linear algebra - Accelerate the calculation of inv(X'*X)*Q*inv(X'*X) in Matlab? -

i have calculate newey-west standard errors large multiple regression models.

the final step of calculation obtain

nwse = sqrt(diag(n.*inv(x'*x)*q*inv(x'*x))); 

this file exchange contribution implements

nwse = sqrt(diag(n.*((x'*x)\q/(x'*x))));   

this looks reasonable, in case (5000x5000 sparse q , x'*x) it's far slow needs (about 30secs, have repeat 1 million different models). ideas how make line faster?

please note need diagonal, not entire matrix , both q , (x'*x) positive-definite.

i believe can save lot of computation time explicitly doing lu factorization, [l, u, p, q] = lu(x'*x); , use factors when doing calculations. also, since x constant 100 models, pre-calculating x'*x save time.

note in case, time demanding operation might sqrt-function.

% constant every 100 models or so: m = x'*x; [l, u, p, q] = lu(m);  % now, guess should quite bit faster (i might have messed order): nwse = sqrt(diag(n.*(q * ( u \ (l \ (p * q))) * q * (u \ (l \ p))))); 

the first 2 terms commonly used:

l lower triangular matrix

u upper triangular matrix

now, p , q bit more uncommon.

p row permutation matrix used obtain numerical stability. there not performance gain in doing [l, u, p] = lu(m) compared [l, u] = lu(m) sparse matrices.

q offers significant performance gain. q column permutation matrix used reduce amount of fill when doing factorization.

note [l, u, p, q] = lu(m) valid syntax sparse matrices.

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as why using full pivoting described above should faster:

try following see purpose of column permutation matrix q. easier work elements aligned around diagonal.

s = sprand(100,100,0.01); [l, u, p] = lu(s); spy(l) figure spy(u) 

now, compare this:

[ll, uu, pp, qq] = lu(s); spy(ll); figure spy(uu); 

^{unfortunately, don't have matlab here right now, can't guarantee put arguments in correct order, think it's correct.}