Polynomials and Fast Fourier Transform Chapter 30, pp.823-848 new edition

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Polynomials and Fast Fourier TransformChapter
30, pp.823-848 new edition
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Polynomials

• Polynomial in coefficient representation
• A(x ) a0 a1x a2 x2 an-2 xn-2 an-1 xn-1
• Operations over polynomials
• polynomial degree n highest nonzero coeff
• addition O(n)
• multiplication O(n2)!!! Bad -- too slow!
• evaluation (finding the value in a point) O(n)
  !!! Good --
• Horners rule stack-based
• A(x0 ) a0 x0 (a1 x0 (a2 x0 (an-2 x0
  (an-1)))
• Point-value representation
• (x0 ,y0 ), (x1 ,y1 ), , (xn-1 ,yn-1 ) - n
  point-values are sufficient

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Point-value representation

• Interpolation getting polynomial coefficients
  from point-value representation
• Theorem (unique interpolation)
• for any set of distinct n point-value pairs,
• ?! polynomial of degree less than n
• Operations over polynomials
• addition O(n)
• multiplication O(n)!!! Good --
• how?
• evaluation (finding the value in a point)
  O(n2)!!! Bad -- too slow!
• IDEA Use both representations!!!

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Simple Transformations

• Coefficient gt point-value (evaluation)
• just O(n) per point
• O(n2)!!! Bad -- too slow!
• Point-value gt coefficient (interpolation)
• Lagranges formula
• n-1
• ? yk ? ?(x-xj) / ?(xk-xj)
• k0 j?k
  j?k
• O(n2)!!! Bad -- too slow!
• GOAL
• both transformations in O(n log n)

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O(n log n) Multiplication

• Double-degree bound
• 2n point-value pairs
• O(n)
• Evaluate
• Compute point point-value representations using
  FFT
• in (2n)-roots of unity
• O(n log n)
• Point-wise multiply
• Multiply the values for each of 2n points
• O(n)
• Interpolate
• Compute coefficient representation of product
  using FFT
• O(n log n)

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Complex Roots of Unity

• Point-value representation in complex roots of 1
• DFT discrete Fourier Transform
• Complex n-th root of 1 wn 1
• Complex numbers i ? -1
• The principal n-th root is
• wn1 e2? i /n cos(2?/n) i sin (2?/n)
• n roots of n-th power
• wn0 1, wn1 principal , wn2 wnwn, ,
  wnn-1
• Properties
• Cancellation wdndk wnk
• Halving if n is even then the squares of nth
  roots are n/2-roots
• Summation n-1
• ? (wnk ) j 0
  
• j0
  

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Fast Fourier Transform

• Problem
• Given polynomial
• A(x ) a0 a1x a2 x2 an-2 xn-2 an-1 xn-1
• Find values in roots of unity
• FFT divide and conquer
• A0(x ) a0 a1x a2 x2 an-2 xn/2-1
• A1(x ) a1 a3x a5 x2 an-1 xn/2-1
• A(x) A0(x2 ) xA1(x2)
• Recursive procedure
• Evaluate A0(x) and A1(x) in points (wn0 )2 ,
  (wn1 )2 ,, (wnn-1 )2
• n/2 roots each
• Combine the results
• T(n) 2T(n/2) ?(n) ?(n log n) (master
  theorem)

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Inverse FFT Interpolation

• y (y0 y1 y2...yn-1), a (a0 a1 a2...an-1),
  x (x0 x1 x2xn-1)
• Vn Vn (x) Vandermonde matrix
• y a Vn (x)
• Replace x (x0 x1 x2xn-1) gt wn (wn0 w n1
  w n2 wnn-1)
• DFT y a Vn (wn)
• a y Vn-1 (wn)

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Efficient FFT