Add fft, add header info

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mbinary 2019-06-11 16:26:24 +08:00
parent df3f138550
commit b7e3bb5470
15 changed files with 436 additions and 17 deletions

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Some pictures and ideas are from `<<Introduction to Algotithm>>`
# Notes
* [docs](./docs)
* [algorithm-general.md](./docs/algorithm-general.md)
* [b-tree.md](./docs/b-tree.md)
* [fib-heap.md](./docs/fib-heap.md)
* [graph.md](./docs/graph.md)
* [hashTable.md](./docs/hashTable.md)
* [red-black-tree.md](./docs/red-black-tree.md)
* [sort.md](./docs/sort.md)
* [src](./docs/src)
* [string-matching.md](./docs/string-matching.md)
* [tree.md](./docs/tree.md)
[Click here](./docs) to view notes
# Index
* [.](.)
* [LICENSE](./LICENSE)
* [README.md](./README.md)
* [_config.yml](./_config.yml)
* [backtracking](./backtracking)
* [dataStructure](./dataStructure)
* [allOone](./dataStructure/allOone)
* [LRU](./dataStructure/LRU)
* [bTree.py](./dataStructure/bTree.py)
* [binaryHeap.py](./dataStructure/binaryHeap.py)
* [binaryHeap1.py](./dataStructure/binaryHeap1.py)
@ -41,6 +28,7 @@ Some pictures and ideas are from `<<Introduction to Algotithm>>`
* [graph](./dataStructure/graph)
* [hashTable.py](./dataStructure/hashTable.py)
* [huffman](./dataStructure/huffman)
* [insert_remove_getRandom.py](./dataStructure/insert_remove_getRandom.py)
* [intervalTree.py](./dataStructure/intervalTree.py)
* [leftHeap.py](./dataStructure/leftHeap.py)
* [linkedList.py](./dataStructure/linkedList.py)
@ -56,23 +44,40 @@ Some pictures and ideas are from `<<Introduction to Algotithm>>`
* [winnerTree.py](./dataStructure/winnerTree.py)
* [divideAndConquer](./divideAndConquer)
* [min_distance_of_n_points.py](./divideAndConquer/min_distance_of_n_points.py)
* [docs](./docs)
* [README.md](./docs/README.md)
* [_config.yml](./docs/_config.yml)
* [algorithm-general.md](./docs/algorithm-general.md)
* [b-tree.md](./docs/b-tree.md)
* [dft.md](./docs/dft.md)
* [fib-heap.md](./docs/fib-heap.md)
* [graph.md](./docs/graph.md)
* [hashTable.md](./docs/hashTable.md)
* [red-black-tree.md](./docs/red-black-tree.md)
* [sort.md](./docs/sort.md)
* [src](./docs/src)
* [string-matching.md](./docs/string-matching.md)
* [tree.md](./docs/tree.md)
* [dynamicProgramming](./dynamicProgramming)
* [Vec2d.hs](./dynamicProgramming/Vec2d.hs)
* [last-stone-weight.py](./dynamicProgramming/last-stone-weight.py)
* [lcs.py](./dynamicProgramming/lcs.py)
* [matrixChainMultiply.py](./dynamicProgramming/matrixChainMultiply.py)
* [max-len-of-repeated-subarray.py](./dynamicProgramming/max-len-of-repeated-subarray.py)
* [splitStripe.hs](./dynamicProgramming/splitStripe.hs)
* [splitStripe.py](./dynamicProgramming/splitStripe.py)
* [stoneGame.py](./dynamicProgramming/stoneGame.py)
* [testVec2d.hs](./dynamicProgramming/testVec2d.hs)
* [wildcard_matching.py](./dynamicProgramming/wildcard_matching.py)
* [graph](./graph)
* [cloneGraph.cpp](./graph/cloneGraph.cpp)
* [dfs.py](./graph/dfs.py)
* [isBipartGraph.py](./graph/isBipartGraph.py)
* [math](./math)
* [README.md](./math/README.md)
* [convertWeight.py](./math/convertWeight.py)
* [fastPow.py](./math/fastPow.py)
* [fft.py](./math/fft.py)
* [fibonacci](./math/fibonacci)
* [numWeight](./math/numWeight)
* [numberTheory](./math/numberTheory)
* [numericalAnalysis](./math/numericalAnalysis)
* [permute](./math/permute)
@ -100,6 +105,7 @@ Some pictures and ideas are from `<<Introduction to Algotithm>>`
* [README.md](./string/README.md)
* [manacher.py](./string/manacher.py)
* [markov.py](./string/markov.py)
* [min-window-substring.py](./string/min-window-substring.py)
* [rabin_karp.py](./string/rabin_karp.py)
* [rotate.py](./string/rotate.py)
* [src](./string/src)

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''' mbinary
#########################################################################
# File : lru_allone.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-05-23 23:50
# Description:
#########################################################################
'''
from allOne import allOne
'''In this implementation, the lru doesn't use some funcs of allOne,

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''' mbinary
#########################################################################
# File : lru_orderedDict.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-05-23 20:13
# Description:
#########################################################################
'''
class LRUCache(object):
def __init__(self, capacity):

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''' mbinary
#########################################################################
# File : binaryHeap1.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
'''
from functools import partial
class heap:
def __init__(self,lst,reverse = False):

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''' mbinary
#########################################################################
# File : circularQueue.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
'''
class MyCircularQueue:
def __init__(self, k):

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''' mbinary
#########################################################################
# File : linkedList.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
'''
class node:
def __init__(self,val,follow=None):
self.val = val

270
docs/dft.md Normal file
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<!-- TOC -->
- [0.1. 定义](#01-定义)
- [0.1.1. 连续](#011-连续)
- [0.1.2. 离散](#012-离散)
- [0.2. 性质](#02-性质)
- [0.2.1. 分离性](#021-分离性)
- [0.2.2. 位移定理](#022-位移定理)
- [0.2.3. 周期性](#023-周期性)
- [0.2.4. 共轭对称性](#024-共轭对称性)
- [0.2.5. 旋转性](#025-旋转性)
- [0.2.6. 加法定理](#026-加法定理)
- [0.2.7. 平均值](#027-平均值)
- [0.2.8. 相似性定理](#028-相似性定理)
- [0.2.9. 卷积定理](#029-卷积定理)
- [0.2.10. 相关定理](#0210-相关定理)
- [0.2.11. Rayleigh 定理](#0211-rayleigh-定理)
- [0.3. 快速傅里叶变换](#03-快速傅里叶变换)
- [0.3.1. 复数中的单位根](#031-复数中的单位根)
- [0.3.2. 快速傅里叶变换的计算](#032-快速傅里叶变换的计算)
- [0.4. 代码](#04-代码)
- [0.5. 参考](#05-参考)
<!-- /TOC -->
图像处理中, 为了方便处理,便于抽取特征,数据压缩等目的,常常要将图像进行变换。
一般有如下变换方法
1. 傅立叶变换Fourier Transform
2. 离散余弦变换Discrete Cosine Transform
3. 沃尔希-哈德玛变换Walsh-Hadamard Transform
4. 斜变换Slant Transform
5. 哈尔变换Haar Transform
6. 离散K-L变换Discrete Karhunen-Leave Transform
7. 奇异值分解SVD变换Singular-Value Decomposition
8. 离散小波变换Discrete Wavelet Transform
这篇文章介绍一下傅里叶变换
## 0.1. 定义
### 0.1.1. 连续
积分形式
如果一个函数的绝对值的积分存在,即
![](https://latex.codecogs.com/gif.latex?&space;\int_{-\infty}&space;^\infty&space;|h(t)|dt<\infty&space;)
并且函数是连续的或者只有有限个不连续点,则对于 x 的任何值, 函数的傅里叶变换存在
- 一维傅里叶变换
![](https://latex.codecogs.com/gif.latex?&space;H(f)=\int_{-\infty}&space;^\infty&space;h(t)e^{-j2\pi&space;ft}dt&space;)
- 一维傅里叶逆变换
![](https://latex.codecogs.com/gif.latex?&space;H(f)=\int_{-\infty}&space;^\infty&space;h(t)e^{j2\pi&space;ft}dt&space;)
同理多重积分
### 0.1.2. 离散
实际应用中,多用离散傅里叶变换 DFT.
- 一维傅里叶变换
![](https://latex.codecogs.com/gif.latex?&space;F(u)=\sum_{x=0}&space;^{N-1}&space;f(x)e^{\frac{-2\pi&space;j}{N}&space;ux}&space;)
- 一维傅里叶逆变换
![](https://latex.codecogs.com/gif.latex?&space;f(x)=\frac{1}{N}\sum_{u=0}&space;^{N-1}&space;F(u)e^{\frac{2\pi&space;j}{N}&space;ux}&space;)
需要注意的是, 逆变换乘以 ![](https://latex.codecogs.com/gif.latex?\frac{1}{N}) 是为了**归一化**,这个系数可以随意改变, 即可以正变换乘以 ![](https://latex.codecogs.com/gif.latex?\frac{1}{N}), 逆变换就不乘,或者两者都乘以![](https://latex.codecogs.com/gif.latex?\frac{1}{\sqrt{N}})等系数。
- 二维傅里叶变换
![](https://latex.codecogs.com/gif.latex?&space;F(u,v)=\frac{1}{N}\sum_{x=0}^{N-1}\sum_{y=0}&space;^{N-1}&space;f(x,y)e^{\frac{-2\pi&space;j}{N}&space;(ux+vy)}&space;)
- 二维傅里叶逆变换
![](https://latex.codecogs.com/gif.latex?&space;f(x,y)=\frac{1}{N}\sum_{u=0}^{N-1}\sum_{v=0}&space;^{N-1}&space;F(u,v)e^{\frac{2\pi&space;j}{N}&space;(ux+vy)}&space;)
幅度
![](https://latex.codecogs.com/gif.latex?&space;|F(u,v)|&space;=&space;\sqrt{real(F)^2+imag(F)^2}&space;)
相位
![](https://latex.codecogs.com/gif.latex?&space;arctan{\frac{imag(F)}{real(F)}}&space;)
对于图像的幅度谱显示,由于 |F(u,v)| 变换范围太大,一般显示 ![](https://latex.codecogs.com/gif.latex?D=&space;log(|F(u,v)+1))
`<=>` 表示傅里叶变换对
![](https://latex.codecogs.com/gif.latex?&space;f(x)<=>F(u)\\&space;f(x,y)<=>F(u,v)&space;)
f,g,h 对应的傅里叶变换 F,G,H
![](https://latex.codecogs.com/gif.latex?F^*) 表示 ![](https://latex.codecogs.com/gif.latex?F) 的共轭
## 0.2. 性质
### 0.2.1. 分离性
![](https://latex.codecogs.com/gif.latex?&space;\begin{aligned}&space;&F(x,v)=\sum_{y=0}&space;^{N-1}&space;f(x,y)e^{\frac{-2\pi&space;j}{N}&space;vy}\\&space;&F(u,v)=\frac{1}{N}\sum_{x=0}^{N-1}F(x,v)e^{\frac{-2\pi&space;j}{N}ux}&space;\end{aligned}&space;)
进行多维变换时,可以依次对每一维进行变换。 下面在代码中就是这样实现的。
### 0.2.2. 位移定理
![](https://latex.codecogs.com/gif.latex?&space;f(x,y)e^{\frac{2\pi&space;j}{N}(u_0x+v_0y)}&space;<=>F(u-u_0,v-v_0)&space;)
![](https://latex.codecogs.com/gif.latex?&space;f(x-x_0,y-y_0)<=>F(u,v)e^{\frac{-2\pi&space;j}{N}(ux_0+vy_0)}&space;)
### 0.2.3. 周期性
![](https://latex.codecogs.com/gif.latex?&space;F(u,v)&space;=&space;F(u+N,v+N)&space;)
### 0.2.4. 共轭对称性
![](https://latex.codecogs.com/gif.latex?F(u,v)&space;=&space;F^*(-u,-v))
a)偶分量函数在变换中产生偶分量函数;
b)奇分量函数在变换中产生奇分量函数;
c)奇分量函数在变换中引入系数-j;
d)偶分量函数在变换中不引入系数.
### 0.2.5. 旋转性
if ![](https://latex.codecogs.com/gif.latex?&space;f(r,\theta)<=>F(\omega,\phi)&space;)
then ![](https://latex.codecogs.com/gif.latex?f(r,\theta+t)<=>F(\omega,\phi+t)&space;)
### 0.2.6. 加法定理
1.
![](https://latex.codecogs.com/gif.latex?&space;Fourier[f+g]=Fourier[f]+Fourier[g]&space;)
2.
![](https://latex.codecogs.com/gif.latex?&space;af(x,y)<=>aF[u,v]&space;)
### 0.2.7. 平均值
![](https://latex.codecogs.com/gif.latex?&space;\frac{1}{N^2}\sum_{x=0}^{N-1}\sum_{y=0}&space;^{N-1}&space;f(x,y)&space;=&space;\frac{1}{N}F(0,0)&space;)
### 0.2.8. 相似性定理
尺度变换
![](https://latex.codecogs.com/gif.latex?&space;f(ax,by)<=>\frac{F(\frac{u}{a},\frac{v}{b})}{ab}&space;)
### 0.2.9. 卷积定理
卷积定义
1d
![](https://latex.codecogs.com/gif.latex?&space;f*g&space;=&space;\frac{1}{M}\sum_{m=0}^{M-1}f(m)g(x-m)&space;)
2d
![](https://latex.codecogs.com/gif.latex?&space;f(x,y)*g(x,y)&space;=&space;\frac{1}{MN}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}f(m,n)g(x-m,y-n)&space;)
卷积定理
![](https://latex.codecogs.com/gif.latex?&space;f(x,y)*g(x,y)&space;<=>&space;F(u,v)G(u,v)&space;)
![](https://latex.codecogs.com/gif.latex?&space;f(x,y)g(x,y)<=>F(u,v)*G(u,v)&space;)
离散卷积
![](https://latex.codecogs.com/gif.latex?&space;\sum_{i=0}^{N-1}x(iT)h[(k-i)T]&space;<=>&space;X(\frac{n}{NT})H(\frac{n}{NT})&space;)
即两个周期为 N 的抽样函数, 他们的卷积的离散傅里叶变换等于他们的离散傅里叶变换的卷积
卷积的应用:
去除噪声, 特征增强
两个不同周期的信号卷积需要周期扩展的原因:如果直接进行傅里叶变换和乘积,会产生折叠误差(卷绕)。
### 0.2.10. 相关定理
下面用![](https://latex.codecogs.com/gif.latex?\infty) 表示相关。
相关函数描述了两个信号之间的相似性,其相关性大小有相关系数衡量
- 相关函数的定义
离散
![](https://latex.codecogs.com/gif.latex?f(x,y)\quad&space;\infty&space;\quad&space;g(x,y)&space;=&space;\frac{1}{MN}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}f^*(m,n)g(x+m,y+n)&space;)
连续
![](https://latex.codecogs.com/gif.latex?z(t)&space;=&space;\int_{-\infty}^{\infty}x^*(\tau)&space;h(t+\tau)d\tau)
- 定理
![](https://latex.codecogs.com/gif.latex?&space;f(x,y)\quad&space;\infty&space;\quad&space;g(x,y)<=>F^*(u,v)G(u,v)&space;)
### 0.2.11. Rayleigh 定理
能量变换
对于有限区间非零函数 f(t), 其能量为
![](https://latex.codecogs.com/gif.latex?&space;E&space;=&space;\int_{-\infty}^{\infty}|f(t)|^2dt&space;)
其变换函数与原函数有相同的能量
![](https://latex.codecogs.com/gif.latex?&space;\int_{-\infty}^{\infty}|f(t)|^2dt&space;=&space;\int_{-\infty}^{\infty}|F(u)|^2dt&space;)
## 0.3. 快速傅里叶变换
由上面离散傅里叶变换的性质易知,直接计算 1维 dft 的时间复杂度维 ![](https://latex.codecogs.com/gif.latex?O(N^2))。
利用到单位根的对称性,快速傅里叶变换可以达到 ![](https://latex.codecogs.com/gif.latex?O(nlogn))的时间复杂度。
### 0.3.1. 复数中的单位根
我们知道, 在复平面,复数 ![](https://latex.codecogs.com/gif.latex?cos\theta&space;+i\&space;sin\theta)k可以表示成 ![](https://latex.codecogs.com/gif.latex?e^{i\theta}) 可以对应一个向量。![](https://latex.codecogs.com/gif.latex?\theta)即为幅角。
在**单位圆**中 ,单位圆被分成 ![](https://latex.codecogs.com/gif.latex?\frac{2\pi}{\theta}) 份, 由单位圆的对称性
![](https://latex.codecogs.com/gif.latex?&space;e^{i\theta}&space;=&space;e^{i(\theta+2\pi)}&space;)
现在记 ![](https://latex.codecogs.com/gif.latex?n&space;=\frac{&space;2\pi&space;}{\theta}) 即被分成 n 份,幅度角为正且最小的向量称为 n 次单位向量, 记为![](https://latex.codecogs.com/gif.latex?\omega&space;_n)
其余的 n-1 个向量分别为 ![](https://latex.codecogs.com/gif.latex?\omega_{n}^{2},\omega_{n}^{3},\ldots,\omega_{n}^{n}) ,它们可以由复数之间的乘法得来 ![](https://latex.codecogs.com/gif.latex?w_{n}^{k}=w_{n}^{k-1}\cdot&space;w_{n}^{1}\&space;(2&space;\leq&space;k&space;\leq&space;n))。
单位根的性质
1. 这个可以用 e 表示出来证明
![](https://latex.codecogs.com/gif.latex?&space;\omega_{2n}^{2k}=\omega_{n}^{k}&space;)
2. 可以写成三角函数证明
![](https://latex.codecogs.com/gif.latex?&space;\omega_{n}^{k+\frac{n}{2}}=-\omega_{n}^{k}&space;)
容易看出 ![](https://latex.codecogs.com/gif.latex?w_{n}^{n}=w_{n}^{0}=1)。
对于![](https://latex.codecogs.com/gif.latex?w_{n}^{k}) , 它事实上就是 ![](https://latex.codecogs.com/gif.latex?e^{\frac{2\pi&space;i}{n}k}) 。
### 0.3.2. 快速傅里叶变换的计算
下面的推导假设 ![](https://latex.codecogs.com/gif.latex?n=2^k),以及代码实现 FFT 部分也是 如此。
利用上面的对称性,
将傅里叶计算进行奇偶分组
![](https://latex.codecogs.com/gif.latex?&space;\begin{aligned}&space;F(u)&=\sum_{i=0}^{n-1}\omega_n&space;^{iu}&space;a^i\\&space;&=&space;\sum_{i=0}^{\frac{n}{2}-1}\omega_n&space;^{2iu}&space;a^{2i}+\sum_{i=0}^{\frac{n}{2}-1}\omega_n&space;^{(2i+1)u}&space;a^{2i+1}\\&space;&=\sum_{i=0}^{\frac{n}{2}-1}\omega_{\frac{n}{2}}&space;^{iu}&space;a^{2i}+\omega_n^u\sum_{i=0}^{\frac{n}{2}-1}\omega_{\frac{n}{2}}&space;^{iu}&space;a^{2i+1}\\&space;&&space;=&space;F_{even}(u)+\omega_n^u&space;F_{odd}(u)&space;\end{aligned}&space;)
![](https://latex.codecogs.com/gif.latex?F_{even})表示将 输入的次序中偶数点进行 Fourier 变换, ![](https://latex.codecogs.com/gif.latex?F_{odd}) 同理,这样就形成递推公式。
现在还没有减少计算量,下面通过将分别计算的 奇项,偶项利用起来,只计算 前 ![](https://latex.codecogs.com/gif.latex?\frac{n}{2}-1)项,后面的一半可以利用此结果马上算出来。每一次可以减少一半的计算量。
对于 ![](https://latex.codecogs.com/gif.latex?\frac{n}{2}\leq&space;i+\frac{n}{2}\leq&space;n-1)
![](https://latex.codecogs.com/gif.latex?&space;\begin{aligned}&space;F(\omega_{n}^{i+\frac{n}{2}})&=F_{even}(\omega_{n}^{2i+n})+\omega_{n}^{i+\frac{n}{2}}\cdot&space;F_{odd}(\omega_{n}^{2i+n})\\&space;&=F_{even}(\omega_{\frac{n}{2}}^{i+\frac{n}{2}})+\omega_{\frac{n}{2}}^{i+\frac{n}{2}}\cdot&space;F_{odd}(\omega_{\frac{n}{2}}^{i+\frac{n}{2}})\\&space;&&space;=F_{even}(\omega_{\frac{n}{2}}^{i})-\omega_{\frac{n}{2}}^{i}\cdot&space;F_{odd}(\omega_{\frac{n}{2}}^{i})&space;\end{aligned}&space;)
现在很清楚了,在每次计算 a[0..n-1] 的傅里叶变换F[0..n-1],分别计算出奇 odd[0..n/2-1]偶even[0..n/2-1](可以递归地进行),
那么傅里叶变换为:
![](https://latex.codecogs.com/gif.latex?&space;F[i]&space;=&space;\begin{cases}&space;even[i]+&space;\omega^i&space;\cdot&space;odd[i],&space;\quad&space;i<\frac{n}{2}\\&space;even[i]-&space;\omega^i&space;\cdot&space;odd[i],&space;\quad&space;else&space;\end{cases}&space;)
## 0.4. 代码
下面是 python 实现
一维用 FFT 实现, 不过 只实现了 2 的幂。/ 对于非 2 的幂,用 FFT 实现有点困难,还需要插值,所以我 用![](https://latex.codecogs.com/gif.latex?O(n^2)) 直接实现。
二维的 DFT利用 分离性,直接调用 一维 FFT。
[GitHub](https://github.com/mbinary/algorithm)
```python
import numpy as np
def _fft(a, invert=False):
N = len(a)
if N == 1:
return [a[0]]
elif N & (N - 1) == 0: # O(nlogn), 2^k
even = _fft(a[::2], invert)
odd = _fft(a[1::2], invert)
i = 2j if invert else -2j
factor = np.exp(i * np.pi * np.arange(N // 2) / N)
prod = factor * odd
return np.concatenate([even + prod, even - prod])
else: # O(n^2)
w = np.arange(N)
i = 2j if invert else -2j
m = w.reshape((N, 1)) * w
W = np.exp(m * i * np.pi / N)
return np.concatenate(np.dot(W, a.reshape(
(N, 1)))) # important, cannot use *
def fft(a):
'''fourier[a]'''
n = len(a)
if n == 0:
raise Exception("[Error]: Invalid length: 0")
return _fft(a)
def ifft(a):
'''invert fourier[a]'''
n = len(a)
if n == 0:
raise Exception("[Error]: Invalid length: 0")
return _fft(a, True) / n
def fft2(arr):
return np.apply_along_axis(fft, 0,
np.apply_along_axis(fft, 1, np.asarray(arr)))
def ifft2(arr):
return np.apply_along_axis(ifft, 0,
np.apply_along_axis(ifft, 1, np.asarray(arr)))
def test(n=128):
print('\nsequence length:', n)
print('fft')
li = np.random.random(n)
print(np.allclose(fft(li), np.fft.fft(li)))
print('ifft')
li = np.random.random(n)
print(np.allclose(ifft(li), np.fft.ifft(li)))
print('fft2')
li = np.random.random(n * n).reshape((n, n))
print(np.allclose(fft2(li), np.fft.fft2(li)))
print('ifft2')
li = np.random.random(n * n).reshape((n, n))
print(np.allclose(ifft2(li), np.fft.ifft2(li)))
if __name__ == '__main__':
for i in range(1, 3):
test(i * 16)
```
## 0.5. 参考
- [万寿红老师课件]()
- [一小时学会快速傅里叶变换 Fast Fourier Transform](https://zhuanlan.zhihu.com/p/31584464)
- [快速傅里叶变换FFT算法【详解】](https://www.cnblogs.com/ECJTUACM-873284962/p/6919424.html)

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/* mbinary
#########################################################################
# File : cloneGraph.cpp
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
*/
class Solution {
public:
map<Node*,Node*> st;

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@ -1,3 +1,14 @@
''' mbinary
#########################################################################
# File : fft.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-06-11 12:48
# Description:
#########################################################################
'''
import numpy as np

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''' mbinary
#########################################################################
# File : nega.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-06-02 23:15
# Description:
#########################################################################
'''
def nega(n:int,base=-2:int)->:list:
'''return list of num, the first is the highest digit'''
if base>-2:

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''' mbinary
#########################################################################
# File : test_token_scanner.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
'''
import unittest
from token_scanner import gen_token

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''' mbinary
#########################################################################
# File : genExpr.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
'''
from random import randint

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''' mbinary
#########################################################################
# File : work_dispatch.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
'''
'''
设有n件工作要分配给n个人去完成将工作i分配给第j个人所需费用为c_ij 试设计一个算法为每个人分配1件不同的工作并使总费用达到最小
'''

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/* mbinary
#########################################################################
# File : quickSort.c
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
*/
int partition(int *arr,int i,int j)
{
int pivot = arr[j],p=i,q=j;

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''' mbinary
#########################################################################
# File : codecogs.py
# Author: mbinary
# Mail: zhuheqin1@gmail.com
# Blog: https://mbinary.xyz
# Github: https://github.com/mbinary
# Created Time: 2019-04-16 09:41
# Description:
#########################################################################
'''
import os
import re
import sys