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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
+
+(with many rows truncated)
+
+|   a |             h |           err |
+| 1.0 |           1.0 |           1.0 |
+| 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 |                  1.0d0 |                  1.0d0 |
+| 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 |