update docs

docs/manual/build/html/_sources/docs/inputfile.rst.txt

@@ -8,22 +8,38 @@
 Input files
 ===============================
 1. Parameter file (param.in)
-
-The variables for which default value is not given are mandatory.
+
+  - The format of the parameter file is the "name = value" pair format.
+
+  - ``#`` means a comment. In other words, a ``#`` character and the subsequent part in a line will be ignored.
+
+  - A blank line is skipped.
+
+  - The variables for which default value is not given are mandatory.
+
+.. code-block::
+
+   # character means comment
+
+   statistics = "fermion"  # stats.
+   beta = 100.0            # inv. temp.
+
+   ### continued ###
+
 
 * INPUT/OUTPUT
 
   .. csv-table::
      :header-rows: 1
      :widths: 1,1,2,4
 
-     Name, Type, Default value, Description
-     statistics, String, ---, Choose "fermion" or "boson".
-     beta, Double, ---, The inverse temperature
-     filein_G, String, Gtau.in, The name of an input file for Green's function.
-     column, Integer, 1, The column number where the values of G(tau) are stored in the filein_G file.
-     fileout_spec, String, spectrum.out, The name of output file of spectrum.
-       
+     Name,          Type,    Default value, Description
+     statistics,    String,  ---,           Choose "fermion" or "boson".
+     beta,          Double,  ---,           The inverse temperature
+     filein_G,      String,  Gtau.in,       The name of an input file for Green's function.
+     column,        Integer, 1,             The column number where the values of G(tau) are stored in the ``filein_G file``.
+     fileout_spec,  String,  spectrum.out,  The name of output file of spectrum.
+     fileout_pade,  String,  pade.out,      The name of output file of spectrum calculated by the Pade approximant.
 
 
 * OMEGA
@@ -66,6 +82,19 @@ The variables for which default value is not given are mandatory.
      maxiteration, Integer,1000,	The maximum number of iterations.
      printlevel, Integer,2,	"0; minimum, 1; moderate, 2; verbose."
 
+* SpM-Pade
+
+  .. csv-table::
+     :header-rows: 1
+     :widths: 1,1,2,4
+
+     Name, Type, Default value, Description
+     PadeEta, Double, 0.0, The weight of the spectrum by the Pade approximant.
+     filein_Gsigma, String,  Gtau.in,       The name of input file for errors of Green's function.
+     column_sigma,  Integer, 1,             The column number where the errors of G(tau) are stored in the ``filein_Gsigma`` file.
+     g_sigma,       Double,  inf,           "The errors of G(tau). If this is inf (default value), the errors will be loaded from a file specified by ``filein_Gsigma``."
+     NSamplePade, Integer, 30, The number of random samples (with noise) will be used to calculate the expected value and the standard deviation  of the spectrum by the Pade approximant.
+
 2. Green's function (Gtau.in)
 
    In SPM, the values of Green's function is only used for calculation,

docs/manual/build/html/_sources/docs/outputfile.rst.txt

@@ -17,6 +17,7 @@ List of files
 
    output
    ├── spectrum.dat
+   ├── pade.dat
    ├── lambda_dep.dat
    ├── find_lambda_opt.dat
    ├── SV.dat
@@ -54,7 +55,7 @@ Some comments may be given at the header of files (lines beginning with '#').
     * 1st column: :math:`\omega_i`
     * 2nd column: :math:`\rho(\omega_i)`
 
-    ex.) calculated results for fermion sample outputted as  ``samles/fermion/spectrum.dat``.
+    ex.) calculated results for fermion sample outputted as  ``samples/fermion/output/spectrum.dat``.
 
     ::
 
@@ -64,37 +65,47 @@ Some comments may be given at the header of files (lines beginning with '#').
        -3.98400e+00 0.00000e+00
        -3.97600e+00 0.00000e+00
 
+* *spectrum.dat*
+
+    Real frequency spectrum :math:`\rho(\omega)` calculated by the Pade approximant.
+
+    * 1st column: :math:`\omega_i`
+    * 2nd column: :math:`\rho(\omega_i)` from G(tau) in ``filein_G``.
+    * 3rd column: The expectation value of :math:`\rho(\omega_i)` over the generated ``NSamplePade`` samples with independent noise.
+    * 4th column: The sample standard deviation of :math:`\rho(\omega_i)` over the generated ``NsamplePade`` samples with independent noise.
+
 * *lambda_dep.dat*
 
     Several quantities as a function of lambda.
 
     * 1st column: :math:`\lambda`
     * 2nd column: Square error in the SV basis, :math:`\| \bm{y}'-S\bm{x}' \|_2^2` (the first term in :math:`F`)
     * 3rd column: Square error computed in the original basis, :math:`\| \bm{y}-K\bm{x} \|_2^2`
-    * 4th column: L1 norm of :math:`\bm{x}'`, :math:`\| \bm{x}' \|_1` (the second term in :math:`F`)
-    * 5th column: not used
+    * 4th column: L0 norm of :math:`\bm{x}'`, :math:`\| \bm{x}' \|_0` (the number of non-zero components in :math:`\bm{x}'`)
+    * 5th column: L1 norm of :math:`\bm{x}'`, :math:`\| \bm{x}' \|_1` (the second term in :math:`F`)
+    * 6th column: not used
 
-    ex.) calculated results for fermion sample outputted as  ``samles/fermion/lambda_dep.dat``.
+    ex.) calculated results for fermion sample outputted as  ``samples/fermion/output/lambda_dep.dat``.
 
     ::
 
-       1.00000e+02 1.09110e-01 1.13080e-01 4.81361e-02 0.00000e+00
-       6.30957e+01 1.08854e-01 1.12824e-01 4.81376e-02 0.00000e+00
-       3.98107e+01 1.08752e-01 1.12722e-01 4.81386e-02 0.00000e+00
-       2.51189e+01 1.08710e-01 1.12679e-01 4.81394e-02 0.00000e+00
-       1.58489e+01 1.08609e-01 1.12578e-01 4.81459e-02 0.00000e+00
+       1.00000e+02 1.09110e-01 1.13080e-01 2 4.81361e-02 0.00000e+00
+       6.30957e+01 1.08854e-01 1.12824e-01 2 4.81376e-02 0.00000e+00
+       3.98107e+01 1.08752e-01 1.12722e-01 2 4.81386e-02 0.00000e+00
+       2.51189e+01 1.08710e-01 1.12679e-01 2 4.81394e-02 0.00000e+00
+       1.58489e+01 1.08609e-01 1.12578e-01 2 4.81459e-02 0.00000e+00
 
 
 * *find_lambda_opt.dat*
 
-    Auxiliary data that are used to determine the optimal value of lambda. See ref.?? for details.
+    Auxiliary data that are used to determine the optimal value of lambda. See our original paper `Phys. Rev. E 95, 061302(R) (2017) <https://doi.org/10.1103/PhysRevE.95.061302>`_ for details.
 
     * 1st column: :math:`\log_{10}(\lambda)`
     * 2nd column: :math:`\log_{10}(f(x)/\chi^2)`
     * 3rd column: :math:`\log_{10}(\chi^2)`
     * 4th column: :math:`\log_{10}(f(x))`
 
-    ex.) calculated results for fermion sample outputted as  ``samles/fermion/find_lambda_opt.dat``.
+    ex.) calculated results for fermion sample outputted as  ``samples/fermion/output/find_lambda_opt.dat``.
 
     ::
 
@@ -111,7 +122,7 @@ Some comments may be given at the header of files (lines beginning with '#').
     * 1st column: index :math:`l` (starting from 0)
     * 2nd column: The singular value :math:`S_l` in descending order
 
-    ex.) calculated results for fermion sample outputted as  ``samles/fermion/find_lambda_opt.dat``.
+    ex.) calculated results for fermion sample outputted as  ``samples/fermion/output/SV.dat``.
 
     ::
 
@@ -138,7 +149,7 @@ Some comments may be given at the header of files (lines beginning with '#').
     * 9th column: sum of spectrum, :math:`\langle V\bm{x}' \rangle`
     * 10th column: negative weight in the spectrum :math:`V\bm{x}'`
 
-    ex.) calculated results for fermion sample outputted as  ``samles/fermion/lambda_opt/iter.dat``.
+    ex.) calculated results for fermion sample outputted as  ``samples/fermion/output/lambda_opt/iter.dat``.
 
     ::
 
@@ -157,7 +168,7 @@ Some comments may be given at the header of files (lines beginning with '#').
     * 3rd column: :math:`z'_l` (**must be sparse**)
     * 4th column: :math:`V^{\rm t}\bm{z}`
 
-    ex.) calculated results for fermion sample outputted as  ``samles/fermion/lambda_opt/x_sv.dat``.
+    ex.) calculated results for fermion sample outputted as  ``samples/fermion/output/lambda_opt/x_sv.dat``.
 
     ::
 
@@ -177,7 +188,7 @@ Some comments may be given at the header of files (lines beginning with '#').
     * 3rd column: :math:`V\bm{z}'` (final result for a given lambda when non-negativity is not imposed)
     * 4th column: :math:`\bm{z}` (**final result** for a given lambda when non-negativity is imposed)
 
-    ex.) calculated results for fermion sample outputted as  ``samles/fermion/lambda_opt/x_tw.dat``.
+    ex.) calculated results for fermion sample outputted as  ``samples/fermion/output/lambda_opt/x_tw.dat``.
 
     ::
 
@@ -197,7 +208,7 @@ Some comments may be given at the header of files (lines beginning with '#').
     * 3rd column: :math:`S\bm{x}'`
     * 4th column: :math:`S\bm{z}'`
 
-    ex.) calculated results for fermion sample outputted as  ``samles/fermion/lambda_opt/y_sv.dat``.
+    ex.) calculated results for fermion sample outputted as  ``samples/fermion/output/lambda_opt/y_sv.dat``.
 
     ::
 
@@ -216,7 +227,7 @@ Some comments may be given at the header of files (lines beginning with '#').
     * 3rd column: :math:`U S \bm{x}'`
     * 4th column: :math:`U S \bm{z}'`
 
-    ex.) calculated results for fermion sample outputted as ``samles/fermion/lambda_opt/y_tw.dat``.
+    ex.) calculated results for fermion sample outputted as ``samples/fermion/output/lambda_opt/y_tw.dat``.
 
     ::
 

docs/manual/build/html/_sources/docs/tutorials.rst.txt

@@ -10,10 +10,15 @@ Tutorials
 
 Sample data are provided both for fermionic and bosonic cases:
 
-    - ``samples/fermion``  # a sample for fermionic spectrum (data in the article)
+    - ``samples/fermion``  # a sample for fermionic spectrum (data in the SpM article)
+    - ``samples/fermion_twopeak``  # another sample for fermionic spectrum (data in the SpM-Pade article)
     - ``samples/boson``  # a sample for bosonic spectrum
 
-Here, an explanation is given for the fermionic case.
+
+Fermionic system
+~~~~~~~~~~~~~~~~~
+
+Here, an explanation is given for the fermionic case (``samples/fermion``).
 
 Script
 ----------------------------------
@@ -103,7 +108,7 @@ In the directory ``samples/fermion/``, just type the command
 
 ::
 
-    $ build_directory/src/SpM.out -i param.in
+    $ build_directory/src/SpM.out param.in
 
 Results
 -------
@@ -167,3 +172,41 @@ Let us look at some graphs below.
 
     The light blue points show the input data :math:`G'_l` transformed into the SVD basis, and red circles are data used in computing :math:`\rho(\omega)` above.
     The blue points show, for comparison, the exact :math:`G'_l` without noise, which is provided in the file ``Gtau.in.sv_basis`` (not output of the ``SpM`` program).
+
+
+Bosonic system
+~~~~~~~~~~~~~~~~~~~
+
+Sample files are available at ``samples/boson``.
+
+Users can deal with a bosonic system by only changing ``statistics`` parameter to ``"boson"`` from ``"fermion"`` in the parameter file.
+
+
+SpM-Pade method
+~~~~~~~~~~~~~~~~~
+
+Sample files are available at ``samples/fermion_twopeak``.
+
+The SpM-Pade method is another SpM AC method, where unphysical oscillation in the SpM spectrum is reduced by using spectra reconstructed by the Pade AC method.
+
+The cost function is
+
+.. math::
+
+   L_\text{SpM-Pade}(\boldsymbol{\rho}) = L_\text{SpM}(\boldsymbol{\rho}) + \frac{\eta}{2} \sum_i w_i \left(\rho(\omega_i) - \rho^\text{Pade}(\omega_i)\right)^2,
+
+where
+
+.. math:: 
+
+   w_i = \left(1.0 + \left(\frac{\sigma^\text{Pade}(\omega_i)}{\rho^\text{Pade}(\omega_i)}\right)^2\right)^{-1}
+
+and :math:`\rho^\text{Pade}(\omega_i)` and :math:`\sigma^\text{Pade}(\omega_i)` are the expectation value and standard deviation of the spectrum reconstructed by the Pade AC method from ``NSamplePade`` Green's functions with independent Gaussian noise.
+The strength of the Gaussian noise is specified by ``g_sigma`` or the pair of ``filein_Gsigma`` and ``column_sigma`` parameters (see :ref:`inputfiles` for details).
+The coefficient of the Pade weight, :math:`eta`, is specified by the ``PadeEta`` parameter.
+If ``PadeEta = 0``, the original SpM method will be used.
+
+.. image:: figs/spectrum_spmpade.png
+
+The blue curve and red curve show the spectrum reconstructed by the SpM method (:math:`\eta=0`) and the SpM-Pade method (:math:`\eta=1`), respectively.
+The black dashed curve shows the exact spectrum.

docs/manual/build/html/_sources/index.rst.txt

@@ -15,13 +15,26 @@ License
 --------------
 This package is distributed under GNU General Public License version 3 (GPL v3).
 
-We kindly ask you to cite the article
+When you publish a publication that includes results obtained using this package,
+we kindly ask you to cite the following articles
 
-    J. Otsuki, M. Ohzeki, H. Shinaoka, K. Yoshimi,
-    "*Sparse modeling approach to analytical continuation of imaginary-time quantum Monte Carlo data*"
-    `Phys. Rev. E 95, 061302(R) (2017) <https://doi.org/10.1103/PhysRevE.95.061302>`_.
+- SpM method
 
-in publications that includes results obtained using this package.
+   J. Otsuki, M. Ohzeki, H. Shinaoka, K. Yoshimi,
+   "*Sparse modeling approach to analytical continuation of imaginary-time quantum Monte Carlo data*"
+   `Phys. Rev. E 95, 061302(R) (2017) <https://doi.org/10.1103/PhysRevE.95.061302>`_.
+
+- SpM package
+
+   K. Yoshimi, J. Otsuki, Y. Motoyama, M. Ohzeki, and H. Shinaoka,
+   "*SpM: Sparse modeling tool for analytic continuation of imaginary-time Green’s function*"
+   `Comput. Phys. Commun. 244, 319-323 (2019) <https://www.sciencedirect.com/science/article/abs/pii/S0010465519302103>`_.
+
+- SpM-Pade method (if you use)
+
+   Y. Motoyama, K. Yoshimi, and J. Otsuki,
+   "*Robust analytic continuation combining the advantages of the sparse modeling approach and Pade approximation*",
+   arXiv:2109.XXXXX.
 
 
 Contents