High-resolution Interferometric Diagnostics for Ultrashort Pulses

3.5 Main algorithmfor j = 1,2,...,k . I then vertically concatenate the equations for all the shears into a single setB (k ) ¯φ(k ) = ¯F (k ) (3.22)where B (k ) and ¯F (k ) are the vertical concatenations of {W j G j } and {W j ¯Γ j } respectively, for all j =1,2,...,k :⎡ ⎤ ⎡ ⎤W 1 G 1W 1 ¯Γ 1B (k ) W 2 G 2=, ¯F (k ) W 2 ¯Γ 2=. (3.23)..⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦W k G k W k ¯Γ kThe number of rows of B (k ) and ¯F (k ) is equal to the total number of “observations”kP = KN− |C j |, (3.24)j =1whilst B has N columns. Equation (3.22) can then be solved in the least-squares sense via a suitablelinear algebra method. The normal equation matrix (B (k ) ) T B (k ) is sparse — an element nmis nonzero only if frequencies ω n and ω m are connected by one or more shears. It is thereforebanded, with nonzero elements positioned above or below the main diagonal by no more thanmax{|C j |}. Equation (3.22) can therefore be solved by efficient LU factorisation and backsubstitutionroutines for banded matrices. For example, the LAPACK routines are DGBTRF and DGBTRS.For the numerical and experimental results presented below, I used MATLAB, achieving a computationtime of 40 ms for a 1000-point inversion on a laptop computer.Elementary linear algebra shows that the covariance, expressed as a matrix, is equal to theinverse of the normal equations matrix:V (k ) = (B (k ) ) T B (k ) −1. (3.25)77