{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Tutoraggio 3"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"addpath(\"./functions\");"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Esercizio 1\n",
"\n",
"Sia assegnata la funzione $f(x) = \\sin(x) - \\sqrt{x}$ nell'intervallo $[0, \\frac{\\pi}{2}]$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Punto a\n",
"\n",
"Si determina il polinomio di interpolazione della funzione $f$, che si ottiene dalla formula di Lagrange relativa ai nodi $x_0 = 0$, $x_1 = h$, $x_2 = 2h$, $x_3 = 3h$, $x_4 = \\frac{\\pi}{2}$, con $h = \\frac{\\pi}{8}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Per una retta sono necessari due nodi, tre per una parabola, quattro per una cubica, cinque per una quadrica."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/plain": [
"<IPython.core.display.Image object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"f = @(x) sin(x) - x.^(1/2);\n",
"\n",
"h = pi/8;\n",
"x = [0, h, 2*h, 3*h, pi/2];\n",
"y = f(x);\n",
"\n",
"% si cerca un polinomio di quarto grado (num punti -1) tale che:\n",
"% p_4 (x(k)) = y(k), in altri termini si vuole l'uguaglianza punto per punto\n",
"% dei punti dati\n",
"% p = L_1 + L_2 + L_3 + L_4 + L_5\n",
"% dove per il k-esimo polinomio L_k vale:\n",
"% L_k(x(j)) = y(k) * delta(j,k)\n",
"% delta vale 1 se j = k, 0 altrimenti\n",
"\n",
"figure\n",
"t = linspace(0, pi/2);\n",
"y_lag = interp_lagrange(x, y, t);\n",
"plot( x, y, \"*\", ...\n",
" t, y_lag);\n",
"box off\n",
"legend(\"nodi\", \"interpolante\");"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Punto b\n",
"\n",
"b) si costruisce (sia mediante il metodo delle equazioni normali che mediante il metodo QRLS) il polinomio di terzo grado che approssima ai minimi quadrati il set di dati ottenuto campionando la funzione $f$ nei punti `0:pi/24:pi/2`."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"x = (0: pi/24 : pi/2);\n",
"y = f(x);\n",
"\n",
"% si cerca p_3 tale che p sia:\n",
"% p_3(t) = a*t^3 + b*t^2 + c*t + d\n",
"% p_3(x) ~~ y <==> [x.^3, x.^2, x, 1] * [a, b, c, d]' ~~ y\n",
"% in modo che ||Y - y||_2 sia minima\n",
"\n",
"p_enls = NEmethod(x', y', 3);\n",
"p_qrls = QRmethod(x', y', 3);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Punto c\n",
"\n",
"c) si confrontano i grafici dei polinomi ottenuti ai punti a) e b) con quello di $f$."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/plain": [
"<IPython.core.display.Image object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x = linspace(0, pi/2);\n",
"y = f(x);\n",
"\n",
"y_enls = polyval(p_enls, x);\n",
"y_qrls = polyval(p_qrls, x);\n",
"\n",
"figure\n",
"plot( t, f(t), ...\n",
" t, y_lag, ...\n",
" t, y_enls, ...\n",
" t, y_qrls, \"--\")\n",
"box off\n",
"legend(\"f\", \"Lagrange\", \"Minimi quadrati - EN\", \"Minimi quadrati - QR\");"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"La funzione $f$ per la sua struttura (regolare e con unico flesso, ovvero un unico cambio di concavità, il punto di flesso corrisponde allo zero della derivata seconda) si presta ad un buon grado di approssimazione anche con un polinomio di terzo grado. Il risultato che si ottiene è un sistema sovradeterminato stabile, che dà lo stesso risultato sia con EN che QR. Se il sistema non fosse stato particolarmente stabile probabilmente il metodo delle equazioni normali non sarebbe riuscito ad ottenere un'approssimazione abbastanza accurata."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"err_inf_lag = 0.10564\n",
"err_inf_enls = 0.076735\n",
"err_inf_qrls = 0.076735\n"
]
}
],
"source": [
"err_inf_lag = max(abs(y - y_lag))\n",
"err_inf_enls = max(abs(y - y_enls))\n",
"err_inf_qrls = max(abs( y - y_qrls))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Si nota che l'errore assoluto in norma infinito commesso è addirittura minore nel caso dei minimi quadrati."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Esercizio 2\n",
"\n",
"Si considerino i punti del piano di coordinate:\n",
"\n",
"$$\n",
"\\begin{array}{|c|c|c|c|c|c|}\n",
"\\hline & i=1 & i=2 & i=3 & i=4 & i=5 \\\\\n",
"\\hline x_{i} & 0.0004 & 0.2507 & 0.5008 & 2.0007 & 8.0013 \\\\\n",
"\\hline y_{i} & 0.0007 & 0.0162 & 0.0288 & 0.0309 & 0.0310 \\\\\n",
"\\hline\n",
"\\end{array}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Punto a\n",
"\n",
"Determinare i polinomi di approssimazione ai minimi quadrati di grado 1 e 2 rappresentarli in un grafico insieme ai dati $(x_i, y_i), i = 1, \\dotsc, 5$"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/plain": [
"<IPython.core.display.Image object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x = [0.0004 0.2507 0.5008 2.0007 8.0013]';\n",
"y = [0.0007 0.0162 0.0288 0.0309 0.0310]';\n",
"\n",
"% grafico dei punti forniti\n",
"figure\n",
"plot(x,y,\"*\");\n",
"xlim([0 9])\n",
"box off"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Né un polinomio di grado 1 né uno di grado 2 hanno grandi possibilità di approssimare il polinomio dato in modo accurato. Le basi esponenziali potenzialmente possono fornire la migliore approssimazione."
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [],
"source": [
"% lineare -- grado 1 -- a*x + b\n",
"V = [x, ones(5,1)];\n",
"f_1 = (V' * V) \\ (V' * y);\n",
"\n",
"% quadratica -- grado 2 -- a*x^2 + b*x + c\n",
"V = [x.^2, x, ones(5,1)];\n",
"%Va = vander(x) % solo le ultime tre colonne\n",
"%V = Va(:,5)\n",
"f_2 = (V' * V) \\ (V' * y);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Punto b\n",
"\n",
"Determinare, nel senso dei minimi quadrati, i coefficienti a, b, c della seguente\n",
"approssimazione con basi esponenziali:\n",
"\n",
"$$y = a + be^{-x} + ce^{-2x} $$\n",
"\n",
"Rappresentare l’approssimante ottenuto in un grafico, insieme ai dati $(x_i, y_i), i = 1, \\dotsc, 5$"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [],
"source": [
"% i coefficienti vanno in ordine crescente in f_3 in base all'ordine\n",
"% stabilito nella matrice di Vandermonde V\n",
"V = [exp(-2*x), exp(-x), ones(5,1)];\n",
"a = (V' * V) \\ (V' * y);\n",
"f_3 = @(x) a(1)*exp(-2*x) + a(2)*exp(-x) + a(3);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Punto c\n",
"\n",
"Quale tra le tre approssimazioni ottenute ai punti precedenti risulta migliore?\n",
"Confrontare gli errori\n",
"\n",
"$$\n",
"E_{j}=\\sum_{i=1}^{5}\\left(f_{j}\\left(x_{i}\\right)-y_{i}\\right)^{2}, \\quad j=1,2,3\n",
"$$\n",
"\n",
"dove $f_1$ e $f_2$ denotano i polinomi di approssimazione di grado 1 e 2 determinati al punto a), mentre $f_3$ denota l’approssimazione con basi esponenziali determinata al punto b)."
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/plain": [
"<IPython.core.display.Image object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"t = linspace(0, 9);\n",
"figure\n",
"plot( x, y, \"*\", ...\n",
" t, polyval(f_1, t), ...\n",
" t, polyval(f_2, t), ...\n",
" t, f_3(t));\n",
"legend(\"punti\", \"lineare\", \"quadratica\", \"esponenziale\", ...\n",
" \"location\", \"southoutside\");\n",
"xlim([0 9])\n",
"box off"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Per fornire una valutazione più formale dei risultati ottenuti si calcolano le norme al quadrato"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"E_1 = 4.848327762313228e-04\n",
"E_2 = 2.364635594024985e-04\n",
"E_3 = 1.224973312890181e-05\n"
]
}
],
"source": [
"format long\n",
"E_1 = norm(polyval(f_1, x) - y, 2)^2\n",
"E_2 = norm(polyval(f_2, x) - y, 2)^2\n",
"E_3 = norm(f_3(x) - y, 2)^2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"L'errore che si commette con l'esponenziale è 20 volte inferiore a quello commesso con la parabola. L'errore commesso con la parabola è pari alla metà di quello commesso con la retta. Anche dal punto di vista qualitativo, osservando il grafico, si nota che la migliore approssimazione si ha proprio con l'esponenziale."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Octave",
"language": "octave",
"name": "octave"
},
"language_info": {
"file_extension": ".m",
"help_links": [
{
"text": "GNU Octave",
"url": "https://www.gnu.org/software/octave/support.html"
},
{
"text": "Octave Kernel",
"url": "https://github.com/Calysto/octave_kernel"
},
{
"text": "MetaKernel Magics",
"url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
}
],
"mimetype": "text/x-octave",
"name": "octave",
"version": "5.2.0"
}
},
"nbformat": 4,
"nbformat_minor": 4
}