Pade_approximants

Aim

The purpose is to perform an analytical continuation of a (Green's) function in the complex plane.

What is contained?

• Both a Python and a Fortran program.
• Parameter file pade.par (with standard settings) for the Fortran program, in the tests folder.
• Test models, located in the tests folder.

Test models

• Sm7
• betheU0
• haldane_model
• two-poles_wc1_dw0.5

Each test model folder contains:

• Very short description of the test model, in file README.md.
• Exact solution on the real axis, in file exact.dat.
• Input data file pade.in.
• Parameter file pade.par for the Fortran program.

Python notebook

• The Python notebook is short and easy to use.
• The analytical continuation can be done by either Beach's matrix formulation, Thiele's recursive algorithm or a nonlinear Least Square minimization.
• The matrix formulation by Beach is here calculated using double precision, which usually is too little.
• (Green's) function data somewhere in the complex plane (e.g. at Matsubara frequency points) is used as input.
• (Green's) function data somewhere else in the complex plane is the output.

Fortran program

• The analytical continuation uses Beach's matrix formulation.
• A more low-level control of the inversion routines in Beach's matrix formulation is possible. For example, both double and quadruple precision inversion routines exist.
• The Fortran program prints more information during execution than the Python script.

How to use

• A parameter input file pade.par has to exist in the simulation directory.
• A input file pade.in has to exist in the simulation directory. Four columns are expected: Re[zin], Im[zin], Re[f(zin)] and Im[f(zin)] where zin are the input points and f(zin) the corresponding function values.
• Execute the binary pade_approximants
• The following output files will be generated:
  • pade_info, gives information about the performed continuations
  • pade_fout_all, all the Pade approximants, evaluated on the zout points
  • pade_fout, the Pade approximant average
  • pade_A, first column and -1/pi times the third column of pade_fout

Parameters in pade.par

Below follows a description about each line in the input parameter file pade.par.

1. Works as a header (this line can be left empty).

2. Settings for the output real-axis mesh. Three parameters: wmin, wmax, Nw

3. Distance above the real-axis for the output mesh.

4. Lowest index in the file pade.in to use for the continuations. Three parameters: nminstart, nminfinish, nminstep. For example, input: 0 6 2 means the first, third, fifth and seventh point in pade.in are all used as starting points for the continuations. Mirror symmetry $f(z)^* = f(z^*)$ can be enforced by using mirror points, which are added with nmin < 0.

5. Specify how many input points to use in the continuations. Three parameters: Mmin, Mmax, Mstep.

6. Specify how many Pade approximant coefficients to use in the continuations. Three parameters: Nmin, Nmax, Nstep.

7. Whether to only consider continuations with equally many input points as coefficients. Fortran boolean.

8. Select analytical continuation method. 0: Beach's matrix formulation

9. If to print the poles of the Pade approximants. Fortran boolean.

10. The numerical precision in the inversion routine. 64: double precision, 128: quadruple precision.

11. Select algorithm for solving matrix equation.

    • 0: Let LAPACK solve equation $A x = b$ in LS sense
    • 1: Let LAPACK solve normal equation: $A^{\dagger} A x = A^{\dagger} b$
    • 2: explicit SVD

12. Exclude Pade approximants having positive imaginary part on at least one output point.

13. Averaging criteria parameters for the Pade approximants on the real axis. Two parameters: c1v, c2v. A continuation is included in the average if it fulfills two criteria. The two criteria sort out Pade approximant outliers at the real-axis. Each Pade approximant, on the real-axis, is compared with the other Pade approximants. The differences to all the other Pade approximants are summed, giving a distance measure for each continuation.

    • criterion 1: Continuation has a distance measure smaller than c1v times the average distance measure.
    • criterion 2: Continuation belongs to the 100 c2v procent lowest continuations (in terms of the distance measure).

Compile

• Change directory to the fortran folder.
• Copy the example Makefile: Makefile_example to Makefile and adjust it to fit to the current machine.
• The program requires standard LAPACK to be installed on the computer.
• Access to modified LAPACK library using quadruple precision is required. Change directory to the quad/zgels/zgels_quad folder, copy the example Makefile: Makefile_example to Makefile and adjust it to fit to the current machine. Then run make to generate the library libzgelsquad.a.
• Go back to the fortran folder and run make. The binary pade_approximants should be created.

Possible improvements

• The parameters below are on the 2-do list to implement, but not of great importance.
  • 0 # Impose spectral symmetry. 0: no, 1: even, 2: odd
  • .false. # Shift real part of input data by Re[f(z_inf)] before the analytical continuation.
  • 128 # Determining the precision of all variables, except those involved in the inversion. This would enable use of only double precision variables, which would give a speed-up (at the prize of some precision).
• Any suggestion, idea or problem can conveniently be written in the github Issues section.

Notes

• Comparing ZGELSD with ZGELS (using double precision), the spectra shows more features using ZGELS. ZGELSD sometimes gives too smooth spectra, with many features washed out or absent. ZGELS assumes full rank and gives more spectral features. Usually Beach's matrix is rank deficient. For rank deficient matrices it is common to also minimize the norm of the solution vector. ZGELSD uses SVD. Using rcond=-1 and rcond=10^(-40) gives similarly smooth spectra for the Sm7 test model, but the estimated effective rank became very different. Hence, the different output spectra seems originate from the different algorithms in ZGELS and ZGELSD. In the folder tests/Sm7/solving_Beach_system spectra for varous setups are shown to illustrate the differences.
• Speed: For the Sm7 test model a simululation took 44 seconds with quadruple precision in the inversion routines and 15 seconds with double precision in the inversion routines, using the Fortran program. The same setup with the python notebook, but using only double precision numerics, took 1.5 seconds. The slow performance in the Fortran case is because of a few conversions inbetween quadruple and double variables and some calculations still done in quadruple precision. But this enables comparisons of analytical continuation spectra where only the inversion precision is varied. Small note, the simulation time using double precision in Fortran was independent of if used ZGELS or ZGELSD.
• Support for MPACK's arbitrary precsion is removed to simplify compilation. For high precision routines, instead use e.g. the Mathematica software.