true1 = Integrate @ f @ x D , x D ;
true = Integrate @ f @ x D , 8 x, 0, 1 <D N x @ 0 D = a;
n = 20;
x @ n D = b;
h = H b - a L n;
Do @ x @ i + 1 D = x @ i D + h, 8 i, 0, n - 2 <D ; sum = 0.0;
Do @ sum += f @ x @ i DD , 8 i, 1, n - 1 <D ; trap = 0.5 * h * H f @ a D + 2.0 * sum + f @ b DL err = Abs @ trap - true D
1.71828 1.71864 0.00035796
w @ 0 D = 1.; w @ n D = 1.;
Do @ w @ i D = 4., 8 i, 1, n - 1, 2 <D Do @ w @ i D = 2., 8 i, 2, n - 2, 2 <D sum = 0.0;
Do @ sum += w @ i D * f @ x @ i DD , 8 i, 0, n <D simps = h * sum 3.
err = Abs @ simps - true D 1.71828
5.96448´ 10
-8ticks2 = 88 0, 0.05, 0.1, 0.15, 0.2 < , 8 0.001, 0.002, 0.003, 0.004, 0.005 << ;
prange = 88 0, 0.2 < , 8 0, 0.005 << ;
nmax = 200;
Do @ x @ n D = b;
h = H b - a L n; hh @ n D = h;
Do @ x @ i + 1 D = x @ i D + h, 8 i, 0, n - 2 <D ; sum = 0.0;
Do @ sum += f @ x @ i DD , 8 i, 1, n - 1 <D ; trap = 0.5 * h * H f @ a D + 2.0 * sum + f @ b DL ;
errtrap @ n D = Abs @ trap - true D ; w @ 0 D = 1.; w @ n D = 1.;
Do @ w @ i D = 4., 8 i, 1, n - 1, 2 <D ; Do @ w @ i D = 2., 8 i, 2, n - 2, 2 <D ; sum = 0.0;
Do @ sum += w @ i D * f @ x @ i DD , 8 i, 0, n <D ; simps = h * sum 3.;
errsimps @ n D = Abs @ simps - true D , 8 n, 4, nmax, 2 <D
pl1 = ListPlot @ Table @8 hh @ n D , errtrap @ n D< , 8 n, 4, nmax, 2 <D ,
PlotStyle ® RGBColor @ 1, 0, 0 D , PlotRange ® prange, Ticks ® ticks2 D ; pl2 = ListPlot @ Table @8 hh @ n D , 1000. * errsimps @ n D< , 8 n, 4, nmax, 2 <D ,
PlotStyle ® RGBColor @ 0, 0, 1 D , PlotRange ® prange, Ticks ® ticks2 D ; Show @
pl1, pl2 D
0 0.05 0.1 0.15 0.2
0.001 0.002 0.003 0.004 0.005
0.001 0.002 0.003 0.004 0.005
H* SECONDA FUNZIONE * L
f @ x_ D := x * Sin @ x D ; Plot @ f @ x D , 8 x, 0, 4. Pi <D
2 4 6 8 10 12
-10 -5 5
a = 0.; b = 4. * Pi;
true1 = Integrate @ f @ x D , x D ;
true = Integrate @ f @ x D , 8 x, 0, b <D N -12.5664
x @ 0 D = a;
n = 200;
x @ n D = b;
h = H b - a L n;
Do @ x @ i + 1 D = x @ i D + h, 8 i, 0, n - 2 <D ; sum = 0.0;
Do @ sum += f @ x @ i DD , 8 i, 1, n - 1 <D ; trap = 0.5 * h * H f @ a D + 2.0 * sum + f @ b DL err = Abs @ trap - true D
-12.5622 0.00413444
w @ 0 D = 1.; w @ n D = 1.;
Do @ w @ i D = 4., 8 i, 1, n - 1, 2 <D Do @ w @ i D = 2., 8 i, 2, n - 2, 2 <D sum = 0.0;
Do @ sum += w @ i D * f @ x @ i DD , 8 i, 0, n <D simps = h * sum 3.
err = Abs @ simps - true D
ticks2 = 88 0, 0.05, 0.1, 0.15, 0.2 < , 8 0.01, 0.02, 0.03, 0.04 << ; prange = 88 0, 0.15 < , 8 0, 0.04 << ;
nmax = 1000;
Do @ x @ n D = b;
h = H b - a L n; hh @ n D = h;
Do @ x @ i + 1 D = x @ i D + h, 8 i, 0, n - 2 <D ; sum = 0.0;
Do @ sum += f @ x @ i DD , 8 i, 1, n - 1 <D ; trap = 0.5 * h * H f @ a D + 2.0 * sum + f @ b DL ;
errtrap @ n D = Abs @ trap - true D ; w @ 0 D = 1.; w @ n D = 1.;
Do @ w @ i D = 4., 8 i, 1, n - 1, 2 <D ; Do @ w @ i D = 2., 8 i, 2, n - 2, 2 <D ; sum = 0.0;
Do @ sum += w @ i D * f @ x @ i DD , 8 i, 0, n <D ; simps = h * sum 3.;
errsimps @ n D = Abs @ simps - true D , 8 n, 4, nmax, 2 <D
pl1 = ListPlot @ Table @8 hh @ n D , errtrap @ n D< , 8 n, 4, nmax, 2 <D ,
PlotStyle ® RGBColor @ 1, 0, 0 D , PlotRange ® prange, Ticks ® ticks2 D ; pl2 = ListPlot @ Table @8 hh @ n D , 1000. * errsimps @ n D< , 8 n, 4, nmax, 2 <D ,
PlotStyle ® RGBColor @ 0, 0, 1 D , PlotRange ® prange, Ticks ® ticks2 D ; Show @
pl1, pl2 D
0 0.05 0.1 0.15
0.01 0.02 0.03 0.04
0 0.05 0.1 0.15 0.01
0.02 0.03 0.04
