Date post: | 06-Jun-2018 |
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Ejemplo Polinomios de Legendre
Tomamos una función
f@x_D = Sin@3 xD + Sqrt@2 + xDx + 2 + sinH3 xL
Plot@f@xD, 8x, -2, 2<D
-2 -1 1 2
0.5
1.0
1.5
2.0
2.5
Sabemos que
f[x]=S aL PLHxL dónde aL = H2 L + 1L �2 Ù-1
1f HxL PLHxL â x
a@L_D :=H2 L + 1L
2à
-1
1
f@xD LegendreP@L, xD âx
8a@0D, a@1D, a@2D< �� N
81.39872, 1.3978, -0.0316861<For@i = 0, i < 20, i++, aa@iD = a@iD �� N; Print@aa@iDDD
1.39872
1.3978
-0.0316861
-1.05932
-0.000959976
0.180572
-0.0000435402
-0.0123628
-2.28268´10-6
0.000460618
-1.2903´10-7
-0.0000107173
-7.67854´10-9
1.74448´10-7
-1.1394´10-9
-1.15484´10-7
0.
0.
0.
-0.00238037
Aproximacion de M términos.
Aprox@M_, x_D := âL=0
M
aa@LD LegendreP@L, xD
Aprox@3, xD-0.529658 I5 x
3- 3 xM - 0.0158431 I3 x
2- 1M + 1.3978 x + 1.39872
2 Legendre.nb
Ap0
Plot@8f@xD, Aprox@0, xD<, 8x, -2, 2<, PlotStyle ® 88Thick, Green<, 8Thick, Red<<D
-2 -1 1 2
0.5
1.0
1.5
2.0
2.5
Ap1
Plot@8f@xD, Aprox@1, xD<, 8x, -2, 2<, PlotStyle ® 88Thick, Green<, 8Thick, Red<<D
-2 -1 1 2
-1
1
2
3
4
Legendre.nb 3
Ap3
Plot@8f@xD, Aprox@3, xD<, 8x, -2, 2<, PlotStyle ® 88Thick, Green<, 8Thick, Red<<D
-2 -1 1 2
-2
2
4
6
Ap9
Plot@8f@xD, Aprox@9, xD<, 8x, -2, 2<, PlotStyle ® 88Thick, Green<, 8Thick, Red<<D
-2 -1 1 2
-2
-1
1
2
3
4
4 Legendre.nb
Ap19
Plot@8f@xD, Aprox@19, xD<, 8x, -3, 3<, PlotStyle ® 88Thick, Green<, 8Thick, Red<<D
-3 -2 -1 1 2 3
-4
-2
2
4
6
App19-cerca
Plot@8f@xD, Aprox@19, xD<, 8x, -1.1, 1.1<, PlotStyle ® 88Thick, Green<, 8Thick, Red<<D
-1.0 -0.5 0.5 1.0
0.5
1.0
1.5
2.0
2.5
3.0
Legendre.nb 5