In this thesis three methods LMGS, TQR and GR are applied to solve an inverseeigenvalue problem. We list the numerical results and compare the accuracy of the computed Jacobi matrix $T$ and the associated orthogonal matrix $Q$, wherethe columns of $Q^T$ are the eigenvectors of $T$. In the application of this inverse eigenvalue problem, the Fourier coefficients of $h(x)=e^x$ relative to the orthonormal polynomials associatedwith $T$ are evaluated, and these values are used to compute the least squarescoefficients of $h$ relative to the Chebyshev polynomials. We list thesenumerical results and compare them as our conclusion.