fix bug in which err = 0
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@ -45,7 +45,6 @@ Computing $\epsilon_{\text{mac}}$ for single precision numbers
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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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| 1.0 | 5.9604645e-08 | 0.0 |
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(with many rows truncated)
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@ -54,14 +53,14 @@ Computing $\epsilon_{\text{mac}}$ for single precision numbers
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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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| 1.0 | 5.9604645e-08 | 0.0 |
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$\epsilon_{\text{mac}}$ \approx 5.9604 \cdot 10^{-8}
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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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@ -75,62 +74,6 @@ Computing $\epsilon_{\text{mac}}$ for double precision numbers:
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:domain-values domain-values)))
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#+END_SRC
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#+RESULTS:
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| a | h | err |
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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 | 9.765625d-4 | 9.765625d-4 |
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| 1.0d0 | 4.8828125d-4 | 4.8828125d-4 |
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| 1.0d0 | 2.44140625d-4 | 2.44140625d-4 |
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| 1.0d0 | 1.220703125d-4 | 1.220703125d-4 |
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| 1.0d0 | 6.103515625d-5 | 6.103515625d-5 |
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| 1.0d0 | 3.0517578125d-5 | 3.0517578125d-5 |
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| 1.0d0 | 1.52587890625d-5 | 1.52587890625d-5 |
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| 1.0d0 | 7.62939453125d-6 | 7.62939453125d-6 |
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| 1.0d0 | 3.814697265625d-6 | 3.814697265625d-6 |
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| 1.0d0 | 1.9073486328125d-6 | 1.9073486328125d-6 |
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| 1.0d0 | 9.5367431640625d-7 | 9.5367431640625d-7 |
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| 1.0d0 | 4.76837158203125d-7 | 4.76837158203125d-7 |
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| 1.0d0 | 2.384185791015625d-7 | 2.384185791015625d-7 |
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| 1.0d0 | 1.1920928955078125d-7 | 1.1920928955078125d-7 |
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| 1.0d0 | 5.960464477539063d-8 | 5.960464477539063d-8 |
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| 1.0d0 | 2.9802322387695313d-8 | 2.9802322387695313d-8 |
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| 1.0d0 | 1.4901161193847656d-8 | 1.4901161193847656d-8 |
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| 1.0d0 | 7.450580596923828d-9 | 7.450580596923828d-9 |
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| 1.0d0 | 3.725290298461914d-9 | 3.725290298461914d-9 |
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| 1.0d0 | 1.862645149230957d-9 | 1.862645149230957d-9 |
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| 1.0d0 | 9.313225746154785d-10 | 9.313225746154785d-10 |
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| 1.0d0 | 4.656612873077393d-10 | 4.656612873077393d-10 |
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| 1.0d0 | 2.3283064365386963d-10 | 2.3283064365386963d-10 |
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| 1.0d0 | 1.1641532182693481d-10 | 1.1641532182693481d-10 |
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| 1.0d0 | 5.820766091346741d-11 | 5.820766091346741d-11 |
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| 1.0d0 | 2.9103830456733704d-11 | 2.9103830456733704d-11 |
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| 1.0d0 | 1.4551915228366852d-11 | 1.4551915228366852d-11 |
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| 1.0d0 | 7.275957614183426d-12 | 7.275957614183426d-12 |
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| 1.0d0 | 3.637978807091713d-12 | 3.637978807091713d-12 |
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| 1.0d0 | 1.8189894035458565d-12 | 1.8189894035458565d-12 |
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| 1.0d0 | 9.094947017729282d-13 | 9.094947017729282d-13 |
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| 1.0d0 | 4.547473508864641d-13 | 4.547473508864641d-13 |
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| 1.0d0 | 2.2737367544323206d-13 | 2.2737367544323206d-13 |
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| 1.0d0 | 1.1368683772161603d-13 | 1.1368683772161603d-13 |
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| 1.0d0 | 5.684341886080802d-14 | 5.684341886080802d-14 |
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| 1.0d0 | 2.842170943040401d-14 | 2.842170943040401d-14 |
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| 1.0d0 | 1.4210854715202004d-14 | 1.4210854715202004d-14 |
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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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| 1.0d0 | 1.1102230246251565d-16 | 0.0d0 |
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(with many rows truncated)
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| a | h | err |
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| 1.0d0 | 0.5d0 | 0.5d0 |
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@ -148,9 +91,8 @@ Computing $\epsilon_{\text{mac}}$ for double precision numbers:
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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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| 1.0d0 | 1.1102230246251565d-16 | 0.0d0 |
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Thus, $\epsilon_{\text{mac}}$ \approx 1.1102 \cdot 10^{-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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@ -204,9 +146,9 @@ Thus, $\epsilon_{\text{mac}}$ \approx 1.1102 \cdot 10^{-16}
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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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(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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@ -225,9 +167,9 @@ Thus, $\epsilon_{\text{mac}}$ \approx 1.1102 \cdot 10^{-16}
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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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(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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@ -246,8 +188,8 @@ Thus, $\epsilon_{\text{mac}}$ \approx 1.1102 \cdot 10^{-16}
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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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(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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Binary file not shown.
@ -1,4 +1,4 @@
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% Created 2023-09-25 Mon 09:52
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% Created 2023-09-27 Wed 10:09
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% Intended LaTeX compiler: pdflatex
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\documentclass[11pt]{article}
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\usepackage[utf8]{inputenc}
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@ -29,15 +29,16 @@
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\setlength\parindent{0pt}
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\section{Question One}
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\label{sec:orga203815}
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\label{sec:orga21c813}
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Computing \(\epsilon_{\text{mac}}\) for single precision numbers
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\begin{verbatim}
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(load "../cl/lizfcm.asd")
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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 1.0
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(lambda (x) x))))
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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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@ -52,24 +53,25 @@ a & h & err\\[0pt]
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1.0 & 0.25 & 0.25\\[0pt]
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1.0 & 0.125 & 0.125\\[0pt]
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1.0 & 0.0625 & 0.0625\\[0pt]
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1.0 & 0.03125 & 0.03125\\[0pt]
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1.0 & 1.9073486e-06 & 1.9073486e-06\\[0pt]
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1.0 & 9.536743e-07 & 9.536743e-07\\[0pt]
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1.0 & 4.7683716e-07 & 4.7683716e-07\\[0pt]
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1.0 & 2.3841858e-07 & 2.3841858e-07\\[0pt]
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1.0 & 1.1920929e-07 & 1.1920929e-07\\[0pt]
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1.0 & 5.9604645e-08 & 0.0\\[0pt]
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\end{tabular}
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\end{center}
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\(\epsilon_{\text{mac}}\) \(\approx\) 5.9604 \(\cdot\) 10\textsuperscript{-8}
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\(\epsilon_{\text{mac single precision}}\) \(\approx\) 1.192(10\textsuperscript{-7})
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\section{Question Two}
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\label{sec:orgdd79be1}
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\label{sec:org06c4a23}
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Computing \(\epsilon_{\text{mac}}\) for double precision numbers:
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\begin{verbatim}
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(let ((domain-values (lizfcm.approx:compute-maceps 1.0d0
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(lambda (x) x))))
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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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@ -94,14 +96,13 @@ a & h & err\\[0pt]
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1.0d0 & 8.881784197001252d-16 & 8.881784197001252d-16\\[0pt]
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1.0d0 & 4.440892098500626d-16 & 4.440892098500626d-16\\[0pt]
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1.0d0 & 2.220446049250313d-16 & 2.220446049250313d-16\\[0pt]
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1.0d0 & 1.1102230246251565d-16 & 0.0d0\\[0pt]
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\end{tabular}
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\end{center}
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Thus, \(\epsilon_{\text{mac}}\) \(\approx\) 1.1102 \(\cdot\) 10\textsuperscript{-16}
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Thus, \(\epsilon_{\text{mac double precision}}\) \(\approx\) 2.220 \(\cdot\) 10\textsuperscript{-16}
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\section{Question Three - |v|\textsubscript{2}}
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\label{sec:org04608d9}
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\label{sec:orgf181ba7}
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\begin{verbatim}
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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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@ -123,7 +124,7 @@ x & y & 2norm\\[0pt]
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\end{center}
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\section{Question Four - |v|\textsubscript{1}}
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\label{sec:orgfb57f3a}
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\label{sec:org2196087}
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\begin{verbatim}
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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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@ -145,7 +146,7 @@ x & y & 1norm\\[0pt]
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\end{center}
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\section{Question Five - |v|\textsubscript{\(\infty\)}}
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\label{sec:org7bbdf04}
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\label{sec:org11b8894}
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\begin{verbatim}
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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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@ -166,7 +167,7 @@ x & y & infty-norm\\[0pt]
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\end{center}
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\section{Question Six - ||v - u|| via |v|\textsubscript{2}}
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\label{sec:orge36996c}
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\label{sec:orga2324b2}
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\begin{verbatim}
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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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@ -192,7 +193,7 @@ v1 & v2 & 2-norm\\[0pt]
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\end{center}
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\section{Question Seven - ||v - u|| via |v|\textsubscript{1}}
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\label{sec:orgd1577f0}
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\label{sec:org388fbc7}
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\begin{verbatim}
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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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@ -218,7 +219,7 @@ v1 & v2 & 1-norm-d\\[0pt]
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\end{center}
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\section{Question Eight - ||v - u|| via |v|\textsubscript{\(\infty\)}}
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\label{sec:org2661676}
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\label{sec:org6e77f76}
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\begin{verbatim}
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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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@ -8,4 +8,5 @@
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(setf h (/ h 2)
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err (abs (- (funcall f (+ a h))
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(funcall f a))))
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when (> err 0)
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collect (list a h err))))
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Block a user