Updated 2017-11-10 20:54:33 by gold

## Babylonian doubling and halving algorithm for reciprocals , eTCL demo example calculator, numerical analysis edit

gold Here is some eTCL starter code for Babylonian doubling and halving algorithm for reciprocals in calculator shell. For the Babylonian scribes, the standard table gave reciprocals of regular numbers between 2 and 1_21 (decimal 81) as reciprocal pairs. There were a number of ways the reciprocals could be calculated, but the easiest way is called doubling and halving. An eTCL calculator was adapted to output the decimal equivalent of reciprocal pairs.with contraints, mainly to study target number and factor selection on reciprocal algorithms. Additional console programs below are used to check or improve subroutine.

In the cuneiform math problems and coefficient lists on clay tablets, there are coefficient numbers which were used in determining the amount of materials and the daily work rates of the workers. In most cases, the math problem is how the coefficient was used in estimating materials, work rates, and math problems. One difficulty is determining the effective magnitude or power of the number coefficient in the base 60 notation. In cuneiform, numbers in base 60 are written using a relative notation. For example, 20 could represent either 20*3600,20,20/60, 20/3600, or even 1/20. The basic dimensions and final tallies were presented in the cuneiform accounts on clay tablets, but some calculations and some units were left off the tablet. On the particular algorithms that use factoring, the relative value or uncertain magnitude makes a big difference because the factors of 20 and 1/20 are not the same. At least one approach for the modern reader and using modern terminology is to develop the implied algebraic equations and decimal equivalents from the cuneiform numbers. Then the eTCL calculator can be run over a number of testcases to validate the algebraic equations.

The doubling and halving algorithm for reciprocals was loaded into a console program below. Taking a known reciprocal pair of n and 1/n, the double target number has a (1/2)*reciprocal of the original in proportion. From modern notation, the new pair of reciprocals are 2*n and 1/(2*n). Taking a known reciprocal pair of n and 1/n, the triple target number has a (1/3)*reciprocal of the original in proportion. Using modern notation, the new pair of reciprocals are 3*n and 1/(3*n). Generally, the notation is a*n and 1/(a*n), for a as a constant. The resulting reciprocals from doubling and halving procedure in eTCL code seemed to work as a robust solution for positive integers, floating point numbers, and decimal fractions.
```         # take a known reciprocal pair of 2 and .5, set result from proportions
set target_a 2.
set reciprocal .5
set doubling_halving_reciprocal [* [/ \$target_a \$target_number  ] \$reciprocal  ]```

## Pseudocode Section edit

```           # using pseudocode for Babylonian doubling and halving algorithm
# for reciprocals
# irregular defined as not in standard B. table of reciprocals
supplied values
lower limit
upper limit
start number
factor
standard table  N <  81
1,2,4,8,16,32,64
3,6,12,24,48
5,10,20,40,80
9,18,36,72
doubling and halving algorithm
reciprocal pairs starting with 1
1   1
2   0.5
4   0.25
8   0.125
reciprocal pairs starting with 3
3   0.333
6   0.1666
12   0.08333
24   0.04166
Late Babylonian interest 1> N > 2
need to compare result with TCL calculation
check in error subroutine
```

### Testcases Section

In planning any software, it is advisable to gather a number of testcases to check the results of the program. The math for the testcases can be checked by pasting statements in the TCL console. Aside from the TCL calculator display, when one presses the report button on the calculator, one will have console show access to the capacity functions (subroutines).

#### Testcase 1

table 1printed in tcl wiki format
quantity value comment, if any
1:testcase_number
1.0 :lower limit
81.0 :upper limit
2.0 :doubling factor (usually 2. )
0.5 :optional: halving factor (usually 0.5 ):
1.0 :function
1.0 :percentage error
0.0 :initial reciprocal

doubling_halving_algorithmprinted in tcl wiki format
quantity value factor comment, if any
line number target number factor reciprocal solution
1 2.0 0.5 answer for reciprocal 0.5 1/n comparison with tcl math
2 4.0 0.25 answer for reciprocal 0.25 1/n comparison with tcl math
3 8.0 0.125 answer for reciprocal 0.125 1/n comparison with tcl math
4 16.0 0.0625 answer for reciprocal 0.0625 1/n comparison with tcl math
5 32.0 0.03125 answer for reciprocal 0.03125 1/n comparison with tcl math
6 64.0 0.015625 answer for reciprocal 0.015625 1/n comparison with tcl math
7 128.0 0.0078125 answer for reciprocal 0.0078125 1/n comparison with tcl math

### References:

• Interpretation of reverse algorithms in several mesopotamian texts, Christine Proust
• A Geometric Algorithm with Solutions to Quadratic Equations
• in a Sumerian Juridical Document from Ur III Umma
• Joran Friberg, Chalmers University of Technology, Gothenburg, Sweden
• google search engine <Trapezoid area bisection>
• Wikipedia search engine <Trapezoid area >
• mathworld.wolfram.com, Trapezoid and right trapezoid
• Mathematical Treasure: Old Babylonian Area Calculation, uses ancient method
• Frank J. Swetz , Pennsylvania State University
• Wikipedia, see temple of Edfu, area method used as late as 200 BC in Egypt.
• Oneliner's Pie in the Sky
• One Liners
• Category Algorithm
• [Babylonian Number Series and eTCL demo example calculator]
• Brahmagupta Area of Cyclic Quadrilateral and eTCL demo example calculator
• Gauss Approximate Number of Primes and eTCL demo example calculator
• Land surveying in ancient Mesopotamia, M. A. R. Cooper
• [Sumerian Approximate Area Quadrilateral and eTCL Slot Calculator Demo Example , numerical analysis]
• Thomas G. Edwards, Using the Ancient Method of False Position to Find Solutions
• Joy B. Easton, rule of double false position
• Vera Sanford, rule of false position
• www.britannica.com, topic, mathematics trapezoid
• [Sumerian Equivalency Values, Ratios, and the Law of Proportions with Demo Example Calculator]
• Babylonian Sexagesimal Notation for Math on Clay Tablets in Console Example
• Babylonians Tracked Jupiter With Advanced Tools: Trapezoids, Michael Greshko, news.nationalgeographic.com
• Geometry in Babylonian Astronomy, Cluster of Excellence Topology, Humboldt University of Berlin
• Mathieu Ossendrijver: „Ancient Babylonian astronomers calculated Jupiter’s position
• from the area under a time-velocity graph“, in: Science, January 29, 2016.
• Late Babylonian Field Plans in the British Museum, books.google.com/books
• Karen Rhea Nemet-Nejat
• Late Babylonian Surface Mensuration Author(s): Marvin A. Powell Source: jstor
• translation: trapezoid in two babylonian astronomical cuneiform
• texts for jupiter (act 813 & act 817) from the seleucid era , 310 BC -75 AD
• Otto Neugebauer, Astronomical Cuneiform Texts, 3 Vols.
• Lund Humphreys, London, 1955:405,430-31.
• DeSegnac, MS 3908 A RE-CONSTRUCTION, D.A.R. DeSegnac
• A draft for an essay
• DeSegnac, MENTAL COMPUTING OF THREE ARCHAIC
• MESOPOTAMIAN PUZZLES W 20044, 35, W 20044, 20 & W 20214, essay draft
• DeSegnac, HARMONY OF NUMBERS I and II, D.A.R. DeSegnac, A draft for an essay
• Asger Aaboe, Some Seleucid Mathematical Tables
• (Extended Reciprocals and Squares of Regular Numbers)

## Appendix Code edit

### appendix TCL programs and scripts

```        # pretty print from autoindent and ased editor
# Babylonian doubling and halving algorithm for reciprocals  calculator
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# gold on TCL WIKI, 2nov2017
package require Tk
package require math::numtheory
namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
set tcl_precision 17
frame .frame -relief flat -bg aquamarine4
pack .frame -side top -fill y -anchor center
set names {{} {lower limit :} }
lappend names {upper limit:}
lappend names {doubling factor (usually 2. ) : }
lappend names {optional: halving factor (usually 0.5 ): }
lappend names {function : }
lappend names {percentage error: }
lappend names {initial reciprocal :}
foreach i {1 2 3 4 5 6 7 8} {
label .frame.label\$i -text [lindex \$names \$i] -anchor e
entry .frame.entry\$i -width 35 -textvariable side\$i
set msg "Calculator for doubling and halving algorithm for reciprocals
from TCL WIKI,
written on eTCL "
tk_messageBox -title "About" -message \$msg }
proc ::tcl::mathfunc::precision {precision float}  {
#  tcl:wiki:Floating-point formatting, [AM]
set x [ format "%#.5g" \$float ]
return \$x
}
#proc errorx always returns a positive error.
#Normally assume \$aa is human estimate,
#assume \$bb is divinely exact.
proc errorx  {aa bb} {expr { \$aa > \$bb ?   ((\$aa*1.)/\$bb -1.)*100. : ((\$bb*1.)/\$aa -1.)*100.}}
proc doubling_halving_algorithm { target_number } {
set doubling_halving_reciprocal 1.
set target_a 2.
set reciprocal .5
set doubling_halving_reciprocal [* [/ \$target_a \$target_number  ] \$reciprocal  ]
return \$doubling_halving_reciprocal
}
proc calculate {     } {
global side1 side2 side3 side4 side5
global side6 side7 side8
global lower_limit upper_limit doubling_factor
global testcase_number
incr testcase_number
set side1 [* \$side1 1. ]
set side2 [* \$side2 1. ]
set side3 [* \$side3 1. ]
set side4 [* \$side4 1. ]
set side5 [* \$side5 1. ]
set side6 [* \$side6 1. ]
set side7 [* \$side7 1. ]
set side8 [* \$side8 1. ]
set doubling_halving_reciprocal 1.
set doubling_factor \$side3
# begin doubling_halving_algorithm
set lower_limit \$side1
set upper_limit \$side2
}
proc fillup {aa bb cc dd ee ff gg hh} {
.frame.entry1 insert 0 "\$aa"
.frame.entry2 insert 0 "\$bb"
.frame.entry3 insert 0 "\$cc"
.frame.entry4 insert 0 "\$dd"
.frame.entry5 insert 0 "\$ee"
.frame.entry6 insert 0 "\$ff"
.frame.entry7 insert 0 "\$gg"
.frame.entry8 insert 0 "\$hh"
}
proc clearx {} {
foreach i {1 2 3 4 5 6 7 8 } {
.frame.entry\$i delete 0 end } }
proc reportx {} {
global side1 side2 side3 side4 side5
global side6 side7 side8
global lower_limit upper_limit doubling_factor
global testcase_number
console show;
puts "%|table \$testcase_number|printed in| tcl wiki format|% "
puts "&| quantity| value| comment, if any|& "
puts "&| \$testcase_number:|testcase_number | |&"
puts "&| \$side1 :|lower limit  |   |&"
puts "&| \$side2 :|upper limit | |& "
puts "&| \$side3 :|doubling factor (usually 2. ) | |& "
puts "&| \$side4 :|optional: halving factor (usually 0.5 ): | |&"
puts "&| \$side5 :|answers: trial | |&"
puts "&| \$side6 :|function |  |&"
puts "&| \$side7 :|percentage error |  |&"
puts "&| \$side8 :|initial reciprocal |  |&"
puts " reciprocal [ doubling_halving_algorithm 10 ] "
set counter 1
set keeper 1
set reciprocal 1.
puts "%|doubling_halving_algorithm|printed in| tcl wiki format| || |% "
puts "&| quantity| value| factor |comment, if any|||&"
puts "&| line number | target number | factor |  reciprocal solution |||&"
set target_number 1.
while { \$keeper < \$upper_limit  } {
set keeper [* \$keeper \$doubling_factor ]
set reciprocal [* \$reciprocal [/ 1. \$doubling_factor ] ]
puts "&| \$counter | \$keeper  | \$reciprocal  | answer for reciprocal |
[/ 1. \$keeper ]  | 1/n comparison with tcl math |&   "
incr counter
}

}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { set side8 0 ; calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 1.  81.0 2.0 0.5  4.0   1.0   1.0    0.5}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 1.0 81.0 2.0 0.5  9.0   1.0  0.138   2.1}
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 1.0 81.0 2.0 0.5  81.0  1.0  0.5     1.1}
::ttk::button .clearallx -text clear -command {clearx }
::ttk::button .cons -text report -command { reportx }
::ttk::button .exit -text exit -command {exit}
pack  .clearallx .cons .about .exit .test4 .test3 .test2   -side bottom -in .buttons
grid .frame .buttons -sticky ns -pady {0 10}
. configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
wm title . "Babylonian Doubling and Halving Algorithm for Reciprocals Calculator"     ```

### Pushbutton Operation

For the push buttons, the recommended procedure is push testcase and fill frame, change first three entries etc, push solve, and then push report. Report allows copy and paste from console.

For testcases in a computer session, the eTCL calculator increments a new testcase number internally, eg. TC(1), TC(2) , TC(3) , TC(N). The testcase number is internal to the calculator and will not be printed until the report button is pushed for the current result numbers. The current result numbers will be cleared on the next solve button. The command { calculate; reportx } or { calculate ; reportx; clearx } can be added or changed to report automatically. Another wrinkle would be to print out the current text, delimiters, and numbers in a TCL wiki style table as
```  puts " %| testcase \$testcase_number | value| units |comment |%"
puts " &| volume| \$volume| cubic meters |based on length \$side1 and width \$side2   |&"  ```

## Console program for doubling_halving reciprocal algorithm  edit

```                  # autoindent from ased editor
# console program for doubling_halving reciprocal algorithm
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# TCL WIKI , 30jan2017
console show
package require math::numtheory
namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
proc doubling_halving_algorithm { target_number } {
global side1 side2 side3 side4 side5
global side6 side7 side8
set doubling_halving_reciprocal 1.
set target_a 2.
set reciprocal .5
set doubling_halving_reciprocal [* [/ \$target_a \$target_number  ] \$reciprocal  ]
#puts "doubling_halving \$doubling_halving_reciprocal "
return \$doubling_halving_reciprocal
}
set doubling_halving_reciprocal 1.
# begin doubling_halving_algorithm
# set doubling_halving_reciprocal [ doubling_halving_algorithm \$target_number ]
set counter 1
puts "%|doubling_halving_algorithm|printed in| tcl wiki format| || |% "
puts "&| quantity| value| factor |comment, if any|||&"
puts "&| line number | target number | factor |  reciprocal solution |||&"
set target_number 1.
while { \$counter < 50.  } {
set target_number [* 2. \$target_number ]
puts "&| \$counter |\$target_number  | [ doubling_halving_algorithm \$target_number ] | answer for reciprocal |[/ 1.

\$target_number ] | 1/n comparison with tcl math |&   "
incr counter
} ```
doubling_halving_algorithmprinted in tcl wiki format 1/n tcl math
quantity value factor comment, if any
line number target number factor reciprocal solution
1 2.0 0.5 answer for reciprocal 0.5 comparison with tcl math
2 4.0 0.25 answer for reciprocal 0.25 comparison with tcl math
3 8.0 0.125 answer for reciprocal 0.125 comparison with tcl math
4 16.0 0.0625 answer for reciprocal 0.0625 comparison with tcl math
5 32.0 0.03125 answer for reciprocal 0.03125 comparison with tcl math
6 64.0 0.015625 answer for reciprocal 0.015625 comparison with tcl math
7 128.0 0.0078125 answer for reciprocal 0.0078125 comparison with tcl math
8 256.0 0.00390625 answer for reciprocal 0.00390625 comparison with tcl math
9 512.0 0.001953125 answer for reciprocal 0.001953125 comparison with tcl math
10 1024.0 0.0009765625 answer for reciprocal 0.0009765625 comparison with tcl math

doubling_halving_algorithmprinted in tcl wiki format 1/n tcl math
quantity value factor comment, if any
line number target number factor reciprocal solution
1 0.02 50.0 answer for reciprocal 50.0 comparison with tcl math
2 0.04 25.0 answer for reciprocal 25.0 comparison with tcl math
3 0.08 12.5 answer for reciprocal 12.5 comparison with tcl math
4 0.16 6.25 answer for reciprocal 6.25 comparison with tcl math
5 0.32 3.125 answer for reciprocal 3.125 comparison with tcl math
6 0.64 1.5625 answer for reciprocal 1.5625 comparison with tcl math
7 1.28 0.78125 answer for reciprocal 0.78125 comparison with tcl math
8 2.56 0.390625 answer for reciprocal 0.390625 comparison with tcl math
9 5.12 0.1953125 answer for reciprocal 0.1953125 comparison with tcl math
10 10.24 0.09765625 answer for reciprocal 0.09765625 comparison with tcl math

tripling_taking 1/3 printed in tcl wiki format 1/n tcl math
quantity value factor comment, if any
line number target number factor reciprocal solution
1 3.0 0.3333333333333333 answer for reciprocal 0.3333333333333333 1/n comparison with tcl math
2 9.0 0.1111111111111111 answer for reciprocal 0.1111111111111111 1/n comparison with tcl math
3 27.0 0.037037037037037035 answer for reciprocal 0.037037037037037035 1/n comparison with tcl math
4 81.0 0.012345679012345678 answer for reciprocal 0.012345679012345678 1/n comparison with tcl math
5 243.0 0.00411522633744856 answer for reciprocal 0.00411522633744856 1/n comparison with tcl math
6 729.0 0.0013717421124828531 answer for reciprocal 0.0013717421124828531 1/n comparison with tcl math
7 2187.0 0.0004572473708276177 answer for reciprocal 0.0004572473708276177 1/n comparison with tcl math
8 6561.0 0.00015241579027587258 answer for reciprocal 0.00015241579027587258 1/n comparison with tcl math
9 19683.0 5.080526342529086e-5 answer for reciprocal 5.080526342529086e-5 1/n comparison with tcl math
10 59049.0 1.6935087808430286e-5 answer for reciprocal 1.6935087808430286e-5 1/n comparison with tcl math