***Math proc templates for Prime Number generation and TCL demo example, numerical analysis***

This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks,[gold] 30Apr2021
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<<TOC>>
*** Introduction***
----
[gold] Here are some  Math proc templates for Prime Number generation. These Math proc templates are intended to be compatible with the format of existing subroutines in the TCLLIB:math library. The new Math proc should be checked in TCLLIB::math local copy before submission to TCLLIB ticket pipeline. Morfing these math proc templates into Prime Number generation subroutines should be feasible. I have found it easier and more tidy to start with a working TCL templates and morf a few lines into a new purpose.If one completes and tests  the new TCL subroutine in a console program, then usually one can load  the new TCL subroutine into a graphical user interface (gui).
----
***prime number formula '''< 16n+15 >'''  ***
----
Maybe coincidence, but '''sequence {7 11 13 14 }''' has been found on Sumerian  '''Tablet AO8862'''. The On-Line Encyclopedia of Integer Sequences '''A014311 Oeis {7 11 13 14}''' cited sequence property as Numbers n such that '''<16n+15>''' are prime. '''Tablet YBC10722''' uses the input quantity '''15''' with the sequence '''{7 11 13 14}'''!  [KWJ] in [Monster Prime Predicting Formula] has posted on the Wiki for this formula '''< 16n+15 >'''.
----
This prime number formula '''< 16n+15 >''' as a prime number filtering mechanism produces some primes and some duds.  [KWJ] reported the prime number formula '''< 16n+15 >''' had  an efficiency of '''0.166''' for filtering primes out of the numberline,for the  data set, '''<PLT500K>'''. For N > 2, about half of numbers on the numberline are even, which could never be defined as primes. If the formula added two even numbers, then only even numbers would result in the sum with no primes produced. If the formula added an  even number and odd number, then some  odd numbers might  result with some primes produced. 
--------
***Prime Numbers on Clay Tablets  ***
----

In some sense, the Plimpton 322  clay tablet is also generating a set of prime numbers in the available triplet collection on the clay tablet. The Babylonian regular numbers in base_60 between 0 and 60 have a limited set of triplets that satisfy 1< X-1/X < 1+sqrt(2) which appears to be the role of Plimpton 322. The limit 1< X-1/X < 1+sqrt(2) may explain the 1<X<2 limit on the Late  Babylonian many place reciprocal tablets. 
----
From the Plimpton 322 tablet,  Fernando Quinonez reported that the prime numbers of the set d  (column d) are coincedent with Gauss formula < 4n + 1>  and that prime numbers of set a (column a) are mixed forms,  including  the Gauss formula <4n + 1> and another formula < 4n + 3>. On the Plimpton 322 tablet, 8 of 15 readable numbers were prime numbers. The percentage of prime numbers was expr { (8./15.) * 100.}, 53.3 percent rounded.
----
While the modern mathematicians are trying to '''find''' prime numbers in base_10, the Babylonian mathematicians in base_60 were trying to '''avoid''' prime  numbers. The primes are the Achilles heel of the Babylonian multiplication system using reciprocals in base_60. The number set in the sequence '''{7 11 13 14}''' was especially foul to the Babylonian multiplication system, although the Babylonian mathematicians did use the set '''{7 11 13 14}''' in arithmetic or "advanced non-standard" calculations.  One trick that  the Babylonian mathematicians was to multiply an odd number or prime number by 2 or 3 at one early stage of the calculations and then divide out respectfully the 2 or 3 at a later stage of the calculations. For example, 7*2 converts an overtly prime number to a number 14 with factors 2*7.  However, this multiplication trick with 2 or 3 as factors  is confusing in the extant school and library tablets, so sometimes the modern reader is stressed by a missing factor of 2/3 after corrupted or dropped tablet lines.
----
%|table 1 Percentage Primes in Modern Base_10 and Babylonian Base_60 ||printed in| tcl wiki format|%  
&| quantity| value|  value|  comment, if any|& 
&| set_number:|percent of prime numbers in set | percent primes in set of 10 over / set of N | |&
&| set 10 :|44.44 |1.0    |  Modern base_10|&
&| set 20 :|42.10 |1.05    | Archaic IndoEuropean base_20|&
&| set 30 :|34.48 |1.28    | |&
&| set 40 :|30.76 |1.44   ||&
&| set 50 :|30.61 |1.45    ||&
&| set 60 :|28.81 |1.54    | Babylonian base_60 |&
&| set 70 :|27.53 |1.61   ||&
&| set 80 :|27.84 |1.59    ||&
&| set 90 :|26.96 |1.64    ||&
&| set 100 :|25.25 |1.76    ||&
&| set 110 :|26.60 |1.67  |  |&
&| set 120 :|25.21 |1.76  |  |&
 

----
======
             ;# using TCLLIB math prime subroutines
             ;# COUNTING  SETS OF  10 WITH  PERCENTAGE  OF  PRIMES RELATIVE  TO  BASE_10
             puts "%| set_number:|percent of prime numbers in set | |%"
             foreach value {10 20 30 40 50 60 70 80 90 100 110 120} {
             puts "&| set $value :|[ expr {  [llength [ primesLowerThan [- $value 1 ] ]]*100./  [- $value 1 ]  }] |[ expr { ([llength [ primesLowerThan [- 10 1 ] ]]*100./  [- 10 1 ] )/( [llength [ primesLowerThan [- $value 1 ] ]]*100./  [- $value 1 ])  }]  |  |&" }
======
--------
*** Applications for proc ::math::quantity_into_ratios  ***
---- 
----
[gold] 6/3/2021. Professor Joran Firberg outlined a Babylonian partner or brother inheritance problem for 7 brothers sharing the available silver of 100 shekels on tablet, IM31210P2, ref Friberg, A Remarkable Collection of Babylonian Mathematical Texts. Professor Firberg gives the solution as a  series of integer numbers < 16   14   12   10   8   6   4 > with the commentary on the Babylonian false shares algorithm. One can see that the sum of the shares sequence as expr { 16  + 14  + 12  + 10  +  8  + 6 +  4 } returns 70 silver shekels. For the TCL calculations, the problem constraints developed a normalized sequence of brother shares or multiple ratios,  starting with the eldest brother with largest share as normalized 1.  Each younger brother receives a share less or subtracted as rational fraction (1/8 ). In other words, subtract expr { 1/8. }  = 0.125 shekel for the next younger brothers share. The second eldest brother would receive  a normalized share of expr { 1.-1/8. }  = 0.875 shekels, Base_10 decimal.  The normalized sequence for each brothers share is < 1. 0.875 .75 .625 .5 .375 .25 > decimal notation. The sum of the normalized sequence for each brothers share is  expr {  1. + 0.875 +  .75 +  .625 +  .5 +  .375 + .25 }, 4.375. As an alternate algebraic gist of the false shares method, the equation < $false_share * 4.375 = 70 > transposes to $false_share = expr { 70/4.375 }, 16. Use the $false_share of 16 to multiply the normalized sequence for brother shares  and the share of eldest brother falls out as expr { 16*1 } = 16, and so on with the younger brothers.  Using some alternate calculations on market rates, I posted a Math proc Template called ::math::quantity_into_ratios on the wiki. For the initial available silver of 70 pieces,  eval  < ::math::quantity_into_ratios  70 1. 0.875 .75 .625 .5 .375 .25 > returns < 16.0 14.0 12.0 10.0 8.0 6.0 4.0 > silver pieces. As a check, the sum of all brothers shares is expr {16.0 + 14.0 + 12.0 + 10.0 + 8.0 + 6.0 + 4.0 }, 70. 
----
[gold] 6/3/2021. Professor Joran Firberg offered some commentary on a Babylonian partner or brother inheritance problem for 10 brothers sharing the available silver of 100 shekels on tablet, IM31210P7, ref Friberg, A Remarkable Collection of Babylonian Mathematical Texts.  The Babylonian mathematicians are showing considerable sophistication is solving some of the constraints with the arithmetic mean. The Babylonian text reported that total of first 3 shares was  46.8 shekels. Collateral inference was that the next to eldest brother share  was expr { 46.8 / 3 } = 15.6 shekels. This problem lacked some parts of a complete answer on the tablet, but believe that the available tablet content can be loaded into a provisional TCL solution in Base_10 decimal. For the TCL calculations, the problem constraints developed a normalized sequence of brother shares or multiple ratios,  starting with the eldest brother with largest share as normalized 1.  Each younger brother receives a share less or subtracted as   fraction 0.09302 decimal. In other words, subtract   0.093023   for the next younger brothers share. The second eldest brother would receive  a normalized share of expr { 1.- 0.09302  }  = 0.906977 , Base_10 decimal. The normalized sequence for each brothers share is < 1. 0.90698   0.8139  0.72094  0.62792  0.5349  0.44188  0.34886  0.25584  0.16282   > decimal notation. The sum of the normalized sequence for each brothers share is  expr {  1. + 0.90698 +  0.8139 + 0.72094 + 0.62792 + 0.5349 + 0.44188  0.34886 + 0.25584 + 0.16282    }, 5.8141. As an alternate algebraic gist of the false shares method, the equation < $false_share * 5.8141 = 100 > transposes to $false_share = expr { 100/5.8141 }, 17.1995, rounded to 17.2 shekels. Using some alternate calculations on market rates, I posted a Math proc Template called ::math::quantity_into_ratios on the wiki. For the initial available silver of 100 pieces,  eval  < ::math::quantity_into_ratios  100 1. 0.90698   0.8139  0.72094  0.62792  0.5349  0.44188  0.34886  0.25584  0.16282   > returns < 17.1997  15.5998  13.9988  12.3999  10.80006  9.20014  7.60022  6.0003  4.40038  2.80046   > silver pieces. As a check, the sum of all brothers shares is expr {17.1997 + 15.5998 + 13.9988 + 12.3999  +10.80006 + 9.20014 + 7.60022  + 6.0003 + 4.40038 + 2.80046}, 99.9997, rounds to 100. 

 
----
======
            ;# code snippet for printing IM31210P7 sequence
            set share 1.
            set sum 1.
            set answer [list  ]
            foreach val {1 2 3 4 5 6 7 8 9 }  {
            set share [ expr { $share - 0.09302 } ]
            set sum [ expr { $sum + $share  } ]
            lappend answer $share
            }
            puts " answer sequence =  1.  $answer "
            puts " sum $sum "
======
----

*** Quadratic Equation Solution from Muller's method***
----
Console wrapper for 2nd order quadratic_solution proc returns the real solution  1 or real solutions 1&2  from   quadratic coefficients. Looking for real roots only here, discriminant => 0.  Boiled down from evaluating discriminant, there are three possible outcomes from 2nd order quadratic equation as follows: 1) discriminant $discriminant_2 > 0 has two real roots. 2) discriminant $discriminant_2 == 0 has one real root. 3) discriminant $discriminant_2 < 0 has two complex non-real roots. There are some 2nd order solutions that have no real solutions. Note that Muller's method or rearranged 2C formula avoids significant figure problem (somewhat). TCL may use math ops to eval rational fractions as inputs to subroutine for more significant figures. First argument is 3 execution options.  option == 1 , return positive solution,  option == 2 , return negative and positive solutions , option == 3 , return  negative and positive solutions with diagnostics. Need some more testcases for working envelope. Should be loadable into a graphical user interface gui.
----
Possible sources of error in numerical analysis. References reported potential error in significant figures from subtracting nearly equal quantities in formula. Also, when the two real roots are very close, cancellation may occur between the terms b2 and 4ac of the discriminant. Possibly diagnostics in subroutine should flag when these conditions are approached?
----
*** Conclusions ***
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*** Subroutine Template 1, quantity_into_multiple_ratios ***
----
----
======
        # pretty print from autoindent and ased editor
        # quantity_into_ratios for math::stats? package
        # working under TCL version 8.6
        # console program written on Windows 10 on TCL
        # gold on TCL WIKI, 2Mar2017
        package require Tk
        console show
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        #
        # ::math::quantity_into_ratios --
        #
        # Return the division of quantity by  two or more given ratios
        #
        # Arguments:
        #    val  first value is quantity
        #    args other values are two or more given ratios
        #
        # Results: parts of quantity divided  by ratios
        #    
        proc ::math::quantity_into_ratios {quantity args} {
            set sum 0.
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            foreach val $args {
                set sum [ expr { $sum + $val } ]
            }
            foreach val $args {
                lappend answer [ expr { $quantity * ($val/$sum) } ]
            }
            set answer
        }
        # end of working deck
        # add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for quantity_into_ratios  proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for quantity_into_ratios proc"
        puts " Return the quantities from   list of ratios"
        puts "  [ ::math::quantity_into_ratios  84 2 2 2 ] "  
        puts " for  ::math::quantity_into_ratios 84 2 2 2  returns 28.0 28.0 28.0 "
        puts " for  ::math::quantity_into_ratios 84 1 2 4 returns [ ::math::quantity_into_ratios 84 1 2 4  ] "
        puts "  for  ::math::quantity_into_ratios 84 1 2 4   returns 12.0 24.0 48.0 "    
        puts " [ ::math::quantity_into_ratios 84 ] <<  for  ::math::quantity_into_ratios 84  returns zero "      
        #end of deck
======
----
----
*** Printout for quantity_into_ratios proc***
----
======
Console wrapper for quantity_into_ratios proc
 Return the quantities from   list of ratios
  28.0 28.0 28.0 
 for  ::math::quantity_into_ratios 84 2 2 2  returns 28.0 28.0 28.0 
 for  ::math::quantity_into_ratios 84 1 2 4 returns 12.0 24.0 48.0 
  for  ::math::quantity_into_ratios 84 1 2 4   returns 12.0 24.0 48.0 
 0 <<  for  ::math::quantity_into_ratios 84  returns zero 
======
 
----
*** Subroutine Template 2, sqrt_sum_of_squares ***
----
======
        # pretty print from autoindent and ased editor
        # easy eye calculator, large black type on green
        # used as testbed for one liners programs
        # written on Windows 10 on TCL
        # working under TCL version 8.6
        # gold on TCL WIKI , 10Sep2020
        package require Tk
        package require math::numtheory
        package require math::constants
        package require math::trig
        package require math
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::trig math::constants }
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        #
        #  sqrt_sum_of_squares --
        #
        # Return the sqrt_sum_of_squares by  one or more
        #
        # Arguments:
        #
        #    args other values are one or more
        #
        # Results: sqrt_sum_of_squares
        #
        proc math::sqrt_sum_of_squares { args} {
            set sum 0.
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            foreach val $args {
                set sum [ expr { $sum + $val*$val } ]
            }
            set answer [ expr { sqrt($sum) } ]
        }
        #puts "  ::math::sqrt_sum_of_squares (  2 2 2  )  answer [sqrt 12 ]
        #  3.4641016151377544
        #puts " [ ::math::sqrt_sum_of_squares   1 2 4  ]  "
        #puts " [ ::math::sqrt_sum_of_squares 2 ] for ( ) returns zero "
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for sqrt_sum_of_squares "}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  sqrt_sum_of_squares  2 2 2 equals [ math::sqrt_sum_of_squares  2 2 2 ]"
        puts "  sqrt_sum_of_squares  1 1 1 equals [ math::sqrt_sum_of_squares  1 1 1 ]"
        puts "  sqrt_sum_of_squares  1 1  equals [ math::sqrt_sum_of_squares  1 1  ]"
        puts "  sqrt_sum_of_squares  1  equals [ math::sqrt_sum_of_squares  1  ]"
        puts "  sqrt_sum_of_squares  0  equals [ math::sqrt_sum_of_squares  0  ]"
======
----
*** output for sqrt_sum_of_squares***
----
======
  sqrt_sum_of_squares  2 2 2 equals 3.4641016151377544
  sqrt_sum_of_squares  1 1 1 equals 1.7320508075688772
  sqrt_sum_of_squares  1 1  equals 1.4142135623730951
  sqrt_sum_of_squares  1  equals 1.0
  sqrt_sum_of_squares  0  equals 0.0
======
----
----
 
*** Pseudocode, Equations, and Wiki Page Checklist ***
====== 
     #pseudocode can be developed from rules of thumb.
     #pseudocode: some problems can be solved by proportions (rule of three), to some order of magnitude
     #pseudocode: enter quantity1,  quantity2, quantity3 and expected output (quantity4) for testcases.
     #pseudocode: enter time in years, number of remaining items
     #pseudocode: output fraction of (remaining items) over (items at time zero)
     #pseudocode: ouput remaining items as fraction or percent
     #pseudocode: output fraction of (quantity4 ) over ( quantity1 at time zero)
     #pseudocode: output fraction of (quantity2) * (quantity3 ) over (quantity1 at time zero)
     #pseudocode: outputs should be in compatible units.
     #pseudocode: rules of thumb can be 3 to 15 percent off, partly since g..in g..out.
     #pseudocode: need test cases > small,medium, giant
     #pseudocode: need testcases within range of expected operation.
     #pseudocode: are there any cases too small or large to be solved?
     
====== 
----
*** Subroutine Template 3, geometric_mean ***
----       
----
======
        # pretty print from autoindent and ased editor
        # geometric_mean for math::stats? package
        # working under TCL version 8.5.6 and eTCL 1.0.1
        # console program written on Windows XP on eTCL
        # gold on TCL WIKI, 2Mar2017
        package require Tk
        console show
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        console show
        # ::math::geometric_mean --
        #
        # Return the geometric_mean of two or more values
        #
        # Arguments:
        #    val  first value
        #    args other values
        #
        # Results:
        #    geometric_mean  arithmetic geometric_mean value
        proc ::math::geometric_mean { args } {
            set product 1.
            set N [ expr { [ llength $args ] } ]
            foreach val $args {
                set product [ expr { $product*$val } ]
            }
            set exponent [ expr { 1./$N } ]
            set geometric_mean [ expr { $product**$exponent } ]
            #puts " $N   $exponent $product  $args "
            set geometric_mean
        }
        ;# end of working deck
        ;# add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for ::math::geometric_mean  proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for ::math::geometric_mean proc"
        puts " Return the quantities from   list of ratios"
        puts "  ::math::geometric_mean ( 1 2 3 4  )  answer  2.2133638  "
        puts " [ ::math::geometric_mean 1 2 3 4  ]   answer  2.2133638  "
        
======
----
----
*** Subroutine Template 4, list of normalized_set ***
---- 
        ;# pretty print from autoindent and ased editor
        ;# normalized_set  for math::stats? package
        ;# working under TCL version 8.6
        ;# console program written on Windows 10 on TCL
        ;# gold on TCL Club, 26Jun2021
        package require Tk
        console show
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        ;# adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        ;#
        ;# ::math::normalized_set  --
        ;# 
        ;# Return the  list of normalized_set by  division of selected quantity 
        ;# Arguments:
        ;#    val  first value is selected quantity
        ;#    args other values are numbers to be normalized
        ;#
        ;#  Exemption 1, if input list has 1  or zero entries, return 0
        ;#       >> args of one element, meaning no quanities,  return 0
        ;#  Exemption 2, if selected quantity has zero entries,  return 0
        ;#       >> remove possibility of division by zero 
        ;# Results: set of normalized numbers divided  by selected quantity
        ;#    
        ;# $sum_? reserved for sums and averages of quantites, not used now
        ;#  
        proc ::math::normalized_set  {quantity args} {
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            if { $N == 0  } { return 0 }
            if { $quantity == 0  } { return 0 }
            set quantity [ expr { $quantity * 1. } ]
            set sum_entries 0.
            set sum_normalized 0.
            foreach val $args {
                set sum_entries [ expr { $sum_entries + $val } ]
                set sum_normalized [ expr { $sum_normalized + $val/$quantity } ]
            }
            set mean [ expr { $sum_entries * 0.5 } ]
            set average_normalized [ expr { $sum_normalized * 0.5 } ]
            ;# remove comment for sum and mean reports in printout
            ;# puts " sum_entries $sum_entries entries mean $mean sum_normalized $sum_normalized  normalized mean $average_normalized"
            foreach val $args {
                lappend answer [ expr { $val * (1./$quantity) } ]
            }
            return $answer
        }
        ;# check with printout >>>  puts "check with printout $quantity $args answer $answer"
        ;#  end of working deck
        ;#  add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for normalized_set   proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for normalized_set  proc"
        puts " ************************** "
        puts " Begin test suite "
        puts " ************************** "
        puts " Return the quantities from   list of normalized_set by division of selected quantity"
        puts "  [ ::math::normalized_set   2 2 2 2 ] "  
        puts " for  ::math::normalized_set  2 2 2 2  returns 1.0 1.0 1.0 "
        puts " for  ::math::normalized_set  2 1 2 4 returns [ ::math::normalized_set  2 1 2 4  ] "
        puts " for  ::math::normalized_set  2 1 2 4   returns 0.5 2.0 2.0 "
        puts " for  ::math::normalized_set  2 1 2 3 4 5 6 7 8 9 10   returns [ ::math::normalized_set  2 1 2 3 4 5 6 7 8 9 10 ]"
        puts " for  ::math::normalized_set  10. 1 2 3 4 5 6 7 8 9 10   returns [ ::math::normalized_set  2 1 2 3 4 5 6 7 8 9 10 ]"
        puts " [ ::math::normalized_set  2 ] <<  for args set of 1 element ::math::normalized_set  2  returns zero "   
        puts " [ ::math::normalized_set  0 ] <<  for  ::math::normalized_set  0  returns zero "    
        ;# end of deck   

----
----
***  Test Suite printout for list of normalized_set ***
---- 

======
 Console wrapper for normalized_set  proc
 ************************** 
 Begin test suite 
 ************************** 
 Return the quantities from   list of normalized_set by division of selected quantity
 sum_entries 6.0 entries mean 3.0 sum_normalized 3.0  normalized mean 1.5
  1.0 1.0 1.0 
 for  ::math::normalized_set  2 2 2 2  returns 1.0 1.0 1.0 
 sum_entries 7.0 entries mean 3.5 sum_normalized 3.5  normalized mean 1.75
 for  ::math::normalized_set  2 1 2 4 returns 0.5 1.0 2.0 
 for  ::math::normalized_set  2 1 2 4   returns 0.5 2.0 2.0 
 sum_entries 55.0 entries mean 27.5 sum_normalized 27.5  normalized mean 13.75
 for  ::math::normalized_set  2 1 2 3 4 5 6 7 8 9 10   returns 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
 sum_entries 55.0 entries mean 27.5 sum_normalized 27.5  normalized mean 13.75
 for  ::math::normalized_set  10. 1 2 3 4 5 6 7 8 9 10   returns 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
 0 <<  for args set of 1 element ::math::normalized_set  2  returns zero 
 0 <<  for  ::math::normalized_set  0  returns zero 
(System32) 1 % 
======  
----
*** Possible Errors, Predefined Entries, Break Lines, Possible Wild_Cards in TCLLIB::math local copy ***
----
======
                # Arguments:
                #     limit1            The limit for the largest prime to be included in the estimate
                #
                if { $limit1 <= 1 } {
                        return -code error "The limit must be larger than 1"
                }
                if { $limit2 <= 1 } {
                        return -code error "The upper limit must be larger than 1"
                }
                if { $n == 0 } {return 0}
                if { $n < 1  } {return 0}
                if { $n == 1 } {return 1}
                if {$n<2}      then {return -code return 2}
                if {$n<4}      then {return -code return 4}
                if {$n<9}      then {return -code return 9}
                #    forgivable error but no entry forwarded for calculation
                if { [ isprime $limit3 ] < 1 } {
                        break
                }
                # Add the additional procedures to TCLLIB::math local copy
                #
                proc ::math::numtheory::call {args} {uplevel catch [list $args]}
                proc ::math::numtheory::mathx { args } { set tcl_precision 17; puts [ expr [ expr { $args } ] ] }
                ;# translating Fortran and Qbasic control flow statements
                proc ::math::numtheory::stop {} { return }
                proc ::math::numtheory::end {}  { exit }
                proc ::math::numtheory::pause {} { wait .00010}
                ;# set comment lines from Fortran
                proc ::math::numtheory::C {args} { set C # }
                proc ::math::numtheory::REM {args}  { set REM # }     ;# attack of bots from Qbasic
                proc ::math::numtheory::write {args}  { puts $args }
                proc ::math::numtheory::print {args}  { puts $args }  ;# attack of bots from Quickbasic
                proc ::math::numtheory::SUBROUTINE {args}  { proc $args }
                proc ::math::numtheory::SUB  {args}  { proc $args }   ;# attack of bots from Quickbasic
                ;# namespace eval ::math::constants {
                variable constants {
                        pi        3.14159265358979323846   "ratio of circle circumference and diameter"
                        e         2.71828182845904523536   "base for natural logarithm"
                        ln10      2.30258509299404568402   "natural logarithm of 10"
                        phi       1.61803398874989484820   "golden ratio"
                        phi_2    0.61803398874989484820   "golden ratio conjugate"
                tcl_precision_1    17    "tcl_precision number"

                        )   set result {}
                        ;# set list for calculation results
                        lappend result $number
                }
                return $result
======

----
***Testcases Section***  

In planning any software, it is advisable to gather a number of testcases to check the results of the program.
----
**** Testcase 1 ****
----  
%|table testcase  1 | |printed in| tcl wiki format|%  
&| quantity| value| comment, if any|& 
&| set_number:|percent of prime numbers in set | percent primes in set of 10 over / set of N | |&
&| set 10 :|44.44 |1.0    |  Modern base_10|&
&| set 20 :|42.10 |1.05    | Archaic IndoEuropean base_20|&
&| set 30 :|34.48 |1.28    | |&
&| set 40 :|30.76 |1.44   ||&
&| set 50 :|30.61 |1.45    ||&
&| set 60 :|28.81 |1.54    | Babylonian base_60 |&
&| set 70 :|27.53 |1.61   ||&
&| set 80 :|27.84 |1.59    ||&
&| set 90 :|26.96 |1.64    ||&
&| set 100 :|25.25 |1.76    | |&
&| set 110 :|26.60 |1.67  |  |&
&| set 120 :|25.21 |1.76  |    |&


----
***Screenshots Section***
----
***Screenshots Section***
****figure 1a. Console wrapper for prime_formula proc ****
[Prime_Number_generation]
----
****figure 1b, Console wrapper for sqrt_sum_of_squares ****
[console_wrapper]
----
****figure 1c, Console wrapper for sqrt_sum_of_squares ****
----
[PERCENTAGE_PRIMES_RELATIVE]
----
----
****figure 1d, Console wrapper for normalized_set ****
----
[Math_proc_templates_normalized_set]
----
****figure 1e, Market_Rate_Problem  ****
----
[Market_Rate_Problem]
----
****figure 1e, Gauss Approximate Number of Primes  ****
----
[gauss approximation of number of primes etcl caluclator PNG]
****figure 2.****
[Gauss Approximate Number of Primes and eTCL demo example calculator graph approx no primes]
****figure 3.****
[Gauss Approximate Number of Primes and eTCL demo example graphr]
****figure 4.****
[Gauss Approximate Number of Primes and eTCL demo example grapSSS]
****figure 5.****
[Gauss Approximate Number of Primes and eTCL LEGENDRE APPROXIMATION]
****figure 6.****
[Gauss Approximate Number of Primes and eTCL demo example absolute error]
****figure 7.****
[Gauss Approximate Number of Primes and eTCL large scale absolute error]
****figure 8.****
[Gauss Approximate Number of Primes and eTCL demo example relative error]
****figure 9.****
 
****figure 10.****
[Gauss Approximate Number of Primes regular legendre error]
****figure 11.****
[Gauss Approximate Number of Primes modified legrendre error]
****figure 12.****
[Gauss Approximate Number of Primes riemann traces]
----
***References:***
   *   [Primes]
   *   [Oneliner's Pie in the Sky] 
   *   [One Liners] 
   *   [Comparing Tcl and Python]
   *   [Primes Sieve - on a resizable canvas] from  [uniquename]
   *   [TCLLIB] ::math::numtheory 
   *   https://en.wikipedia.org/wiki/Prime-counting_function
   *   http://sweet.ua.pt/tos/primes.html#i
   *   http://mathworld.wolfram.com/PrimeCountingFunction.html
   *   http://www.math.uni-bonn.de/people/jbuethe/topics/AnalyticPiX.html
   *   http://primes.utm.edu/notes/rh.html
   *   http://numbers.computation.free.fr/Constants/Primes/pixtable.html
   *   http://numbers.computation.free.fr/Constants/constants.html
   *   on line prime number generator, see
   *   http://www.miniwebtool.com/list-of-prime-numbers/?to=50
   *   Mathematician pair find prime numbers aren't as random as thought, March 15, 2016 by Bob Yirka,
   *   ref Robert Lemke Oliver and Kannan Soundararajan, Stanford University 
   *   google < Prime number theorem Wikipedia >
   *   Probability that a number is prime, John D. Cook, 6 October 2010 
   *   Probability of Next Prime Digit, Kevin's Math Page, Kevin Gong,March 17th, 2016
   *   Mathematicians Discover Prime Conspiracy, Erica Klarreich , March 13, 2016, Quanta Magazine
   *   Prime Probability: Catch the Wave, Michael M. Ross , Excel spreadsheet, Jan2008
   *   Prime Numbers and Their Statistical Properties,Dr. Phillip M. Feldman
   *   19TH CENTURY MATHEMATICS - GAUSS, The Story of Mathematics 
   *   A Conjecture about the Density of Prime Numbers,  
   *   Lincoln Almir AMARANTE RIBEIRO, Departamento de Física, 
   *   Universidade Federal de Minas Gerais, good explanation 
   *   Dan Rockmore Stalking the Riemann Hypothesis,  The Quest to Find the Hidden Law of Prime Numbers
   *   J. Pintz. On Legendre's prime number formula. Amer. Math. Monthly 87 (1980)
   *   Dusart, Pierre. Estimates of some functions over primes without r.h 
   *   Ref. proc sum_primes_to by [dkf]
   *   from  [Comparing Tcl and Python] on TCL WIKI
   *   Ref. Sieve of Eratosthenes « Python recipes « = code.activestate.com/recipes/117119/ 
----
 
----
 
***References:***
   * Wikipedia search engine < TCL programs  >
   * Wikipedia search engine < TCL introduction  >
   * Wikipedia search engine <  prime number >
   * Google search engine < prime number formulas  >
   * [Seaching for Babylonian Triplets Slot Calculator Example]
   * [Gauss Approximate Number of Primes and eTCL demo example calculator, numerical analysis]    
   * [KWJ] in [Monster Prime Predicting Formula] 
----
   * http://scidiv.bcc.ctc.edu/math/pythagoras.html
   * http://www.saltire.com/applets/pythag/incircle.html
   * http://math.uic.edu/~fields/puzzle/triples.html
   * http://rle-vlsi.mit.edu/people/gjcoram/triples.html
   * http://bsuvc.bsu.edu/home/00lcsapp/maths460.html
   * http://www.math.utah.edu/~alfeld/teaching/pt.html
   * http://138.25.60.81/numericon/triples.html
   * http://www.geom.umn.edu/~demo5337/Group3/ti4.html
   * http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Babylonian_Pythagoras.html
   * Daniel F. Mansfield , N.J. Wildberger.
   * Plimpton 322 is Babylonian exact sexagesimal trigonometry.
   * Historia Mathematica, August 2017 DOI: 10.1016/j.hm.2017.08.001 
   * Newsweek, Babylonian Tablet Could Hold Mathematical 
   * Secrets For Today's Researchers,  Josh Lowe ,  8/25/17 


----
    * Unexpected biases in the distribution of consecutive primes
    * Robert J. Lemke Oliver, Kannan Soundararajan ,Department of Mathematics, Stanford University
    * The Last Digit of Prime Numbers - Numberphile on Youtube
    * Lecture on “Rock, paper, scissors in probability,  Tadashi Tokieda
    * Hardy-Littlewood conjectures.
    * MATHEMATICS SCIENCE, A new pattern in primes, Shubashree Desikan, Hindu Times
    * en.wikipedia.org wiki search on Tadashi_Tokieda
    * https://www.ncbi.nlm.nih.gov articles/PMC4978288/
    * Distribution of the units digit of primes, C. M. Ko, National Central University
    * www.wired.com tadashi-tokieda-finds-startling-objects-in-plain-sight/
----
*** Updating Plimpton 322, Babylonian tablet References: ***
----
    * The Algorithm of Factor 12:
    * Generating the Information Carved on the Plimpton 322 Tablet.
    * September 20, 2011,  Anthony Phillips, Stony Brook University
    * Fernando Quinonez, mentioned available prime numbers on tablet.
    * Robert Creighton Buck. Sherlock Holmes in Babylon. Am. Math. Mon., 87:335 – 345, 1980.
    * Buck cited prime numbers on Plimpton 322.
    * Victor Katz, Robin Wilson - Sherlock Holmes in Babylon and Other Tales of Mathematical History (Spectrum) (2004)
    * The Plimpton 322 Tablet and the Babylonian Method of Generating  Pythagorean Triples
    * by Abdulrahman Ali Abdulaziz, date 2010-03-31
    * Words and Pictures: New Light on Plimpton 322
    * ELEANOR ROBSON, American Mathematical Monthly 109 (2002), 105-120
    * A history of mathematics, Carl Boyer, page 36-37 
    * Note. Some above  B. math references are successfully locked behind paywalls.
---- 
======
;# References 
;# current TCLLIB routines in Math Module
math::numtheory::numberPrimesGauss N
math::numtheory::numberPrimesLegendre N
math::numtheory::numberPrimesLegendreModified N
math::numtheory::differenceNumberPrimesLegendreModified lower upper
math::numtheory::listPrimePairs lower upper step
math::numtheory::listPrimeProgressions lower upper step
======
----

----
**Appendix Code**

***appendix TCL programs and scripts ***
----
*** Subroutine Template 4, prime_formula ***
----
====== 
        # pretty print from autoindent and ased editor
        # easy eye format, large black type on green
        # used as testbed for one liners programs
        # written on Windows 10 on TCL
        # working under TCL version 8.6
        # gold on TCL WIKI , 5/15/2021
        package require Tk
        package require math::numtheory
        package require math::constants
        package require math::trig
        package require math
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::trig math::constants }
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        #
        #  prime_formula --
        #
        # Return the possible primes in list by  one or more
        #
        # Arguments:
        #
        #    args other values are one or more
        #
        # Results: prime_formula
        #
        proc prime_formula { args} {
            set list_new [list  ]
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            foreach val $args {
                lappend list_new [ expr {  16*$val + 15. } ]
            }
            return $list_new
        }
        # end of initial deck
        # add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for prime_formula proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for prime_formula proc"
        puts " Return the possible primes in list "
        puts " formula < 7 11 13 14 >   equals possible primes? [  prime_formula 7 11 13 14 ]"
                             
======
----
----
*** Subroutine Template 5, Combined Market Rate formula ***
----
----
======
        ;# pretty print from autoindent and ased editor
        ;# easy eye console, large black type on green
        ;# used as testbed for one liners programs
        ;# written on Windows 10 on TCL
        ;# working under TCL version 8.6
        ;# gold on TCL WIKI , 10Sep2020
        ;# Babylonian Combined Market Rate V6
        package require Tk
        package require math::numtheory
        package require math::constants
        package require math::trig
        package require math
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::trig math::constants }
        ;# adapted from tcl Stats 2011-05-22, arithmetic mean  <RLE>
        ;# ::math::combined_market_rate --
        ;#
        ;# Return the combined_market_rate by  one,two, or more given rates
        ;# market rate defined as quantity per price
        ;# or 1 over (price per quantity)
        ;#
        ;# Arguments:
        ;#
        ;#    args  values are one, two, or more given rates
        ;#
        ;# Results: combined_market_rate
        ;#  works for positive numbers, negative numbers,
        ;#  and mixed positive & negative numbers.
        ;#  arg of zero returns zero
        ;#  arg of null returns zero
        ;#  filter foreach drops irregular zero elements from argument
        proc ::math::combined_market_rate { args} {
            set sum 0.
            set N [ expr { [ llength $args ] } ]
            if { $N == 0 } { return 0 }
            if { $N == 1 || [ lindex $args 0 ] == 0 } { return 0 }
            set res {};set counter2 0;
            ;# filter foreach drops irregular zero elements
            foreach item $args {if {$item != 0 } {incr counter2 1; lappend res $item } }
            set counter 0
            foreach val $res {
                set sum [ expr { $sum + 1./$val } ]
                incr counter 1
            }
            set combined_market_rate1 [ expr { 1./(($sum*1.)/$counter2) } ]
            return $combined_market_rate1
        }
        ;# various testcases on combined_market_rate
        ;# puts  [::math::combined_market_rate  0.5524 0.4807 0.42918 0.47846 ]
        ;#  answer 0.48131    "
        ;# puts   [ ::math::combined_market_rate .1 .2 .3 .4 ]
        ;#  answer 0.192
        ;# ::math::combined_market_rate -.1 -.2 -.3 -.4
        ;# answer -0.192, correct
        ;# operator math formula follows
        ;# check [/ 1. [/ [+ [/ 1. -.1] [/ 1. -.2] [/ 1. -0.3 ] [/ 1. -0.4] ] 4. ] ]
        ;# returns  -0.192, correct
        ;# puts " [ ::math::combined_market_rate .1 ] "
        ;# :math::combined_market_rate -.1 -.2  .3 .4
        ;# answer -0.4363636363636364
        ;# operator math formula follows
        ;# set check [/ 1. [/ [+ [/ 1. -.1] [/ 1. -.2] [/ 1.  0.3 ] [/ 1. 0.4] ] 4. ] ]
        ;# check equals    -0.4363636363636364
        ;# puts " for (::math::combined_market_rates .1)
        ;# returns .1 "
        ;# ::math::combined_market_rate  {}
        ;# null returns zero, correct.
        ;# ::math::combined_market_rate 0
        ;# arg 0 returns zero, correct.
        ;# addition dated 24sep2018
        ;# added filter foreach to remove zero's
        ;# irregular zeros,
        ;# test on zero's 0.1 0.0 0.0 0.2 0.3 0.4
        ;# returns 0.192 , correct
        ;# test on zero's   0.1 0.0 0.0 0.2 0.3 0.4
        ;# returns 0.192 , correct      
        # end of initial deck
        # add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for combined market rate proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for combined market rate proc"
        puts " Return the combined market rate from  list "
        puts " formula < 0.5524 0.4807 0.42918 0.47846  >   equals market rate? [ ::math::combined_market_rate  0.5524 0.4807 0.42918 0.47846 ]"
        puts " formula <  7. 11. 13. 14.  >   equals market rate? [ ::math::combined_market_rate 7. 11. 13. 14.  ]"
====== 
----
----
*** Subroutine Template 6,  Quadratic Equation Solution, Muller's method***
----
======
        # pretty print from autoindent and ased editor
        # quadratic_solution for math::stats? package
        # working under TCL version 8.6
        # console program written on Windows 10 on TCL
        # gold on TCL Club, 2Mar2021
        # note rearranged 2C formula avoids significant figure problem
        package require Tk
        console show
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        #
        # ::math::quadratic_solution  
        #
        # Return the possible real solutions of quadratic equation from 3 coefficients
        #
        # Arguments:
        #    val  first value is option
        #    args other values are 3 coefficients of quadratic equation
        #
        # Results: 2 solutions, positive and negative
        #    looking for real roots only here, discriminant => 0  
        # option == 1 , return positive solution   
        # option == 2 , return negative and positive solutions 
        # option == 3 , return diagnostics
        proc ::math::quadratic_solution {option args} {
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            set option_1 $option
            set answer [list ]
            set a [lindex  $args 0 ]
            set b [lindex  $args 1 ]
            set c [lindex  $args 2 ]         
            set denom_term_1 [ expr { $b*$b-4.*$a*$c } ] 
            set discriminant_2 [ expr { sqrt ($denom_term_1) } ]  
            ;# looking for real roots only here, discriminant => 0 
            if { $discriminant_2 < 0  } { return 0 }
            set solution_negative [ expr { ( 2.*$c)/( $discriminant_2 - $b) } ]
            set solution_positive [ expr { ( 2.*$c)/( -1.*$discriminant_2 - $b) } ]
            if { $option_1 == 1  } { lappend answer $solution_positive }
            if { $option_1 == 2  } { lappend answer  $solution_negative $solution_positive } 
            if { $option_1 == 3  } {
            lappend answer  $solution_negative $solution_positive
            puts " printout  diagnostics "
            puts " option number $option_1 ???  =? 3 returns  diagnostics "
            puts " a [lindex   $args 0  ] "
            puts " b  [lindex   $args 1  ] "
            puts " c [lindex   $args 2  ]  " 
            puts " discriminant_2 $discriminant_2 "
            if { $discriminant_2 > 0  } {
            puts "discriminant $discriminant_2 > 0 has two real roots "}
            if { $discriminant_2 == 0  } {
            puts "discriminant $discriminant_2 > 0 has one real root "}
            if { $discriminant_2 < 0  } {
            puts "discriminant $discriminant_2 < 0 has two complex non-real roots  "}                          }  
            return $answer          
        }
        # end of working deck
        # add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for quadratic_solution  proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for quadratic_solution proc"
        puts " Return the quadratic_solution 1 or 1&2  from   quadratic coefficients"
        puts " for  ::math::quadratic_solution  1 1 1.5 -1. returns  [ ::math::quadratic_solution  1 1 1.5 -1. ] "  
        puts " for  ::math::quadratic_solution 2 1 1.5 -1.  returns  [ ::math::quadratic_solution 2 1 1.5 -1.] "
        puts " for  ::math::quadratic_solution  1 1 2.66666 -1 returns [ ::math::quadratic_solution 1 1 2.66666 -1  ] "
        puts " for  ::math::quadratic_solution  2 1 2.66666 -1 returns [ ::math::quadratic_solution 2 1 2.66666 -1  ] "
        puts " for  ::math::quadratic_solution  1 200. -1.5e-12 -1 returns [ ::math::quadratic_solution 1 1 200. -1.5e-12  ] "
        puts " for  ::math::quadratic_solution  2 200. -1.5e-12 -1 returns [ ::math::quadratic_solution 2 1 200. -1.5e-12  ] "
        puts " for  ::math::quadratic_solution 1  returns zero returns [ ::math::quadratic_solution 1 ]  " 
        puts " option 3 prints out quadratic diagnostics  " 
        puts " for  ::math::quadratic_solution  3 200. -1.5e-12 -1 & diagnostics returns [ ::math::quadratic_solution 3 1 200. -1.5e-12  ] "
        puts "  TCL may use math ops to eval rational fractions as inputs to subroutine for more significant figures. "
        puts " for  ::math::quadratic_solution  3 1 <3./2.> -1. returns  [ ::math::quadratic_solution  3 1 [/ 3. 2.] -1. ] "
        puts " for  ::math::quadratic_solution  3 1 <8./3.> -1 returns [ ::math::quadratic_solution 3 1 [/ 8. 3.] -1  ] "
        #end of deck
======
----
*** Printout for Subroutine Template 6,  Quadratic Equation Solution, Muller's method***
======
Console wrapper for quadratic_solution proc
 Return the quadratic_solution 1 or 1&2  from   quadratic coefficients
 for  ::math::quadratic_solution  1 1 1.5 -1. returns  0.5 
 for  ::math::quadratic_solution 2 1 1.5 -1.  returns  -2.0 0.5 
 for  ::math::quadratic_solution  1 1 2.66666 -1 returns 0.3333340000012 
 for  ::math::quadratic_solution  2 1 2.66666 -1 returns -2.9999940000012004 0.3333340000012 
 for  ::math::quadratic_solution  1 200. -1.5e-12 -1 returns 7.4999999999999996e-15 
 for  ::math::quadratic_solution  2 200. -1.5e-12 -1 returns -105.553116266496 7.4999999999999996e-15 
 for  ::math::quadratic_solution 1  returns zero returns 0  
 option 3 prints out quadratic diagnostics  
 printout  diagnostics 
 option number 3 ???  =? 3 returns  diagnostics 
 a 1 
 b  200. 
 c -1.5e-12  
 discriminant_2 200.00000000000003 
discriminant 200.00000000000003 > 0 has two real roots 
 for  ::math::quadratic_solution  3 200. -1.5e-12 -1 & diagnostics returns -105.553116266496 7.4999999999999996e-15 
  TCL may use math ops to eval rational fractions as inputs to subroutine for more significant figures. 
 printout  diagnostics 
 option number 3 ???  =? 3 returns  diagnostics 
 a 1 
 b  1.5 
 c -1.  
 discriminant_2 2.5 
discriminant 2.5 > 0 has two real roots 
 for  ::math::quadratic_solution  3 1 <3./2.> -1. returns  -2.0 0.5 
 printout  diagnostics 
 option number 3 ???  =? 3 returns  diagnostics 
 a 1 
 b  2.6666666666666665 
 c -1  
 discriminant_2 3.3333333333333335 
 discriminant 3.3333333333333335 > 0 has two real roots 
 for  ::math::quadratic_solution  3 1 <8./3.> -1 returns -2.9999999999999987 0.33333333333333331 
(System32) 1 % 
======
---- 
**Hidden Comments Section**

<<discussion>>
Please place any comments here, Thanks, [gold] 30Apr2021

                              
<<categories>> Numerical Analysis | Toys | Calculator | Mathematics| Example| Toys and Games | Games | Application | GUI
----
<<categories>> Development | Concept| Algorithm