new table macro, notes for 9/11
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@ -7,6 +7,7 @@
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((:module "utils"
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:components
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((:file "within-range" :depends-on ("package"))
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(:file "table" :depends-on ("package"))
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(:file "package")))
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(:module "approx"
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:components
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@ -21,7 +22,8 @@
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:lizfcm)
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:components ((:module "tests"
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:components
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((:file "approx" :depends-on ("suite"))
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((:file "table" :depends-on ("suite"))
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(:file "approx" :depends-on ("suite"))
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(:file "suite"))))
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:perform (asdf:test-op (o c) (uiop:symbol-call
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:fiveam :run!
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@ -5,4 +5,5 @@
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(x1 (- x delta))
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(y2 (apply f (list x2)))
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(y1 (apply f (list x1))))
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(/ (- y2 y1) (- x2 x1))))
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(/ (- y2 y1)
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(- x2 x1))))
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@ -1,4 +1,5 @@
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(in-package :cl-user)
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(defpackage lizfcm.utils
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(:use :cl)
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(:export :within-range-p))
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(:export :within-range-p
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:table))
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11
cl/src/utils/table.lisp
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11
cl/src/utils/table.lisp
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@ -0,0 +1,11 @@
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(in-package :lizfcm.utils)
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(defmacro table ((&key headers domain-order domain-values) &body body)
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`(cons
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,headers
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(mapcar (lambda (tuple)
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(destructuring-bind ,domain-order tuple
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(append tuple
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(list
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,@body))))
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,domain-values)))
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@ -15,9 +15,10 @@
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:description "derivative at is within bounds"
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(let ((f (lambda (x) (* x x)))
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(x 2)
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(accepted-delta 0.02)
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(f-prime-at-x 4)
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(delta 0.01))
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(is (within-range-p
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(derivative-at f x delta)
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f-prime-at-x
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0.1))))
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accepted-delta))))
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31
cl/tests/table.lisp
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31
cl/tests/table.lisp
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@ -0,0 +1,31 @@
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(defpackage lizfcm/tests.table
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(:use :cl
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:fiveam
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:lizfcm.utils
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:lizfcm/tests)
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(:export :approx-suite))
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(in-package :lizfcm/tests.table)
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(def-suite table-suite
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:in lizfcm-test-suite)
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(in-suite table-suite)
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(defun fib (n)
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(cond ((< n 2) n)
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(t (+ (fib (- n 1)) (fib (- n 2))))))
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(test table-of-fib-vals
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:description "table generates correctly"
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(let* ((headers '("n" "fib(n)"))
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(n-values '((1) (2) (3) (4)))
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(expected `(("n" "fib(n)")
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(1 ,(fib 1))
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(2 ,(fib 2))
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(3 ,(fib 3))
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(4 ,(fib 4))))
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(tabled (lizfcm.utils:table (:headers headers
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:domain-order (n)
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:domain-values n-values)
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(fib n))))
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(is (equal expected tabled))))
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93
notes/Sep-11.org
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93
notes/Sep-11.org
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@ -0,0 +1,93 @@
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#+TITLE: Errors
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#+AUTHOR: Elizabeth Hunt
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#+STARTUP: entitiespretty fold inlineimages
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#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,landscape]{geometry}
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#+LATEX: \setlength\parindent{0pt}
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#+OPTIONS: toc:nil
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* Errors
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$x,y \in \mathds{R}$, using y as a way to approximate x. Then the
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absolute error of in approximating x w/ y is $e_{abs}(x, y) = |x-y|$.
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and the relative error is $e_{rel}(x, y) = \frac{|x-y|}{|x|}$
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Table of Errors
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#+BEGIN_SRC lisp :results table
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(load "../cl/lizfcm.asd")
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(ql:quickload 'lizfcm)
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(defun eabs (x y) (abs (- x y)))
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(defun erel (x y) (/ (abs (- x y)) (abs x)))
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(defparameter *u-v* '(
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(1 0.99)
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(1 1.01)
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(-1.5 -1.2)
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(100 99.9)
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(100 99)
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))
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(lizfcm.utils:table (:headers '("u" "v" "e_{abs}" "e_{rel}")
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:domain-order (u v)
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:domain-values *u-v*)
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(fround (eabs u v) 4)
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(fround (erel u v) 4))
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#+END_SRC
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#+RESULTS:
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| u | v | e_{abs} | e_{rel} |
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| 1 | 0.99 | 0.0 | 0.0 |
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| 1 | 1.01 | 0.0 | 0.0 |
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| -1.5 | -1.2 | 0.0 | 0.0 |
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| 100 | 99.9 | 0.0 | 0.0 |
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| 100 | 99 | 0.0 | 0.0 |
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Look at $u \approx 0$ then $v \approx 0$, $e_{abs}$ is better error since $e_{rel}$ is high.
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* Vector spaces & measures
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Suppose we want solutions fo a linear system of the form $Ax = b$, and we want to approximate $x$,
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we need to find a form of "distance" between vectors in $\mathds{R}^n$
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** Vector Distances
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A norm on a vector space $|| v ||$ is a function from $\mathds{R}^n$ such that:
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1. $||v|| \geq 0$ for all $v \in \mathds{R}^n$ and $||v|| = \Leftrightarrow v = 0$
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2. $||cv|| = |c| ||v||$ for all $c \in \mathds{R}, v \in \mathds{R}^n$
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3. $||x + y|| \leq ||x|| + ||y|| \forall x,y \in \mathds{R}^n$
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*** Example norms:
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$||v||_2 = || [v_1, v_2, \dots v_n] || = (v_1^2 + v_2^2 + \dots + v_n^2)^{}^{\frac{1}{2}}$
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$||v||_1 = |v_1| + |v_2| + \dots + |v_n|$
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$||v||_{\infty} = \text{max}(|v_i|)$ (most restriction)
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p-norm:
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$||v||_p = \sum_{i=1}^{h} (|v_i|^p)^{\frac{1}{p}}$
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** Length
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The length of a vector in a given norm is $||v|| \forall v \in \mathds{R}^n$
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All norms on finite dimensional vectors are equivalent. Then exist constants
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$\alpha, \beta > 0 \ni \alpha ||v||_p \leq ||v||_q \leq \beta||v||_p$
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** Distance
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Let $u,v$ be vectors in $\mathds{R}^n$ then the distance is $||u - v||$ by some norm:
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$e_{abs} = d(v, u) = ||u - v||$
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The relative errors is:
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$e_{rel} = \frac{||u - v||}{||v||}$
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** Approxmiating Solutions to $Ax = b$
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We define the residual vector $r(x) = b - Ax$
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If $x$ is the exact solution, then $r(x) = 0$.
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Then we can measure the "correctness" of the approximated solution on the norm of the
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residual. We want to minimize the norm.
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But, $r(y) = b - Ay \approx 0 \nRightarrow y \equiv x$, if $A$ is not invertible.
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