DOCSLIB.ORG

Weird Multiplication

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Weird Multiplication An Introduction to Various Multiplication Strategies Lynn West Bellevue, NE In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2011 Multiplication is one of the four basic operations of elementary arithmetic and is commonly defined as repeated addition. However, while this definition applies to whole number multiplication, some math researchers argue that it falls short for multiplication of fractions and other kinds of numbers. These mathematicians prefer to define multiplication as the scaling of one number by another, or as the process by which the product of two numbers is computed (Princeton University Wordnet, 2010). Despite the controversy, multiplication, by any definition, is an essential skill to students preparing for life in the mathematical world of the 21st century. It is an important tool in solving real-life problems and builds a firm foundation for proportional reasoning, algebraic thinking, and higher-level math. The standard algorithm for teaching the multiplication of larger numbers in this country is known as long multiplication and was originally brought to Europe by the Arabic-speaking people of Africa. In long multiplication, one multiplies the multiplicand by each digit of the multiplier and then adds up all the appropriately shifted results. This method requires memorization of the basic multiplication facts. However, a wide variety of efficient, alternative algorithms exist. Many students find these methods appealing and easier to navigate, even to the point of preferring them to the more traditional algorithm. Finger Multiplication Some of the oldest methods of multiplication involved finger calculations. One such method is believed to have come out of Italy and was widely used throughout medieval Europe (Rouse Ball, 1960 p. 189). The algorithm is fairly simple and can be used to calculate the product of two single digit numbers between five and nine. In order to use this method, one must understand that the closed fist represents five, and each raised finger adds one to that value. Thus, to determine the appropriate number of fingers to be raised on each hand, subtract five from each factor. For example, to find the product of 8 × 7, use the following steps: 1. Raise 3 fingers on the left hand; 8 - 5 = 3 2. Raise 2 fingers on the right hand; 7 - 5 = 2 3. Multiply each raised finger by 10; 5 × 10 = 50 4. Multiply the number of fingers in the down position on the left hand by the number of fingers in the down position on the right hand; 2 × 3 = 6 5. Add the two numbers; 50 + 6 = 56 × 8 7 Therefore, 8 × 7 = 56, but why does the algorithm work? Let’s rewrite the preceding process as an algebraic equation substituting x for eight and y for seven: 10[(x - 5) + (y - 5)] + [(10 - x)(10 - y)] Multiply the number of raised fingers (x-5 = 10x - 50 + 10y -50 + 100 -10x -10y + xy and y-5) by ten, and add the product of the = 10x - 50 + 10y -50 + 100 -10x -10y + xy fingers in the down position (10-x)(10-y). = xy Since all the terms cancel except for x and y, the equation gives the product of x and y. One advantage of this method is that it does not require memorization of multiplication facts beyond 5 × 5, so it is an effective tool for students who have not yet mastered the entire multiplication table. Introducing this algorithm to more advanced students offers them the opportunity to develop an understanding of the multiplication process and make connections to prior learning. Using mathematical reasoning to validate the algorithm fosters the development of conceptual understanding, an important component of proficiency (NCTM, 2000). Area Model of Multiplication All students need to be able to make connections between mathematical ideas and previously learned concepts in order to build new understandings. The area model of multiplication is an algorithm that uses multiple representations to explain the multiplication process, and can help students make connections to algebra and algebraic thinking. One can represent the multiplication of 14 × 12 as an area problem by drawing a rectangle with height 12 and width 14 on plain paper (or on graph paper to show the intermediate steps or in the case of multiplication by fractions). The area model is an application of the distributive property. 14 × 12 10 × 10 10 × 4 = [(10 + 2) × 10] + [(10 + 2) × 4] = 100 = 40 = (10 × 10) + (2 × 10) + (10 × 4) + (2 × 4) = 100 + 20 + 40 + 8 = 100 + 60 + 6 = 1= 2 × 10 = 20 2 × 4 = 8 The area model has theoretical limitations and cannot easily be used with irrational numbers; however, it is an excellent tool in helping students to establish a fundamental understanding of a variety of basic math concepts. For instance, it can be an effective tool in helping students to understand the concept of multiplication involving negative numbers. In the following example, we can represent the multiplication of 14 × 12 as (20 - 6) × (20 - 8): 400 -120 -160 48 This model clearly illustrates that the product of a positive number and a negative number is negative, while the product of two negative numbers must be positive. The area model also highlights the Distributive Property of Multiplication and using expanded notation. By allowing students to demonstrate graphically that 14 × 12 is the same as 12 × 14, the Commutative Property of Multiplication can be illustrated as well. In more advanced classes, this algorithm can be used to assist visual learners with the development of a relational understanding of polynomial multiplication and factoring. Lattice Multiplication Lattice multiplication, also known as sieve multiplication or the jalousia (gelosia) method, dates back to 10th century India and was introduced into Europe by Fibonacci in the 14th century (Carroll & Porter, 1998). It is algorithmically identical to the traditional long multiplication method, but breaks the process into smaller steps. For example, to multiply 453 × 25: 1. Draw a grid that has as many rows and columns as the multiplicand and the multiplier. 2. Draw a diagonal through each box from upper right corner to lower left corner. 3. Write the multipliers across the top and down the right side, lining up the digits with the boxes. 4. Record each partial product as a two-digit number with the tens digit in the upper left triangle and ones digit in the lower right triangle. (If the product does not have a tens digit, record a zero in the tens triangle.) 5. When all multiplications are complete, sum the numbers along the diagonals 6. Carry double digits to the next place, and record the answer. Therefore, 453 × 25 = 11,325. While some would argue that this algorithm ignores place value, it is easy to see that the diagonals actually represent the places that the digits occupy. Therefore, this multiplication essentially represents: (20 × 400) + (20 × 50) + (20 × 3) + (5 × 400) + (5 × 50) + (5 × 3) = 11,325 Multiplication of numbers beyond the single digits relies on three steps: multiplying, regrouping, and adding. The lattice method does each of these steps separately, so students are able to focus on the meaning of each part of the process. This method provides students with a structure for thinking about and recording their work. Lattice multiplication can also easily be extended to multiply decimal fractions and polynomials. Line Multiplication Another algorithm that is sometimes introduced to elementary school children is referred to as line multiplication. This algorithm presents students with a graphic representation of multiplication and can visually enhance their understanding of the multiplication process. Suppose you want to multiply 22 × 13: 1. First, draw two sets of vertical lines, two on the left and two on the right, to represent 22 (red lines). Next, draw two sets of horizontal lines, one on the top and three on the bottom, to represent 13 (blue lines). 2. Notice there are four sets of intersecting points (highlighted). To find the product, count up the intersection points in each of the highlighted sets and add diagonally. (Tanton, 2010) As with the traditional long multiplication algorithm, when a multiplication problem calls for regrouping, digits must be carried to the next place. Consider the multiplication problem 246 × 32. The answer, 6 thousands, 16 hundreds, 26 tens, and 12 ones is correct; however, numbers greater than or equal to 10 must be regrouped to write the answer in its standard form. Add the intersection points diagonally and then regroup numbers greater or equal to 10. 16,000 + 1,600 + 260 + 12 = 7,000 + 2600 + 260 + 12 = 7,000 + 800 + 160 + 12 = 7,000 + 800 + 70 + 2 = 7,872 (Tanton, 2010) Therefore, 246 × 32 = 7,872. This method works because diagonals of intersections of lines serve as placeholders (ones, 10’s, 100’s, etc.) and the number of points at each intersection represents the product of the number of lines. This is very similar to an area model of multiplication. When multiplying two two-digit numbers, as with 22 × 13, note that the problem can be rewritten as (20 + 2) × (10 + 3). 22 × 13 = (20 + 2) × (10 + 3) = (20 × 10) + (20 × 3) + (2 × 10) + (2 ×3) = 200 + 60 + 20 + 6 = 200 + 80 + 6 = 286 Note that the four sets of intersecting points are added diagonally. This serves to collect the points by place value (highlighted). This particular algorithm can be introduced to enhance the visual learner’s understanding of the multiplication process; however, all students should be exposed to a variety of strategies, models, and representations. Students should be able to explain the methods they use and understand there are multiple methods of efficiently solving any particular problem (NCTM, 2000).
Load more

Recommended publications
  • Computation of 2700 Billion Decimal Digits of Pi Using a Desktop Computer
    Computation of 2700 billion decimal digits of Pi using a Desktop Computer Fabrice Bellard Feb 11, 2010 (4th revision) This article describes some of the methods used to get the world record of the computation of the digits of π digits using an inexpensive desktop computer. 1 Notations We assume that numbers are represented in base B with B = 264. A digit in base B is called a limb. M(n) is the time needed to multiply n limb numbers. We assume that M(Cn) is approximately CM(n), which means M(n) is mostly linear, which is the case when handling very large numbers with the Sch¨onhage-Strassen multiplication [5]. log(n) means the natural logarithm of n. log2(n) is log(n)/ log(2). SI and binary prefixes are used (i.e. 1 TB = 1012 bytes, 1 GiB = 230 bytes). 2 Evaluation of the Chudnovsky series 2.1 Introduction The digits of π were computed using the Chudnovsky series [10] ∞ 1 X (−1)n(6n)!(A + Bn) = 12 π (n!)3(3n)!C3n+3/2 n=0 with A = 13591409 B = 545140134 C = 640320 . It was evaluated with the binary splitting algorithm. The asymptotic running time is O(M(n) log(n)2) for a n limb result. It is worst than the asymptotic running time of the Arithmetic-Geometric Mean algorithms of O(M(n) log(n)) but it has better locality and many improvements can reduce its constant factor. Let S be defined as n2 n X Y pk S(n1, n2) = an . qk n=n1+1 k=n1+1 1 We define the auxiliary integers n Y2 P (n1, n2) = pk k=n1+1 n Y2 Q(n1, n2) = qk k=n1+1 T (n1, n2) = S(n1, n2)Q(n1, n2).
    [Show full text]
  • Course Notes 1 1.1 Algorithms: Arithmetic
    CS 125 Course Notes 1 Fall 2016 Welcome to CS 125, a course on algorithms and computational complexity. First, what do these terms means? An algorithm is a recipe or a well-defined procedure for performing a calculation, or in general, for transform- ing some input into a desired output. In this course we will ask a number of basic questions about algorithms: • Does the algorithm halt? • Is it correct? That is, does the algorithm’s output always satisfy the input to output specification that we desire? • Is it efficient? Efficiency could be measured in more than one way. For example, what is the running time of the algorithm? What is its memory consumption? Meanwhile, computational complexity theory focuses on classification of problems according to the com- putational resources they require (time, memory, randomness, parallelism, etc.) in various computational models. Computational complexity theory asks questions such as • Is the class of problems that can be solved time-efficiently with a deterministic algorithm exactly the same as the class that can be solved time-efficiently with a randomized algorithm? • For a given class of problems, is there a “complete” problem for the class such that solving that one problem efficiently implies solving all problems in the class efficiently? • Can every problem with a time-efficient algorithmic solution also be solved with extremely little additional memory (beyond the memory required to store the problem input)? 1.1 Algorithms: arithmetic Some algorithms very familiar to us all are those those for adding and multiplying integers. We all know the grade school algorithm for addition from kindergarten: write the two numbers on top of each other, then add digits right 1-1 1-2 1 7 8 × 2 1 3 5 3 4 1 7 8 +3 5 6 3 7 914 Figure 1.1: Grade school multiplication.
    [Show full text]
  • A Scalable System-On-A-Chip Architecture for Prime Number Validation
    A SCALABLE SYSTEM-ON-A-CHIP ARCHITECTURE FOR PRIME NUMBER VALIDATION Ray C.C. Cheung and Ashley Brown Department of Computing, Imperial College London, United Kingdom Abstract This paper presents a scalable SoC architecture for prime number validation which targets reconfigurable hardware This paper presents a scalable SoC architecture for prime such as FPGAs. In particular, users are allowed to se- number validation which targets reconfigurable hardware. lect predefined scalable or non-scalable modular opera- The primality test is crucial for security systems, especially tors for their designs [4]. Our main contributions in- for most public-key schemes. The Rabin-Miller Strong clude: (1) Parallel designs for Montgomery modular arith- Pseudoprime Test has been mapped into hardware, which metic operations (Section 3). (2) A scalable design method makes use of a circuit for computing Montgomery modu- for mapping the Rabin-Miller Strong Pseudoprime Test lar exponentiation to further speed up the validation and to into hardware (Section 4). (3) An architecture of RAM- reduce the hardware cost. A design generator has been de- based Radix-2 Scalable Montgomery multiplier (Section veloped to generate a variety of scalable and non-scalable 4). (4) A design generator for producing hardware prime Montgomery multipliers based on user-defined parameters. number validators based on user-specified parameters (Sec- The performance and resource usage of our designs, im- tion 5). (5) Implementation of the proposed hardware ar- plemented in Xilinx reconfigurable devices, have been ex- chitectures in FPGAs, with an evaluation of its effective- plored using the embedded PowerPC processor and the soft ness compared with different size and speed tradeoffs (Sec- MicroBlaze processor.
    [Show full text]
  • Version 0.5.1 of 5 March 2010
    Modern Computer Arithmetic Richard P. Brent and Paul Zimmermann Version 0.5.1 of 5 March 2010 iii Copyright c 2003-2010 Richard P. Brent and Paul Zimmermann ° This electronic version is distributed under the terms and conditions of the Creative Commons license “Attribution-Noncommercial-No Derivative Works 3.0”. You are free to copy, distribute and transmit this book under the following conditions: Attribution. You must attribute the work in the manner specified by the • author or licensor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial. You may not use this work for commercial purposes. • No Derivative Works. You may not alter, transform, or build upon this • work. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this license impairs or restricts the author’s moral rights. For more information about the license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ Contents Preface page ix Acknowledgements xi Notation xiii 1 Integer Arithmetic 1 1.1 Representation and Notations 1 1.2 Addition and Subtraction 2 1.3 Multiplication 3 1.3.1 Naive Multiplication 4 1.3.2 Karatsuba’s Algorithm 5 1.3.3 Toom-Cook Multiplication 6 1.3.4 Use of the Fast Fourier Transform (FFT) 8 1.3.5 Unbalanced Multiplication 8 1.3.6 Squaring 11 1.3.7 Multiplication by a Constant 13 1.4 Division 14 1.4.1 Naive
    [Show full text]
  • 1 Multiplication
    CS 140 Class Notes 1 1 Multiplication Consider two unsigned binary numb ers X and Y . Wewanttomultiply these numb ers. The basic algorithm is similar to the one used in multiplying the numb ers on p encil and pap er. The main op erations involved are shift and add. Recall that the `p encil-and-pap er' algorithm is inecient in that each pro duct term obtained bymultiplying each bit of the multiplier to the multiplicand has to b e saved till all such pro duct terms are obtained. In machine implementations, it is desirable to add all such pro duct terms to form the partial product. Also, instead of shifting the pro duct terms to the left, the partial pro duct is shifted to the right b efore the addition takes place. In other words, if P is the partial pro duct i after i steps and if Y is the multiplicand and X is the multiplier, then P P + x Y i i j and 1 P P 2 i+1 i and the pro cess rep eats. Note that the multiplication of signed magnitude numb ers require a straight forward extension of the unsigned case. The magnitude part of the pro duct can b e computed just as in the unsigned magnitude case. The sign p of the pro duct P is computed from the signs of X and Y as 0 p x y 0 0 0 1.1 Two's complement Multiplication - Rob ertson's Algorithm Consider the case that we want to multiply two 8 bit numb ers X = x x :::x and Y = y y :::y .
    [Show full text]
  • Divide-And-Conquer Algorithms
    Chapter 2 Divide-and-conquer algorithms The divide-and-conquer strategy solves a problem by: 1. Breaking it into subproblems that are themselves smaller instances of the same type of problem 2. Recursively solving these subproblems 3. Appropriately combining their answers The real work is done piecemeal, in three different places: in the partitioning of problems into subproblems; at the very tail end of the recursion, when the subproblems are so small that they are solved outright; and in the gluing together of partial answers. These are held together and coordinated by the algorithm's core recursive structure. As an introductory example, we'll see how this technique yields a new algorithm for multi- plying numbers, one that is much more efficient than the method we all learned in elementary school! 2.1 Multiplication The mathematician Carl Friedrich Gauss (1777–1855) once noticed that although the product of two complex numbers (a + bi)(c + di) = ac bd + (bc + ad)i − seems to involve four real-number multiplications, it can in fact be done with just three: ac, bd, and (a + b)(c + d), since bc + ad = (a + b)(c + d) ac bd: − − In our big-O way of thinking, reducing the number of multiplications from four to three seems wasted ingenuity. But this modest improvement becomes very significant when applied recur- sively. 55 56 Algorithms Let's move away from complex numbers and see how this helps with regular multiplica- tion. Suppose x and y are two n-bit integers, and assume for convenience that n is a power of 2 (the more general case is hardly any different).
    [Show full text]
  • Fast Integer Multiplication Using Modular Arithmetic ∗
    Fast Integer Multiplication Using Modular Arithmetic ∗ Anindya Dey Piyush P Kurur,z Computer Science Division Dept. of Computer Science and Engineering University of California at Berkeley Indian Institute of Technology Kanpur Berkeley, CA 94720, USA Kanpur, UP, India, 208016 anindya@eecs.berkeley.edu ppk@cse.iitk.ac.in Chandan Saha Ramprasad Saptharishix Dept. of Computer Science and Engineering Chennai Mathematical Institute Indian Institute of Technology Kanpur Plot H1, SIPCOT IT Park Kanpur, UP, India, 208016 Padur PO, Siruseri, India, 603103 csaha@cse.iitk.ac.in ramprasad@cmi.ac.in Abstract ∗ We give an N · log N · 2O(log N) time algorithm to multiply two N-bit integers that uses modular arithmetic for intermediate computations instead of arithmetic over complex numbers as in F¨urer's algorithm, which also has the same and so far the best known complexity. The previous best algorithm using modular arithmetic (by Sch¨onhageand Strassen) has complexity O(N · log N · log log N). The advantage of using modular arithmetic as opposed to complex number arithmetic is that we can completely evade the task of bounding the truncation error due to finite approximations of complex numbers, which makes the analysis relatively simple. Our algorithm is based upon F¨urer'salgorithm, but uses FFT over multivariate polynomials along with an estimate of the least prime in an arithmetic progression to achieve this improvement in the modular setting. It can also be viewed as a p-adic version of F¨urer'salgorithm. 1 Introduction Computing the product of two N-bit integers is nearly a ubiquitous operation in algorithm design.
    [Show full text]
  • Chap03: Arithmetic for Computers
    CHAPTER 3 Arithmetic for Computers 3.1 Introduction 178 3.2 Addition and Subtraction 178 3.3 Multiplication 183 3.4 Division 189 3.5 Floating Point 196 3.6 Parallelism and Computer Arithmetic: Subword Parallelism 222 3.7 Real Stuff: x86 Streaming SIMD Extensions and Advanced Vector Extensions 224 3.8 Going Faster: Subword Parallelism and Matrix Multiply 225 3.9 Fallacies and Pitfalls 229 3.10 Concluding Remarks 232 3.11 Historical Perspective and Further Reading 236 3.12 Exercises 237 CMPS290 Class Notes (Chap03) Page 1 / 20 by Kuo-pao Yang 3.1 Introduction 178 Operations on integers o Addition and subtraction o Multiplication and division o Dealing with overflow Floating-point real numbers o Representation and operations x 3.2 Addition and Subtraction 178 Example: 7 + 6 Binary Addition o Figure 3.1 shows the sums and carries. The carries are shown in parentheses, with the arrows showing how they are passed. FIGURE 3.1 Binary addition, showing carries from right to left. The rightmost bit adds 1 to 0, resulting in the sum of this bit being 1 and the carry out from this bit being 0. Hence, the operation for the second digit to the right is 0 1 1 1 1. This generates a 0 for this sum bit and a carry out of 1. The third digit is the sum of 1 1 1 1 1, resulting in a carry out of 1 and a sum bit of 1. The fourth bit is 1 1 0 1 0, yielding a 1 sum and no carry.
    [Show full text]
  • Multi-Digit Multiplication Using the Standard Algorithm 5.NBT.B.5 Fluency Mini-Assessment by Student Achievement Partners
    Multi-Digit Multiplication Using the Standard Algorithm 5.NBT.B.5 Fluency Mini-Assessment by Student Achievement Partners OVERVIEW This mini-assessment is designed to illustrate the standard 5.NBT.B.5, which sets an expectation for fluently multiplying multi-digit whole numbers using the standard algorithm. This mini-assessment is designed for teachers to use either in the classroom, for self-learning, or in professional development settings to: • Evaluate students’ progress toward the skills described by 5.NBT.B.5 in order to prepare to teach this material or to check for fluency near the end of the grade; • Gain a better understanding of assessing fluency with multiplying multi-digit whole numbers; • Illustrate CCR-aligned assessment problems; • Illustrate best practices for writing tasks that allow access for all learners; and • Support mathematical language acquisition by offering specific guidance. MAKING THE SHIFTS This mini-assessment attends to focus as it addresses multi-digit multiplication, which is at the heart of the grade 5 standards and a key component of the Major Work of the Grade.1 It addresses coherence across grades because multiplying multi-digit whole numbers sets the stage for multi-digit division and multi-digit decimal operations. Standard 5.NBT.B.5 and this mini-assessment target procedural skill and fluency, one of the three elements of rigor. A CLOSER LOOK Standard 5.NBT.B.5 is a prime example of how “[t]he Standards are not written at uniform grain size” (K–8 Publishers’ Criteria, 5.NBT.B.5. Fluently multiply Spring 2013, p. 18). One cannot address this standard in a single multi-digit whole numbers day, lesson, or unit.
    [Show full text]
  • On-Line Algorithms for Division and Multiplication
    IEEE TRANSACTIONS ON COMPUTERS, VOL. C-26, NO. 7, JULY 1977 681 On-Line Algorithms for Division and Multiplication KISHOR S. TRIVEDI AND MILO0 D. ERCEGOVAC Abstract-In this paper, on-line algorithms for division and subtraction operations always satisfy this property. multiplication are developed. It is assumed that the operands as well Atrubin [3] has developed a right-to-left on-line algorithm as the result flow through the arithmetic unit in a digit-by-digit, most significant digit first fashion. The use of a redundant digit set, for multiplication. In this paper, we will retain our earlier at least for the digits of the result, is required. definition of the on-line property where all of the operands as well as the result digits flow through the arithmetic unit Index Terms-Computer arithmetic, division, multiplication, in a left-to-right digit-by-digit fashion. on-line algorithms, pipelining, radix, redundancy. Consider an mr-digit radix r number N = 1=l nir-i. In the conventional representation, each digit ni can take any I. INTRODUCTION value from the digit set 0,1, - - ,r - 1}. Such representa- IN THIS PAPER we consider problems of divi- tions, which allow only r values in the digit set, are non- sion and multiplication in a computational environ- redundant since there is a unique representation for each ment in which all basic arithmetic algorithms satisfy the (representable) number. By contrast, number systems that "on-line" property. This implies that to generate the jth allow more than r values in the digit set are redundant.
    [Show full text]
  • A Rigorous Extension of the Schönhage-Strassen Integer
    A Rigorous Extension of the Sch¨onhage-Strassen Integer Multiplication Algorithm Using Complex Interval Arithmetic∗ Raazesh Sainudiin Laboratory for Mathematical Statistical Experiments & Department of Mathematics and Statistics University of Canterbury, Private Bag 4800 Christchurch 8041, New Zealand r.sainudiin@math.canterbury.ac.nz Thomas Steinke Harvard School of Engineering and Applied Sciences Maxwell Dworkin, Room 138, 33 Oxford Street Cambridge, MA 02138, USA tsteinke@seas.harvard.edu Abstract Multiplication of n-digit integers by long multiplication requires O(n2) operations and can be time-consuming. A. Sch¨onhage and V. Strassen published an algorithm in 1970 that is capable of performing the task with only O(n log(n)) arithmetic operations over C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixed-point numbers, this results in an O(n(log(n))2+ε)-time algorithm. However, to make this algorithm more efficient and practical we need to make use of hardware-based floating- point numbers. How do we deal with rounding errors? and how do we determine the limits of the fixed-precision hardware? Our solution is to use interval arithmetic to guarantee the correctness of results and deter- mine the hardware’s limits. We examine the feasibility of this approach and are able to report that 75,000-digit base-256 integers can be han- dled using double-precision containment sets. This demonstrates that our approach has practical potential; however, at this stage, our implemen- tation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.
    [Show full text]
  • Fast Division of Large Integers --- a Comparison of Algorithms
    NADA Numerisk analys och datalogi Department of Numerical Analysis Kungl Tekniska Högskolan and Computer Science 100 44 STOCKHOLM Royal Institute of Technology SE-100 44 Stockholm, SWEDEN Fast Division of Large Integers A Comparison of Algorithms Karl Hasselström d98-kha@nada.kth.se TRITA-InsertNumberHere Master’s Thesis in Computer Science (20 credits) at the School of Computer Science and Engineering, Royal Institute of Technology, February 2003 Supervisor at Nada was Stefan Nilsson Examiner was Johan Håstad Abstract I have investigated, theoretically and experimentally, under what circumstances Newton division (inversion of the divisor with Newton’s method, followed by di- vision with Barrett’s method) is the fastest algorithm for integer division. The competition mainly consists of a recursive algorithm by Burnikel and Ziegler. For divisions where the dividend has twice as many bits as the divisor, Newton division is asymptotically fastest if multiplication of two n-bit integers can be done in time O(nc) for c<1.297, which is the case in both theory and practice. My implementation of Newton division (using subroutines from GMP, the GNU Mul- tiple Precision Arithmetic Library) is faster than Burnikel and Ziegler’s recursive algorithm (a part of GMP) for divisors larger than about five million bits (one and a half million decimal digits), on a standard PC. Snabb division av stora heltal En jämförelse av algoritmer Sammanfattning Jag har undersökt, ur både teoretiskt och praktiskt perspektiv, under vilka omstän- digheter Newtondivision (invertering av nämnaren med Newtons metod, följd av division med Barretts metod) är den snabbaste algoritmen för heltalsdivision. Den främsta konkurrenten är en rekursiv algoritm av Burnikel och Ziegler.
    [Show full text]