How to resolve the algorithm Faulhaber's triangle step by step in the REXX programming language

Published on 12 May 2024 09:40 PM
#Rexx

How to resolve the algorithm Faulhaber's triangle step by step in the REXX programming language

Table of Contents

Problem Statement

Named after Johann Faulhaber, the rows of Faulhaber's triangle are the coefficients of polynomials that represent sums of integer powers, which are extracted from Faulhaber's formula:

where

B

n

{\displaystyle B_{n}}

is the nth-Bernoulli number.

The first 5 rows of Faulhaber's triangle, are:

Using the third row of the triangle, we have:

∑

k

1

n

k

2

=

1 6

n +

1 2

n

2

1 3

n

3

{\displaystyle \sum _{k=1}^{n}k^{2}={1 \over 6}n+{1 \over 2}n^{2}+{1 \over 3}n^{3}}

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Faulhaber's triangle step by step in the REXX programming language

Source code in the rexx programming language

Numeric Digits 100
Do r=0 To 20
  ra=r-1
  If r=0 Then
    f.r.1=1
  Else Do
    rsum=0
    Do c=2 To r+1
      ca=c-1
      f.r.c=fdivide(fmultiply(f.ra.ca,r),c)
      rsum=fsum(rsum,f.r.c)
      End
    f.r.1=fsubtract(1,rsum)
    End
  End
Do r=0 To 9
  ol=''
  Do c=1 To r+1
    ol=ol right(f.r.c,5)
    End
  Say ol
  End
Say ''
x=0
Do c=1 To 18
  x=fsum(x,fmultiply(f.17.c,(1000**c)))
  End
Say k(x)
s=0
Do k=1 To 1000
  s=s+k**17
  End
Say s
Exit

fmultiply: Procedure
Parse Arg a,b
Parse Var a ad '/' an
Parse Var b bd '/' bn
If an='' Then an=1
If bn='' Then bn=1
res=(abs(ad)*abs(bd))'/'||(an*bn)
Return s(ad,bd)k(res)

fdivide: Procedure
Parse Arg a,b
Parse Var a ad '/' an
Parse Var b bd '/' bn
If an='' Then an=1
If bn='' Then bn=1
res=s(ad,bd)(abs(ad)*bn)'/'||(an*abs(bd))
Return k(res)

fsum: Procedure
Parse Arg a,b
Parse Var a ad '/' an
Parse Var b bd '/' bn
If an='' Then an=1
If bn='' Then bn=1
n=an*bn
d=ad*bn+bd*an
res=d'/'n
Return k(res)

fsubtract: Procedure
Parse Arg a,b
Parse Var a ad '/' an
Parse Var b bd '/' bn
If an='' Then an=1
If bn='' Then bn=1
n=an*bn
d=ad*bn-bd*an
res=d'/'n
Return k(res)

s: Procedure
Parse Arg ad,bd
s=sign(ad)*sign(bd)
If s<0 Then Return '-'
       Else Return ''

k: Procedure
Parse Arg a
Parse Var a ad '/' an
Select
  When ad=0 Then Return 0
  When an=1 Then Return ad
  Otherwise Do
    g=gcd(ad,an)
    ad=ad/g
    an=an/g
    Return ad'/'an
    End
  End

gcd: procedure
Parse Arg a,b
if b = 0 then return abs(a)
return gcd(b,a//b)

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