An Introduction To The Analysis Algorithms 4E – Michael Soltys

. . , xn) · (y1, y2, . . . , yn) = Pn i=1 xiyi, and the norm of a vector x is defined as ∥x∥= √x · x. Two vectors x, y are orthogonal if x · y = 0. 7.2 Gaussian Elimination Gaussian Elimination is a historic algorithm, just like Euclid’s algorithm (Section 1.1.3). It was first proposed by Isaac Newton ([Gravesande (1752)]), and later refined by Carl Friedrich Gauss.

We say that a matrix is in row-echelon form if it satisfies the following two conditions: (i) if there are non-zero rows, the first non-zero entry of such rows is 1, (the pivot), and (ii) the first non- zero entry of row i + 1 is to the right of the first non-zero entry of row i.

In short, a matrix is in row-echelon form if it looks as follows:   ∗. . . ∗ ∗ ∗. . . ∗ ∗ ∗. . . ∗ ∗ ∗. . . ∗ ∗ ∗. . . ∗ ∗ … ∗. . . ∗ ∗ . . . … … …   (7.1) where the ∗’s indicate arbitrary entries. We define the function Gaussian Elimination, GE : Mn×m −→ Mn×n, to be the function which when given an n × m matrix A as input, it outputs an n × n matrix GE(A), with the property that GE(A)A is in row-echelon form.

We call this property the correctness condition of GE. We show how to compute GE(A), given A. The idea is, of course, that GE(A) is equal to a product of elementary matrices which bring A to row-echelon form. We start by defining elementary matrices. Let Tij be a matrix with zeros everywhere except in the (i, j)th position, where it has a 1.

This page intentionally left blank This page intentionally left blank This page intentionally left blank This page intentionally left blank A n I n t r o d u c t i o n t o t h e Analysis of Algorithms 4th Edition Michael Soltys California State University Channel Islands, USA NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO World Scientific Published by World Scientific Publishing Co.

Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Soltys, Michael, 1971- author Title: An introduction to the analysis of algorithms / Michael Soltys, California State University Channel Islands, USA. Description: 4th edition. | New Jersey : World Scientific, [2026] | Previous edition: 2018.

| Includes bibliographical references and index. Identifiers: LCCN 2025040561 | ISBN 9789819823512 hardcover | ISBN 9789819823529 ebook | ISBN 9789819823536 ebook other Subjects: LCSH: Algorithms–Textbooks Classification: LCC QA9.58 .S63 2026 LC record available at https://lccn.loc.gov/2025040561 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2026 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/14590#t=suppl Desk Editors: Eshak Nabi Akbar Ali/Amanda Yun Typeset by Stallion Press Email: [email protected] Printed in Singapore To my family This page intentionally left blank This page intentionally left blank This page intentionally left blank This page intentionally left blank Preface If he had only learnt a little less, how infinitely better he might have taught much more!

Charles Dickens [Dickens (1854)], pg. 7 This book is a short introduction to algorithms, which are the meth- ods whereby we assign intellectual work to machines. Given a com- putational problem, an algorithm is a procedure to solve it; this procedure is usually implemented in a programming language, such as Python, to be run on a computer. We present two intertwined concepts related to algorithms: design technique, such as Greedy or Dynamic Programming; and analysis, such as performance or cor- rectness.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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