{***************************************************
*   Program to demonstrate Laguerre coefficients   *
* ------------------------------------------------ *
* Reference: BASIC Scientific Subroutines, Vol. II *
* By F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981 [1].*
*                                                  *
*           Pascal Version by J.-P. Moreau, Paris. *
*                     (www.jpmoreau.fr)            *
* ------------------------------------------------ *
* 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. . 
------------------------------------------------------------------------------------}
PROGRAM Laguerre;
Uses WinCrt;

Type
        real_ar = DOUBLE;

Var
        A : Array[0..10] of real_ar;
        B : Array[0..10,0..10] of real_ar;
        n, k : INTEGER;


{**************************************************
* Laguerre polynomial coefficients evaluation by  *
*'* means of recursion relation. The order of the *
* polynomial is n. The coefficients are returned  *
*'* in A(i).                                      *
**************************************************}
PROCEDURE Laguerre_Coeff;
Var i,j : integer;
Begin
  {Establish l0 and l1 coefficients}
  B[0,0]:=1.0 ; B[1,0]:=1.0 ; B[1,1]:=-1.0;
  {Return if order is less than two}
  if n>1 then
  begin
    for i:=2 to n do
    begin
      B[i,0]:=(2*i-1)*B[i-1,0]-(i-1)*(i-1)*B[i-2,0];
      for j:=1 to i do
        {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];
    end;
    for i:=0 to n do A[i]:=B[n,i]
  end
End;


{main program}
BEGIN
  clrscr;
  for n:=2 to 10 do
  begin
    writeln;
    writeln(' Laguerre polynomial coefficients for order ',n);
    writeln;

    Laguerre_Coeff;

    for k:=0 to n do writeln('   A(',k:2,') = ',A[k]:9:0);
    if n<10 then ReadKey
  end;
  ReadKey; DoneWinCrt
END.

{End of file laguerre.pas}