Large matrix solver algorithm (mldivide implementation)

Hello freelancers, and math whizzes I need a matrix solver algorithm figured out ( A.x=y ) The matrix is 3396*3396 sized and the solution is obtained by applying a left division on a vector of 3396 values in matlab its smoothly and efficiently done with the '\' (backslash operator) aka. 'mldivide' X = M[3396*3396] \ Y[3396] but now i need an implementation to be used on a web platform using PHP so i need an efficient algorithm that solves this problem in a time span of < 15 seconds and it should be efficient memory wise. The algorithm should take advantage of the matrix structure ( as described in the attached figure ) further details would be given uppon contact