ECEn 621 Computer Arithmetic. Method Comparisons

Transcription

1 ECEn 61 Computer Arithmetic Chapter 7: Iterative Approximation Slie #1 Metho Comparisons Digit Recurrence Metho Computes Multiplication Division/Square Root Log/Exp Operations Use Shifting Aition Simple igit Multiplication Digit Selection (Table Lookup) Convergence Linear convergence rate One result igit per cycle Iterative Approximation Also calle multiplicative metho Computes Division Square Root Reciprocal Operations Use Aition Multiplication Initial Approximation (Table Lookup) Convergence Quaratic convergence rate Doubles the number of result igits per cycle. Slie # 1

2 Newton-Raphson Metho Newton's metho (also known as the Newton Raphson metho), name after Isaac Newton an Joseph Raphson, is a metho for fining successively better approximations to the roots (or zeroes) of a real-value function. f (x n ) x n x n+1 Slie #3 Reciprocol Compute : R = 1 where is the input. Define a function : f (R) = 1 R Root : f (R) = 0 R = 1 1 <1 1 R < Slie #4

3 Newton-Raphson for Reciprocol Approximation f (R j ) = 1 R j f (R j ) = 1 R j R j +1 = R j f (R j) f (R j ) = R j 1 R j 1 R j = R j + R j R j R j +1 = R j ( R j ) R 0 =1 R 1 = Slie #5 Quaratic Convergence Let error be efine as : e j =1 R j R j = 1 e j R j +1 = R j ( R j ) = 1 e j 1 e j = 1 e j 1+ e j ( ) = 1 e j e j +1 =1 R j +1 =1 1 e j = e j e j +1 = e j is quaratic convergence. Let the initial approximation be such that : e 0 < r k Then after j iterations : e j < r k j = r n where n is the esire accuracy k j = n k j = log n iterations woul be neee. Slie #6 3

4 Example Slie #7 Reciprocal by Multiplicative Normalization Slie #8 4

5 Determine P[ j] for Quaratic Convergence Slie #9 Comparison of the Two Methos Newton-Raphson Multiplicative Normalization R Two epenent multiplies R P k+1 Two inepenent multiplies. Can use one pipeline multiplier Slie #10 5

6 Inaccurate Arithmetic? No Problem Both of these methos are very forgiving... Computation error is ae to algorithmic error, an both are ecrease quaratically the next iteration. Computation error Slie #11 Initial Approximation Slie #13 6

7 Table Lookup 1.xxxxxxxxxxxxxxxxxxxxxxx k-bits Table Size = k k bits approx 1/ 0.1xxxxx (k+1)-bits, error = ± -k Slie #14 Piecewise-Linear Table Lookup 1.xxxxxxxxxxxxxxxxxxxxxxx k-bits p-bits k a b Table Size = k (k+p) bits 0.1xxxxx p approx 1/ Slie #15 + xxxxxxxxxxxxxxx 0.1xxxxxxxxxxxxxxx 7

8 Bipartite Table Lookup 1.xxxxxxxxxxxxxxxxxxxxxxx k-bits a approx 1/ Slie # xxxxxx 0.yyyyyy 0.1zzzzz Using Reuce Multipliers At iteration j, approximation has a precision of k j bits (where error of initial approximation is < k ). OK to keep proucts at this precision. No nee to perform multiplication at full precision. Two alternatives Use a floating-point multiplier that prouces a roune prouct. All proucts are same number of bits. Use an n k rectangular multiplier (smaller than square multiplier). As the precision increases, multiplications are performe by a sequence of several rectangular multiplications. In either case, you must o error propagation analysis to etermine the number of prouct bits neee. Slie #17 8

9 Computing Division from Reciprocal Slie #18 Square-Root Newton-Raphson Metho f (S j ) = S j x Root is at S j = f (S j ) = S j S j +1 = S j f (S j) f (S j ) = S j S j x S j S j +1 = 1 S j + x S j x f (R j ) = 1 R j x f (R j ) = 3 R j Root is at R j = 1 x = R j +1 = R j f (R ) j f (R j ) = R j ( ) R j +1 = R j 3 xr j x x 1 R j x R j 3 Slie # 9

10 Square-Root by Multiplicative Normalization m x P j 1 P j j= 0 j= 0 x m m 1 x m x P j j= 0 S m x x j +1 = x j = 1 x j P j j 0 S j +1 = x j = 1 S j P j j 0 P j +1 =1+ 1 ( 1 x j ) Slie #3 10