# Need help with a fortran-routine that calculates the associated Laguerre function

1. Aug 10, 2011

### d4n1el

Hi!
Im trying to do some rather easy QM-calculations in Fortran.
To do that i need a routine that calculates the generalized Laguerre polynomials.

I just did the simplest implementation of the equation:
$$L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!}$$

I implemented this in the following way:
Code (Text):

SUBROUTINE LAGUERRE(n,l,r,u,arraylength)
Implicit none
INTEGER, INTENT(IN) :: arraylength,n
real(kind=kind(0.d0)), INTENT(IN) :: l
REAL(kind=kind(0.d0)), INTENT(IN),DIMENSION(arraylength) :: r
REAL(kind=kind(0.d0)), INTENT(OUT),DIMENSION(arraylength) :: u
REAL(kind=kind(0.d0)), DIMENSION(arraylength) :: temp
Integer :: m

temp=0.d0
do m=0,n
temp=temp+gamma(real(n)+l+1.0)*(-r)**m/
+        (gamma(real(n)-real(m)+1.0)*gamma(l+real(m)+1.0)
+        *gamma(real(m)+1.0))
end do
u=temp
END SUBROUTINE LAGUERRE

The code is running and i do get results, but when i compare them with the results given in for example Mathematica, they seems quite strange.
So my questions are: Do you see some obvious mistakes? Do you think there are possibilities for numerical errors? (In such a case, do you have any advices on improvments?)

When i have been googling around, it looks like alot of the numerical implementations are using recursive relations. What are the pro's and con's for doing this kind of calculation in such a manner?

2. Aug 12, 2011

### SteamKing

Staff Emeritus
What is gamma? Is it an array, a function, a subroutine?

3. Aug 12, 2011

### uart

It looks like he's using it for factorial so it must the special function gamma (which reduces to factorial for integer values of the argument : gamma(n+1) = n! ).

I just did the simplest implementation of the equation:
$$L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!}$$

I implemented this in the following way:
Code (Text):

SUBROUTINE LAGUERRE(n,l,r,u,arraylength)
Implicit none
INTEGER, INTENT(IN) :: arraylength,n
real(kind=kind(0.d0)), INTENT(IN) :: l
REAL(kind=kind(0.d0)), INTENT(IN),DIMENSION(arraylength) :: r
REAL(kind=kind(0.d0)), INTENT(OUT),DIMENSION(arraylength) :: u
REAL(kind=kind(0.d0)), DIMENSION(arraylength) :: temp
Integer :: m

temp=0.d0
do m=0,n
temp=temp+gamma(real(n)+l+1.0)*(-r)**m/
+        (gamma(real(n)-real(m)+1.0)*gamma(l+real(m)+1.0)
+        *gamma(real(m)+1.0))
end do
u=temp
END SUBROUTINE LAGUERRE

Why are "r" and "u" declared as arrays yet you don't index them like arrays?

Where did the (l+k)! term come from?
Why isn't "r" squared.

Last edited: Aug 12, 2011
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