24/12/2023, 22:11
f[x_] = Sin[x];
{a, b, n} = {-Pi/4, Pi/4, 10};
u = Table[i, {i, a, b, (b - a)/(n + 1)}];
Do[p[x_] = Sum[StringJoin["c", ToString[i]] x^i, {i, 0, n}];
v = Table[p[u[[i]]] - f[u[[i]]] == (-1)^i k, {i, n + 2}];
p[x_] = p[x] /. NSolve[v][[1]];
w = NSolve[{TrigToExp[D[p[x] - f[x], x]] == 0, a < x < b}];
u = Sort@N@Join[{a, b}, w[[All, 1, 2]]], {j, 20}];
Chop@p[x]
Plot[p[x] - f[x], {x, a, b}, AxesLabel -> {"x", "p(x)-f(x)"}]
0.999999999976264 x -
0.16666666589651935 x^3 +
0.008333326335356108 x^5 -
0.0001983867334444072 x^7 +
0.000002713535577336344 x^9
x -
0.16666666666666666 x^3 +
0.008333333333333333 x^5 -
0.0001984126984126984 x^7 +
0.0000027557319223985893 x^9
sin[x_] := Module[{n, pi, y, siny, cosy},
{n, pi, y} = {0, 3.141592653589793, x};
While[y < -pi/4, n = n + 1; y = y + pi/2];
While[y > +pi/4, n = n - 1; y = y - pi/2];
siny = 0.999999999976264 y - 0.16666666589651935 y^3 +
0.008333326335356108 y^5 - 0.0001983867334444072 y^7 +
0.000002713535577336344 y^9;
cosy = 0.9999999999999444 - 0.4999999999935193 y^2 +
0.04166666654402005 y^4 - 0.0013888880398758362 y^6 +
0.000024798929682192885 y^8 - 0.00000027173490012904213 y^10;
If[Mod[n, 4] == 0, siny, If[Mod[n, 4] == 1, -cosy, If[Mod[n, 4] == 2, -siny, cosy]]]]
cos[x_] := Module[{n, pi, y, siny, cosy},
{n, pi, y} = {0, 3.141592653589793, x};
While[y < -pi/4, n = n - 1; y = y + pi/2];
While[y > +pi/4, n = n + 1; y = y - pi/2];
siny = 0.999999999976264 y - 0.16666666589651935 y^3 +
0.008333326335356108 y^5 - 0.0001983867334444072 y^7 +
0.000002713535577336344 y^9;
cosy = 0.9999999999999444 - 0.4999999999935193 y^2 +
0.04166666654402005 y^4 - 0.0013888880398758362 y^6 +
0.000024798929682192885 y^8 - 0.00000027173490012904213 y^10;
If[Mod[n, 4] == 0, cosy, If[Mod[n, 4] == 1, -siny, If[Mod[n, 4] == 2, -cosy, siny]]]]
arctan[x_] := Module[{m, n, pi, y, arctany},
{m, n, pi, y} = {0, 1, 3.141592653589793, x};
If[y < 0, n = -1; y = -x];
If[y > 1, m = 1; y = 1/y];
arctany = 0.9999999146206229 y + 0.000004440916687621521 y^2 -
0.3334229328258918 y^3 + 0.0009160176902325137 y^4 +
0.19467852030293517 y^5 + 0.01810645296260239 y^6 -
0.17583841962361396 y^7 + 0.011728511647071702 y^8 +
0.1956314075888416 y^9 - 0.19507021552110818 y^10 +
0.08234872936245016 y^11 - 0.013684264265137207 y^12;
If[m == 0, n arctany, n (pi/2 - arctany)]]
exp[x_] := Module[{n, log2, y, expy},
{n, log2, y} = {1, 0.6931471805599453, x};
While[y < 0, n = n/2; y = y + log2];
While[y > log2, n = 2 n; y = y - log2];
expy = 0.9999999999999809 + 1.0000000000054843 y +
0.49999999974080833 y^2 + 0.16666667141752853 y^3 +
0.041666622484762016 y^4 + 0.008333568911715338 y^5 +
0.0013881265875033293 y^6 + 0.00019993071014806345 y^7 +
0.00002299592844852003 y^8 + 0.000003910027293592432 y^9;
n expy]
log[x_] := Module[{n, log2, y, logy},
{n, log2, y} = {0, 0.6931471805599453, x};
While[y < 1, n = n - 1; y = 2 y];
While[y > 2, n = n + 1; y = y/2];
logy = -2.758798873492117 + 8.482177884092106 y -
16.404818190077595 y^2 + 25.509315252813245 y^3 -
29.970734938543238 y^4 + 26.57580667306714 y^5 -
17.810152466522702 y^6 + 8.975140918706439 y^7 -
3.350162015718218 y^8 + 0.8989936219430049 y^9 -
0.16408089302388532 y^10 + 0.01824637444011346 y^11 -
0.0009333476666561241 y^12;
n log2 + logy]
25/12/2023, 14:41
26/12/2023, 11:23
log[x_] := Module[{m, n, y, z, logz},
{m, n, y} = {0, 1, x};
While[y < 1, m = m - 1; y = 2 y];
While[y > 2, m = m + 1; y = y/2];
z = {1.00, 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09,
1.10, 1.11, 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19,
1.20, 1.21, 1.22, 1.23, 1.24, 1.25, 1.26, 1.27, 1.28, 1.29,
1.30, 1.31, 1.32, 1.33, 1.34, 1.35, 1.36, 1.37, 1.38, 1.39,
1.40, 1.41, 1.42, 1.43, 1.44, 1.45, 1.46, 1.47, 1.48, 1.49,
1.50, 1.51, 1.52, 1.53, 1.54, 1.55, 1.56, 1.57, 1.58, 1.59,
1.60, 1.61, 1.62, 1.63, 1.64, 1.65, 1.66, 1.67, 1.68, 1.69,
1.70, 1.71, 1.72, 1.73, 1.74, 1.75, 1.76, 1.77, 1.78, 1.79,
1.80, 1.81, 1.82, 1.83, 1.84, 1.85, 1.86, 1.87, 1.88, 1.89,
1.90, 1.91, 1.92, 1.93, 1.94, 1.95, 1.96, 1.97, 1.98, 1.99, 2.00};
logz = {0, 0.00995033, 0.0198026, 0.0295588, 0.0392207, 0.0487902,
0.0582689, 0.0676586, 0.0769610, 0.0861777, 0.0953102, 0.104360,
0.113329, 0.122218, 0.131028, 0.139762, 0.148420, 0.157004,
0.165514, 0.173953, 0.182322, 0.190620, 0.198851, 0.207014,
0.215111, 0.223144, 0.231112, 0.239017, 0.246860, 0.254642,
0.262364, 0.270027, 0.277632, 0.285179, 0.292670, 0.300105,
0.307485, 0.314811, 0.322083, 0.329304, 0.336472, 0.343590,
0.350657, 0.357674, 0.364643, 0.371564, 0.378436, 0.385262,
0.392042, 0.398776, 0.405465, 0.412110, 0.418710, 0.425268,
0.431782, 0.438255, 0.444686, 0.451076, 0.457425, 0.463734,
0.470004, 0.476234, 0.482426, 0.488580, 0.494696, 0.500775,
0.506818, 0.512824, 0.518794, 0.524729, 0.530628, 0.536493,
0.542324, 0.548121, 0.553885, 0.559616, 0.565314, 0.570980,
0.576613, 0.582216, 0.587787, 0.593327, 0.598837, 0.604316,
0.609766, 0.615186, 0.620576, 0.625938, 0.631272, 0.636577,
0.641854, 0.647103, 0.652325, 0.657520, 0.662688, 0.667829,
0.672944, 0.678034, 0.683097, 0.688135, 0.693147};
While[z[[n]] < y, n = n + 1];
m logz[[101]] + logz[[n - 1]] + (y - z[[n - 1]]) (logz[[n]] - logz[[n - 1]])/(z[[n]] - z[[n - 1]])]
Plot[Log[x] - log[x], {x, 10^-3, 10^3}, AxesLabel -> {"x", "ε(x)"}]
03/01/2024, 18:25
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