Inconsistent results in Matrix-vec​tor-multip​lication?

Dear all,

I was analyzing the results of a matlab code where I found a strange inconsistency in a matrix and vector multiplication operation.

I have a matrix as:

M= [ -0.499999999999999,  -0.866025403784439 , 0;   0.866025403784439,   -0.499999999999999, 0;  0 ,   0 ,  1.000000000000000]; 

And a vector as:

V = [ 1.351250000000000;   2.340433653727400;   0];

Now if a simply multiply them I receive the following vector:

RES1 = [-2.702499999999961, 2.565324651410645e-14, 0];

However, if I execute the following command (which should actually reproduce the same results as RES1(2) due to matrix and vector multiplication rule):

RES2 = M(2,:) * V

I receive a different result as expected:

RES2 = 2.575717417130363e-14;

This means that RES1(2) != RES2 which should not happen actually from a mathematical point of view (if I an not wrong…).

Is this because of somewhat precision error?

Thank you for your help,

Kind regards

[Guillaume]

How did you obtain RES1?

M= [ -0.499999999999999,  -0.866025403784439 , 0;   0.866025403784439,   -0.499999999999999, 0;  0 ,   0 ,  1.000000000000000]; 
V = [ 1.351250000000000;   2.340433653727400;   0];

>> format longg
>> RES1 = M*V
RES1 =
         -2.70249999999996
      2.57571741713036e-14
                         0
                         
>> RES2 = M(2, :) * V
RES2 =
      2.57571741713036e-14

>> sum(M(2, :) .* V.')
ans =
      2.57571741713036e-14

Answer 1

[Matt J]

The order of the summation steps in the matrix multiplication may be slightly different when you execute the computation in two different ways, resulting in small floating point approximation differences.

[Guillaume]

Yes, also note that considering the magnitude of the numbers that 2e-14 is for all intent and purpose just 0. See that the two values summed to obtain that result are more or less the opposite of each other:

>> M(2, :) .* V'
ans =
          1.17021682686372          -1.1702168268637      0