CSE 246: Computer Arithmetic Algorithms and Hardware Design

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CSE 246 Computer Arithmetic Algorithms and
Hardware Design
Lecture 6.1 Multiplication Arithmetic

• Instructor
• Prof. Chung-Kuan Cheng

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Topics

• Karatsubas Method (1962)
• Tooms Method (1963)
• Modular Method
• FFT

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Karatsubas Method

• U2nU1U0, V2nV1V0
• UV 22nU1V12n(U1V0U0V1)U0V0
• (22n2n)U1V12n(U1-U0)(V0-V1)(2n1)U0V0

T(2n)lt 3T(n)cn T(2k)ltc(3k-2k) T(n)T(2lgn)ltc(3
lgn-2lgn)lt3cnlg3
lg31.585
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Tooms Method

• U2rnUr2nU1U0
• V2rnVr2nV1V0
• U(x) xrUrxU1U0
• V(x) xrVrxV1V0
• U(x)V(x)W(x) x2rW2rxW1W0
• Set 2r1 equations
• W(0)U(0)V(0)
• W(1)U(1)V(1)
• W(2r)U(2r)V(2r)

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Tooms Method

• T((r1)n)lt (2r1)T(n)cn
• T(n)ltcnlogr1(2r1)ltcn1logr12

Theorem Given egt 0, there exists a
multiplication algorithm such that the number of
elementary operation T(n) needed to multiply two
n-bit numbers satisfies for some constant c(e)
independent of n
T(n)ltc(e)n1e
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Tooms Method

• U(4,13,2)16, V(9,2,5)16
• U(x)4x213x2, V9x22x5
• W(x)U(x)V(x)
• W(0)10, W(1)304,W(2)1980
• W(3)7084,W(4)18526
• W(x) x2rW2rxW1W0

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Tooms Method

• W(x) x2rW2rxW1W0
• Rewrite
• W(x) a2rx2ra1x1a0
• where xkx(x-1)(x-k1)
• W(x1)-W(x)
• 2ra2rx2r-1(2r-1)a2r-1x2r-2a1
• (W(x2)-W(x1))-(W(x1)-W(x))
• 2r(2r-1)a2rx2r-2(2r-1)(2r-2)a2r-1x2r-32a2

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Tooms Method

• W()10, 304, 1980, 7084, 18526
• W()294, 1676, 5104, 11442
• W()1382, 3428, 6338
• W()/2 691, 1714, 3169
• W()/2 1023, 1455
• W()/6 341, 485
• W()/6 144
• W()/24 36
• W(x) 36x4341x3691x2294x110
• (((36(x-3)341)(x-2)691)(x-1)294)x10
• 36x4125x364x269x10

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Tooms Method
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Toom and Cooks Method

• Theorem There is a constant c such that the
  execution time of Toom and Cooks method is less
  than
• cn23.5sqrt(lgn) cycles

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Modular Method (Schonhage)

• Recursive formula q01, qk13qk-1
• Thus, we have qk1/2(3k1)
• Relatively prime pi
• 6qk-1,6qk1,6qk2,6qk3,6qk5,6qk7
• Set six moduli
• mi2pi-1

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Modular Method

• Given U and V, Find WUxV
• Compute uiUmodmi viVmodmi
• Compute wiuixvimodmi
• Recover W
• T(n)O(nlog36)O(n1.631)

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FFT
Given U(t)(u0,u1,uK-1),V(t)(v0,v1,vK-1) Find
P(t)(p0,p1,,pK-1), where ptsum(ijt modK) uivj

• Set wexp(2pi/K), i.e. wK1
• us sum(0lttltK) wstut
• vs sum(0lttltK) wstvt
• U(s)V(s)(u0v0,u1v1,,uK-1vK-1)
• P(s)U(s)V(s), psusvs
• ps sum(0lttltK) wstpt

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FFT

• Kgt 2n-1, unun1uK-10
• vnvn1vK-10
• ptsum(ijt modK)uivj
• utv0ut-1v1u0vt

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FFT (K2k ,t(tk-1,,t0))

• Set A0(tk-1,,t0)ut ,i.e. A0(t)ut
• Set A1(sk-1,tk-2,,t0)
• A0(0,tk-2,,t0)w2k-1sk-1A0(1,tk-2,,t0)
• Set A2(sk-1,sk-2,tk-3,,t0)
• A1(sk-1,0,tk-3,,t0)
• w2k-2(sk-2sk-1)2A1(sk-1,1,tk-3,,t
  0)
• Set Ak(sk-1,sk-2,sk-3,,s0)
• Ak-1(sk-1,,s1,0)
• w(s0s1sk-1)2 Ak-1(sk-1,,s1,1)

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FFT (K2k ,t(tk-1,,t0))

• Replace tk-1 with sk-1
• sk-1 determines w2k-1sk-1
• Replace tk-2 with sk-2
• sk-1,sk-2 determines w2k-2(sk-2sk-1)2
• Replace t0 with s0
• sk-1,sk-2,,s0 determines w(s0s1sk-1)2
• Binary s(s0,s1,,sk-1)2

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FFT (K2k ,t(tk-1,,t0))

• By induction, we have
• Aj(sk-1,,sk-j,tk-j-1,,t0)
• sum(tk-1,,tk-j)w2k-j (sk-j,,sk-1)2
  (tk-1,,tk-j)2ut
• Ak(sk-1,,s0)
• sum(tk-1,,t0) w(s0,,sk-1)2(tk-1,,t0)2ut
• us

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FFT k2

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FFT k2

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FFT k2

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FFT k2

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FFT k3
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FFT k3
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FFT k3
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FFT k3

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FFT

• usu0u1su2s2u2k-1s2k-1
• usu0u2s2u2k-2s2k-2
• u1su3s3u2k-1s2k-1
• us Fe(s2) sFd(s2)
• Fe(s2)u0u2s2u2k-2s2k-2
• Fd(s2)u1u3s2u2k-1s2k-1
• us Fee(s4)s2Fed(s4) sFde(s4) s2Fdd(s4)

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FFT

• usu0u1su2s2u2k-1s2k-1
• us Fee(s4)s2Fed(s4) sFde(s4) s2Fdd(s4)
• us Feee(s8) s4Feed(s8) s2Fede(s8)
  s4Fedd(s8) sFdee(s8)s4Fded(s8)
  s2Fdde(s8) s4Fddd(s8)
• Fxx(s2k-1) Fxxe(s2k) s2k-1Fxxd(s2k)

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FFT

• usu0u1su2s2u3s3u4s4u5s5u6s6u7s7
• us Fe(s2) sFd(s2)
• Fe(s2)u0u2s2u4s4u6s6
• Fd(s2)u1u3s2u5s4u7s6
• Fe(s2)Fee(s4) s2Fed(s4)
• Fee(s4)u0u4s4, Fed(s4)u2u6s6
• Fd(s2)Fde(s4) s2Fdd(s4)
• Fde(s4)u1u5s4, Fdd(s4)u3u7s4
• Fx(sw0)Fx(sw4), Fx(sw2)Fx(sw6),
  Fx(sw)Fx(sw5), Fx(sw3)Fx(sw7)
• xe,d (s0,s1,s2)(-,0,0),(-,0,1),(-,1,0),(-,1
  ,1)
• Fxx(sw0)Fxx(sw2)Fxx(sw4)Fxx(sw6),
  Fxx(sw)Fxx(sw3)Fxx(sw5)Fxx(sw7),
• xxee,ed,de,dd, (s0,s1,s2)(-,-,0),(-,-,1)

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FFT (Inversion)

• ur sum(0ltsltK)wrsus
• sum(0lts,tltK)wrswstut
• sum(0lttltK)utsum(0ltsltK)ws(tr)
• Ku(-r)modK
• sum(0ltsltK)wsjK if jmodK0,
• 0 otherwise.

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FFT

• 2nlt2k glt 4n, K2k
• Precision m 6k
• Let M time of m-bit multiplication
• Total time to multiply n-bit numbers
• O(n)O(Mnk/g)