Numerical Methods Contents - SAM

3 for n = 2 . ^ ( 4 : 1 3 )
4 A = randn ( n , n ) ;
5
6 t1 = 1000;
7 for k =1:K, t i c ;
8 for j = 1 : n−1, A ( : , j +1) = A ( : , j +1) − A ( : , j ) ; end ;
9 t1 = min ( toc , t1 ) ;
10 end
11 t2 = 1000;
12 for k =1:K, t i c ;
13 for i = 1 : n−1, A( i + 1 , : ) = A( i + 1 , : ) − A( i , : ) ; end ;
14 t2 = min ( toc , t2 ) ;
15 end
16 res = [ res ; n , t1 , t2 ] ;
17 end
18
19 figure ; plot ( res ( : , 1 ) , res ( : , 2 ) , ’ r + ’ , res ( : , 1 ) , res ( : , 3 ) , ’m∗ ’ ) ;
20 xlabel ( ’ { \ b f n } ’ , ’ f o n t s i z e ’ ,14) ;
21 ylabel ( ’ { \ b f runetime [ s ] } ’ , ’ f o n t s i z e ’ ,14) ;
22 legend ( ’A ( : , j +1) = A ( : , j +1) − A ( : , j ) ’ , ’A( i + 1 , : ) = A( i + 1 , : ) − A( i , : ) ’ , . . .
23 ’ l o c a t i o n ’ , ’ northwest ’ ) ;
24 p r i n t −depsc2 ’ . . / PICTURES/ a c c e s s r t l i n . eps ’ ;
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26 figure ; loglog ( res ( : , 1 ) , res ( : , 2 ) , ’ r + ’ , res ( : , 1 ) , res ( : , 3 ) , ’m∗ ’ ) ;
25
1.2 Elementary operations
What you should know from linear algebra:
vector space operations in K m,n
dot product: x,y ∈ K n , n ∈ N: x·y := x H y =
(in MATLAB:
dot(x,y))
(addition, multiplication with scalars)
n∑
¯x i y i ∈ K
tensor product: x ∈ K m ,y ∈ K n , n ∈ N: xy H = ( x i ȳ j
)i=1,...,m ∈ K m,n
j=1,...,n
All are special cases of the matrix product:
⎛ ⎞
n∑
A ∈ K m,n , B ∈ K n,k : AB = ⎝ a ij b jl
⎠
“Visualization” of matrix product:
i=1
j=1
i=1,...,m
l=1,...,k
∈ R m,k . (1.2.1)
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27 xlabel ( ’ { \ b f n } ’ , ’ f o n t s i z e ’ ,14) ;
28 ylabel ( ’ { \ b f runetime [ s ] } ’ , ’ f o n t s i z e ’ ,14) ;
29 legend ( ’A ( : , j +1) = A ( : , j +1) − A ( : , j ) ’ , ’A( i + 1 , : ) = A( i + 1 , : ) − A( i , : ) ’ , . . .
30 ’ l o c a t i o n ’ , ’ northwest ’ ) ;
31 p r i n t −depsc2 ’ . . / PICTURES/ a c c e s s r t l o g . eps ’ ;
m
n
=
m
6
5
A(:,j+1) = A(:,j+1) − A(:,j)
A(i+1,:) = A(i+1,:) − A(i,:)
10 0
A(:,j+1) = A(:,j+1) − A(:,j)
A(i+1,:) = A(i+1,:) − A(i,:)
n
k
runetime [s]
4
3
2
10 −1
10 −2
10 −3
1
10 −4
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
n
10 1 n
runetime [s]
k
=
=
10 −5
10 1 10 2 10 3 10 4
✸
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dot product
tensor product
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