Refactor&Feature: add default value controlling decimal length of output for out_mat_hs 1 case and expose user control keyword out_ndigits

[kirk0830]

Motivation

The precision loss of output is a large problem for any postprocessing, also for restarting tasks. When I print out Hamiltonian matrix and overlap, also wavefunctions, I find wavefunctions' orthonormalities are quite poor if I calculate psi@S@psi.conj().T with numpy directly, this is because of loss of numerical precision when outputting H and S matrix.
There is also other user @QG-phy mention the low efficiency of restarting SCF from a converged, output charge density and has addressed the problem raises from the lack of output precision control.
The formatting of output will be a relatively long-term task because there are many output files, we need to refactor them with formatter library one-by-one.
It is also a pre-requisite of implementation of QO, PR #3236

Results

In INPUT, write:

...
out_mat_hs 1
out_ndigits 10

Then find files in OUT.${suffix} directory:

26 (-4.17692579e-01,0.00000000e+00) (-8.25088205e-01,0.00000000e+00) (1.05161523e-16,0.00000000e+00) (1.61710920e-16,0.00000000e+00) (-1.64819880e-16,0.00000000e+00) (-2.44735352e-17,0.00000000e+00) (1.34155519e-16,0.00000000e+00) (-1.55070663e-16,0.00000000e+00) (5.47256130e-17,0.00000000e+00) (1.00917570e-16,0.00000000e+00) (-1.13651916e-16,0.00000000e+00) (5.67914496e-16,0.00000000e+00) (2.61974586e-17,0.00000000e+00) (-6.18508722e-01,0.00000000e+00) (-4.99914119e-01,0.00000000e+00) (-5.17536714e-17,0.00000000e+00) (-1.57853108e-16,0.00000000e+00) (1.24107296e-16,0.00000000e+00) (2.62584160e-17,0.00000000e+00) (-5.66502583e-18,0.00000000e+00) (1.08152054e-17,0.00000000e+00) (1.54453122e-16,0.00000000e+00) (1.51381726e-16,0.00000000e+00) (-2.40438784e-17,0.00000000e+00) (2.89587216e-16,0.00000000e+00) (-3.24278119e-17,0.00000000e+00)
 (8.12531923e-01,0.00000000e+00) (-2.30443779e-17,0.00000000e+00) (2.34782967e-17,0.00000000e+00) (-7.54155828e-17,0.00000000e+00) (8.28061401e-18,0.00000000e+00) (1.41954682e-16,0.00000000e+00) (-1.59716497e-17,0.00000000e+00) (1.80898335e-16,0.00000000e+00) (-2.50044123e-17,0.00000000e+00) (1.26059687e-17,0.00000000e+00) (2.70914304e-16,0.00000000e+00) (3.10683211e-17,0.00000000e+00) (-4.99914119e-01,0.00000000e+00) (7.25966376e-02,0.00000000e+00) (1.18232785e-16,0.00000000e+00) (-6.52689190e-17,0.00000000e+00) (-6.38592875e-17,0.00000000e+00) (4.95509727e-17,0.00000000e+00) (6.54849468e-17,0.00000000e+00) (-9.23721506e-17,0.00000000e+00) (4.68092003e-17,0.00000000e+00) (6.21112850e-17,0.00000000e+00) (6.83453958e-17,0.00000000e+00) (3.29627550e-16,0.00000000e+00) (-1.12214659e-17,0.00000000e+00)
 (3.51881441e-01,0.00000000e+00) (-4.91312989e-17,0.00000000e+00) (-1.81112573e-17,0.00000000e+00) (3.77316936e-01,0.00000000e+00) (1.73425760e-16,0.00000000e+00) (7.49120190e-17,0.00000000e+00) (-4.45517731e-16,0.00000000e+00) (9.69190884e-18,0.00000000e+00) (-2.88825003e-17,0.00000000e+00) (-5.51985961e-17,0.00000000e+00) (6.67473696e-02,0.00000000e+00) (-1.80584078e-16,0.00000000e+00) (-9.08024701e-17,0.00000000e+00) (-2.92935234e-02,0.00000000e+00) (2.85416140e-17,0.00000000e+00) (1.52781036e-16,0.00000000e+00) (-3.73147925e-03,0.00000000e+00) (8.18842099e-17,0.00000000e+00) (-6.74915646e-17,0.00000000e+00) (-7.78273500e-17,0.00000000e+00) (-5.20730475e-17,0.00000000e+00) (1.70044089e-17,0.00000000e+00) (-1.82281472e-17,0.00000000e+00) (-1.48125029e-01,0.00000000e+00)
 (3.51881441e-01,0.00000000e+00) (-4.62916208e-17,0.00000000e+00) (1.45594801e-16,0.00000000e+00) (3.77316936e-01,0.00000000e+00) (1.37365029e-16,0.00000000e+00) (-1.40458468e-16,0.00000000e+00) (1.62052226e-16,0.00000000e+00) (6.67473696e-02,0.00000000e+00) (-8.15349614e-17,0.00000000e+00) (-1.57507894e-17,0.00000000e+00) (-3.30139082e-17,0.00000000e+00) (8.96634679e-17,0.00000000e+00) (-5.43795234e-17,0.00000000e+00) (-2.92935234e-02,0.00000000e+00) (-4.89517098e-17,0.00000000e+00) (4.62954005e-17,0.00000000e+00) (-3.73147925e-03,0.00000000e+00) (9.07477954e-17,0.00000000e+00) (1.41615836e-18,0.00000000e+00) (-5.26244696e-17,0.00000000e+00) (-1.48125029e-01,0.00000000e+00) (-1.29432616e-18,0.00000000e+00) (3.82316857e-17,0.00000000e+00)
 (3.51881441e-01,0.00000000e+00) (-3.63919253e-17,0.00000000e+00) (6.45032529e-17,0.00000000e+00) (3.77316936e-01,0.00000000e+00) (6.08482530e-17,0.00000000e+00) (6.67473696e-02,0.00000000e+00) (1.71374935e-16,0.00000000e+00) (-3.43767260e-17,0.00000000e+00) (1.66083730e-16,0.00000000e+00) (-6.72883257e-17,0.00000000e+00) (-7.60027927e-17,0.00000000e+00) (-1.48467841e-17,0.00000000e+00) (-5.24211568e-17,0.00000000e+00) (-2.92935234e-02,0.00000000e+00) (1.24683489e-17,0.00000000e+00) (-7.93635464e-17,0.00000000e+00) (-3.73147925e-03,0.00000000e+00) (2.77820072e-18,0.00000000e+00) (-1.48125029e-01,0.00000000e+00) (6.31395324e-17,0.00000000e+00) (3.60500390e-18,0.00000000e+00) (3.60361752e-17,0.00000000e+00)
 (7.14941523e-01,0.00000000e+00) (-5.42236655e-17,0.00000000e+00) (-1.34888298e-16,0.00000000e+00) (8.58192570e-18,0.00000000e+00) (-5.02334398e-17,0.00000000e+00) (2.49614467e-16,0.00000000e+00) (-4.62128473e-17,0.00000000e+00) (3.86652462e-02,0.00000000e+00) (-8.00416200e-17,0.00000000e+00) (9.37541535e-18,0.00000000e+00) (-3.73147925e-03,0.00000000e+00) (-7.94449465e-17,0.00000000e+00) (-8.40254289e-18,0.00000000e+00) (1.38783181e-01,0.00000000e+00) (3.16589511e-17,0.00000000e+00) (-2.22262014e-17,0.00000000e+00) (3.13395124e-17,0.00000000e+00) (-4.20721568e-18,0.00000000e+00) (5.58169274e-17,0.00000000e+00) (-4.81116573e-18,0.00000000e+00) (-6.79510076e-03,0.00000000e+00)
 (7.14941523e-01,0.00000000e+00) (-1.00642765e-16,0.00000000e+00) (5.48125341e-17,0.00000000e+00) (1.03169461e-16,0.00000000e+00) (3.86652462e-02,0.00000000e+00) (-9.66065958e-17,0.00000000e+00) (2.65663406e-17,0.00000000e+00) (-9.07205473e-17,0.00000000e+00) (-5.16071590e-17,0.00000000e+00) (6.71661654e-17,0.00000000e+00) (-3.73147925e-03,0.00000000e+00) (7.58946784e-18,0.00000000e+00) (1.05818380e-16,0.00000000e+00) (1.38783181e-01,0.00000000e+00) (-1.22298181e-16,0.00000000e+00) (-3.52636523e-17,0.00000000e+00) (-1.86483006e-17,0.00000000e+00) (-6.79510076e-03,0.00000000e+00) (4.66206933e-17,0.00000000e+00) (6.33174367e-17,0.00000000e+00)
 (7.14941523e-01,0.00000000e+00) (3.86806594e-17,0.00000000e+00) (3.86652462e-02,0.00000000e+00) (-2.46932344e-17,0.00000000e+00) (-7.03799674e-17,0.00000000e+00) (8.17002016e-17,0.00000000e+00) (1.03568124e-16,0.00000000e+00) (9.32395123e-17,0.00000000e+00) (-4.83553962e-17,0.00000000e+00) (8.13158986e-18,0.00000000e+00) (-3.73147925e-03,0.00000000e+00) (2.88397210e-17,0.00000000e+00) (-7.37275055e-18,0.00000000e+00) (1.38783181e-01,0.00000000e+00) (-6.17983799e-18,0.00000000e+00) (-6.79510076e-03,0.00000000e+00) (3.72220069e-17,0.00000000e+00) (-9.21897730e-19,0.00000000e+00) (4.42354539e-17,0.00000000e+00)
 (1.40993516e+00,0.00000000e+00) (1.15010134e-17,0.00000000e+00) (-1.54947658e-16,0.00000000e+00) (3.71359792e-17,0.00000000e+00) (-8.40767021e-17,0.00000000e+00) (2.08532787e-16,0.00000000e+00) (1.00910889e-16,0.00000000e+00) (-2.01421747e-17,0.00000000e+00) (4.88162835e-17,0.00000000e+00) (7.26348366e-17,0.00000000e+00) (2.39615747e-17,0.00000000e+00) (2.56955681e-17,0.00000000e+00) (5.58634025e-17,0.00000000e+00) (-2.74384397e-01,0.00000000e+00) (8.64989909e-18,0.00000000e+00) (-4.55707282e-19,0.00000000e+00) (-5.38476285e-17,0.00000000e+00) (-6.26431660e-17,0.00000000e+00)
 (1.33944855e+00,0.00000000e+00) (-8.93121863e-17,0.00000000e+00) (-2.34545541e-17,0.00000000e+00) (-5.82293858e-17,0.00000000e+00) (7.13201488e-18,0.00000000e+00) (2.52348586e-17,0.00000000e+00) (6.24271783e-17,0.00000000e+00) (-2.22396904e-17,0.00000000e+00) (1.48125029e-01,0.00000000e+00) (-1.20908021e-16,0.00000000e+00) (3.84894091e-18,0.00000000e+00) (6.79510076e-03,0.00000000e+00) (1.30053425e-17,0.00000000e+00) (1.00268940e-01,0.00000000e+00) (-6.56094768e-17,0.00000000e+00) (5.93871716e-17,0.00000000e+00) (-7.44033810e-18,0.00000000e+00)
 (1.33944855e+00,0.00000000e+00) (-5.83188007e-17,0.00000000e+00) (-3.70723821e-17,0.00000000e+00) (-5.24389639e-17,0.00000000e+00) (3.19636366e-17,0.00000000e+00) (-4.78320538e-18,0.00000000e+00) (1.48125029e-01,0.00000000e+00) (-2.69424265e-17,0.00000000e+00) (-6.36714639e-17,0.00000000e+00) (6.79510076e-03,0.00000000e+00) (-1.10859583e-17,0.00000000e+00) (3.44233834e-18,0.00000000e+00) (4.63055983e-17,0.00000000e+00) (1.00268940e-01,0.00000000e+00) (-3.54398883e-18,0.00000000e+00) (5.58702940e-17,0.00000000e+00)
 (1.40993516e+00,0.00000000e+00) (4.87382758e-17,0.00000000e+00) (3.41632308e-16,0.00000000e+00) (3.30901487e-16,0.00000000e+00) (-4.73762487e-17,0.00000000e+00) (5.08626942e-17,0.00000000e+00) (7.84284430e-17,0.00000000e+00) (-3.00730392e-17,0.00000000e+00) (-4.93311981e-18,0.00000000e+00) (1.71036152e-17,0.00000000e+00) (-3.82876333e-17,0.00000000e+00) (1.03681066e-16,0.00000000e+00) (2.15366567e-17,0.00000000e+00) (-2.74384397e-01,0.00000000e+00) (1.77538153e-17,0.00000000e+00)
 (1.33944855e+00,0.00000000e+00) (2.18195434e-18,0.00000000e+00) (-1.17690146e-16,0.00000000e+00) (1.48125029e-01,0.00000000e+00) (-7.18284598e-18,0.00000000e+00) (-3.51349321e-17,0.00000000e+00) (6.79510076e-03,0.00000000e+00) (5.38848182e-17,0.00000000e+00) (-6.49979255e-17,0.00000000e+00) (-1.28647337e-17,0.00000000e+00) (5.11218258e-17,0.00000000e+00) (-3.28411606e-17,0.00000000e+00) (-1.70151931e-17,0.00000000e+00) (1.00268940e-01,0.00000000e+00)
 (-4.17692579e-01,0.00000000e+00) (-8.25088205e-01,0.00000000e+00) (-1.12834151e-16,0.00000000e+00) (-1.63818356e-16,0.00000000e+00) (-6.54312868e-17,0.00000000e+00) (-3.95901744e-17,0.00000000e+00) (-3.14070592e-16,0.00000000e+00) (-1.37203773e-16,0.00000000e+00) (8.96356933e-17,0.00000000e+00) (7.13233165e-17,0.00000000e+00) (1.24319875e-16,0.00000000e+00) (4.25931690e-16,0.00000000e+00) (1.97114950e-16,0.00000000e+00)
 (8.12531923e-01,0.00000000e+00) (7.13225300e-17,0.00000000e+00) (1.10035006e-16,0.00000000e+00) (-2.95921019e-17,0.00000000e+00) (1.08701856e-16,0.00000000e+00) (-4.27833165e-17,0.00000000e+00) (5.36013037e-17,0.00000000e+00) (5.01340934e-17,0.00000000e+00) (4.68101125e-17,0.00000000e+00) (-4.75727268e-17,0.00000000e+00) (3.56720289e-16,0.00000000e+00) (1.30653375e-17,0.00000000e+00)
 (3.51881441e-01,0.00000000e+00) (1.95187055e-16,0.00000000e+00) (8.24277780e-17,0.00000000e+00) (3.77316936e-01,0.00000000e+00) (-2.74198566e-17,0.00000000e+00) (-6.81745348e-17,0.00000000e+00) (9.39386008e-18,0.00000000e+00) (8.24266209e-17,0.00000000e+00) (-8.92105064e-17,0.00000000e+00) (3.80746374e-18,0.00000000e+00) (-6.67473696e-02,0.00000000e+00)
 (3.51881441e-01,0.00000000e+00) (-4.17423451e-17,0.00000000e+00) (-1.15413057e-16,0.00000000e+00) (3.77316936e-01,0.00000000e+00) (2.23341302e-17,0.00000000e+00) (-4.51088323e-17,0.00000000e+00) (4.52757455e-17,0.00000000e+00) (-6.67473696e-02,0.00000000e+00) (1.64796414e-17,0.00000000e+00) (1.30185055e-16,0.00000000e+00)
 (3.51881441e-01,0.00000000e+00) (-7.93982519e-17,0.00000000e+00) (9.32406738e-17,0.00000000e+00) (3.77316936e-01,0.00000000e+00) (-9.45848274e-18,0.00000000e+00) (-6.67473696e-02,0.00000000e+00) (1.37668978e-16,0.00000000e+00) (-5.64400221e-17,0.00000000e+00) (3.96687125e-17,0.00000000e+00)
 (7.14941523e-01,0.00000000e+00) (-2.21504151e-17,0.00000000e+00) (1.40939321e-16,0.00000000e+00) (-1.98881960e-16,0.00000000e+00) (1.01538681e-16,0.00000000e+00) (1.62903751e-16,0.00000000e+00) (-4.50501840e-17,0.00000000e+00) (-3.86652462e-02,0.00000000e+00)
 (7.14941523e-01,0.00000000e+00) (-5.89756016e-17,0.00000000e+00) (-1.65227024e-16,0.00000000e+00) (5.55209679e-17,0.00000000e+00) (-3.86652462e-02,0.00000000e+00) (-1.19134390e-17,0.00000000e+00) (-3.36093295e-17,0.00000000e+00)
 (7.14941523e-01,0.00000000e+00) (-1.13359425e-16,0.00000000e+00) (-3.86652462e-02,0.00000000e+00) (1.24909963e-16,0.00000000e+00) (-2.59027466e-16,0.00000000e+00) (4.63520479e-19,0.00000000e+00)
 (1.40993516e+00,0.00000000e+00) (7.62437225e-17,0.00000000e+00) (3.52814359e-17,0.00000000e+00) (-1.72563685e-16,0.00000000e+00) (5.59719207e-17,0.00000000e+00)
 (1.33944855e+00,0.00000000e+00) (-5.90975926e-17,0.00000000e+00) (-3.06146007e-17,0.00000000e+00) (6.42012408e-17,0.00000000e+00)
 (1.33944855e+00,0.00000000e+00) (-8.39994268e-17,0.00000000e+00) (1.72443739e-16,0.00000000e+00)
 (1.40993516e+00,0.00000000e+00) (6.44337904e-17,0.00000000e+00)
 (1.33944855e+00,0.00000000e+00)

Reminder

• Have you linked an issue with this pull request?
• Have you noticed possible changes of behavior below or in the linked issue?
• Have you explained the changes of codes in core modules of ESolver, HSolver, ElecState, Hamilt, Operator or Psi? (ignore if not applicable)

Linked Issue

Fix #3373

What's changed?

• change the default precision for printing Hamiltonian and overlap matrix
• add one keyword out_ndigits to control the output precision, default is 8.

Any changes of core modules? (ignore if not applicable)

• no.

[WHUweiqingzhou]

This PR introduces a new parameter out_ndigits, which can tune the precision of output. This parameter only supports the output of Hamil and overlap currently. Thanks for your valuable effort.