I'm using CMKL 9.0 TT routines of staggered cosine transforms. But currently can not get results I want. The details are:
The matrix of finite difference of 1d three-point approximations Poisson equation with Neumann boundary conditions is:
Eigenvalues of this matrix is 2-2*cos((i-1)*pi/n), and the Eigenvectors are corresponding to staggered cosine transform. But I found in the example of TT routines d_trig_tforms_bvp.f90 and d_trig_tforms_slae.f90, the eigenvalues seems be given to (2.0D0*dsin(0.25D0*pi*(2*k-1)/n))**2. Am I wrong?
And manuals of different version provide different description to staggered cosine transforms. Should I followthe latest version released in march 2009 to use CMKL 9.0?
Yes, the first eigenvalue is zero. This is something concerning singularity of poisson equation with Neumann boundary condition. That means if u is a solution, u+c is also solution. c is a constant.
With best regards.