8-SageCalculus-checkpoint.ipynb - mathsoftware - A course about LaTeX and SageMath

8-SageCalculus-checkpoint.ipynb (214050B)
      1 {
      2  "cells": [
      3   {
      4    "cell_type": "markdown",
      5    "metadata": {},
      6    "source": [
      7     "# Symbolic expressions\n",
      8     "\n",
      9     "**Reference:** [[1](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html)]\n",
     10     "\n",
     11     "Last time we saw the basics of symbolic expressions:\n",
     12     "* How to define and manipulate symbolic expressions\n",
     13     "* How to introduce new variables (in the Mathematical sense) with `var()`\n",
     14     "* How to solve equations and inequalities\n",
     15     "* Some of the Mathematical constants that are included in Sage, and how to approximate them using `n()`\n",
     16     "\n",
     17     "Here are some examples to remind you of these basic things:"
     18    ]
     19   },
     20   {
     21    "cell_type": "code",
     22    "execution_count": 2,
     23    "metadata": {},
     24    "outputs": [
     25     {
     26      "name": "stdout",
     27      "output_type": "stream",
     28      "text": [
     29       "[\n",
     30       "x == -sqrt(-pi),\n",
     31       "x == sqrt(-pi)\n",
     32       "]\n",
     33       "[\n",
     34       "z == -sqrt(pi + x^2),\n",
     35       "z == sqrt(pi + x^2)\n",
     36       "]\n",
     37       "[[y < -2], [y > 1]]\n",
     38       "2*pi + e is approximately 9.00146713563863\n"
     39      ]
     40     }
     41    ],
     42    "source": [
     43     "var('y', 'z') # Define new variables (x is already defined by Sage)\n",
     44     "f = x^2 + pi\n",
     45     "g = y^2 + y - 2 > 0\n",
     46     "print( solve(f==0, x) )\n",
     47     "print( solve(z^2 - f, z) )\n",
     48     "print( solve(g, y) )\n",
     49     "print( 2*pi + e, \"is approximately\", n(2*pi + e) )"
     50    ]
     51   },
     52   {
     53    "cell_type": "markdown",
     54    "metadata": {},
     55    "source": [
     56     "Now we will see some more details about solving equations and manipulating their solutions."
     57    ]
     58   },
     59   {
     60    "cell_type": "markdown",
     61    "metadata": {},
     62    "source": [
     63     "## Solving equations and inequalities\n",
     64     "\n",
     65     "**Reference** [[1](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html)] for the details of `solve()` and `find_root()`, [[2](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/relation.html#solving)] for examples.\n",
     66     "\n",
     67     "Other than equations and inequalities, we can also solve systems: it is enough to give Sage a list of expressions and a list of variables with respect to which we want to solve. For example the system\n",
     68     "\n",
     69     "\\begin{align*}\n",
     70     "    \\begin{cases}\n",
     71     "        x + y = 2 \\\\\n",
     72     "        2x - y = 6\n",
     73     "    \\end{cases}\n",
     74     "\\end{align*}\n",
     75     "\n",
     76     "Can be solved as"
     77    ]
     78   },
     79   {
     80    "cell_type": "code",
     81    "execution_count": 40,
     82    "metadata": {},
     83    "outputs": [
     84     {
     85      "data": {
     86       "text/plain": [
     87        "[[x == (8/3), y == (-2/3)]]"
     88       ]
     89      },
     90      "execution_count": 40,
     91      "metadata": {},
     92      "output_type": "execute_result"
     93     }
     94    ],
     95    "source": [
     96     "solve([x+y == 2, 2*x - y == 6], [x,y])"
     97    ]
     98   },
     99   {
    100    "cell_type": "markdown",
    101    "metadata": {},
    102    "source": [
    103     "**Exercise.** Find the intersection of the circle of radius $2$ centered in the origin and the parabula of equation $y=x^2-2x^2+1$."
    104    ]
    105   },
    106   {
    107    "cell_type": "markdown",
    108    "metadata": {},
    109    "source": [
    110     "### The set of solutions\n",
    111     "\n",
    112     "One would expect the result of `solve()` to be a list of solutions, but it is actually a list of expressions (technically it is not a list but a different type of Python collection, but this is not so important)"
    113    ]
    114   },
    115   {
    116    "cell_type": "code",
    117    "execution_count": 37,
    118    "metadata": {},
    119    "outputs": [
    120     {
    121      "data": {
    122       "text/plain": [
    123        "x == -3"
    124       ]
    125      },
    126      "execution_count": 37,
    127      "metadata": {},
    128      "output_type": "execute_result"
    129     }
    130    ],
    131    "source": [
    132     "solutions = solve(x^2-9 == 0, x)\n",
    133     "solutions[0] # This is the expression 'x == -3'"
    134    ]
    135   },
    136   {
    137    "cell_type": "markdown",
    138    "metadata": {},
    139    "source": [
    140     "To read the actual solution without the `x ==` part you can use the `rhs()` or `lhs()` functions, which can be applied to any expression containing a relation operator (like `==`, `<`, `>=`...) and return the *right hand side* and *left hand side* of the expression, respectively"
    141    ]
    142   },
    143   {
    144    "cell_type": "code",
    145    "execution_count": 41,
    146    "metadata": {},
    147    "outputs": [
    148     {
    149      "name": "stdout",
    150      "output_type": "stream",
    151      "text": [
    152       "rhs: 2\n",
    153       "lhs: x\n"
    154      ]
    155     }
    156    ],
    157    "source": [
    158     "f =  x == 2\n",
    159     "print(\"rhs:\", f.rhs())\n",
    160     "print(\"lhs:\", f.lhs())"
    161    ]
    162   },
    163   {
    164    "cell_type": "markdown",
    165    "metadata": {},
    166    "source": [
    167     "When you solve an inequality or a system, the set of solutions can be more complicated to describe. In this case the result is a list containing lists of expressions that have to be `True` at the same time. It is easier to explain with an example:"
    168    ]
    169   },
    170   {
    171    "cell_type": "code",
    172    "execution_count": 38,
    173    "metadata": {},
    174    "outputs": [
    175     {
    176      "name": "stdout",
    177      "output_type": "stream",
    178      "text": [
    179       "Simple inequality: [[x < -3], [x > 3]]\n",
    180       "System of inequalities:\n",
    181       " [\n",
    182       "[3 < x, x < 6],\n",
    183       "[x < -3]\n",
    184       "]\n"
    185      ]
    186     }
    187    ],
    188    "source": [
    189     "print(\"Simple inequality:\", solve(x^2-9 > 0, x))\n",
    190     "print(\"System of inequalities:\\n\", solve([x^2-9 > 0, x < 6], x))"
    191    ]
    192   },
    193   {
    194    "cell_type": "markdown",
    195    "metadata": {},
    196    "source": [
    197     "In the last example (system of inequalities), Sage is telling us that the system\n",
    198     "\\begin{align*}\n",
    199     "    \\begin{cases}\n",
    200     "        x^2-9 > 9 \\\\\n",
    201     "        x < 6\n",
    202     "    \\end{cases}\n",
    203     "\\end{align*}\n",
    204     "has two solutions:\n",
    205     "* $x$ is between $3$ and $6$;\n",
    206     "* $x$ is less than $-3$.\n",
    207     "\n",
    208     "Since in Sage (and in Python) expressions can have at most on relational operator like `<`, the first solution requires two expressions to be described. Hence the \"list of lists\".\n"
    209    ]
    210   },
    211   {
    212    "cell_type": "markdown",
    213    "metadata": {},
    214    "source": [
    215     "**Exercise.** In the first exercise you were asked to solve a system of equations, but some of its solutions were complex numbers. Select only the real solutions and print them as pairs $(x,y)$."
    216    ]
    217   },
    218   {
    219    "cell_type": "markdown",
    220    "metadata": {},
    221    "source": [
    222     "When solving a system of equations (not inequalities), you can use the option `solution_dict=True` to have the solutions arranged as a *dictionary*, which is a type of Python collection that we did not treat in this course"
    223    ]
    224   },
    225   {
    226    "cell_type": "code",
    227    "execution_count": 44,
    228    "metadata": {},
    229    "outputs": [
    230     {
    231      "data": {
    232       "text/plain": [
    233        "[{x: 8/3, y: -2/3}]"
    234       ]
    235      },
    236      "execution_count": 44,
    237      "metadata": {},
    238      "output_type": "execute_result"
    239     }
    240    ],
    241    "source": [
    242     "solve([x+y == 2, 2*x - y == 6], [x,y], solution_dict=True)"
    243    ]
    244   },
    245   {
    246    "cell_type": "markdown",
    247    "metadata": {},
    248    "source": [
    249     "### Alternative method for real roots: `find_root()`\n",
    250     "\n",
    251     "The `solve()` method is very useful when solving *symbolic* equations, for example when you have two variables and you want to solve for one of them in terms of the other. However, it does not always find explicit solutions.\n",
    252     "\n",
    253     "When you want to find an explicit, even if approximate, solution, it can be better to use `find_root()`. This function works *numerically*, which means that it finds an approximation of the root. It only works for real solutions and you need to specify an interval where you want the root to be searched:"
    254    ]
    255   },
    256   {
    257    "cell_type": "code",
    258    "execution_count": 52,
    259    "metadata": {},
    260    "outputs": [
    261     {
    262      "name": "stdout",
    263      "output_type": "stream",
    264      "text": [
    265       "Using solve():\n",
    266       " [\n",
    267       "x == -e^x + 10\n",
    268       "]\n",
    269       "Using find_root(): 2.070579904980303\n"
    270      ]
    271     }
    272    ],
    273    "source": [
    274     "f = e^x + x - 10\n",
    275     "print(\"Using solve():\\n\", solve(f, x))\n",
    276     "print(\"Using find_root():\", f.find_root(0,100))"
    277    ]
    278   },
    279   {
    280    "cell_type": "markdown",
    281    "metadata": {},
    282    "source": [
    283     "## Evaluating functions\n",
    284     "\n",
    285     "If an expression contains only one variable you can evaluate it easily, even if it is not a function."
    286    ]
    287   },
    288   {
    289    "cell_type": "code",
    290    "execution_count": 21,
    291    "metadata": {},
    292    "outputs": [
    293     {
    294      "name": "stdout",
    295      "output_type": "stream",
    296      "text": [
    297       "1\n",
    298       "y + 3 > (y + 3)^2\n"
    299      ]
    300     }
    301    ],
    302    "source": [
    303     "var('y')\n",
    304     "f = x^2-3\n",
    305     "g = x > x^2\n",
    306     "\n",
    307     "print(f(2))\n",
    308     "print(g(3+y))"
    309    ]
    310   },
    311   {
    312    "cell_type": "markdown",
    313    "metadata": {},
    314    "source": [
    315     "If an expression contains more than one variable, you can specify a value for each of them and they will be substituted in alphabetic order. You can also specify a value only for some of the variables."
    316    ]
    317   },
    318   {
    319    "cell_type": "code",
    320    "execution_count": 38,
    321    "metadata": {},
    322    "outputs": [
    323     {
    324      "name": "stdout",
    325      "output_type": "stream",
    326      "text": [
    327       "-2 == 0\n",
    328       "3*y == 2\n"
    329      ]
    330     }
    331    ],
    332    "source": [
    333     "var('y','z')\n",
    334     "\n",
    335     "f = y*z^2 - y == z\n",
    336     "print(f(2, 0))\n",
    337     "print(f(z=2))"
    338    ]
    339   },
    340   {
    341    "cell_type": "markdown",
    342    "metadata": {},
    343    "source": [
    344     "## Symbolic computations\n",
    345     "\n",
    346     "Sage can understand and simplify symbolic expressions such as sums (finite or infinite) and products. In the following cell, we compute the following sums using the [`sum()`](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.sum) function:\n",
    347     "\n",
    348     "\\begin{align*}\n",
    349     "    \\begin{array}{llcc}\n",
    350     "        (1) & \\sum_{k=0}^nk                 &=&\\frac{n^2+n}{2}\\\\\n",
    351     "        (2) & \\sum_{k=0}^nk^4               &=&\\frac{6n^5+15n^4+10n^3-n}{30}\\\\\n",
    352     "        (3) & \\sum_{k=0}^n\\binom nk         &=& 2^n\\\\\n",
    353     "        (4) & \\sum_{k=0}^\\infty \\frac1{k^2} &=& \\frac{\\pi^2}{6}\n",
    354     "    \\end{array}\n",
    355     "\\end{align*}"
    356    ]
    357   },
    358   {
    359    "cell_type": "code",
    360    "execution_count": 22,
    361    "metadata": {},
    362    "outputs": [
    363     {
    364      "name": "stdout",
    365      "output_type": "stream",
    366      "text": [
    367       "(1) 1/2*n^2 + 1/2*n\n",
    368       "(2) 1/5*n^5 + 1/2*n^4 + 1/3*n^3 - 1/30*n\n",
    369       "(3) 2^n\n",
    370       "(4) 1/6*pi^2\n"
    371      ]
    372     }
    373    ],
    374    "source": [
    375     "var('k', 'n') # Remember to declare all variables\n",
    376     "\n",
    377     "s = []\n",
    378     "s.append(  sum(k, k, 0, n)  )\n",
    379     "s.append(  sum(k^4, k, 0, n)  )\n",
    380     "s.append(  sum(binomial(n,k), k, 0, n) )\n",
    381     "s.append(  sum(1/k^2, k, 1, infinity)  )\n",
    382     "\n",
    383     "for i in range(len(s)):\n",
    384     "    print(\"({}) {}\".format(i+1, s[i]))"
    385    ]
    386   },
    387   {
    388    "cell_type": "markdown",
    389    "metadata": {},
    390    "source": [
    391     "An alternative notation is `expression.sum(k, a, b)`. There is an analogous [`prod()`](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.prod) for products."
    392    ]
    393   },
    394   {
    395    "cell_type": "markdown",
    396    "metadata": {},
    397    "source": [
    398     "Sometimes Sage tries to keep an expression in its original form without expanding out sums and products. To change this behavior you can use the [`expand()`](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.expand) function:"
    399    ]
    400   },
    401   {
    402    "cell_type": "code",
    403    "execution_count": 30,
    404    "metadata": {},
    405    "outputs": [
    406     {
    407      "name": "stdout",
    408      "output_type": "stream",
    409      "text": [
    410       "(x + 1)^2 - (x - 1)^2\n",
    411       "4*x\n"
    412      ]
    413     }
    414    ],
    415    "source": [
    416     "f = (x+1)^2 - (x-1)^2\n",
    417     "print(f)\n",
    418     "print(f.expand())"
    419    ]
    420   },
    421   {
    422    "cell_type": "markdown",
    423    "metadata": {},
    424    "source": [
    425     "### The Symbolic Ring\n",
    426     "**Reference:** [[3](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/ring.html)]\n",
    427     "\n",
    428     "The symbolic expressions that we have seen so far live in a ring called *symbolic ring* and denoted by `SR` in Sage. This ring works like the ring `ZZ` of integers or `RR` of reals numbers. In particular, you can define matrices and other objects using it as a \"basis\"."
    429    ]
    430   },
    431   {
    432    "cell_type": "code",
    433    "execution_count": 45,
    434    "metadata": {},
    435    "outputs": [
    436     {
    437      "name": "stdout",
    438      "output_type": "stream",
    439      "text": [
    440       "-b*c + a*d\n",
    441       "[(-a, 2)]\n"
    442      ]
    443     }
    444    ],
    445    "source": [
    446     "var('a', 'b', 'c', 'd')\n",
    447     "\n",
    448     "M = matrix([[a,b], [c,d]])\n",
    449     "print(M.determinant())\n",
    450     "\n",
    451     "polring.<x> = SR[]\n",
    452     "f = x^2 + 2*a*x + a^2\n",
    453     "print(f.roots())"
    454    ]
    455   },
    456   {
    457    "cell_type": "markdown",
    458    "metadata": {},
    459    "source": [
    460     "**Exercise.** Compute the eigenvalues of the matrix\n",
    461     "\\begin{align*}\n",
    462     "\\begin{pmatrix}\n",
    463     "\\cos \\alpha & \\sin \\alpha\\\\\n",
    464     "-\\sin\\alpha & \\cos \\alpha\n",
    465     "\\end{pmatrix}\n",
    466     "\\end{align*}"
    467    ]
    468   },
    469   {
    470    "cell_type": "markdown",
    471    "metadata": {},
    472    "source": [
    473     "# Calculus\n",
    474     "**Reference:** [[4](https://doc.sagemath.org/html/en/reference/calculus/index.html)] for an overview, but most functions are described in [[1](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html)]"
    475    ]
    476   },
    477   {
    478    "cell_type": "markdown",
    479    "metadata": {},
    480    "source": [
    481     "## Limits and series\n",
    482     "\n",
    483     "**References:** [[5](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/calculus.html#sage.calculus.calculus.limit)] for limits, [[6](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.series)] for series\n",
    484     "\n",
    485     "You can compute limits"
    486    ]
    487   },
    488   {
    489    "cell_type": "code",
    490    "execution_count": 54,
    491    "metadata": {},
    492    "outputs": [
    493     {
    494      "name": "stdout",
    495      "output_type": "stream",
    496      "text": [
    497       "1\n",
    498       "0\n"
    499      ]
    500     }
    501    ],
    502    "source": [
    503     "f = sin(x)/x\n",
    504     "# print(f(0)) # This one gives an error\n",
    505     "print( f.limit(x=0) )\n",
    506     "\n",
    507     "print( (e^(-x)).limit(x=infinity) )"
    508    ]
    509   },
    510   {
    511    "cell_type": "markdown",
    512    "metadata": {},
    513    "source": [
    514     "**Exercise.** Compute the constant $e$ using a limit."
    515    ]
    516   },
    517   {
    518    "cell_type": "markdown",
    519    "metadata": {},
    520    "source": [
    521     "You can also specify a direction for the limit. If you don't, Sage assumes that you want to take a two-sided limit."
    522    ]
    523   },
    524   {
    525    "cell_type": "code",
    526    "execution_count": 55,
    527    "metadata": {},
    528    "outputs": [
    529     {
    530      "name": "stdout",
    531      "output_type": "stream",
    532      "text": [
    533       "und\n",
    534       "1\n",
    535       "-1\n"
    536      ]
    537     }
    538    ],
    539    "source": [
    540     "f = abs(x)/x  # 1 if x>0, -1 if x<0\n",
    541     "print( f.limit(x=0) ) # undefined\n",
    542     "print( f.limit(x=0, dir=\"+\") )\n",
    543     "print( f.limit(x=0, dir=\"-\") )"
    544    ]
    545   },
    546   {
    547    "cell_type": "markdown",
    548    "metadata": {},
    549    "source": [
    550     "There is also the alternative notation `limit(f, x, dir)` which does the same as `f.limit(x, dir)`."
    551    ]
    552   },
    553   {
    554    "cell_type": "markdown",
    555    "metadata": {},
    556    "source": [
    557     "You can also compute series expansions up to any order. **Watch out:** the notation uses `==` instead of `=` as `limit()` does."
    558    ]
    559   },
    560   {
    561    "cell_type": "code",
    562    "execution_count": 56,
    563    "metadata": {},
    564    "outputs": [
    565     {
    566      "name": "stdout",
    567      "output_type": "stream",
    568      "text": [
    569       "1 + 1*x + 1/2*x^2 + Order(x^3)\n",
    570       "(-2) + 1*x + 1*x^2 + (-1/6)*x^3 + (-1/12)*x^4 + 1/120*x^5 + 1/360*x^6 + Order(x^7)\n",
    571       "1*(x - 1) + (-1/2)*(x - 1)^2 + Order((x - 1)^3)\n"
    572      ]
    573     }
    574    ],
    575    "source": [
    576     "f = e^x\n",
    577     "g = sin(x) - 2*cos(x)\n",
    578     "h = log(x)\n",
    579     "\n",
    580     "print(f.series(x==0, 3))\n",
    581     "print(g.series(x==0, 7))\n",
    582     "print(h.series(x==1, 3))"
    583    ]
    584   },
    585   {
    586    "cell_type": "markdown",
    587    "metadata": {},
    588    "source": [
    589     "## Derivatives\n",
    590     "**References:** [[7](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.derivative)] and [[8](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/functional.html#sage.calculus.functional.derivative)] for derivatives, [[9](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/functions.html#sage.calculus.functions.jacobian)] for the Jacobian matrix and [[10](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.hessian)] for the Hessian."
    591    ]
    592   },
    593   {
    594    "cell_type": "markdown",
    595    "metadata": {},
    596    "source": [
    597     "When computing derivatives, you need to specify with respect to which variables you want to derive, except in case there is only one."
    598    ]
    599   },
    600   {
    601    "cell_type": "code",
    602    "execution_count": 57,
    603    "metadata": {},
    604    "outputs": [
    605     {
    606      "name": "stdout",
    607      "output_type": "stream",
    608      "text": [
    609       "8*y^3\n",
    610       "6*x^2 - 1\n"
    611      ]
    612     }
    613    ],
    614    "source": [
    615     "var('y')\n",
    616     "print( (x^2+2*y^4).derivative(y) ) # Alternative: derivative(f, y)\n",
    617     "print( (2*x^3-x+2).derivative() )"
    618    ]
    619   },
    620   {
    621    "cell_type": "markdown",
    622    "metadata": {},
    623    "source": [
    624     "You can also compute higher order derivatives:"
    625    ]
    626   },
    627   {
    628    "cell_type": "code",
    629    "execution_count": 58,
    630    "metadata": {},
    631    "outputs": [
    632     {
    633      "name": "stdout",
    634      "output_type": "stream",
    635      "text": [
    636       "6*x\n",
    637       "84*x^5*y + 10*y^4 + 24*x^2*y\n",
    638       "1680*x^3 + 48\n"
    639      ]
    640     }
    641    ],
    642    "source": [
    643     "print( (x^3).derivative(x, x) ) # Same as (x^3).derivative(x, 2)\n",
    644     "\n",
    645     "f = x^7*y^2 + x^4*y^2 - 2*x^3 + x^2*y^5 + y + 2\n",
    646     "print( f.derivative(x, x, y) )    # Twice in x, once in y\n",
    647     "print( f.derivative(x, 4, y, 2) ) # 4 times in x, twice in y"
    648    ]
    649   },
    650   {
    651    "cell_type": "markdown",
    652    "metadata": {},
    653    "source": [
    654     "Jacobian and Hessian matrices are also easy to compute:"
    655    ]
    656   },
    657   {
    658    "cell_type": "code",
    659    "execution_count": 59,
    660    "metadata": {},
    661    "outputs": [
    662     {
    663      "name": "stdout",
    664      "output_type": "stream",
    665      "text": [
    666       "[-2*x + 2*y        2*x]\n",
    667       "[         0      3*y^2]\n",
    668       "[     y + 1      x + 1] \n",
    669       "\n",
    670       "[         2   -4*y + 1]\n",
    671       "[  -4*y + 1 -4*x + 6*y]\n"
    672      ]
    673     }
    674    ],
    675    "source": [
    676     "f = (-x^2 + 2*x*y, y^3, x+y+x*y)\n",
    677     "print( jacobian(f, [x,y]), \"\\n\" )\n",
    678     "\n",
    679     "g = x^2 + x*y + y^3 -2*x*y^2 -3\n",
    680     "print( g.hessian() )"
    681    ]
    682   },
    683   {
    684    "cell_type": "markdown",
    685    "metadata": {},
    686    "source": [
    687     "*Note:* the notation `f.jacobian([x,y])` is also valid, but only if you specify that `f` is vector by declaring it as `f = vector([...])`."
    688    ]
    689   },
    690   {
    691    "cell_type": "markdown",
    692    "metadata": {},
    693    "source": [
    694     "## Integrals\n",
    695     "**References:** [[11](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/integration/integral.html)] for symbolic integration and [[12](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/integration.html)] for numerical methods.\n",
    696     "\n",
    697     "You should remember from high school or from your first calculus/analysis course that derivatives are easy, but integrals are hard.\n",
    698     "When using a computer software to solve your integrals, you have two choices:\n",
    699     "\n",
    700     "1. You can try to compute a primitive function exactly, and then (if you are computing a definite integral) substitute the endpoints of your integration interval to get the result. We can call this *symbolic integration*.\n",
    701     "2. You can get an *approximated* result with a *numerical method*. This method always gives some kind of result, but it cannot be used to compute indefinite integrals.\n",
    702     "\n",
    703     "Sage can do both of these things, although people that work in numerical analysis and use often the second method tend to prefer other programs, such as Matlab (or its open-source clone Octave)."
    704    ]
    705   },
    706   {
    707    "cell_type": "markdown",
    708    "metadata": {},
    709    "source": [
    710     "### Symbolic integration\n",
    711     "\n",
    712     "Symbolic integrals work more or less like derivatives. You must specify an integration variable, but the endpoints of the integration interval are optional. If they are not given you get an indefinite integral."
    713    ]
    714   },
    715   {
    716    "cell_type": "code",
    717    "execution_count": 60,
    718    "metadata": {},
    719    "outputs": [
    720     {
    721      "name": "stdout",
    722      "output_type": "stream",
    723      "text": [
    724       "1/2*x^2 - cos(x)\n",
    725       "0\n",
    726       "-1/2*a^2 + 1/2*b^2 + cos(a) - cos(b)\n"
    727      ]
    728     }
    729    ],
    730    "source": [
    731     "var('a', 'b')\n",
    732     "f = x + sin(x)\n",
    733     "print( f.integral(x) ) # Alternative: integral(f, x)\n",
    734     "print( f.integral(x, -10, 10) )\n",
    735     "print( f.integral(x, a, b) )"
    736    ]
    737   },
    738   {
    739    "cell_type": "markdown",
    740    "metadata": {},
    741    "source": [
    742     "Your endpoints can also be $\\pm\\infty$:"
    743    ]
    744   },
    745   {
    746    "cell_type": "code",
    747    "execution_count": 61,
    748    "metadata": {},
    749    "outputs": [
    750     {
    751      "name": "stdout",
    752      "output_type": "stream",
    753      "text": [
    754       "1\n",
    755       "sqrt(pi)\n"
    756      ]
    757     }
    758    ],
    759    "source": [
    760     "print( integral(e^(-x),   x, 0,         infinity) )\n",
    761     "print( integral(e^(-x^2), x, -infinity, infinity) )"
    762    ]
    763   },
    764   {
    765    "cell_type": "markdown",
    766    "metadata": {},
    767    "source": [
    768     "The last function is also an example of an integral that perhaps you might want to compute numerically. In fact:"
    769    ]
    770   },
    771   {
    772    "cell_type": "code",
    773    "execution_count": 65,
    774    "metadata": {},
    775    "outputs": [
    776     {
    777      "name": "stdout",
    778      "output_type": "stream",
    779      "text": [
    780       "1/2*sqrt(pi)*erf(x)\n",
    781       "1/2*sqrt(pi)*erf(2) - 1/2*sqrt(pi)*erf(1)\n"
    782      ]
    783     }
    784    ],
    785    "source": [
    786     "print( integral(e^(-x^2), x) )\n",
    787     "print( integral(e^(-x^2), x, 1, 2) )"
    788    ]
    789   },
    790   {
    791    "cell_type": "markdown",
    792    "metadata": {},
    793    "source": [
    794     "Here `erf(x)` denotes the [error function](https://en.wikipedia.org/wiki/Error_function)."
    795    ]
    796   },
    797   {
    798    "cell_type": "markdown",
    799    "metadata": {},
    800    "source": [
    801     "### Numerical integration\n",
    802     "\n",
    803     "In order to get an explicit value for the computations above, we can use a *numerical* method.\n",
    804     "\n",
    805     "The word \"numerical\" does not have much to do with numbers, but it refers to the fact that we are trying to compute explicit results rather than symbolic or algebraic ones. [Numerical analysis](https://en.wikipedia.org/wiki/Numerical_analysis) is the branch of mathematics that studies methods to approximate computations over the real or complex numbers. With these methods there is usually a trade-off between speed and precision.\n",
    806     "\n",
    807     "The Sage function [`numerical_integral()`](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/integration.html#sage.calculus.integration.numerical_integral) takes as a parameter a real-valued one-variable function and the integration endpoints, and it returns both an approximate value for the integral and an error estimate."
    808    ]
    809   },
    810   {
    811    "cell_type": "code",
    812    "execution_count": 40,
    813    "metadata": {},
    814    "outputs": [
    815     {
    816      "data": {
    817       "text/plain": [
    818        "(0.13525725794999466, 1.5016572202374808e-15)"
    819       ]
    820      },
    821      "execution_count": 40,
    822      "metadata": {},
    823      "output_type": "execute_result"
    824     }
    825    ],
    826    "source": [
    827     "numerical_integral(e^(-x^2), 1, 2)"
    828    ]
    829   },
    830   {
    831    "cell_type": "markdown",
    832    "metadata": {},
    833    "source": [
    834     "The result above means, in symbols\n",
    835     "\\begin{align*}\n",
    836     "\\int_1^2 e^{-x^2}\\mathrm dx = 0.13525725794999466 \\pm 1.5016572202374808\\times 10^{-15}\n",
    837     "\\end{align*}\n",
    838     "\n",
    839     "There is also a [`monte_carlo_integral()`](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/integration.html#sage.calculus.integration.monte_carlo_integral) method for functions with more than one variable."
    840    ]
    841   },
    842   {
    843    "cell_type": "markdown",
    844    "metadata": {},
    845    "source": [
    846     "**Exercise.** Compute the area of the ellipse of equation $y^2+\\left(\\frac x3\\right)^2=1$."
    847    ]
    848   },
    849   {
    850    "cell_type": "markdown",
    851    "metadata": {},
    852    "source": [
    853     "## Differential equations\n",
    854     "**Reference:** [[13](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/desolvers.html)]\n",
    855     "\n",
    856     "A [differential equation](https://en.wikipedia.org/wiki/Differential_equation) is an equation involving an unknwon function and its derivatives. They can be of two kinds: *ordinary* differential equations ([ODE](https://en.wikipedia.org/wiki/Ordinary_differential_equation)) and *partial* differential equations ([PDE](https://en.wikipedia.org/wiki/Partial_differential_equation)). The latter involve multivariate functions and their partial derivatives.\n",
    857     "\n",
    858     "Differential equations are in general hard to solve *exactly* (or *symbolically*): even a simple equation of the form $f'(x)=g(x)$, where $g(x)$ is someknown function, requires solving the integral $\\int g(x)\\mathrm{d}x$ in order to find $f$, which as we know is not always easy!\n",
    859     "\n",
    860     "Theoretical results on differential equations usually ensure the existence and/or uniquess of a solution under certain conditions, but in general they do not give a way to solve them. There exits many methods to find approximate solutions, and some of them are implemented in Sage as well (see [[13](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/desolvers.html)]). However we will focus on the simple ODEs that can be solved exactly.\n",
    861     "\n",
    862     "Let's start with a simple example. Let's find all functions $f(x)$ such that $f'(x)=f(x)$. In order to do so, we need to use the `function()` construct, which allows us to define an \"unknwon\" function inside Sage, like we define variables with `var()`."
    863    ]
    864   },
    865   {
    866    "cell_type": "code",
    867    "execution_count": 4,
    868    "metadata": {},
    869    "outputs": [
    870     {
    871      "data": {
    872       "text/plain": [
    873        "_C*e^x"
    874       ]
    875      },
    876      "execution_count": 4,
    877      "metadata": {},
    878      "output_type": "execute_result"
    879     }
    880    ],
    881    "source": [
    882     "var('x')\n",
    883     "function('f')\n",
    884     "equation = derivative(f(x)) == f(x)\n",
    885     "desolve(equation, f(x)) # f is the unknown function"
    886    ]
    887   },
    888   {
    889    "cell_type": "markdown",
    890    "metadata": {},
    891    "source": [
    892     "As you can expect, they are all the functions $Ce^x$ for some constant $C$. The constant $C$ plays the same role as the constant in the solution of an integral, but in this case Sage writes it explicitly.\n",
    893     "\n",
    894     "We can also specify *initial conditions* for our function. For example we can impose that $f(0)=3$ as follows:"
    895    ]
    896   },
    897   {
    898    "cell_type": "code",
    899    "execution_count": 5,
    900    "metadata": {},
    901    "outputs": [
    902     {
    903      "data": {
    904       "text/plain": [
    905        "3*e^x"
    906       ]
    907      },
    908      "execution_count": 5,
    909      "metadata": {},
    910      "output_type": "execute_result"
    911     }
    912    ],
    913    "source": [
    914     "desolve(equation, f(x), (0,3))"
    915    ]
    916   },
    917   {
    918    "cell_type": "markdown",
    919    "metadata": {},
    920    "source": [
    921     "You can also solve *second order* equations, that is equations where the second derivative also appears. In this case if you want to specify an initial condition you should write the triple of values $(x_0, f(x_0), f'(x_0))$."
    922    ]
    923   },
    924   {
    925    "cell_type": "code",
    926    "execution_count": 6,
    927    "metadata": {},
    928    "outputs": [
    929     {
    930      "data": {
    931       "text/plain": [
    932        "-1/2*I*sqrt(2)*sqrt(pi)*integrate(erf(1/2*I*sqrt(2)*x)*e^(-1/2*x^2), x)"
    933       ]
    934      },
    935      "execution_count": 6,
    936      "metadata": {},
    937      "output_type": "execute_result"
    938     }
    939    ],
    940    "source": [
    941     "equation = derivative(f(x), x, 2) + x*derivative(f(x)) == 1\n",
    942     "desolve(equation, f(x), (0, 0, 0))"
    943    ]
    944   },
    945   {
    946    "cell_type": "markdown",
    947    "metadata": {},
    948    "source": [
    949     "**Exercise.** Use Sage to find out the functions $f(x)$ that satisfy\n",
    950     "\\begin{align*}\n",
    951     "    \\begin{array}{rlcrl}\n",
    952     "        (A) &\n",
    953     "        \\begin{cases}\n",
    954     "            f(0)   &= 1\\\\\n",
    955     "            f'(0)  &= 0\\\\\n",
    956     "            f''(x) &= -f(x)\n",
    957     "        \\end{cases}\n",
    958     "        & \\qquad \\qquad &\n",
    959     "        (B) &\n",
    960     "        \\begin{cases}\n",
    961     "            f(0)   &= 0\\\\\n",
    962     "            f'(0)  &= 1\\\\\n",
    963     "            f''(x) &= -f(x)\n",
    964     "        \\end{cases}\n",
    965     "    \\end{array}\n",
    966     "\\end{align*}"
    967    ]
    968   },
    969   {
    970    "cell_type": "code",
    971    "execution_count": null,
    972    "metadata": {},
    973    "outputs": [],
    974    "source": []
    975   },
    976   {
    977    "cell_type": "markdown",
    978    "metadata": {},
    979    "source": [
    980     "### A real-world example\n",
    981     "\n",
    982     "Differential equations have countless applications in Science, so it would be a shame not to see at least a simple one.\n",
    983     "\n",
    984     "Consider an object moving with constant acceleration $a$. Its velocity at time $t$ is described by the formula $v(t) = v(0) + at$. For example an object falling from the sky has acceleration $g\\sim 9.8 m/s^2$ towards the ground, so its velocity is $v(t) = -gt$.\n",
    985     "\n",
    986     "However in the real world you need to take into account the air's resistance, which depends (among other things) on the velocity of the object. In this case the acceleration $a(t)$ is not constant anymore, and it satisfies an equation of the form $a(t)=-g -kv(t)$, where $k$ is some constant that may depend on the shape and mass of the object (in practice it may be more complicated than this).\n",
    987     "\n",
    988     "Since the acceleration is the derivative of the velocity, we have a differential equation\n",
    989     "\\begin{align*}\n",
    990     "    v'(t) = -g -kv(t)\n",
    991     "\\end{align*}\n",
    992     "and we can try to solve it with Sage!"
    993    ]
    994   },
    995   {
    996    "cell_type": "code",
    997    "execution_count": 7,
    998    "metadata": {},
    999    "outputs": [
   1000     {
   1001      "data": {
   1002       "text/plain": [
   1003        "-98/15*(e^(3/2*t) - 1)*e^(-3/2*t)"
   1004       ]
   1005      },
   1006      "execution_count": 7,
   1007      "metadata": {},
   1008      "output_type": "execute_result"
   1009     }
   1010    ],
   1011    "source": [
   1012     "var('t')\n",
   1013     "function('v')\n",
   1014     "g = 9.8\n",
   1015     "k = 1.5\n",
   1016     "conditions = (0, 0) # Start with velocity 0\n",
   1017     "desolve(derivative(v(t)) == -g -k*v(t), v(t), conditions)"
   1018    ]
   1019   },
   1020   {
   1021    "cell_type": "markdown",
   1022    "metadata": {},
   1023    "source": [
   1024     "If you want to solve this equation symbolically (that is, keeping $g$ and $k$ in symbols) you need to specify that $t$ is the *independent variable* of the equation:"
   1025    ]
   1026   },
   1027   {
   1028    "cell_type": "code",
   1029    "execution_count": 10,
   1030    "metadata": {},
   1031    "outputs": [
   1032     {
   1033      "data": {
   1034       "text/plain": [
   1035        "-(g*e^(k*t) - g)*e^(-k*t)/k"
   1036       ]
   1037      },
   1038      "execution_count": 10,
   1039      "metadata": {},
   1040      "output_type": "execute_result"
   1041     }
   1042    ],
   1043    "source": [
   1044     "var('t', 'g', 'k')\n",
   1045     "function('v')\n",
   1046     "conditions = (0, 0) # Start with velocity 0\n",
   1047     "desolve(derivative(v(t)) == -g -k*v(t), v(t), conditions, ivar=t)"
   1048    ]
   1049   },
   1050   {
   1051    "cell_type": "markdown",
   1052    "metadata": {},
   1053    "source": [
   1054     "# Basic data analysis and visualization\n",
   1055     "\n",
   1056     "## Statistics\n",
   1057     "**References:** [[14](https://doc.sagemath.org/html/en/reference/stats/sage/stats/basic_stats.html)]\n",
   1058     "\n",
   1059     "Sage includes the most basic functions for statistical analysis."
   1060    ]
   1061   },
   1062   {
   1063    "cell_type": "code",
   1064    "execution_count": 20,
   1065    "metadata": {},
   1066    "outputs": [
   1067     {
   1068      "name": "stdout",
   1069      "output_type": "stream",
   1070      "text": [
   1071       "Values:\t [1, 2, 3, 3, -6, -2, 4, -1, 0, 2, 3, -4, 0]\n",
   1072       "Mean:\t\t\t 5/13\n",
   1073       "Median:\t\t\t 1\n",
   1074       "Mode:\t\t\t [3]\n",
   1075       "Standard deviation:\t 2*sqrt(29/13)\n",
   1076       "Variance:\t\t 116/13\n",
   1077       "Moving average (5): [3/5, 0, 2/5, -2/5, -1, 3/5, 8/5, 0, 1/5]\n"
   1078      ]
   1079     }
   1080    ],
   1081    "source": [
   1082     "L = [1, 2, 3, 3, -6, -2, 4, -1, 0, 2, 3, -4, 0]\n",
   1083     "\n",
   1084     "print(\"Values:\\t\", L)\n",
   1085     "\n",
   1086     "print(\"Mean:\\t\\t\\t\",           mean(L))\n",
   1087     "print(\"Median:\\t\\t\\t\",         median(L))\n",
   1088     "print(\"Mode:\\t\\t\\t\",           mode(L))\n",
   1089     "\n",
   1090     "print(\"Standard deviation:\\t\", std(L))\n",
   1091     "print(\"Variance:\\t\\t\",         variance(L))\n",
   1092     "\n",
   1093     "print(\"Moving average (5):\", moving_average(L,5))"
   1094    ]
   1095   },
   1096   {
   1097    "cell_type": "markdown",
   1098    "metadata": {},
   1099    "source": [
   1100     "You can also compare your data to a probability distribution, see [this page](https://doc.sagemath.org/html/en/reference/probability/sage/probability/probability_distribution.html). If you need to do more advanced statistics you should consider using [R](https://www.r-project.org/); you can also use it inside Sage."
   1101    ]
   1102   },
   1103   {
   1104    "cell_type": "markdown",
   1105    "metadata": {},
   1106    "source": [
   1107     "## Plotting\n",
   1108     "**Reference:** [[15](https://doc.sagemath.org/html/en/reference/plotting/index.html)], more specifically the subsection [[16](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot.html)].\n",
   1109     "\n",
   1110     "Some Sage objects can be plotted:"
   1111    ]
   1112   },
   1113   {
   1114    "cell_type": "code",
   1115    "execution_count": 21,
   1116    "metadata": {},
   1117    "outputs": [
   1118     {
   1119      "data": {
   1120       "image/png": 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\n",
   1121       "text/plain": [
   1122        "Graphics object consisting of 1 graphics primitive"
   1123       ]
   1124      },
   1125      "execution_count": 21,
   1126      "metadata": {},
   1127      "output_type": "execute_result"
   1128     }
   1129    ],
   1130    "source": [
   1131     "f = sin(x)\n",
   1132     "plot(f)"
   1133    ]
   1134   },
   1135   {
   1136    "cell_type": "markdown",
   1137    "metadata": {},
   1138    "source": [
   1139     "Sage's plotting functions are based on Python's [matplotlib](https://matplotlib.org/).\n",
   1140     "\n",
   1141     "You can give a number of options to adjust the aspect of your plot, see [here](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot.html#sage.plot.plot.plot). Let's see some of them:"
   1142    ]
   1143   },
   1144   {
   1145    "cell_type": "code",
   1146    "execution_count": 67,
   1147    "metadata": {},
   1148    "outputs": [
   1149     {
   1150      "data": {
   1151       "image/png": 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\n",
   1152       "text/plain": [
   1153        "Graphics object consisting of 1 graphics primitive"
   1154       ]
   1155      },
   1156      "execution_count": 67,
   1157      "metadata": {},
   1158      "output_type": "execute_result"
   1159     }
   1160    ],
   1161    "source": [
   1162     "f = sin(x)\n",
   1163     "plot(f,\n",
   1164     "     -2*pi, 2*pi,                   # bounds for x\n",
   1165     "     ymin  = -0.7, ymax = 0.7,      # bounds for y\n",
   1166     "     color = \"red\",\n",
   1167     "     title = \"The sin function\",\n",
   1168     "    )"
   1169    ]
   1170   },
   1171   {
   1172    "cell_type": "markdown",
   1173    "metadata": {},
   1174    "source": [
   1175     "Some of the options are not described precisely in Sage's documentation, but you can find them on [matplotlib's documentation](https://matplotlib.org/stable/contents.html). You can find many examples online for adjusting your plot as you like!"
   1176    ]
   1177   },
   1178   {
   1179    "cell_type": "markdown",
   1180    "metadata": {},
   1181    "source": [
   1182     "If you need to plot more than one object at the time, you can sum two plots and show them together with `show()`:"
   1183    ]
   1184   },
   1185   {
   1186    "cell_type": "code",
   1187    "execution_count": 36,
   1188    "metadata": {},
   1189    "outputs": [
   1190     {
   1191      "data": {
   1192       "image/png": 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\n",
   1193       "text/plain": [
   1194        "Graphics object consisting of 2 graphics primitives"
   1195       ]
   1196      },
   1197      "metadata": {},
   1198      "output_type": "display_data"
   1199     }
   1200    ],
   1201    "source": [
   1202     "cosine      = plot(cos(x), (x,-pi/2,pi/2), color=\"red\")\n",
   1203     "exponential = plot(exp(x), (x,-2,0.5))\n",
   1204     "\n",
   1205     "show(cosine + exponential)"
   1206    ]
   1207   },
   1208   {
   1209    "cell_type": "markdown",
   1210    "metadata": {},
   1211    "source": [
   1212     "Finally, there are other types of plots that you can use, like [scatter plots](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/scatter_plot.html#sage.plot.scatter_plot.scatter_plot) and [bar charts](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/bar_chart.html#sage.plot.bar_chart.bar_chart). You can also add [text](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/text.html#sage.plot.text.text) to your plot:"
   1213    ]
   1214   },
   1215   {
   1216    "cell_type": "code",
   1217    "execution_count": 53,
   1218    "metadata": {},
   1219    "outputs": [
   1220     {
   1221      "data": {
   1222       "image/png": 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\n",
   1223       "text/plain": [
   1224        "Graphics object consisting of 3 graphics primitives"
   1225       ]
   1226      },
   1227      "metadata": {},
   1228      "output_type": "display_data"
   1229     }
   1230    ],
   1231    "source": [
   1232     "b = bar_chart(range(1,10))\n",
   1233     "s = scatter_plot([(1,5), (4,2), (8,8), (4,7)],\n",
   1234     "                 marker = \"*\",       # symbol\n",
   1235     "                 markersize = 100,\n",
   1236     "                 edgecolor  = \"black\",\n",
   1237     "                 facecolor  = \"red\"\n",
   1238     "                )\n",
   1239     "t = text(\"wow, such plot!\", (1, 8), color=\"black\", fontsize=20)\n",
   1240     "show(b + s + t)"
   1241    ]
   1242   },
   1243   {
   1244    "cell_type": "markdown",
   1245    "metadata": {},
   1246    "source": [
   1247     "## Interpolation\n",
   1248     "**References:** [[17](https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_ring.html#sage.rings.polynomial.polynomial_ring.PolynomialRing_field.lagrange_polynomial)] and [[18](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/interpolation.html)].\n",
   1249     "\n",
   1250     "When you need to work with a discrete set of data, like measurements of real-world quantities, it can be useful to visualize a \"smoothed out\" version of this data, for example by plotting a function that approximates it.\n",
   1251     "\n",
   1252     "One way to do so is finding the lowest-degree polynomial that passes through all your points. This is called [Lagrange Polynomial](https://en.wikipedia.org/wiki/Lagrange_polynomial)."
   1253    ]
   1254   },
   1255   {
   1256    "cell_type": "code",
   1257    "execution_count": 65,
   1258    "metadata": {},
   1259    "outputs": [
   1260     {
   1261      "data": {
   1262       "image/png": 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\n",
   1263       "text/plain": [
   1264        "Graphics object consisting of 3 graphics primitives"
   1265       ]
   1266      },
   1267      "metadata": {},
   1268      "output_type": "display_data"
   1269     }
   1270    ],
   1271    "source": [
   1272     "points = [ (0,1), (1,2), (1.5,0), (2,4), (3,5) ]\n",
   1273     "polring.<x> = QQ[] # you need to specify a polynomial ring\n",
   1274     "lp = polring.lagrange_polynomial(points)\n",
   1275     "show(scatter_plot(points, facecolor=\"red\")\n",
   1276     "     + plot(lp, 0, 3) # slightly different notation for polynomials\n",
   1277     "     + text(lp, (1,8), color=\"black\")\n",
   1278     "    )"
   1279    ]
   1280   },
   1281   {
   1282    "cell_type": "markdown",
   1283    "metadata": {},
   1284    "source": [
   1285     "One can compute the Lagrange Polynomial over any base ring, and it has the advantage that it is a very \"nice\" function (continuous and differentiable as much as you like, with easily computable derivatives and primitives).\n",
   1286     "\n",
   1287     "However, it does not always give you good approximation of your data:"
   1288    ]
   1289   },
   1290   {
   1291    "cell_type": "code",
   1292    "execution_count": 2,
   1293    "metadata": {},
   1294    "outputs": [
   1295     {
   1296      "data": {
   1297       "image/png": 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\n",
   1298       "text/plain": [
   1299        "Graphics object consisting of 2 graphics primitives"
   1300       ]
   1301      },
   1302      "metadata": {},
   1303      "output_type": "display_data"
   1304     }
   1305    ],
   1306    "source": [
   1307     "R = [x/10 for x in range(-10,10)]\n",
   1308     "L = [1/(1+25*x^2) for x in R]\n",
   1309     "points = [(R[i], L[i]) for i in range(len(L))]\n",
   1310     "polring.<x> = RR[]\n",
   1311     "lp = polring.lagrange_polynomial(points)\n",
   1312     "\n",
   1313     "show(plot(lp, -0.82, 0.72) + scatter_plot(points))"
   1314    ]
   1315   },
   1316   {
   1317    "cell_type": "markdown",
   1318    "metadata": {},
   1319    "source": [
   1320     "This particular example is called [Runge's phenomenon](https://en.wikipedia.org/wiki/Runge%27s_phenomenon). For a better approximation you can use a [spline](https://en.wikipedia.org/wiki/Spline_(mathematics)), which is a *piecewise* polynomial function:"
   1321    ]
   1322   },
   1323   {
   1324    "cell_type": "code",
   1325    "execution_count": 90,
   1326    "metadata": {},
   1327    "outputs": [
   1328     {
   1329      "data": {
   1330       "image/png": 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\n",
   1331       "text/plain": [
   1332        "Graphics object consisting of 2 graphics primitives"
   1333       ]
   1334      },
   1335      "metadata": {},
   1336      "output_type": "display_data"
   1337     }
   1338    ],
   1339    "source": [
   1340     "show(plot(spline(points), -1, 1) + scatter_plot(points))"
   1341    ]
   1342   },
   1343   {
   1344    "cell_type": "markdown",
   1345    "metadata": {},
   1346    "source": [
   1347     "A detailed explanation of splines is a good topic for a course of numerical analysis. For this course it is enough that you know that they exist and they can be plotted."
   1348    ]
   1349   }
   1350  ],
   1351  "metadata": {
   1352   "kernelspec": {
   1353    "display_name": "SageMath 9.2",
   1354    "language": "sage",
   1355    "name": "sagemath"
   1356   },
   1357   "language_info": {
   1358    "codemirror_mode": {
   1359     "name": "ipython",
   1360     "version": 3
   1361    },
   1362    "file_extension": ".py",
   1363    "mimetype": "text/x-python",
   1364    "name": "python",
   1365    "nbconvert_exporter": "python",
   1366    "pygments_lexer": "ipython3",
   1367    "version": "3.8.5"
   1368   }
   1369  },
   1370  "nbformat": 4,
   1371  "nbformat_minor": 4
   1372 }
	mathsoftware A course about LaTeX and SageMath
	git clone https://git.tronto.net/mathsoftware
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