How to use mldivide ("\") instead of inv in this formulae

Hi,

I would like to use the mldivide "\" instead of inv in this formulae, but I am not sure how I should do it? I want to replace it for a matter of speed.

    R = alpha * inv(B) * Q' * inv(I-((1-alpha)*P)) * Q;

The sizes of the matrices are the following:

n = total number of elements in a matrix 
m = total number of unique elements in a matrix

P = n x n 
I = n x n (identity matrix) 
Q = n x m 
B = n x m (diagonal matrix, sum of the rows in Q) 
R = m x m

Any help would be greatly appreciated!

Thank you!

[Jean-Philippe]

Suggested Action (from Matlab help)

Instead of multiplying by the inverse, use matrix right division (/) or matrix left division (\). That is:

Replace inv(A)*b with A\b

Replace b*inv(A) with b/A

[Shrinivas Iyengar]

is a typo, right? It would be a $m \times m$ matrix.

Answer 1

[Walter Roberson]

http://www.mathworks.com/help/techdoc/ref/mldivide.html

If A is a square matrix, A\B is roughly the same as inv(A)*B, except it is computed in a different way.

Now your I-((1-alpha)*P) is certainly square, so you should be able to replace inv(I-((1-alpha)*P)) * Q by

(I-((1-alpha)*P)) \ Q

(but see below.)

When we scan further left to inv(B) we encounter a problem: you have B as n x m, not as square, so inv(B) does not exist, which leads to questions about the correctness of your original formula. You can have B on the left side of the "\" operator (the meaning is well defined) but you should reconsider your formula to see whether it makes sense. You might need to put () around portions, for example.

You need to be cautious because MATLAB processes operations of equal priority from left to right. Your formula

alpha * inv(B) * Q' * inv(I-((1-alpha)*P)) * Q

would be interpreted as

(((alpha * inv(B)) * Q') * inv(I-((1-alpha)*P))) * Q

Algebraically, with matrix multiplication, (A * B) * C is the same as A * (B * C), but numerically they are not the same. (alpha * (inv(B) * Q')) differs from ((alpha * inv(B)) * Q') proportional to eps(alpha^max(m,n)) [I think it is -- I could be wrong.] You thus need to consider the order you do the calculations in. For example, perhaps instead of using X * ((I-((1-alpha)*P)) \ Q) you should instead use (X / (I-((1-alpha)*P))) * Q

[Jean-Philippe]

Thank you Walter, I will try your suggestions and I will let you know my result.

[Jean-Philippe]

I had a previous formulae very similar that I resolved like this:

R = alpha * inv(speye(n)-(1-alpha)*P) * Q;

R = (speye(n)-(1-alpha)*P) \ (alpha*Q);

[Jean-Philippe]

Here is what I got:

R = inv(B) * Q' * ((speye(n)-(1-alpha)*P) \ (alpha*Q));

[Walter Roberson]

Is B still n x m ? If so then inv(B) does not exist. pinv(B) would exist but might not be what you want.

Perhaps you want B \ Q' instead of inv(B) * Q'