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diff --git a/homeworks/hw-2.org b/homeworks/hw-2.org new file mode 100644 index 0000000..c570b65 --- /dev/null +++ b/homeworks/hw-2.org @@ -0,0 +1,179 @@ +#+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 | |