Tuesday, July 26, 2016

Legendre polynomial (Legendre function of the first kind)



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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)
D4: =xlLegendrePn(B3,B4)




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


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
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

From the Tutorial -1:









Matlab

Values of Pn(x)
Polynomials 


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