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# Division algorithm

About: Division algorithm is a research topic. Over the lifetime, 703 publications have been published within this topic receiving 8908 citations.

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TL;DR: A class of division methods best suited for use in digital computers with facilities for floating point arithmetic by considering the nature of each quotient digit as generated during the division process is described.

Abstract: This paper describes a class of division methods best suited for use in digital computers with facilities for floating point arithmetic. The division methods may be contrasted with conventional division procedures by considering the nature of each quotient digit as generated during the division process. In restoring division, each quotient digit has one of the values 0,1, . . . , r ?1, for an arbitrary integer radix r. In nonrestoring division, each quotient digit has one of the values ?(r?1),. . ., ?1, +1, . . ., +(r?1). For the division methods described here, each quotient digit has one of the values ?n, ?(n?1), . . ., ?1, 0, 1, . . . n?1, n, where n is an integer such that ?(r ?1)?n?r?1. A method for serial conversion of the quotient digits to conventional (restoring) form is given. Examples of new division procedures for radix 4 and radix 10 are given.

330 citations

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01 Jan 1994

TL;DR: This chapter discusses the theory and implementation of Digit-Recurrence Division, and some of the implementations of this division were described in detail in general comments.

Abstract: 1.General Comments. 2. Division by Digit Recurrence. 3. Theory of Digit-Recurrence Division. 4. Division with Scaling and Prediction. 5. Higher Radix Division. 6. On-the-Fly Conversion and Rounding. 7. Square Root by Digit Recurrence. 8. Implementations of Square Root. Appendices: Restoring and Non-Restoring Division Evaluation of Some Implementations.

326 citations

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01 Jan 1982

TL;DR: In this article, a cohomological interpretation of the complexity of the division algorithm for a fixed number of variables is given, which generalizes row reduction and the euclidean algorithm, in the same way that elimination theory generalizes the determinant and resultant.

Abstract: In this thesis, a division algorithm is studied, following work of Macaulay, Hironaka, Buchberger, and others, which generalizes row reduction and the euclidean algorithm, in the same way that elimination theory generalizes the determinant and the resultant.
The main result is a cohomological interpretation of the complexity of this algorithm, for a fixed number of variables. This follows from a new result on the vanishing of coherent sheaf cohomology, which generalizes previous work by Gotzmann, and Macaulay.
The Hilbert scheme offers a setting in which results about this algorithm can be understood; this relationship is described.
The theory of the division algorithm is related to the problem of manipulating objects in algebraic geometry by computer. The problem of computing coherent sheaf cohomology is considered, as a guiding example.
Finally, explicit equations are given for the Hilbert scheme.

289 citations

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TL;DR: The nature of a class of division techniques which permit the selection of quotient digits in digital division by the inspection of truncated versions of the divisor and partial remainder is reviewed in detail.

Abstract: —The nature of a class of division techniques which permit the selection of quotient digits in digital division by the inspection of truncated versions of the divisor and partial remainder is reviewed in detail. Two types of mechanisms, or so-called model divisions, for the selection of quotient digits are introduced. For both types of techniques, analytic tools are suggested for determining the number of bits which must be inspected as a function of the radix and form of representation of quotient digits. The analysis accounts for the representation of the partial remainder in a redundant form such as the one produced by an adder-subtractor which eliminates carry-borrow propagation.

182 citations

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01 Jan 2006

TL;DR: The author's research focused on the development of a number representation system that allowed for the addition and subtraction of numbers up to and including the number of bits in a discrete-time system.

Abstract: Preface. About the Authors. 1. Introduction. 1.1 Number Representation. 1.2 Algorithms. 1.3 Hardware Platforms. 1.4 Hardware-Software Partitioning. 1.5 Software Generation. 1.6 Synthesis. 1.7 A First Example. 1.7.1 Specification. 1.7.2 Number Representation. 1.7.3 Algorithms. 1.7.4 Hardware Platform. 1.7.5 Hardware-Software Partitioning. 1.7.6 Program Generation. 1.7.7 Synthesis. 1.7.8 Prototype. 1.8 Bibliography. 2. Mathematical Background. 2.1 Number Theory. 2.1.1 Basic Definitions. 2.1.2 Euclidean Algorithms. 2.1.3 Congruences. 2.2 Algebra. 2.2.1 Groups. 2.2.2 Rings. 2.2.3 Fields. 2.2.4 Polynomial Rings. 2.2.5 Congruences of Polynomial. 2.3 Function Approximation. 2.4 Bibliography. 3. Number Representation. 3.1 Natural Numbers. 3.1.1 Weighted Systems. 3.1.2 Residue Number System. 3.2 Integers. 3.2.1 Sign-Magnitude Representation. 3.2.2 Excess-E Representation. 3.2.3 B's Complement Representation. 3.2.4 Booth's Encoding. 3.3 Real Numbers. 3.4 Bibliography. 4. Arithmetic Operations: Addition and Subtraction. 4.1 Addition of Natural Numbers. 4.1.1 Basic Algorithm. 4.1.2 Faster Algorithms. 4.1.3 Long-Operand Addition. 4.1.4 Multioperand Addition. 4.1.5 Long-Multioperand Addition. 4.2 Subtraction of Natural Numbers. 4.3 Integers. 4.3.1 B's Complement Addition. 4.3.2 B's Complement Sign Change. 4.3.3 B's Complement Subtraction. 4.3.4 B's Complement Overflow Detection. 4.3.5 Excess-E Addition and Subtraction. 4.3.6 Sign-Magnitude Addition and Subtraction. 4.4 Bibliography. 5. Arithmetic Operations: Multiplication. 5.1 Natural Numbers Multiplication. 5.1.1 Introduction. 5.1.2 Shift and Add Algorithms. 5.1.2.1 Shift and Add 1. 5.1.2.2 Shift and Add 2. 5.1.2.3 Extended Shift and Add Algorithm: XY t C t D. 5.1.2.4 Cellular Shift and Add. 5.1.3 Long-Operand Algorithm. 5.2 Integers. 5.2.1 B's Complement Multiplication. 5.2.1.1 Mod Bntm B's Complement Multiplication. 5.2.1.2 Signed Shift and Add. 5.2.1.3 Postcorrection B's Complement Multiplication. 5.2.2 Postcorrection 2's Complement Multiplication. 5.2.3 Booth Multiplication for Binary Numbers. 5.2.3.1 Booth-r Algorithms. 5.2.3.2 Per Gelosia Signed-Digit Algorithm. 5.2.4 Booth Multiplication for Base-B Numbers (Booth-r Algorithm in Base B). 5.3 Squaring. 5.3.1 Base-B Squaring. 5.3.1.1 Cellular Carry-Save Squaring Algorithm. 5.3.2 Base-2 Squaring. 5.4 Bibliography. 6 Arithmetic Operations: Division. 6.1 Natural Numbers. 6.2 Integers. 6.2.1 General Algorithm. 6.2.2 Restoring Division Algorithm. 6.2.3 Base-2 Nonrestoring Division Algorithm. 6.2.4 SRT Radix-2 Division. 6.2.5 SRT Radix-2 Division with Stored-Carry Encoding. 6.2.6 P-D Diagram. 6.2.7 SRT-4 Division. 6.2.8 Base-B Nonrestoring Division Algorithm. 6.3 Convergence (Functional Iteration) Algorithms. 6.3.1 Introduction. 6.3.2 Newton-Raphson Iteration Technique. 6.3.3 MacLaurin Expansion-Goldschmidt's Algorithm. 6.4 Bibliography. 7. Other Arithmetic Operations. 7.1 Base Conversion. 7.2 Residue Number System Conversion. 7.2.1 Introduction. 7.2.2 Base-B to RNS Conversion. 7.2.3 RNS to Base-B Conversion. 7.3 Logarithmic, Exponential, and Trigonometric Functions. 7.3.1 Taylor-MacLaurin Series. 7.3.2 Polynomial Approximation. 7.3.3 Logarithm and Exponential Functions Approximation by Convergence Methods. 7.3.3.1 Logarithm Function Approximation by Multiplicative Normalization. 7.3.3.2 Exponential Function Approximation by Additive Normalization. 7.3.4 Trigonometric Functions-CORDIC Algorithms. 7.4 Square Rooting. 7.4.1 Digit Recurrence Algorithm-Base-B Integers. 7.4.2 Restoring Binary Shift-and-Subtract Square Rooting Algorithm. 7.4.3 Nonrestoring Binary Add-and-Subtract Square Rooting Algorithm. 7.4.4 Convergence Method-Newton-Raphson. 7.5 Bibliography. 8. Finite Field Operations. 8.1 Operations in Zm. 8.1.1 Addition. 8.1.2 Subtraction. 8.1.3 Multiplication. 8.1.3.1 Multiply and Reduce. 8.1.3.2 Modified Shift-and-Add Algorithm. 8.1.3.3 Montgomery Multiplication. 8.1.3.4 Specific Ring. 8.1.4 Exponentiation. 8.2 Operations in GF(p). 8.3 Operations in Zp[x]/f (x). 8.3.1 Addition and Subtraction. 8.3.2 Multiplication. 8.4 Operations in GF(pn). 8.5 Bibliography. Appendix 8.1 Computation of fki. 9 Hardware Platforms. 9.1 Design Methods for Electronic Systems. 9.1.1 Basic Blocks of Integrated Systems. 9.1.2 Recurring Topics in Electronic Design. 9.1.2.1 Design Challenge: Optimizing Design Metrics. 9.1.2.2 Cost in Integrated Circuits. 9.1.2.3 Moore's Law. 9.1.2.4 Time-to-Market. 9.1.2.5 Performance Metric. 9.1.2.6 The Power Dimension. 9.2 Instruction Set Processors. 9.2.1 Microprocessors. 9.2.2 Microcontrollers. 9.2.3 Embedded Processors Everywhere. 9.2.4 Digital Signal Processors. 9.2.5 Application-Specific Instruction Set Processors. 9.2.6 Programming Instruction Set Processors. 9.3 ASIC Designs. 9.3.1 Full-Custom ASIC. 9.3.2 Semicustom ASIC. 9.3.2.1 Gate-Array ASIC. 9.3.2.2 Standard-Cell-Based ASIC. 9.3.3 Design Flow in ASIC. 9.4 Programmable Logic. 9.4.1 Programmable Logic Devices (PLDs). 9.4.2 Field Programmable Gate Array (FPGA). 9.4.2.1 Why FPGA? A Short Historical Survey. 9.4.2.2 Basic FPGA Concepts. 9.4.3 XilinxTM Specifics. 9.4.3.1 Configurable Logic Blocks (CLBs). 9.4.3.2 Input/Output Blocks (IOBs). 9.4.3.3 RAM Blocks. 9.4.3.4 Programmable Routing. 9.4.3.5 Arithmetic Resources in Xilinx FPGAs. 9.4.4 FPGA Generic Design Flow. 9.5 Hardware Description Languages (HDLs). 9.5.1 Today's and Tomorrow's HDLs. 9.6 Further Readings. 9.7 Bibliography. 10. Circuit Synthesis: General Principles. 10.1 Resources. 10.2 Precedence Relation and Scheduling. 10.3 Pipeline. 10.4 Self-Timed Circuits. 10.5 Bibliography. 11 Adders and Subtractors. 11.1 Natural Numbers. 11.1.1 Basic Adder (Ripple-Carry Adder). 11.1.2 Carry-Chain Adder. 11.1.3 Carry-Skip Adder. 11.1.4 Optimization of Carry-Skip Adders. 11.1.5 Base-Bs Adder. 11.1.6 Carry-Select Adder. 11.1.7 Optimization of Carry-Select Adders. 11.1.8 Carry-Lookahead Adders (CLAs). 11.1.9 Prefix Adders. 11.1.10 FPGA Implementation of Adders. 11.1.10.1 Carry-Chain Adders. 11.1.10.2 Carry-Skip Adders. 11.1.10.3 Experimental Results. 11.1.11 Long-Operand Adders. 11.1.12 Multioperand Adders. 11.1.12.1 Sequential Multioperand Adders. 11.1.12.2 Combinational Multioperand Adders. 11.1.12.3 Carry-Save Adders. 11.1.12.4 Parallel Counters. 11.1.13 Subtractors and Adder-Subtractors. 11.1.14 Termination Detection. 11.1.15 FPGA Implementation of the Termination Detection. 11.2 Integers. 11.2.1 B's Complement Adders and Subtractors. 11.2.2 Excess-E Adders and Subtractors. 11.2.3 Sign-Magnitude Adders and Subtractors. 11.3 Bibliography. 12 Multipliers. 12.1 Natural Numbers. 12.1.1 Basic Multiplier. 12.1.2 Sequential Multipliers. 12.1.3 Cellular Multiplier Arrays. 12.1.3.1 Ripple-Carry Multiplier. 12.1.3.2 Carry-Save Multiplier. 12.1.3.3 Figures of Merit. 12.1.4 Multipliers Based on Dissymmetric Br Bs Cells. 12.1.5 Multipliers Based on Multioperand Adders. 12.1.6 Per Gelosia Multiplication Arrays. 12.1.6.1 Introduction. 12.1.6.2 Adding Tree for Base-B Partial Products. 12.1.7 FPGA Implementation of Multipliers. 12.2 Integers. 12.2.1 B's Complement Multipliers. 12.2.2 Booth Multipliers. 12.2.2.1 Booth-1 Multiplier. 12.2.2.2 Booth-2 Multiplier. 12.2.2.3 Signed-Digit Multiplier. 12.2.3 FPGA Implementation of the Booth-1 Multiplier. 12.3 Bibliography. 13. Dividers. 13.1 Natural Numbers. 13.2 Integers. 13.2.1 Base-2 Nonrestoring Divider. 13.2.2 Base-B Nonrestoring Divider. 13.2.3 SRT Dividers. 13.2.3.1 SRT-2 Divider. 13.2.3.2 SRT-2 Divider with Carry-Save Computation of the Remainder. 13.2.3.3 FPGA Implementation of the Carry-Save SRT-2 Divider. 13.2.4 SRT-4 Divider. 13.2.5 Convergence Dividers. 13.2.5.1 Newton-Raphson Divider. 13.2.5.2 Goldschmidt Divider. 13.2.5.3 Comparative Data Between Newton-Raphson (NR) and Goldschmidt (G) Implementations. 13.3 Bibliography. 14 Other Arithmetic Operators. 14.1 Base Conversion. 14.1.1 General Base Conversion. 14.1.2 BCD to Binary Converter. 14.1.2.1 Nonrestoring 2p Subtracting Implementation. 14.1.2.2 Shift-and-Add BCD to Binary Converter. 14.1.3 Binary to BCD Converter. 14.1.4 Base-B to RNS Converter. 14.1.5 CRT RNS to Base-B Converter. 14.1.6 RNS to Mixed-Radix System Converter. 14.2 Polynomial Computation Circuits. 14.3 Logarithm Operator. 14.4 Exponential Operator. 14.5 Sine and Cosine Operators. 14.6 Square Rooters. 14.6.1 Restoring Shift-and-Subtract Square Rooter (Naturals). 14.6.2 Nonrestoring Shift-and-Subtract Square Rooter (Naturals). 14.6.3 Newton-Raphson Square Rooter (Naturals). 14.7 Bibliography. 15. Circuits for Finite Field Operations. 15.1 Operations in Zm. 15.1.1 Adders and Subtractors. 15.1.2 Multiplication. 15.1.2.1 Multiply and Reduce. 15.1.2.2 Shift and Add. 15.1.2.3 Montgomery Multiplication. 15.1.2.4 Modulo (Bk2c) Reduction. 15.1.2.5 Exponentiation. 15.2 Inversion in GF(p). 15.3 Operations in Zp[x]/f (x). 15.4 Inversion in GF(pn). 15.5 Bibliography. 16. Floating-Point Unit. 16.1 Floating-Point System Definition. 16.2 Arithmetic Operations. 16.2.1 Addition of Positive Numbers. 16.2.2 Difference of Positive Numbers. 16.2.3 Addition and Subtraction. 16.2.4 Multiplication. 16.2.5 Division. 16.2.6 Square Root. 16.3 Rounding Schemes. 16.4 Guard Digits. 16.5 Adder-Subtractor. 16.5.1 Alignment. 16.5.2 Additions. 16.5.3 Normalization. 16.5.4 Rounding. 16.6 Multiplier. 16.7 Divider. 16.8 Square Root. 16.9 Comments. 16.10 Bibliography. Index.

169 citations