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    "<center><h2> Performance Tuning Example</h2></center>\n",
    "<p>\n",
    "A Basic tuning program showing memory allocation. (In memory of Paul Fink, the original author.)"
   ]
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      "iter  ener      err \n",
      "  1  1.0000000  0.3034969 \n",
      "  2  0.9217880  0.0572816 \n",
      "  3  0.9171967  0.0156207 \n",
      "  4  0.9169062  0.0040152 \n",
      "  5  0.9168841  0.0011571 \n",
      "  6  0.9168824  0.0003081 \n",
      "  7  0.9168823  0.0000879 \n",
      "  8  0.9168823  0.0000238 \n",
      "  9  0.9168823  0.0000067 \n",
      " 10  0.9168823  0.0000018 \n",
      " 11  0.9168823  0.0000005 \n",
      " time =  0:00:00.777358\n"
     ]
    }
   ],
   "source": [
    "\"\"\" From \"COMPUTATIONAL PHYSICS\" & \"COMPUTER PROBLEMS in PHYSICS\"\n",
    "    by RH Landau, MJ Paez, and CC Bordeianu (deceased).\n",
    "    Copyright R Landau, Oregon State Unv, MJ Paez, Univ Antioquia, \n",
    "    C Bordeianu, Univ Bucharest, 2021. \n",
    "    Please respect copyright & acknowledge our work.\"\"\"\n",
    "\n",
    "# Tune.pynb: Basic tuning program showing memory allocation\n",
    " \n",
    "import datetime;  from numpy import zeros;  from math import (sqrt, pow)\n",
    "\n",
    "Ldim = 251;         iter = 0;          step = 0.   \n",
    "diag  = zeros((Ldim, Ldim), float);    coef = zeros( (Ldim), float)\n",
    "sigma = zeros((Ldim), float);          ham = zeros( (Ldim, Ldim), float)\n",
    "t0 = datetime.datetime.now()                            # Initialize time\n",
    "\n",
    "for i in range(1, Ldim):                             # Set up Hamiltonian\n",
    "    for j in range(1, Ldim):\n",
    "        if (abs(j - i) >10): ham[j, i] = 0.  \n",
    "        else : ham[j, i] = pow(0.3, abs(j - i) )    \n",
    "    ham[i,i] = i ;            coef[i] = 0.; \n",
    "    \n",
    "coef[1] = 1.;                 err = 1.;         iter = 0 ;\n",
    "print(\"iter  ener      err \")\n",
    "while (iter  < 15 and err > 1.e-6):       # Compute current E & normalize\n",
    "    iter = iter + 1; ener = 0. ; ovlp = 0.;\n",
    "    for i in range(1, Ldim):\n",
    "        ovlp = ovlp + coef[i]*coef[i]   \n",
    "        sigma[i] = 0.\n",
    "        for j in range(1, Ldim): sigma[i] = sigma[i] + coef[j]*ham[j][i]\n",
    "        ener = ener + coef[i]*sigma[i]  \n",
    "    ener = ener/ovlp\n",
    "    for i in range(1, Ldim):\n",
    "        coef[i] = coef[i]/sqrt(ovlp)    \n",
    "        sigma[i] = sigma[i]/sqrt(ovlp)\n",
    "    err = 0.;   \n",
    "    for i in range(2, Ldim):                                     # Update\n",
    "        step = (sigma[i]  -  ener*coef[i])/(ener - ham[i, i])\n",
    "        coef[i] = coef[i] + step\n",
    "        err = err +  step*step   \n",
    "    err = sqrt(err) \n",
    "    print(\" %2d  %9.7f  %9.7f \"%(iter, ener, err))\n",
    "    \n",
    "delta_t = datetime.datetime.now()  -  t0                   # Elapsed time\n",
    "print(\" time = \", delta_t)"
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