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