Operations by matrix disease: 1.000.. - 1 is not equal to zero

Goodmorning everyone, I'm having some trouble with my CLA code. The product delta = P*beta - m_origin must be equal to [0; 0; 0] because P*beta is equal to [1; 2; 3] and m_orign is [1; 2; 3] but the difference between them, in my code, is'nt equal to [0; 0; 0] but to [2.2204e-16; 0; 0]. I've just checked if the first element of the arrays is approximated but this is'nt the case. Thansk to everyone.

n_asset = 3;
C = [1, 0.5, 0; 0.5, 1, 0; 0, 0, 1];
A = ones(n_asset,1)';
P = [C, A'];
m_orig = [1; 2; 3];
beta = [-1.428571428571429; 0.571428571428571; 0.857142857142857; 2.142857142857143];    
delta = P*beta-m_orig 

[Scott MacKenzie]

It is not true that

P*beta is equal to [1; 2; 3]

Here's a little modification I made to your code to illustrate what is going on:

format longg

x = P*beta

y = m_orig

delta = x-y

Output:

x =

0.999999999999999

2

3

y =

1

2

3

delta =

-8.88178419700125e-16

-8.88178419700125e-16

0

Answer 1

[Steven Lord]

Welcome to the world of floating-point arithmetic.

Please try this little experiment. Find something to write with and something to write on (ideally compatible things; pencil and paper not pencil and whiteboard.)

Step 1: Using long division (like you learned in school) divide 1 by 7. Call the result x. You are allowed to write as many decimal places of the result as you want, but only those you explicitly write can be used in step 2. No indicating a set of repeated digits to get "an infinite" number of places.

Step 2: Multiply x by 7. Call the result y.

In exact arithmetic we know (1/7)*7 is exactly 1. But x is not one seventh. It is slightly smaller than one seventh because you rounded off one seventh to fit it into x. Therefore y will not be 1, it will be slightly smaller than 1.

The numbers in your beta variable are not exactly multiples of one seventh.

beta = [-1.428571428571429; 0.571428571428571; 0.857142857142857; 2.142857142857143];
q = 7*beta

q = 4×1

-10.0000 4.0000 6.0000 15.0000

format longg
beta - [-10; 4; 6; 15]/7 % not all 0's

ans = 4×1

-4.44089209850063e-16 -4.44089209850063e-16 -1.11022302462516e-16 0

Those differences between the elements of beta and multiples of one seventh are small, but they have an impact on the calculations performed using beta.

Answer 2

[Leonardo Coccia]

Thanks everyone. I solved:

delta = round(P*beta, 12)-m_origin;