Table Of Content

C++ MATHEMATICAL OBJECTS in Computational Tools in a Unified Object-Oriented Approach (cid:39)(cid:44)(cid:37)(cid:52)(cid:49)(cid:37)(cid:50)(cid:3)(cid:10)(cid:3)(cid:44)(cid:37)(cid:48)(cid:48)(cid:19)(cid:39)(cid:54)(cid:39) (cid:49)(cid:88)(cid:80)(cid:72)(cid:85)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:86)(cid:76)(cid:86)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:54)(cid:70)(cid:76)(cid:72)(cid:81)(cid:87)(cid:76)(cid:192)(cid:70)(cid:3)(cid:38)(cid:82)(cid:80)(cid:83)(cid:88)(cid:87)(cid:76)(cid:81)(cid:74) Aims and scope: (cid:54)(cid:70)(cid:76)(cid:72)(cid:81)(cid:87)(cid:76)(cid:192)(cid:70)(cid:3)(cid:70)(cid:82)(cid:80)(cid:83)(cid:88)(cid:87)(cid:76)(cid:81)(cid:74)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:81)(cid:88)(cid:80)(cid:72)(cid:85)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:86)(cid:76)(cid:86)(cid:3)(cid:83)(cid:85)(cid:82)(cid:89)(cid:76)(cid:71)(cid:72)(cid:3)(cid:76)(cid:81)(cid:89)(cid:68)(cid:79)(cid:88)(cid:68)(cid:69)(cid:79)(cid:72)(cid:3)(cid:87)(cid:82)(cid:82)(cid:79)(cid:86)(cid:3)(cid:73)(cid:82)(cid:85)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:86)(cid:70)(cid:76)(cid:72)(cid:81)(cid:70)(cid:72)(cid:86)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:72)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:17)(cid:3) (cid:55)(cid:75)(cid:76)(cid:86)(cid:3)(cid:86)(cid:72)(cid:85)(cid:76)(cid:72)(cid:86)(cid:3)(cid:68)(cid:76)(cid:80)(cid:86)(cid:3)(cid:87)(cid:82)(cid:3)(cid:70)(cid:68)(cid:83)(cid:87)(cid:88)(cid:85)(cid:72)(cid:3)(cid:81)(cid:72)(cid:90)(cid:3)(cid:71)(cid:72)(cid:89)(cid:72)(cid:79)(cid:82)(cid:83)(cid:80)(cid:72)(cid:81)(cid:87)(cid:86)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:86)(cid:88)(cid:80)(cid:80)(cid:68)(cid:85)(cid:76)(cid:93)(cid:72)(cid:3)(cid:86)(cid:87)(cid:68)(cid:87)(cid:72)(cid:16)(cid:82)(cid:73)(cid:16)(cid:87)(cid:75)(cid:72)(cid:16)(cid:68)(cid:85)(cid:87)(cid:3)(cid:80)(cid:72)(cid:87)(cid:75)(cid:82)(cid:71)(cid:86)(cid:3)(cid:82)(cid:89)(cid:72)(cid:85)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:90)(cid:75)(cid:82)(cid:79)(cid:72)(cid:3) (cid:86)(cid:83)(cid:72)(cid:70)(cid:87)(cid:85)(cid:88)(cid:80)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:86)(cid:72)(cid:3)(cid:192)(cid:72)(cid:79)(cid:71)(cid:86)(cid:17)(cid:3)(cid:44)(cid:87)(cid:3)(cid:90)(cid:76)(cid:79)(cid:79)(cid:3)(cid:76)(cid:81)(cid:70)(cid:79)(cid:88)(cid:71)(cid:72)(cid:3)(cid:68)(cid:3)(cid:69)(cid:85)(cid:82)(cid:68)(cid:71)(cid:3)(cid:85)(cid:68)(cid:81)(cid:74)(cid:72)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:72)(cid:91)(cid:87)(cid:69)(cid:82)(cid:82)(cid:78)(cid:86)(cid:15)(cid:3)(cid:80)(cid:82)(cid:81)(cid:82)(cid:74)(cid:85)(cid:68)(cid:83)(cid:75)(cid:86)(cid:15)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:75)(cid:68)(cid:81)(cid:71)(cid:69)(cid:82)(cid:82)(cid:78)(cid:86)(cid:17)(cid:3) (cid:57)(cid:82)(cid:79)(cid:88)(cid:80)(cid:72)(cid:86)(cid:3)(cid:76)(cid:81)(cid:3)(cid:87)(cid:75)(cid:72)(cid:82)(cid:85)(cid:92)(cid:15)(cid:3)(cid:76)(cid:81)(cid:70)(cid:79)(cid:88)(cid:71)(cid:76)(cid:81)(cid:74)(cid:3)(cid:71)(cid:76)(cid:86)(cid:70)(cid:85)(cid:72)(cid:87)(cid:76)(cid:86)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:87)(cid:72)(cid:70)(cid:75)(cid:81)(cid:76)(cid:84)(cid:88)(cid:72)(cid:86)(cid:15)(cid:3)(cid:81)(cid:88)(cid:80)(cid:72)(cid:85)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:68)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86)(cid:15)(cid:3)(cid:80)(cid:88)(cid:79)(cid:87)(cid:76)(cid:86)(cid:70)(cid:68)(cid:79)(cid:72)(cid:3)(cid:87)(cid:72)(cid:70)(cid:75)(cid:81)(cid:76)(cid:84)(cid:88)(cid:72)(cid:86)(cid:15)(cid:3) (cid:83)(cid:68)(cid:85)(cid:68)(cid:79)(cid:79)(cid:72)(cid:79)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:71)(cid:76)(cid:86)(cid:87)(cid:85)(cid:76)(cid:69)(cid:88)(cid:87)(cid:72)(cid:71)(cid:3)(cid:68)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86)(cid:15)(cid:3)(cid:68)(cid:86)(cid:3)(cid:90)(cid:72)(cid:79)(cid:79)(cid:3)(cid:68)(cid:86)(cid:3)(cid:68)(cid:83)(cid:83)(cid:79)(cid:76)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:86)(cid:72)(cid:3)(cid:80)(cid:72)(cid:87)(cid:75)(cid:82)(cid:71)(cid:86)(cid:3)(cid:76)(cid:81)(cid:3)(cid:80)(cid:88)(cid:79)(cid:87)(cid:76)(cid:16)(cid:71)(cid:76)(cid:86)(cid:70)(cid:76)(cid:83)(cid:79)(cid:76)(cid:81)(cid:68)(cid:85)(cid:92)(cid:3)(cid:192)(cid:72)(cid:79)(cid:71)(cid:86)(cid:15)(cid:3) (cid:68)(cid:85)(cid:72)(cid:3)(cid:90)(cid:72)(cid:79)(cid:70)(cid:82)(cid:80)(cid:72)(cid:17)(cid:3)(cid:55)(cid:75)(cid:72)(cid:3)(cid:76)(cid:81)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:70)(cid:82)(cid:81)(cid:70)(cid:85)(cid:72)(cid:87)(cid:72)(cid:3)(cid:85)(cid:72)(cid:68)(cid:79)(cid:16)(cid:90)(cid:82)(cid:85)(cid:79)(cid:71)(cid:3)(cid:72)(cid:91)(cid:68)(cid:80)(cid:83)(cid:79)(cid:72)(cid:86)(cid:3)(cid:76)(cid:86)(cid:3)(cid:75)(cid:76)(cid:74)(cid:75)(cid:79)(cid:92)(cid:3)(cid:72)(cid:81)(cid:70)(cid:82)(cid:88)(cid:85)(cid:68)(cid:74)(cid:72)(cid:71)(cid:17)(cid:3)(cid:55)(cid:75)(cid:76)(cid:86)(cid:3)(cid:86)(cid:72)(cid:85)(cid:76)(cid:72)(cid:86)(cid:3)(cid:76)(cid:86)(cid:3)(cid:80)(cid:72)(cid:68)(cid:81)(cid:87)(cid:3) (cid:87)(cid:82)(cid:3)(cid:68)(cid:83)(cid:83)(cid:72)(cid:68)(cid:79)(cid:3)(cid:87)(cid:82)(cid:3)(cid:86)(cid:87)(cid:88)(cid:71)(cid:72)(cid:81)(cid:87)(cid:86)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:85)(cid:72)(cid:86)(cid:72)(cid:68)(cid:85)(cid:70)(cid:75)(cid:72)(cid:85)(cid:86)(cid:3)(cid:76)(cid:81)(cid:3)(cid:80)(cid:68)(cid:87)(cid:75)(cid:72)(cid:80)(cid:68)(cid:87)(cid:76)(cid:70)(cid:86)(cid:15)(cid:3)(cid:72)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:15)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:70)(cid:82)(cid:80)(cid:83)(cid:88)(cid:87)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:68)(cid:79)(cid:3)(cid:86)(cid:70)(cid:76)(cid:72)(cid:81)(cid:70)(cid:72)(cid:17) Editors (cid:38)(cid:75)(cid:82)(cid:76)(cid:16)(cid:43)(cid:82)(cid:81)(cid:74)(cid:3)(cid:47)(cid:68)(cid:76) (cid:41)(cid:85)(cid:112)(cid:71)(cid:112)(cid:85)(cid:76)(cid:70)(cid:3)(cid:48)(cid:68)(cid:74)(cid:82)(cid:88)(cid:79)(cid:113)(cid:86) School of Computing and Applied Mathematics and Mathematical Sciences Systems Laboratory University of Greenwich Ecole Centrale Paris Editorial Advisory Board (cid:48)(cid:68)(cid:85)(cid:78)(cid:3)(cid:36)(cid:76)(cid:81)(cid:86)(cid:90)(cid:82)(cid:85)(cid:87)(cid:75) (cid:51)(cid:72)(cid:87)(cid:72)(cid:85)(cid:3)(cid:45)(cid:76)(cid:80)(cid:68)(cid:70)(cid:78) Mathematics Department School of Computing Strathclyde University University of Leeds (cid:55)(cid:82)(cid:71)(cid:71)(cid:3)(cid:36)(cid:85)(cid:69)(cid:82)(cid:74)(cid:68)(cid:86)(cid:87) (cid:55)(cid:68)(cid:78)(cid:68)(cid:86)(cid:75)(cid:76)(cid:3)(cid:46)(cid:68)(cid:78)(cid:82) Institute for Computational Department of Computer Science Engineering and Sciences The University of Electro-Communications The University of Texas at Austin (cid:51)(cid:72)(cid:87)(cid:72)(cid:85)(cid:3)(cid:48)(cid:82)(cid:81)(cid:78) (cid:38)(cid:85)(cid:68)(cid:76)(cid:74)(cid:3)(cid:38)(cid:17)(cid:3)(cid:39)(cid:82)(cid:88)(cid:74)(cid:79)(cid:68)(cid:86) Department of Mathematical Sciences Computer Science Department University of Delaware University of Kentucky (cid:41)(cid:85)(cid:68)(cid:81)(cid:70)(cid:82)(cid:76)(cid:86)(cid:16)(cid:59)(cid:68)(cid:89)(cid:76)(cid:72)(cid:85)(cid:3)(cid:53)(cid:82)(cid:88)(cid:91) (cid:44)(cid:89)(cid:68)(cid:81)(cid:3)(cid:42)(cid:85)(cid:68)(cid:75)(cid:68)(cid:80) ONERA Department of Mathematical Sciences University of Bath (cid:36)(cid:85)(cid:87)(cid:75)(cid:88)(cid:85)(cid:3)(cid:40)(cid:17)(cid:51)(cid:17)(cid:3)(cid:57)(cid:72)(cid:79)(cid:71)(cid:80)(cid:68)(cid:81) Institute of Mathematics and Computing Science University of Groningen (cid:51)(cid:85)(cid:82)(cid:83)(cid:82)(cid:86)(cid:68)(cid:79)(cid:86)(cid:3)(cid:73)(cid:82)(cid:85)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:86)(cid:72)(cid:85)(cid:76)(cid:72)(cid:86)(cid:3)(cid:86)(cid:75)(cid:82)(cid:88)(cid:79)(cid:71)(cid:3)(cid:69)(cid:72)(cid:3)(cid:86)(cid:88)(cid:69)(cid:80)(cid:76)(cid:87)(cid:87)(cid:72)(cid:71)(cid:3)(cid:87)(cid:82)(cid:3)(cid:82)(cid:81)(cid:72)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:86)(cid:72)(cid:85)(cid:76)(cid:72)(cid:86)(cid:3)(cid:72)(cid:71)(cid:76)(cid:87)(cid:82)(cid:85)(cid:86)(cid:3)(cid:68)(cid:69)(cid:82)(cid:89)(cid:72)(cid:3)(cid:82)(cid:85)(cid:3)(cid:71)(cid:76)(cid:85)(cid:72)(cid:70)(cid:87)(cid:79)(cid:92)(cid:3)(cid:87)(cid:82)(cid:29) CRC Press, Taylor & Francis Group (cid:23)(cid:87)(cid:75)(cid:15)(cid:3)(cid:41)(cid:79)(cid:82)(cid:82)(cid:85)(cid:15)(cid:3)(cid:36)(cid:79)(cid:69)(cid:72)(cid:85)(cid:87)(cid:3)(cid:43)(cid:82)(cid:88)(cid:86)(cid:72) (cid:20)(cid:16)(cid:23)(cid:3)(cid:54)(cid:76)(cid:81)(cid:74)(cid:72)(cid:85)(cid:3)(cid:54)(cid:87)(cid:85)(cid:72)(cid:72)(cid:87) (cid:47)(cid:82)(cid:81)(cid:71)(cid:82)(cid:81)(cid:3)(cid:40)(cid:38)(cid:21)(cid:36)(cid:3)(cid:23)(cid:37)(cid:52) (cid:56)(cid:46) Published Titles A Concise Introduction to Image Processing using C++ Meiqing Wang and Choi-Hong Lai Decomposition Methods for Differential Equations: Theory and Applications Juergen Geiser Grid Resource Management: Toward Virtual and Services Compliant Grid Computing Frédéric Magoulès, Thi-Mai-Huong Nguyen, and Lei Yu Introduction to Grid Computing Frédéric Magoulès, Jie Pan, Kiat-An Tan, and Abhinit Kumar Mathematical Objects in C++: Computational Tools in a Unified Object-Oriented Approach Yair Shapira Numerical Linear Approximation in C Nabih N. Abdelmalek and William A. Malek Numerical Techniques for Direct and Large-Eddy Simulations Xi Jiang and Choi-Hong Lai Parallel Algorithms Henri Casanova, Arnaud Legrand, and Yves Robert Parallel Iterative Algorithms: From Sequential to Grid Computing Jacques M. Bahi, Sylvain Contassot-Vivier, and Raphael Couturier C++ MATHEMATICAL OBJECTS in Computational Tools in a Unified Object-Oriented Approach Yair Shapira CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4398-1147-4 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Shapira, Yair, 1960- Mathematical objects in C++ : computational tools in a unified object-oriented approach / Yair Shapira. p. cm. -- (CHAPMAN & HALL/CRC numerical analysis and scientific computing) Includes bibliographical references and index. ISBN-13: 978-1-4398-1147-4 (alk. paper) ISBN-10: 1-4398-1147-4 (alk. paper) 1. Numerical analysis--Data processing. 2. C++ (Computer program language) I. Title. II. Series. QA297.S464 2010 518.0285’5133--dc22 2009007343 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Part I Numbers∗ 1 1 Natural Numbers 5 1.1 The Need for Natural Numbers ........................... 5 1.2 Mathematical Induction ................................. 5 1.3 Unboundedness ......................................... 6 1.4 Infinity ................................................ 6 1.5 Adding Natural Numbers ................................ 7 1.6 Recursion .............................................. 8 1.7 The Addition Function .................................. 8 1.8 Stack of Calls .......................................... 9 1.9 Multiplying Natural Numbers ............................ 9 1.10 One and Zero .......................................... 11 1.11 Decimal Representation ................................. 12 1.12 Binary Representation ................................... 13 1.13 Prime Numbers ........................................ 14 1.14 Prime Factors .......................................... 14 1.15 Mathematical Induction in Proofs ......................... 15 1.16 The Greatest Common Divisor ........................... 16 1.17 Least Common Multiple ................................. 17 1.18 The Factorial Function .................................. 18 1.19 Exercises .............................................. 18 2 Integer Numbers 21 2.1 Negative Numbers ...................................... 21 2.2 The Prisoner Problem ................................... 22 2.3 The Integer Numbers ................................... 23 2.4 The Number Axis ...................................... 23 2.5 Angles of Numbers ...................................... 24 2.6 Exercises .............................................. 25 3 Rational Numbers 27 3.1 Rational Numbers ...................................... 27 3.2 The Unique Form ....................................... 28 3.3 Adding Rational Numbers ............................... 28 3.4 Multiplying Rational Numbers ........................... 28 3.5 Periodic Decimal Representation .......................... 29 3.6 Diverging Series ........................................ 31 3.7 The Harmonic Series .................................... 32 3.8 Converging Series ....................................... 32 3.9 Finite Power Series ..................................... 33 3.10 Infinite Power Series .................................... 34 3.11 Periodic Decimal Fractions ............................... 35 3.12 Exercises .............................................. 36 4 Real Numbers 37 4.1 The Square Root of 2 ................................... 37 4.2 The Least-Upper-Bound Axiom .......................... 39 4.3 The Real Numbers .......√............................... 40 4.4 Decimal Representation of 2 ............................ 41 4.5 Irrational Numbers ..................................... 42 4.6 Transcendental Numbers ................................ 43 4.7 The Natural Exponent .................................. 43 4.8 Exercises .............................................. 45 5 Complex Numbers 47 5.1 The Imaginary Number ................................. 48 5.2 The Number Plane ..................................... 50 5.3 Sine and Cosine ........................................ 51 5.4 Adding Complex Numbers ............................... 51 5.5 Multiplying Complex Numbers ........................... 52 5.6 The Sine and Cosine Theorems ........................... 53 5.7 Exercises .............................................. 54 Part II Geometrical Objects∗ 57 6 Euclidean Geometry 61 6.1 Points and Lines ........................................ 61 6.2 Rays and Intervals ...................................... 62 6.3 Comparing Intervals .................................... 62 6.4 Ratios between Intervals ................................. 63 6.5 Angles ................................................ 66 6.6 Comparing Angles ...................................... 66 6.7 Corresponding and Alternate Angles ...................... 69 6.8 The Reversed Corresponding-Angle Theorem ............... 70 6.9 Parallel Lines – The Uniqueness Theorem .................. 71 6.10 Triangles .............................................. 73 6.11 Similar and Identical Triangles ........................... 74 6.12 Isosceles Triangles ...................................... 76 6.13 Pythagoras’ Axiom ..................................... 79 6.14 Sum of Edges .......................................... 80 6.15 The Longer Edge ....................................... 81 6.16 Tales’ Theorem ......................................... 82 6.17 The Reversed Tales’ Theorem ............................ 84 6.18 Circles ................................................ 85 6.19 Tangents .............................................. 89 6.20 Properties of the Tangent ................................ 91 6.21 Exercises .............................................. 93 7 Analytic Geometry 95 7.1 God and the Origin ..................................... 95 7.2 Numbers and Points .................................... 96 7.3 Lines – Sets of Points ................................... 97 7.4 Hierarchy of Objects .................................... 98 7.5 Half-Planes ............................................ 98 7.6 Angles ................................................ 99 7.7 Triangles .............................................. 100 7.8 Circles ................................................ 101 7.9 Exercises .............................................. 101 Part III Composite Mathematical Objects 103 8 Sets 107 8.1 Alice in Wonderland .................................... 107 8.2 Sets and Containers ..................................... 107 8.3 Russell’s Paradox ....................................... 108 8.4 The Empty Set ......................................... 109 8.5 Sets and Natural Numbers ............................... 109 8.6 The Order of the Natural Numbers ....................... 109 8.7 Mappings and Cardinality ............................... 110 8.8 Ordered Sets and Sequences .............................. 111 8.9 Infinite Sets ............................................ 111 8.10 Enumerable Sets ........................................ 112 8.11 The Set of Integer Numbers .............................. 114 8.12 Product of Sets ......................................... 114 8.13 Equivalence of Sets ..................................... 115 8.14 The Set of Rational Numbers ............................ 117 8.15 Arbitrarily Long Finite Sequences ........................ 118 8.16 Function Sets .......................................... 119 8.17 Cardinality of Function Sets ............................. 119 8.18 Nonenumerable Sets .................................... 120 8.19 Cardinality of the Real Axis ............................. 121 8.20 Cardinality of the Plane ................................. 122 8.21 Cardinality of the Multidimensional Space ................. 123 8.22 Larger Cardinalities ..................................... 124 8.23 Sets of Zero Measure .................................... 125 8.24 Cantor’s Set ........................................... 127

ebook img

Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) PDF

637 Pages·2009·1.9 MB·English
by  Yair Shapira
Save to my drive
Quick download
Download

Download Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) PDF Free - Full Version

by Yair Shapira| 2009| 637 pages| 1.9| English

Download Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) by Yair Shapira in PDF format completely FREE. No registration required, no payment needed. Get instant access to this valuable resource on PDFdrive.to!

Free Download PDF

About Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing)

No description available for this book.

Detailed Information

Author:Yair Shapira
Publication Year:2009
Pages:637
Language:English
File Size:1.9
Format:PDF
Price:FREE
Download Free PDF

Safe & Secure Download - No registration required

Why Choose PDFdrive for Your Free Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) Download?

  • 100% Free: No hidden fees or subscriptions required for one book every day.
  • No Registration: Immediate access is available without creating accounts for one book every day.
  • Safe and Secure: Clean downloads without malware or viruses
  • Multiple Formats: PDF, MOBI, Mpub,... optimized for all devices
  • Educational Resource: Supporting knowledge sharing and learning

Free Books Similar to Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing)

Frequently Asked Questions

Is it really free to download Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) PDF?

Yes, on https://PDFdrive.to you can download Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) by Yair Shapira completely free. We don't require any payment, subscription, or registration to access this PDF file. For 3 books every day.

How can I read Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) on my mobile device?

After downloading Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) PDF, you can open it with any PDF reader app on your phone or tablet. We recommend using Adobe Acrobat Reader, Apple Books, or Google Play Books for the best reading experience.

Is this the full version of Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing)?

Yes, this is the complete PDF version of Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) by Yair Shapira. You will be able to read the entire content as in the printed version without missing any pages.

Is it legal to download Mathematical Objects in C++: Computational Tools in A Unified Object-Oriented Approach (Chapman & Hall Crc Numerical Analysis and Scientific Computing) PDF for free?

https://PDFdrive.to provides links to free educational resources available online. We do not store any files on our servers. Please be aware of copyright laws in your country before downloading.

The materials shared are intended for research, educational, and personal use in accordance with fair use principles.

We use cookies

We use cookies to ensure you get the best browsing experience on our website. By clicking "Accept Cookies", you agree that we can store cookies on your device in accordance with our Terms and Privacy.