|\frac{h}{2} f''(u)| \leq M h^1$
+
+$M = \frac{1}{2}|f'(u)|$
+
+*** Another approximation
+
+$\text{err} = |f'(a) - \frac{f(a) - f(a - h)}{h}|$
+
+$= f'(a) - \frac{1}{h}(f(a) - (f(a) + f'(a)(a - (a - h)) + \frac{1}{2}f''(a)(a-(a-h))^2 + \cdots))$
+
+$= |f'(a) - \frac{1}{h}(f'(a) + \frac{1}{2}f''(a)h)|$
+
diff --git a/notes/Sep-20.org b/notes/Sep-20.org
new file mode 100644
index 0000000..ba067bb
--- /dev/null
+++ b/notes/Sep-20.org
@@ -0,0 +1,21 @@
+* Review & Summary
+Approx f'(a) with
+
++ forward difference $f'(a) \approx \frac{f(a+h) - f(a)}{h}$
+
++ backward difference $f'(a) \approx \frac{f(a) - f(a-h)}{h}$
+
++ central difference $f'(a) \approx \frac{f(a+h) - f(a-h)}{2h}$
+
+** Taylor Series
+given $C = \frac{1}{2}(|f''(\xi)|) \cdot h^1$
+
+with f.d. $e_{\text{abs}} \leq Ch^1$
+
+b.d.  $e_{\text{abs}} \leq Ch^1$
+
+c.d. $e_{\text{abs}} \leq Ch^2$
+
+$e_{\text{abs}} \leq Ch^r$
+
+$log(e(h)) \leq log(ch^r) = log(C) + log(h^r) = log(C) + rlog(h)$
diff --git a/notes/Sep-22.org b/notes/Sep-22.org
new file mode 100644
index 0000000..b631e3b
--- /dev/null
+++ b/notes/Sep-22.org
@@ -0,0 +1,45 @@
+* regression
+consider the generic problem of fitting a dataset to a linear polynomial
+
+given discrete f: x \rightarrow y
+
+interpolation: y = a + bx
+
+[[1 x_0]            [[y_0]
+ [1 x_1]  \cdot [[a]  =   [y_1]
+ [1 x_n]]   [b]]     [y_n]]
+
+consider p \in col(A)
+
+then y = p + q for some q \cdot p = 0
+
+then we can generate n \in col(A) by $Az$ and n must be orthogonal to q as well
+
+(Az)^T \cdot q = 0 = (Az)^T (y - p)
+
+0 = (z^T A^T)(y - Ax)
+  = z^T (A^T y - A^T A x)
+  = A^T Ax
+  = A^T y
+
+
+A^T A = [[n+1      \Sigma_{n=0}^n x_n]
+        [\Sigma_{n=0}^n x_n  \Sigma_{n=0}^n x_n^2]]
+  
+A^T y = [[\Sigma_{n=0}^n y_n]
+       [\Sigma_{n=0}^n x_n y_n]]
+
+a_11 = n+1
+a_12 = \Sigma_{n=0}^n x_n
+a_21 = a_12
+a_22 = \Sigma_{n=0}^n x_n^2
+b_1 = \Sigma_{n=0}^n y_n
+b_2 = \Sigma_{n=0}^n x_n y_n
+
+then apply this with:
+
+log(e(h)) \leq log(C) + rlog(h)
+ 
+* homework 3:
+
+two columns \Rightarrow coefficients for linear regression
diff --git a/notes/Sep-25.org b/notes/Sep-25.org
new file mode 100644
index 0000000..b2d63e3
--- /dev/null
+++ b/notes/Sep-25.org
@@ -0,0 +1,48 @@
+ex: erfc(x) = \int_{0}^x (\frac{2}{\sqrt{pi}})e^{-t^2 }dt
+ex: IVP \frac{dP}{dt} = \alpha P - \beta P^2
+        P(0) = P_0
+
+Explicit Euler Method
+
+$\frac{P(t + \Delta t) - P(t)}{\Delta t} \approx \alpha P(t) - \beta P^2(t)$
+
+From 0 \rightarrow T
+P(T) \approx n steps
+
+* Steps
+** Calculus: defference quotient
+$f'(a) \approx \frac{f(a+h) - f(a)}{h}$
+
+** Test.
+Roundoff for h \approx 0
+
+** Calculus: Taylor Serioes w/ Remainder
+$e_{abs}(h) \leq Ch^r$
+
+(see Sep-20 . Taylor Series)
+
+* Pseudo Code
+#+BEGIN_SRC python
+  for i in range(n):
+    a12 = a12 + x[i+1]
+    a22 = a22 + x[i+1]**2
+  a21 = a12
+  b1 = y[0]
+  b2 = y[0] * x[0]
+  for i in range(n):
+    b1 = b1 + y[i+1]
+    b2 = b2 + y[i+1]*x[i+1]
+  detA = a22*a11 - a12*a21
+  c = (a22*b1 - a12*b2) / detA
+  d = (-a21 * b1 + a11 * b2) / detA
+
+  return (c, d)
+#+END_SRC
+
+* Error
+We want
+$e_k = |df(h_kk) - f'(a)|$
+
+$= |df(h_k) - df(h_m) + df(h_m) - f'(a)|$
+
+$\leq |df(h_k) - df(h_m)| + |df(h_m) - f'(a)|$ and $|df(h_m) - f'(a)|$ is negligible