(50g) Bernoulli polynomials

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(50g) Bernoulli polynomials - peacecalc - 12-17-2018 11:31 AM

Hello HP50g-fans,

I made a program for Bernoulli-polynomials ( = BP), it works with three equations:

\[ B_o(x) = 1 ~~~~~~~ \text{start}\]

\[ B_n(x) = n \cdot \int B_{n-1}(t)dt ~~~~~~~ \text{recursive definition}\]

\[ \int_0^1 B_n(t)dt = 0 ~~~~~~~~ \text{ calculate the constant from second equation}\]

Therefore we have polynomials it is easy to program and make use of CAS of the HP50g.
The program "BCALC" has to possibilities to work:

a) with one parameter "N", then it will calculate all BP from 0 to N, but it only stores the last N'th BP.

b) with two parameters "K" to "N". Let us say you calculated BP for N = 5, then you decide to calculate further on to N = 7. If you don't delete the global variables "BN1" and "BN2", you can input 6 and 7 and you get BP for N=7 from N = 5 as first equation.
That is a advantage in time, because the CAS operations are very slow.

To let the program know how many parameters I input, I use a list with one or two elements.

The program uses only one integration because the second one (stored in "BN2") prepares the BP for "N+1" (again time saving). I only make comments for the first part, I hope the rest of the program is understandable for you.

Code:

%%HP: T(2)A(R)F(.);
« OBJ->  
  IF 2. ==
  THEN \-> K N
    « K N FOR J 
            BN2 J * 'C' + EVAL DUP 'BN1' STO   @@RCL the prepared BP 
                                                            @@ and multiplied with J 
                                                            @@ add the constant "C" and stored
                                                            @@in BN1 temporally
            INTVX EVAL DUP 'BN2' STO            @@ integrated BP and stored
                                                            @@in BN2 temporally
            'X=1' SUBST EVAL 0 = 'C' ISOL DUP @@ set the upper limit 1 and calculate
                                                            @@ the constant "C"
            BN1 SWAP SUBST EVAL FDISTRIB   @@ RCL the BN1 und substitute the "C"
            'BN1' STO                                   @@ with its number and store it in BN1
            BN2 SWAP SUBST EVAL 'BN2' STO   @@ RCL the BN2 und substitute the "C"
                                                            @@ with its number and store it in BN2
                                                            @@ as the prepared BP of J + 1
            NEXT
    »
  ELSE \-> N
    « B0 INTVX 'C' + DUP 'BN1' STO 
      INTVX DUP 'BN2' STO 
      'X=1' SUBST EVAL 0 = 'C' ISOL DUP
      BN1 SWAP SUBST EVAL 'BN1' STO 
      BN2 SWAP SUBST EVAL 'BN2' STO

      IF N 1 > 
      THEN 
          2 N FOR J 
               BN2 J * 'C' + EVAL DUP 
               'BN1' STO INTVX EVAL DUP 
               'BN2' STO 'X=1' SUBST EVAL 0 = 'C' ISOL DUP 
               BN1 SWAP SUBST EVAL FDISTRIB 
               'BN1' STO 
               BN2 SWAP SUBST EVAL 
               'BN2' STO
        NEXT
      END
    »
  END
»


Enjoy it and tell me your improvements and critics.

SORRY SORRY SORRY, dear Mr. Admin move this thread to "General Software Libary".

greetings
peacecalc