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