Problems "GSL"

  1. Using GSL special functions and gnuplot, reproduce the (first) illustration plot of the Ai(x) and Bi(x) functions from the Wikipedia article "Airy functions".
  2. Consider the following system of linear equations in the matrix form, 
    

    
[ 6.13  -2.90  5.86 ]
 [x0]   [6.23]

    

    
[ 8.08  -6.31  -3.89 ]
 [x1]  =  [5.37]

    

    
[ -4.36  1.00  0.19 ]
 [x2]   [2.29]

    1. Solve it using GSL.

      Hint: you can solve linear systems using one of the many methods, for example,

      • Householder solver for linear systems, 
        gsl_linalg_HH_solve
      • LU-decomposition, 
        gsl_linalg_LU_decomp
gsl_linalg_LU_solve
      • QR-decomposition, 
        gsl_linalg_QR_decomp
gsl_linalg_QR_solve
        See also:

        GSL manual → Linear Algebra → Householder solver for linear systems;
        GSL manual → Linear Algebra → LU Decomposition;
        GSL manual → Linear Algebra → QR Decomposition;
        GSL manual → Linear Algebra → Linear Algebra Examples.

    2. Check that you have the correct solution by multiplying the original matrix and the solution and comparing with the right-hand-side.
  3. (Optional) Compute the eigenvalues and eigenvectors of 4-th order Hilbert matrix, Hij = 1/(i + j + 1).