1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
|
\documentclass[border=10pt,varwidth]{standalone}
\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{times}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{lipsum}
\usepackage{amscd}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{textcomp}
\usepackage{txfonts}
\usepackage[all]{xy}
\usepackage{paralist}
\usepackage[colorlinks=true]{hyperref}
\usepackage{array}
\usepackage{tikz}
\usepackage{slashed}
\usepackage{pdfpages}
\usepackage{cite}
\usepackage{url}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{tikz}
\usetikzlibrary{arrows,matrix,positioning}
\usetikzlibrary{overlay-beamer-styles}
\usetikzlibrary{matrix.skeleton}
\usetikzlibrary{automata,positioning}
\usepackage{listings}
\usepackage{multirow}
\usepackage{color}
\begin{document}
$
A=
\begin{bmatrix}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{bmatrix},
B=
\begin{bmatrix}
B_{11} & B_{12}\\
B_{21} & B_{22}
\end{bmatrix},
C=
\begin{bmatrix}
C_{11} & C_{12}\\
C_{21} & C_{22}
\end{bmatrix}
$
\medskip
$
A \cdot B = C
$
\medskip
$
C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\
C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\
C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\
C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
$
\medskip
\begin{math}
\begin{aligned}
\text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
\text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\
\text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
\text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\
\text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\
\text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12})) \\
\text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
\end{aligned}
\end{math}
\medskip
\begin{math}
\begin{aligned}
C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
C_{21} &= \text{II} + \text{IV} \\
C_{12} &= \text{III} + \text{V}\\
C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
\end{aligned}
\end{math}
\medskip
\begin{math}
\begin{aligned}
C_{11} &= \text{II} + \text{IV} \\
C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22})C_{21} \\
C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\
C_{11} &= A_{11}B_{11} + A_{12}B_{21}
\end{aligned}
\end{math}
\section{Winograd}
$
x_1 y_1 + x_2 y_2 = (x_1 +y_2)(y_1 + x_2)-x_1 x_2 - y_1 y_2
$
$
x = (x_1, \cdots, x_n), y=(y_1, \cdots, y_n)
$
\[
\xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j}
\]
\[
\eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j}
\]
\[
\langle x,y \rangle =
\begin{cases}
\displaystyle \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if $n$ is even}\\
\displaystyle \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if $n$ is odd}
\end{cases}
\]
\end{document}
|
