#+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 "../cl/lizfcm.asd")
  (ql:quickload :lizfcm)

  (let ((domain-values (lizfcm.approx:compute-maceps 1.0
                                                     (lambda (x) x))))
    (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 |
| 1.0 |           0.5 |           0.5 |
| 1.0 |          0.25 |          0.25 |
| 1.0 |         0.125 |         0.125 |
| 1.0 |        0.0625 |        0.0625 |
| 1.0 | 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 |
| 1.0 | 5.9604645e-08 |           0.0 |

$\epsilon_{\text{mac}}$ \approx 5.9604 \cdot 10^{-8}

* Question Two
Computing $\epsilon_{\text{mac}}$ for double precision numbers:

#+BEGIN_SRC lisp :session t :results table
  (let ((domain-values (lizfcm.approx:compute-maceps 1.0d0
                                                     (lambda (x) x))))
    (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 |
| 1.0d0 | 1.1102230246251565d-16 |                  0.0d0 |

Thus, $\epsilon_{\text{mac}}$ \approx 1.1102 \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 |