The Matrix Inverse

The Inverse Hilbert Matrix

    The formula for the elements of the inverse Hilbert matrix   of order  n  is known to be

        

which can be expressed using binomial coefficients

         .

When exact computations are needed these formulas should be used instead of using a subroutine or built in procedure for computing the inverse of  .  
 

Application to Continuous Least Squares Approximation

    The continuous least squares approximation to a function   on the interval [0,1] for the set of functions    can solved by using the normal equations

(1)           for   .

Where the inner product is .  Solve the linear system (1) for the coefficients and construct the approximation function
        
        .