Most programmers are familiar with the inexactness of floating point calculations in a binary processor.

The classic example being:

0.1 + 0.2 =  0.30000000000000004

In many situations the amount of error in such calculations is very small and can be overlooked or eliminated with rounding.

There are pathological problems however, where seemingly simple, straight-forward calculations are extremely sensitive to even tiny amounts of imprecision.

This task's purpose is to show how your language deals with such classes of problems.


A sequence that seems to converge to a wrong limit.

Consider the sequence:

  v1 = 2 
  v2 = -4 
  vn = 111   -   1130   /   vn-1   +   3000  /   (vn-1 * vn-2) 


As   n   grows larger, the series should converge to   6   but small amounts of error will cause it to approach   100.



Task 1

Display the values of the sequence where   n =   3, 4, 5, 6, 7, 8, 20, 30, 50 & 100   to at least 16 decimal places.

    n = 3     18.5
    n = 4      9.378378
    n = 5      7.801153
    n = 6      7.154414
    n = 7      6.806785
    n = 8      6.5926328
    n = 20     6.0435521101892689
    n = 30     6.006786093031205758530554
    n = 50     6.0001758466271871889456140207471954695237
    n = 100    6.000000019319477929104086803403585715024350675436952458072592750856521767230266



Task 2

The Chaotic Bank Society   is offering a new investment account to their customers.

You first deposit   $e - 1   where   e   is   2.7182818...   the base of natural logarithms.

After each year, your account balance will be multiplied by the number of years that have passed, and $1 in service charges will be removed.

So ...

 after 1 year, your balance will be multiplied by 1 and $1 will be removed for service charges.
 after 2 years your balance will be doubled and $1 removed.
 after 3 years your balance will be tripled and $1 removed.
  ... 
 after 10 years, multiplied by 10 and $1 removed, and so on. 


What will your balance be after   25   years?

   Starting balance: $e-1
   Balance = (Balance * year) - 1 for 25 years
   Balance after 25 years: $0.0399387296732302



Task 3, extra credit

Siegfried Rump's example.   Consider the following function, designed by Siegfried Rump in 1988.

  f(a,b) = 333.75b6 + a2( 11a2b2 - b6 - 121b4 - 2 ) + 5.5b8 + a/(2b) 
  compute    f(a,b)    where    a=77617.0    and    b=33096.0 
  f(77617.0, 33096.0)   =   -0.827396059946821 


Demonstrate how to solve at least one of the first two problems, or both, and the third if you're feeling particularly jaunty.



See also;
   Floating-Point Arithmetic   Section 1.3.2 Difficult problems.


Imports System.Globalization
Imports System.Numerics
Imports Numerics
 
#Const USE_BIGRATIONAL = True
#Const BANDED_ROWS = True
#Const INCREASED_LIMITS = True
 
#If Not USE_BIGRATIONAL Then
' Mock structure to make test code work.
Structure BigRational
    Overrides Function ToString() As String
        Return "NOT USING BIGRATIONAL"
    End Function
    Shared Narrowing Operator CType(value As BigRational) As Decimal
        Return -1
    End Operator
End Structure
#End If
 
Module Common
    Public Const FMT_STR = "{0,4}   {1,-15:G9}   {2,-24:G17}   {3,-32}   {4,-32}"
    Public ReadOnly Property Headings As String =
        String.Format(CultureInfo.InvariantCulture,
                      FMT_STR,
                      {"N", "Single", "Double", "Decimal", "BigRational (rounded as decimal)"})
 
    <Conditional("BANDED_ROWS")>
    Sub SetConsoleFormat(n As Integer)
        If n Mod 2 = 0 Then
            Console.BackgroundColor = ConsoleColor.Black
            Console.ForegroundColor = ConsoleColor.White
        Else
            Console.BackgroundColor = ConsoleColor.White
            Console.ForegroundColor = ConsoleColor.Black
        End If
    End Sub
 
    Function FormatOutput(n As Integer, x As (sn As Single, db As Double, dm As Decimal, br As BigRational)) As String
        SetConsoleFormat(n)
        Return String.Format(CultureInfo.CurrentCulture, FMT_STR, n, x.sn, x.db, x.dm, CDec(x.br))
    End Function
 
    Sub Main()
        WrongConvergence()
        Console.WriteLine()
        ChaoticBankSociety()
 
        Console.WriteLine()
        SiegfriedRump()
 
        SetConsoleFormat(0)
    End Sub
End Module