6. Output files¶

After computation is done, numerical results are stored in directory output.

6.1. List of files¶

output
├── spectrum.dat
├── pade.dat
├── lambda_dep.dat
├── find_lambda_opt.dat
├── SV.dat
├── lambda_opt
│   ├── iter.dat
│   ├── x_sv.dat
│   ├── x_tw.dat
│   ├── y_sv.dat
│   └── y_tw.dat
├── lambda
│   ├── lambda_1.00e+00
│   │   ├── iter.dat
│   │   ├── x_sv.dat
│   │   ├── x_tw.dat
│   │   ├── y_sv.dat
│   │   └── y_tw.dat
│   ├── lambda_1.58e+00
│   │   ├── ...
│   │   ├── ...
│   (for all values of lambda)

The four files just below the output directory summarize the results. More detailed results for a given value lambda are stored in directory lambda_* In particular, the directory lambda_opt is for the optimal values of lambda, and the other directories ending with numbers, e.g. lambda_1.00e+00, contain results for the specific value of lambda.

6.2. Details of each files¶

Each files stores list of real numbers separated with a white space. Some comments may be given at the header of files (lines beginning with ‘#’).

spectrum.dat
Real frequency spectrum $\rho(\omega)$ for the optimal value of $\lambda$ . This is the final solution.
- 1st column: $\omega_i$
- 2nd column: $\rho(\omega_i)$
ex.) calculated results for fermion sample outputted as samples/fermion/output/spectrum.dat.
# lambda=1.585e-02 (l=19) -4.00000e+00 0.00000e+00 -3.99200e+00 0.00000e+00 -3.98400e+00 0.00000e+00 -3.97600e+00 0.00000e+00
spectrum.dat
Real frequency spectrum $\rho(\omega)$ calculated by the Pade approximant.
- 1st column: $\omega_i$
- 2nd column: $\rho(\omega_i)$ from G(tau) in filein_G.
- 3rd column: The expectation value of $\rho(\omega_i)$ over the generated NSamplePade samples with independent noise.
- 4th column: The sample standard deviation of $\rho(\omega_i)$ over the generated NsamplePade samples with independent noise.
lambda_dep.dat
Several quantities as a function of lambda.
- 1st column: $\lambda$
- 2nd column: Square error in the SV basis, $\| \bm{y}'-S\bm{x}' \|_2^2$ (the first term in $F$ )
- 3rd column: Square error computed in the original basis, $\| \bm{y}-K\bm{x} \|_2^2$
- 4th column: L0 norm of $\bm{x}'$ , $\| \bm{x}' \|_0$ (the number of non-zero components in $\bm{x}'$ )
- 5th column: L1 norm of $\bm{x}'$ , $\| \bm{x}' \|_1$ (the second term in $F$ )
- 6th column: not used
ex.) calculated results for fermion sample outputted as samples/fermion/output/lambda_dep.dat.
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 our original paper Phys. Rev. E 95, 061302(R) (2017) for details.
- 1st column: $\log_{10}(\lambda)$
- 2nd column: $\log_{10}(f(x)/\chi^2)$
- 3rd column: $\log_{10}(\chi^2)$
- 4th column: $\log_{10}(f(x))$
ex.) calculated results for fermion sample outputted as samples/fermion/output/find_lambda_opt.dat.
# log(x) diff log(y) log(f(x)) [all in log10 scale] 2.00000e+00 1.92865e-16 -9.621340e-01 -9.621340e-01 1.80000e+00 -8.94525e-02 -9.631547e-01 -1.052607e+00 1.60000e+00 -1.79519e-01 -9.635618e-01 -1.143080e+00 1.40000e+00 -2.69822e-01 -9.637317e-01 -1.233554e+00
SV.dat
Singular values of the kernel $K(\tau, \omega)$ .
- 1st column: index $l$ (starting from 0)
- 2nd column: The singular value $S_l$ in descending order
ex.) calculated results for fermion sample outputted as samples/fermion/output/SV.dat.
0 1.07898e+02 1 9.75830e+01 2 6.93652e+01 3 5.27590e+01 4 3.57046e+01
iter.dat
Several quantities as a function of the iteration number. The 2nd–6th quantities should be small.
- 1st column: iteration (starting from 1)
- 2nd column: $\| \bm{x}'(n) - \bm{x}'(n-1) \|_1$
- 3rd column: primary residual error for the constraint $\bm{z}'=\bm{x}', \| \bm{z}'-\bm{x}' \|_2$
- 4th column: dual residual error for the constraint $\bm{z}'=\bm{x}'$ , $|| \bm{z}'(n) - \bm{z}'(n-1) ||_2$
- 5th column: primary residual error for the constraint $\bm{z}=V\bm{x}'$ , $\| \bm{z}-V\bm{x}' \|_2$
- 6th column: dual residual error for the constraint $\bm{z}=V\bm{x}'$ , $\| \bm{z}(n) - \bm{z}(n-1) \|_2$
- 7th column: Square error in the SV basis (the quantity of 2nd column in lambda_dep.dat)
- 8th column: $\| \bm{x} \|_1'$ (the quantity of 4th column in lambda_dep.dat)
- 9th column: sum of spectrum, $\langle V\bm{x}' \rangle$
- 10th column: negative weight in the spectrum $V\bm{x}'$
ex.) calculated results for fermion sample outputted as samples/fermion/output/lambda_opt/iter.dat.
# iter diff(x,x_old) res1_pri res1_dual res2_pri res2_dual RMSE L1_norm sum(x) negative_weight 1 3.854841e-03 2.119714e-03 1.369725e-04 5.289078e-04 4.523259e-04 5.618012e-05 9.396320e-02 1.000546e+00 3.481495e-03 2 2.399056e-03 1.526513e-03 3.273800e-05 4.131063e-04 6.433491e-04 5.652371e-05 9.203257e-02 1.000546e+00 2.717878e-03 3 2.166172e-03 1.024876e-03 1.232067e-05 4.091785e-04 6.751225e-04 5.686296e-05 9.031303e-02 1.000546e+00 3.234967e-03 4 1.425573e-03 8.163075e-04 4.198961e-06 3.879910e-04 4.450555e-04 5.711656e-05 8.963409e-02 1.000546e+00 3.396003e-03
x_sv.dat
spectrum in the SV basis (solution of the minimization problem)
- 1st column: The singular value index $l$
- 2nd column: $x'_l$
- 3rd column: $z'_l$ (must be sparse)
- 4th column: $V^{\rm t}\bm{z}$
ex.) calculated results for fermion sample outputted as samples/fermion/output/lambda_opt/x_sv.dat.
0 -4.61300e-02 -4.61300e-02 -4.61492e-02 1 1.14335e-06 0.00000e+00 -5.52982e-06 2 -6.65757e-03 -6.65757e-03 -6.69485e-03 3 -2.29934e-05 0.00000e+00 -1.16960e-05 4 -3.21754e-03 -3.21754e-03 -3.16677e-03
x_tw.dat
spectrum rho(omega) in the real-frequency domain
- 1st column: $\omega_i$
- 2nd column: $V\bm{x}'$
- 3rd column: $V\bm{z}'$ (final result for a given lambda when non-negativity is not imposed)
- 4th column: $\bm{z}$ (final result for a given lambda when non-negativity is imposed)
ex.) calculated results for fermion sample outputted as samples/fermion/output/lambda_opt/x_tw.dat.
-4.00000e+00 8.98641e-04 3.95646e-03 0.00000e+00 -3.99200e+00 8.73662e-04 3.79795e-03 0.00000e+00 -3.98400e+00 8.41470e-04 3.64070e-03 0.00000e+00 -3.97600e+00 8.02778e-04 3.48473e-03 0.00000e+00 -3.96800e+00 7.58271e-04 3.33005e-03 0.00000e+00
y_sv.dat
input data transformed into the SV basis
- 1st column: The singular value index $l$
- 2nd column: $y'_l$ computed by $U^{\rm t} \bm{y}$
- 3rd column: $S\bm{x}'$
- 4th column: $S\bm{z}'$
ex.) calculated results for fermion sample outputted as samples/fermion/output/lambda_opt/y_sv.dat.
0 -4.97866e+00 -4.97734e+00 -4.97734e+00 1 1.16952e-04 1.11571e-04 0.00000e+00 2 -4.63728e-01 -4.61804e-01 -4.61804e-01 3 -1.28769e-03 -1.21311e-03 0.00000e+00 4 -1.13754e-01 -1.14881e-01 -1.14881e-01

y_tw.dat

input data $G(\tau)$ in the imaginary-time domain

1st column: $\tau_i$

2nd column: $y_i=G(\tau_i)$ (input data)

3rd column: $U S \bm{x}'$

4th column: $U S \bm{z}'$

ex.) calculated results for fermion sample outputted as samples/fermion/output/lambda_opt/y_tw.dat.
0.00000e+00 5.00734e-01 5.00687e-01 5.00294e-01
2.49938e-04 4.90130e-01 4.90315e-01 4.89958e-01
4.99875e-04 4.81350e-01 4.80277e-01 4.79953e-01
7.49813e-04 4.71476e-01 4.70561e-01 4.70268e-01
9.99750e-04 4.61762e-01 4.61154e-01 4.60890e-01

Gnuplot script files are given in the sample directory. In the output directory, type
gnuplot path_to_SpM/samples/plt/*
and in lambda_* directory, type
gnuplot path_to_SpM/samples/plt/lambda_fix/*

6. Output files¶

6.1. List of files¶

6.2. Details of each files¶

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