Calculus.ipynb

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    "# Calculus\n",
    "\n",
    "**Problem:** Calculate [derivative](https://en.wikipedia.org/wiki/Derivative) and [integral](https://en.wikipedia.org/wiki/Integral) of a curve, given the vectors `x` and `y`."
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    "⎕IO←0\n",
    "x←0.25×⍳100\n",
    "f←⍟1+x ⋄ g←10-⍨(x*0.5)+x÷2 ⋄ h←10×1○x÷5\n",
    "]plot (f g h h)x"
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    "## Derivative\n",
    "\n",
    "The derivative will be calculated as a simple ratio of the difference between two consequtive values. The differences will be calculated using the n-wise reduction operator ([dyadic `/`](https://help.dyalog.com/latest/index.htm#Language/Primitive%20Operators/Reduce%20N%20Wise.htm))."
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    "dy←2-/f ⋄ dx←2-/x ⋄ d←dy÷dx\n",
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    "We better define a function:"
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     },
     "execution_count": 74,
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   ],
   "source": [
    "D←÷⍥(2-/⊢)\n",
    "]plot ((f D x)(g D x)(h D x)) x"
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    "### Noise\n",
    "\n",
    "The problem of calculating the derivative is more complicated if the function is not smooth. For example:"
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   "source": [
    "n←f+0.1-⍨0.2×?0⍴⍨≢f\n",
    "]plot (n (n D x)) x"
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    "In such a case, it would be convenient to first perform some smoothing of the function, for example calculating a running average:"
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   "source": [
    "ns xs←10(+/÷⊣)¨n x\n",
    "]plot (ns (ns D xs)) xs"
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    "S←(+/÷⊣)¨\n",
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    "## Integral\n",
    "\n",
    "To calculate the value of the integral we need to add, each step, the increment of the x value multiplied by the current value of the function at that point. Several options can be used to estimate the value of the function (previous point, next point, average, ...). Here, the average of the value at the previous and next point is used."
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    "I←+\\(¯2-/⊢)×2(+/÷⊣)⊣\n",
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       "<desc>Points follow ...</desc>\n",
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       " <g stroke-width=\"3\" stroke=\"Maroon\" stroke-linejoin=\"round\" stroke-linecap=\"round\" stroke-dasharray=\"9 12\" fill=\"none\" >\n",
       "  <path d=\"M3602 657L3566 687 3531 717 3495 747 3460 777 3424 807 3388 837 3353 867 3317 896 3282 926 3246 955 3211 984 3175 1014 3139 1043 3104 1072 3068 1100 3033 1129 2997 1157 2962 1186 2926 1214 2890 1242 2855 1270 2819 1298 2784 1326 2748 1353 2713 1381 2677 1408 2641 1435 2606 1463 2570 1489 2535 1516 2499 1543 2463 1570 2428 1596 2392 1623 2357 1649 2321 1675 2286 1701 2250 1727 2214 1753 2179 1778 2143 1804 2108 1829 2072 1854 2037 1879 2001 1904 1965 1928 1930 1953 1894 1977 1859 2001 1823 2025 1788 2049 1752 2072 1716 2095 1681 2118 1645 2141 1610 2163 1574 2185 1538 2207 1503 2229 1467 2250 1432 2272 1396 2293 1361 2313 1325 2334 1289 2354 1254 2373 1218 2393 1183 2412 1147 2430 1112 2449 1076 2466 1040 2484 1005 2501 969 2518 934 2535 898 2551 863 2567 827 2582 791 2597 756 2612 720 2626 685 2639 649 2652 613 2664 578 2676 542 2686 507 2696 471 2705 436 2713 400 2720\" />\n",
       " </g>\n",
       "<desc>Reset to original origin</desc>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "]plot ((n I x)(0,⊃10 I/⍤S n x)) x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9caf1806",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Dyalog APL",
   "language": "apl",
   "name": "dyalog-kernel"
  },
  "language_info": {
   "file_extension": ".apl",
   "mimetype": "text/apl",
   "name": "APL"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}