#+TITLE: Errors
#+AUTHOR: Elizabeth Hunt
#+STARTUP: entitiespretty fold inlineimages
#+LATEX_HEADER: \notindent \notag  \usepackage{amsmath} \usepackage[a4paper,margin=1in,landscape]{geometry}
#+LATEX: \setlength\parindent{0pt}
#+OPTIONS: toc:nil

* Errors
$x,y \in \mathds{R}$, using y as a way to approximate x. Then the
absolute error of in approximating x w/ y is $e_{abs}(x, y) = |x-y|$.

and the relative error is $e_{rel}(x, y) = \frac{|x-y|}{|x|}$

Table of Errors

#+BEGIN_SRC lisp :results table
  (load "../cl/lizfcm.asd")
  (ql:quickload 'lizfcm)

  (defun eabs (x y) (abs (- x y)))
  (defun erel (x y) (/ (abs (- x y)) (abs x)))

  (defparameter *u-v* '(
                        (1    0.99)
                        (1    1.01)
                        (-1.5 -1.2)
                        (100  99.9)
                        (100  99)
                        ))

  (lizfcm.utils:table (:headers '("u" "v" "e_{abs}" "e_{rel}")
                       :domain-order (u v)
                       :domain-values *u-v*)
    (eabs u v)
    (erel u v))
#+END_SRC

#+RESULTS:
|    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.

* Vector spaces & measures
Suppose we want solutions fo a linear system of the form $Ax = b$, and we want to approximate $x$,
we need to find a form of "distance" between vectors in $\mathds{R}^n$

** Vector Distances
A norm on a vector space $|| v ||$ is a function from $\mathds{R}^n$ such that:

1. $||v|| \geq 0$ for all $v \in \mathds{R}^n$ and $||v|| = \Leftrightarrow v = 0$
2. $||cv|| = |c| ||v||$ for all $c \in \mathds{R}, v \in \mathds{R}^n$
3. $||x + y|| \leq ||x|| + ||y|| \forall x,y \in \mathds{R}^n$

*** Example norms:
$||v||_2 = || [v_1, v_2, \dots v_n] || = (v_1^2 + v_2^2 + \dots + v_n^2)^{}^{\frac{1}{2}}$

$||v||_1 = |v_1| + |v_2| + \dots + |v_n|$

$||v||_{\infty} = \text{max}(|v_i|)$ (most restriction)

p-norm:
$||v||_p = \sum_{i=1}^{h} (|v_i|^p)^{\frac{1}{p}}$

** Length
The length of a vector in a given norm is $||v|| \forall v \in \mathds{R}^n$

All norms on finite dimensional vectors are equivalent. Then exist constants
$\alpha, \beta > 0 \ni \alpha ||v||_p \leq ||v||_q \leq \beta||v||_p$

** Distance
Let $u,v$ be vectors in $\mathds{R}^n$ then the distance is $||u - v||$ by some norm:
$e_{abs} = d(v, u) = ||u - v||$

The relative errors is:

$e_{rel} = \frac{||u - v||}{||v||}$


** Approxmiating Solutions to $Ax = b$
We define the residual vector $r(x) = b - Ax$

If $x$ is the exact solution, then $r(x) = 0$.

Then we can measure the "correctness" of the approximated solution on the norm of the
residual. We want to minimize the norm.

But, $r(y) = b - Ay \approx 0 \nRightarrow y \equiv x$, if $A$ is not invertible.