How fast can we add (or subtract) two numbers n and m?

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1 Addition and Subtraction How fast do we add (or subtract) two numbers n and m? How fast can we add (or subtract) two numbers n and m? Definition. Let A(d) denote the maximal number of steps required to add two numbers with apple d bits. Theorem. A(d) d. Theorem. S(d) d.

2 Multiplication How fast do we multiply two numbers n and m? How fast can we multiply two numbers n and m? Definition. Let M(d) denote the number of steps required to multiply two numbers with apple d bits. Theorem. M(d) d 2. Can we do better? Yes How can we see easily that something better is possible?

3 Suppose M(d) d 1.5. Let d be large, and let " > 0. Let n and m have apple d bits, and write n = a n 2 r + b n and m = a m 2 r + b m, where r = bd/2c and the a j and b j are integers with b j < 2 r. From Attempt 2 Definition. Let M(d) denote the number of steps required to multiply two numbers with apple d bits. nm = a n a m 2 2r + (a n +b n )(a m +b m ) a n a m b n b m 2 r +b n b m, deduce M(d) apple 3M(r +2)+O(r) apple (3 + ")M(r +2). Hence, M(d) apple (3 + ") s M (d +2 s+2 4)/2 s. Take s = blog b 2 dc c C (with C big). Then 2 s d/2 C+1. Conclude, M(d) (3 + ") log 2 d = d log(3+")/ log 2.

4 Theorem. M(d) d 2. b c Conclude, M(d) (3 + ") log 2 d = d log(3+")/ log 2. log 3 log 2 = Theorem. M(d) d HW: Due September January 317 (Friday) Page 3, Problems 1 and 2 Page 5, unnumbered homework (first set) (you may use (log 5/ log 3) + " instead of log 5/ log 3)

5 Idea for Doing Better Let n and m have apple d bits, and write n = a n 2 r + b n and m = a m 2 r + b m, where r = bd/2c and the a j and b j are integers with b j < 2 r. From nm = a n a m 2 2r + (a n +b n )(a m +b m ) a n a m b n b m 2 r +b n b m, deduce M(d) apple 3M(r +2)+O(r) apple (3 + ")M(r +2). apple Think in terms of writing n = a n 2 2r + b n 2 r + c n and m = a m 2 2r + b m 2 r + c m, where r = bd/3c. b c How many multiplications does it take to expand nm?

6 Theorem. For every " > 0, wehavem(d) " d 1+". Theorem. M(d) d (log d)loglogd. Theorem. Given distinct numbers x 0,x 1,...,x k and numbers y 0,y 1,...,y k, there is a unique polynomial f of degree apple k such that f(x j )=y j for all j. apple Lagrange Interpolation: f(x) = kx Y x x j y i i=0 0applejapplek j6=i x i x j

7 Theorem. For every " > 0, wehavem(d) " d 1+". Suppose n and m have apple kr digits. Write n = k 1 X a u 2 ur and m = u=0 k 1 u=0 k 1 X b v 2 vr. X Xk 1 Then nm = f(2 r ), where X f(x) = X a u x u b v x. v Compute the 2k 1 numbers y j = f(j), for 0 apple j apple 2k 2, apple v=0 v=0 apple apple

8 apple Theorem. For every " > 0, wehavem(d) " d 1+". Lagrange Interpolation: Suppose n and m have apple kr digits. kx Y Write x x j f(x) = k 1 k x1 X n = a u 2 ur i=0 0applejapplek i x j X and m j6=i= b v 2 vr. u=0 k 1 X Xk 1 Then nm = f(2 r ), where X f(x) = X a u x u b v x. v X X u=0 Compute the 2k 1 numbers y j = f(j), for 0 apple j apple 2k 2, using 2k 1 multiplications of two apple r + c k digit numbers. apple Compute the coe cients of f(x) expanded, using Lagrange interpolation, in O k (r) steps. v=0 apple 0 v=0 apple apple y i

9 Theorem. For every " > 0, wehavem(d) " d 1+". Suppose n and Xm have apple kr digits. Write X kx 1 kx 1 n = a u 2 ur and m = b v 2 vr. u=0 k 1 k 1 Then nm = f(2 r X ), where X f(x) = X a u x u b v x. v X X apple apple Compute the 2k 1 numbers y j = f(j), for 0 apple j apple 2k 2, using 2k 1 multiplications of two apple r + c k digit numbers. Compute the coe cients of f(x) expanded, using Lagrange interpolation, in O k (r) steps. u=0 v=0 v=0 0 Deduce M(kr) apple apple (2k 1)M(r + c k )+c kr so that M(d) (2k 1) log k d d log(2k 1)/ log k.

10 Division Problem: Given two positive integers n and m, determine the quotient q and the remainder r when n is divided by m. These should be integers satisfying n = mq + r and 0 apple r<m. Definition. Let M 0 (d) denote an upper bound on the number of steps required to multiply two numbers with apple d bits. Let D 0 (d) denote an upper bound on the number of steps required to obtain q and r given n and m each have apple d binary digits. apple Theorem. Suppose M 0 (d) has the form df (d) where f(d) is an increasing function of d. Then D 0 (d) M 0 (d).

11 Problem: Given two positive integers n and m, determine the quotient q and the remainder r when n is divided by m. These should be integers satisfying n = mq + r and 0 apple r<m. We need only compute 1/m to su cient accuracy. Suppose n and m have apple s digits. If 1/m = 0.d 1 d 2 d 3 d 4... (base 2) with d 1,...,d s known, then n m = 1 2 s(n d 1d 2...d s )+, where 0 apple apple 1. Write this in the form n m = 1 2 s(q0 2 s + q 00 )+, so n = mq where 0 apple 0 < 2m. Tryq = q 0 and q = q 0 +1.

12 Problem: Given two positive integers n and m, determine the quotient q and the remainder r when n is divided by m. These should be integers satisfying n = mq + r and 0 apple r<m. We need only compute 1/m to su cient accuracy. Suppose n and m have apple s digits. If 1/m = 0.d 1 d 2 d 3 d 4... (base 2) with d 1,...,d s known, then n m = 1 2 s(n d 1d 2...d s )+, where 0 apple apple 1. Write this in the form n m = 1 2 s(q0 2 s + q 00 )+, so n = mq where 0 apple 0 < 2m. Tryq = q 0 and q = q 0 +1.

13 Newton s Method Say we want to compute 1/m. Take a function f(x) which has root 1/m. If x 0 is an approximation to the root, then how can we get a better approximation? Take f(x) =m 1/x. Starting with x 0 = x 0, this leads to the approximations x n+1 =2x n mx 2 n.

14 Newton s Method Say we want to compute 1/m. Take a function f(x) which has root 1/m. If x 0 is an approximation to the root, then how can we get a better approximation? Take f(x) =m 1/x. Starting with x 0 = x 0, this leads to the approximations x n+1 =2x n mx 2 n. Note that if x n =(1 ")/m, then x n+1 =(1 " 2 )/m.

15 Algorithm from Knuth, Vol. 2, pp Algorithm R. Let v in binary be v = (0.v 1 v 2 v 3...) 2, with v 1 = 1. The algorithm outputs z satisfying z 1/v apple 2 n. z 2 [0, 2] R1. [Initialize] Set z 1 4 b32/(4v 1 +2v 2 + v 3 )c and k 0. R2. [Newton iteration] (At this point, z apple 2 has the binary form (. ) 2 with 2 k +1 places after the radix point.) Calculate z 2 exactly. Then calculate V k z 2 exactly, where V k = (0.v 1 v 2...v 2 k+1 +3) 2. Then set z 2z V k z 2 + r, where 0 apple r<2 2k+1 1 is added if needed to round up z so that it is a multiple of 2 2k+1 1. Finally, set k k+1. R3. [End Test] If 2 k <n, go back to step R2; otherwise the algorithm terminates.

MATH Dr. Halimah Alshehri Dr. Halimah Alshehri

MATH Dr. Halimah Alshehri Dr. Halimah Alshehri MATH 1101 haalshehri@ksu.edu.sa 1 Introduction To Number Systems First Section: Binary System Second Section: Octal Number System Third Section: Hexadecimal System 2 Binary System 3 Binary System The binary