Polynomial and FFT
Topics
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1.
2.
3.
4.
5.
Problem
Representation of polynomials
The DFT and FFT
Efficient FFT implementations
Conclusion
Problem
Representation of Polynomials
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Definition 2 For the polynomial (1), we
have two ways of representing it:
Coefficient Representation——
(秦九韶算法）Horner’s rule
The coefficient representation is convenient for
certain operations on polynomials. For example,
the operation of evaluating the polynomial A(x) at
a given point x0
Coefficient Representation——
adding and multiplication
Point-value Representation
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By Horner’s rule, it takes Θ(n2) time to get a
point-value representation of polynomial (1).
If we choose xk cleverly, the complexity
reduces to n log n.
Definition 3 The inverse of evaluation. The process
of determining the coefficient form of a polynomial
from a point value representation is called
interpolation.
Does the interpolation uniquely determine
a polynomial? If not, the concept of
interpolation is meaningless.
Uniqueness of Interpolation
Lagrange Formula
We can compute the coefficients of A(x) by (4) in time Θ(n2).
拉格朗日［Lagrange, Joseph Louis，1736-1813
●法国数学家。
● 涉猎力学，著有分析力学。
● 百年以来数学界仍受其理论影响。
Virtues of point value
representation
Fast multiplication of polynomials
in coefficient form
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Can we use the linear-time
multiplication method for polynomials in
point-value form to expedite polynomial
multiplication in coefficient form?
Basic idea of multiplication
Basic idea of multiplication
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If we choose “complex roots of unity”
as the evaluation points carefully, we
can produce a point-value
representation by taking the Discrete
Fourier Transform of a coefficient vector.
The inverse operation interpolation, can
be performed by taking the inverse DFT
of point value pairs.
Complex Roots of Unity
Additive Group
Properties of Complex Roots
Fourier Transform
Now consider generalization to the case of a discrete
function :
Discrete Fourier Transform
Idea of Fast Fourier Transform
Recursive FFT
Complexity of FFT
Property 4 By divide-and-conquer method, the time
cost of FFT is T(n) = 2T(n/2)+Θ(n) =Θ(n log n).
Interpolation
Proof
DFTn vs DFT-1n
Efficient FFT Implementation
Butterfly Operation
Iterative-FFT
ITERATIVE-FFT (a)
1 BIT-REVERSE-COPY (a, A)
2 n ← length[a]
// n is a power of 2.
3 for s ← 1 to lg n
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do m ← 2s
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ωm ← e2πi/m
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for k ← 0 to n - 1 by m
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do ω ← 1
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for j ← 0 to m/2 - 1
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do t ← ωA[k + j + m/2]
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u ← A[k + j]
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A[k + j] ← u + t
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A[k + j + m/2] ← u - t
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ω←ωω
conclusion
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Fourier analysis is not limited to 1dimensional data. It is widely used in
image processing to analyze data in 2
or more dimensions.
Cooley and Tukey are widely credited
with devising the FFT in the 1960’s.