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See also Legendre polynomials with Xnumbers add-in
Excel formulas and VBA functions
D3:
=IF(ISEVEN(B3),(-1)^(B3/2)*FACTDOUBLE(B3-1)/FACTDOUBLE(B3),0)
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D4: =xlLegendrePn(B3,B4)
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Leave cell I4 blank
Ranges I5:K5 and N5:P5 are merged
I6: =IF(H6=0,1,IF(H6=1,$I$2,(2*H6-1)/H6*$I$2*I5-(H6-1)/H6*I4))
J6: =xlLegendrePn(H6,$I$2)
K6: =Poly_Legendre($I$2,H6)
M6: =H6
N6: =(-1)^M6*I6
O6: =xlLegendrePn(M6,-$I$2)
P6: =Poly_Legendre(-$I$2,M6)
MyExcelRoutines
This function returns only the value of the polynomial. The computation of the first derivative will be the object of a separate function.
Function xlLegendrePn(n, x)
Dim i
Dim P0, P1, Pn, Pn_1, Pn_2
P0 = 1: P1 = x
Pn_1 = P1: Pn_2 = P0
If (n <> Int(n) Or n < 0) Then
xlLegendrePn = "** n **"
ElseIf n = 0 Then
xlLegendrePn = 1
ElseIf n = 1 Then
xlLegendrePn = x
ElseIf x = 1 Then
xlLegendrePn = 1
ElseIf x = -1 Then
xlLegendrePn = (-1) ^ n
Else
For i = 2 To n
Pn = (2 * i - 1) / i * x * Pn_1 _
- (i - 1) / i * Pn_2
Pn_2 = Pn_1
Pn_1 = Pn
Next i
xlLegendrePn = Pn
End If
End Function
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Xnumbers add-in
Poly_Legendre is an array function that returns both the values of the polynomial and its first derivative calculated by the subroutine EvalLegendre.
Function Poly_Legendre(x, Optional n)
Dim Pol#, Dpol#, k&, z#
If IsMissing(n) Then k = 1 Else k = n
z = x
Call EvalLegendre(k, z, Pol, Dpol)
Poly_Legendre = PasteVector_(Array(Pol, Dpol))
End Function
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Sub EvalLegendre(n&, x#, Pol#,
Dpol#)
' Rutina para calcular el
polinomio ortonormal de Legendre de orden n y su derivada en x
' Los polinomios de Legendre son
un caso especial de los de Jacobi con a = b = 0
' Pol valor del polinomio en x;
DPol valor de la derivada del polinomio en x
' Bibliografia: Abramowitz M et al.; "Handbook
of Mathematical Functions...",Dover
'
Press et al.; "Numerical recipies in fotran77", Cambridge U
Press
'mod. 12.4.04 VL
Dim
k&, p#(0 To 2), dp#(0 To 2)
If n = 0
Then
Pol =
1
Dpol =
0
ElseIf n =
1 Then
Pol =
x
Dpol =
1
Else
p(0) =
1
p(1) =
x
If
Abs(x - 1) < 0.1 Or Abs(x + 1) < 0.1 Then
dp(0) = 0
dp(1) = 1
For k = 1 To n - 1
p(2) = ((2 * k + 1) * x *
p(1) - k * p(0)) / (k + 1)
'Polinomio de orden k+1 en x
dp(2)
= ((2 * k + 1) * (p(1) + x * dp(1)) - k * dp(0)) / (k + 1) 'Derivata del
polinomio di ' ordine k+1 in x .VL
p(0) = p(1)
p(1) = p(2)
dp(0) = dp(1)
dp(1) = dp(2)
Next
Pol = p(2)
Dpol = dp(2)
Else
For k = 1 To n - 1
p(2) = ((2 * k + 1) * x *
p(1) - k * p(0)) / (k + 1)
' Polinomio de orden k+1 ' ' en x
p(0)
= p(1)
' (***)
p(1) = p(2)
Next
Pol = p(2)
Dpol = n * (x * p(2) - p(0)) / (x ^ 2 - 1) ' Derivada del polinomio de orden k+1
en x
End If
End If
End Sub
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From the Tutorial -1:
Matlab
Values of Pn(x)
Polynomials
