'****************************************************
'*   Program to demonstrate Laguerre coefficients   *
'* ------------------------------------------------ *
'* Reference: BASIC Scientific Subroutines, Vol. II *
'* by F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981.    *
'*                                                  *
'* ------------------------------------------------ *
'* SAMPLE RUN:                                      *
'*                                                  *
'* Laguerre polynomial coefficients for order 2     *
'*                                                  *
'*  A( 0) =         2                               *
'*  A( 1) =        -4                               *
'*  A( 2) =         1                               *
'*                                                  *
'* Laguerre polynomial coefficients for order 3     *
'*                                                  *
'*  A( 0) =         6                               *
'*  A( 1) =       -18                               *
'*  A( 2) =         9                               *
'*  A( 3) =        -1                               *
'*                                                  *
'* Laguerre polynomial coefficients for order 4     *
'*                                                  *
'*  A( 0) =        24                               *
'*  A( 1) =       -96                               *
'*  A( 2) =        72                               *
'*  A( 3) =       -16                               *
'*  A( 4) =         1                               *
'*                                                  *
'* Laguerre polynomial coefficients for order 5     *
'*                                                  *
'*  A( 0) =       120                               *
'*  A( 1) =      -600                               *
'*  A( 2) =       600                               *
'*  A( 3) =      -200                               *
'*  A( 4) =        25                               *
'*  A( 5) =        -1                               *
'*                                                  *
'* Laguerre polynomial coefficients for order 6     *
'*                                                  *
'*  A( 0) =       720                               *
'*  A( 1) =     -4320                               *
'*  A( 2) =      5400                               *
'*  A( 3) =     -2400                               *
'*  A( 4) =       450                               *
'*  A( 5) =       -36                               *
'*  A( 6) =         1                               *
'*                                                  *
'* Laguerre polynomial coefficients for order 7     *
'*                                                  *
'*  A( 0) =      5040                               *
'*  A( 1) =    -35280                               *
'*  A( 2) =     52920                               *
'*  A( 3) =    -29400                               *
'*  A( 4) =      7350                               *
'*  A( 5) =      -882                               *
'*  A( 6) =        49                               *
'*  A( 7) =        -1                               *
'*                                                  *
'* Laguerre polynomial coefficients for order 8     *
'*                                                  *
'*  A( 0) =     40320                               *
'*  A( 1) =   -322560                               *
'*  A( 2) =    564480                               *
'*  A( 3) =   -376320                               *
'*  A( 4) =    117600                               *
'*  A( 5) =    -18816                               *
'*  A( 6) =      1568                               *
'*  A( 7) =       -64                               *
'*  A( 8) =         1                               *
'*                                                  *
'* Laguerre polynomial coefficients for order 9     *
'*                                                  *
'*  A( 0) =    362880                               *
'*  A( 1) =  -3265920                               *
'*  A( 2) =   6531840                               *
'*  A( 3) =  -5080320                               *
'*  A( 4) =   1905120                               *
'*  A( 5) =   -381024                               *
'*  A( 6) =     42336                               *
'*  A( 7) =     -2592                               *
'*  A( 8) =        81                               *
'*  A( 9) =        -1                               *
'*                                                  *
'* Laguerre polynomial coefficients for order 10    *
'*                                                  *
'*  A( 0) =   3628800                               *
'*  A( 1) = -36288000                               *
'*  A( 2) =  81648000                               *
'*  A( 3) = -72576000                               *
'*  A( 4) =  31752000                               *
'*  A( 5) =  -7620480                               *
'*  A( 6) =   1058400                               *
'*  A( 7) =    -86400                               *
'*  A( 8) =      4050                               *
'*  A( 9) =      -100                               *
'*  A(10) =         1                               *
'*                                                  *
'****************************************************
' Laguerre Polynomials 
' --------------------
'
' Laguerre polynomials are defined over the interval 0 <= x <= inf. The weighting 
' function is the exponential w(x) = exp(-x): 
'
'          inf.
'          Sum exp(-x) Ln(x) Lm(x) dx = 0   for n <> m     (3.9.6)  
'           0
'	  	                      = 1   for n = m
'
' The recursion relation is 
'
'          (n + 1) Ln+l(x) = (2n + 1 - x) Ln(x) - nLn(x)   (3.9.7) 
'
'           where L0(x) = 1 
'	          L1(x) = 1 - x 
'
' Simply by changing a few lines in the LEGNDRE subroutine, a program for evaluating 
' Laguerre polynomial coefficients can be generated [see program Laguerre). 
'
' Note that the Laguerre polynomia1 coefficients are aIl integers, alternate in sign, 
' and have a leading coefficient  an = (-1)^n. . 
'------------------------------------------------------------------------------------
defint i-n
defdbl a-h,o-z
dim A(10),B(10,10)
  cls
  for n=2 to 10
    print
    print " Laguerre polynomial coefficients for order ";n
    print
    gosub 1000
    for k=0 to n
      print "   A(";k;") = ";A(k)
    next k
    print
    if n<10 then input r$
  next n
  print
end

'**************************************************
'* Laguerre polynomial coefficients evaluation by *
'* means of recursion relation. The order of the  *
'* polynomial is n. The coefficients are returned *
'* in A(i).                                       *
'**************************************************
1000 'Establish l0 and l1 coefficients
  B(0,0)=1 : B(1,0)=1 : B(1,1)=-1
  'Return if order is less than two
  if n<2 then return
  for i=2 to n
    B(i,0)=(2*i-1)*B(i-1,0)-(i-1)*(i-1)*B(i-2,0)
    for j=1 to i
      'Basic recursion relation
      B(i,j)=(2*i-1)*B(i-1,j)-B(i-1,j-1)-(i-1)*(i-1)*B(i-2,j)
    next j
  next i
  for i=0 to n
    A(i)=B(n,i)
  next i
return

'End of file Laguerre.bas