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Notes on the analysis of multiplication algorithms.. Dr. M. Sakalli, Marmara University

Notes on the analysis of multiplication algorithms.. Dr. M. Sakalli, Marmara University. Integer Multiplication MIT notes and wikipedia. Example.. Classic High school math.. Let g = A|B and h = C|D where A,B,C and D are n/2 bit integers

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Notes on the analysis of multiplication algorithms.. Dr. M. Sakalli, Marmara University

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  1. Notes on the analysis of multiplication algorithms.. Dr. M. Sakalli, Marmara University

  2. Integer Multiplication MIT notes and wikipedia Example.. Classic High school math.. Let g = A|B and h = C|D where A,B,C and D are n/2 bit integers Simple Method: gh = (2n/2A+B)(2n/2C+D) same as given above. 4 multiplication routines. XY = (2n)AC+2n/2(AD+BC) + BD and carriages c. Long multiplication:rj = c + Σk = i-jgj hk Running Time Recurrence T(n) < 4T(n/2) + 100n, 100 multiplications.??, In-place??.. T(n) = q(n2) Provided that neither c nor the total sum exceed log space, indeed, a simple inductive argument shows that the carry c and the total sum for rican never exceed n and 2n: <<?? 2lgn respectively. Space efficiency: S(n)=O(loglog(N)), (loglog(N)). N=gh.

  3. Integer Multiplication MIT notes and wikipedia Pseudo code: Log space multiplication algorithm, multiply(g[0..n-1], h[0..n-1]) // Arrays representing to the binary representations x ← 0 for i= 0 : 2n-1 for j= 0 : i k ← i - j x ← x + (g[j] × h[k]) r[i] ← x mod 2 x ← floor(x/2) //I think this is carriage return. Last bit if 1.. end end Lattice method,Muhammad ibn Musa al-Khwarizmi. Gauss's complex multiplication algorithm.

  4. Integer Multiplication MIT notes and wikipedia Karatsuba’s algorithm: Polynomial extensions.. g=g110n/2+ g2 h=h110n/2+ h2 gh=g1 h110n+ (g1h2 + g2h1)10n/2+ g2h2 (g1h2+ g2h1) = (g1 + h1)(g2 + h2) - (g2h2+ g1h1), f(n) = 4sums+1 more final sum = 5n, n>2, suppose it is a constant 100n, and some carriages. XY = (2n/2+2n)AC+2n/2(A-B)(C-D) + (2n/2+1) BD A(n) = 3A(n/2)+5n, A(n) <O(n lg 3) ≈q(n1.6) Base value 7, when n<2,

  5. Karatsuba (g, h : n-digit integer; n : integer) // return (2n)-digit integer is a, b, c, d; // (n/2)-digit integer U, V, W; //n-digit integer; begin if n == 1 then return g(0)*h(0); ???? else g1  g(n-1) ... g(n/2); g2  g(n/2-1) ... g(0); h1  h(n-1) ... h(n/2); h2  h(n/2-1) ... h(0); U  Karatsuba ( g1, h1, n/2 ); V  Karatsuba ( g2, h2, n/2 ); W  Karatsuba ( g1+g2, h1+h2, n/2 ); return  U*10n + (W-U-V)*10^n/2 + V; end if; end Karatsuba; FFT and Fast Matrix multiplication.

  6. Quarter square multiplier 1980, Everett L. Johnson: gh = {(g + h)2 - (g - h)2}/4= {(g2 + 2hg+ h2) - (g2 - 2hg+ h2) }/4 Think hardware implementation, with a lookup table (converter), the difficulty is that summation of the two numbers each 8bits, will require at least 9 bits, when squared, 18 bits wide.. But if divided by 2 before squared, (discarding remainder when n is odd) . Table lookup from 0 to .. 9+9, from 0 … to 81. O(3n), working S(n) = (n). i.e. 7 by 3, observe that the sum and difference are 10 and 4 respectively. Looking both of those values up on the table yields 25 and 4, the difference of which is 21.

  7. Russian (Egyptian) Peasant’s binary multiplication Shift and add.. In-place algorithm, may be implemented and 2n space.. Try complex examples. 11 3, in binary 1011 11 011 5 6, 101 110 110 2 12, 10 1100 1 24, 1 11000 11000.. = 100001 T(n) = q(n)+O(n2), think about this?.. Why.. S(n) = q(loglog(n)) which is the carriage. If invertible? Division potential question.

  8. Matrix multiplication, 8 multiplications, O(n3) Pseudo code for MM. MM(A, B) for i ← 1 : N for j ← 1 : N C(i, j) ← 0; for k ← 1 : N C(i, j) ← C(i, j) + A(i, k) * B(k, j) end, end, end Time complexity of this algo is n3 multiplications and additions. Can we do better using divide and conquer?.. Subdividing matrices into four sub-matrices. T(n) = b, n2, T(n) = 8T(n/2) + cn2, n>2, which has T(n) = O(n??)

  9. Strassen’s Algorithm • Strassen: 7 multiplies, 18 additions • T(n) = b, n2, • T(n) = 7T(n/2) + (7m+18s)n2, n>2, which has T(n) = O(n2.81) • 7n2(1/4+1/16+…) • Strassen-Winograd: 7 multiplies, 15 additions • Coppersmith-Winograd, O(n2.376) (not easily implementable) • In practice faster (not large hidden constants) for relatively smaller n~64, and stable but demonstrated that for some matrices (Strassen and Strassen-Winograd) are too unstable.

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