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#+TITLE: HW 02
#+AUTHOR: Elizabeth Hunt
#+STARTUP: entitiespretty fold inlineimages
#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX: \setlength\parindent{0pt}
#+OPTIONS: toc:nil
* Question One
Computing $\epsilon_{\text{mac}}$ for single precision numbers
#+BEGIN_SRC lisp :session t :results table
(load "../lizfcm.asd")
(ql:quickload :lizfcm)
(let ((domain-values (lizfcm.approx:compute-maceps (lambda (x) x)
1.0
1.0)))
(lizfcm.utils:table (:headers '("a" "h" "err")
:domain-order (a h err)
:domain-values domain-values)))
#+END_SRC
#+RESULTS:
| a | h | err |
| 1.0 | 0.5 | 0.5 |
| 1.0 | 0.25 | 0.25 |
| 1.0 | 0.125 | 0.125 |
| 1.0 | 0.0625 | 0.0625 |
| 1.0 | 0.03125 | 0.03125 |
| 1.0 | 0.015625 | 0.015625 |
| 1.0 | 0.0078125 | 0.0078125 |
| 1.0 | 0.00390625 | 0.00390625 |
| 1.0 | 0.001953125 | 0.001953125 |
| 1.0 | 0.0009765625 | 0.0009765625 |
| 1.0 | 0.00048828125 | 0.00048828125 |
| 1.0 | 0.00024414063 | 0.00024414063 |
| 1.0 | 0.00012207031 | 0.00012207031 |
| 1.0 | 6.1035156e-05 | 6.1035156e-05 |
| 1.0 | 3.0517578e-05 | 3.0517578e-05 |
| 1.0 | 1.5258789e-05 | 1.5258789e-05 |
| 1.0 | 7.6293945e-06 | 7.6293945e-06 |
| 1.0 | 3.8146973e-06 | 3.8146973e-06 |
| 1.0 | 1.9073486e-06 | 1.9073486e-06 |
| 1.0 | 9.536743e-07 | 9.536743e-07 |
| 1.0 | 4.7683716e-07 | 4.7683716e-07 |
| 1.0 | 2.3841858e-07 | 2.3841858e-07 |
| 1.0 | 1.1920929e-07 | 1.1920929e-07 |
(with many rows truncated)
| a | h | err |
| 1.0 | 0.5 | 0.5 |
| 1.0 | 0.25 | 0.25 |
| 1.0 | 0.125 | 0.125 |
| 1.0 | 0.0625 | 0.0625 |
| 1.0 | 0.03125 | 0.03125 |
| 1.0 | 1.9073486e-06 | 1.9073486e-06 |
| 1.0 | 9.536743e-07 | 9.536743e-07 |
| 1.0 | 4.7683716e-07 | 4.7683716e-07 |
| 1.0 | 2.3841858e-07 | 2.3841858e-07 |
| 1.0 | 1.1920929e-07 | 1.1920929e-07 |
$\epsilon_{\text{mac single precision}}$ \approx 1.192(10^-7)
* Question Two
Computing $\epsilon_{\text{mac}}$ for double precision numbers:
#+BEGIN_SRC lisp :session t :results table
(let ((domain-values (lizfcm.approx:compute-maceps (lambda (x) x)
1.0d0
1.0d0)))
(lizfcm.utils:table (:headers '("a" "h" "err")
:domain-order (a h err)
:domain-values domain-values)))
#+END_SRC
(with many rows truncated)
| a | h | err |
| 1.0d0 | 0.5d0 | 0.5d0 |
| 1.0d0 | 0.25d0 | 0.25d0 |
| 1.0d0 | 0.125d0 | 0.125d0 |
| 1.0d0 | 0.0625d0 | 0.0625d0 |
| 1.0d0 | 0.03125d0 | 0.03125d0 |
| 1.0d0 | 0.015625d0 | 0.015625d0 |
| 1.0d0 | 0.0078125d0 | 0.0078125d0 |
| 1.0d0 | 0.00390625d0 | 0.00390625d0 |
| 1.0d0 | 0.001953125d0 | 0.001953125d0 |
| 1.0d0 | 7.105427357601002d-15 | 7.105427357601002d-15 |
| 1.0d0 | 3.552713678800501d-15 | 3.552713678800501d-15 |
| 1.0d0 | 1.7763568394002505d-15 | 1.7763568394002505d-15 |
| 1.0d0 | 8.881784197001252d-16 | 8.881784197001252d-16 |
| 1.0d0 | 4.440892098500626d-16 | 4.440892098500626d-16 |
| 1.0d0 | 2.220446049250313d-16 | 2.220446049250313d-16 |
Thus, $\epsilon_{\text{mac double precision}}$ \approx 2.220 \cdot 10^{-16}
* Question Three - |v|_2
#+BEGIN_SRC lisp :session t
(let ((vs '((1 1) (2 3) (4 5) (-1 2)))
(2-norm (lizfcm.vector:p-norm 2)))
(lizfcm.utils:table (:headers '("x" "y" "2norm")
:domain-order (x y)
:domain-values vs)
(funcall 2-norm (list x y))))
#+END_SRC
| x | y | 2norm |
| 1 | 1 | 1.4142135 |
| 2 | 3 | 3.6055512 |
| 4 | 5 | 6.4031243 |
| -1 | 2 | 2.236068 |
* Question Four - |v|_1
#+BEGIN_SRC lisp :session t
(let ((vs '((1 1) (2 3) (4 5) (-1 2)))
(1-norm (lizfcm.vector:p-norm 1)))
(lizfcm.utils:table (:headers '("x" "y" "1norm")
:domain-order (x y)
:domain-values vs)
(funcall 1-norm (list x y))))
#+END_SRC
| x | y | 1norm |
| 1 | 1 | 2 |
| 2 | 3 | 5 |
| 4 | 5 | 9 |
| -1 | 2 | 3 |
* Question Five - |v|_{\infty}
#+BEGIN_SRC lisp :session t
(let ((vs '((1 1) (2 3) (4 5) (-1 2))))
(lizfcm.utils:table (:headers '("x" "y" "max-norm")
:domain-order (x y)
:domain-values vs)
(lizfcm.vector:max-norm (list x y))))
#+END_SRC
| x | y | infty-norm |
| 1 | 1 | 1 |
| 2 | 3 | 3 |
| 4 | 5 | 5 |
| -1 | 2 | 2 |
* Question Six - ||v - u|| via |v|_{2}
#+BEGIN_SRC lisp :session t
(let ((vs '((1 1) (2 3) (4 5) (-1 2)))
(vs2 '((7 9) (2 2) (8 -1) (4 4)))
(2-norm (lizfcm.vector:p-norm 2)))
(lizfcm.utils:table (:headers '("v1" "v2" "2-norm-d")
:domain-order (v1 v2)
:domain-values (mapcar (lambda (v1 v2)
(list v1 v2))
vs
vs2))
(lizfcm.vector:distance v1 v2 2-norm)))
#+END_SRC
| v1 | v2 | 2-norm |
| (1 1) | (7 9) | 10.0 |
| (2 3) | (2 2) | 1.0 |
| (4 5) | (8 -1) | 7.2111025 |
| (-1 2) | (4 4) | 5.3851647 |
* Question Seven - ||v - u|| via |v|_{1}
#+BEGIN_SRC lisp :session t
(let ((vs '((1 1) (2 3) (4 5) (-1 2)))
(vs2 '((7 9) (2 2) (8 -1) (4 4)))
(1-norm (lizfcm.vector:p-norm 1)))
(lizfcm.utils:table (:headers '("v1" "v2" "1-norm-d")
:domain-order (v1 v2)
:domain-values (mapcar (lambda (v1 v2)
(list v1 v2))
vs
vs2))
(lizfcm.vector:distance v1 v2 1-norm)))
#+END_SRC
| v1 | v2 | 1-norm-d |
| (1 1) | (7 9) | 14 |
| (2 3) | (2 2) | 1 |
| (4 5) | (8 -1) | 10 |
| (-1 2) | (4 4) | 7 |
* Question Eight - ||v - u|| via |v|_{\infty}
#+BEGIN_SRC lisp :session t
(let ((vs '((1 1) (2 3) (4 5) (-1 2)))
(vs2 '((7 9) (2 2) (8 -1) (4 4))))
(lizfcm.utils:table (:headers '("v1" "v2" "max-norm-d")
:domain-order (v1 v2)
:domain-values (mapcar (lambda (v1 v2)
(list v1 v2))
vs
vs2))
(lizfcm.vector:distance v1 v2 'lizfcm.vector:max-norm)))
#+END_SRC
| v1 | v2 | max-norm-d |
| (1 1) | (7 9) | -6 |
| (2 3) | (2 2) | 1 |
| (4 5) | (8 -1) | 6 |
| (-1 2) | (4 4) | -2 |
|
