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Babylonian Cubic Equation Problem and eTCL demo example calculator, numerical analysis
gold Here is some eTCL starter code for Babylonian Cubic Equation Problem.
Based on the Friberg discussion of tablet, the Babylonian algorithm for a cubic equation was loaded into an eTCL calculator. The Babylonians did not use algebra notation, so the reader will have to bear some anachronisms in the eTCL psuedocode. The tablet has a set of line by line calculations which effectively have functions for the length, front, and depth of a room constrained, multiplied, and rescaled to equal the volume of a room. Taking the length, front, and depth of a room as three constrained functions in the variable a, the product of three functions scale_constant* length(a)* front(a)* depth(a) equals 1 volume unit. For restating the problem in a computer algorithm, the room dimensions will be each in cubits and the volume unit will be in volume sars, 1 volume sar = 144 cubic cubits. In the original problem, the room dimensions were given as 3 different length units and seems unnecessarily complicated. Once the problem is set up, the Babylonians had a lookup table for n*n*(n-1) to solve for the variable a. The eTCL calculator will have to use an iterative solution for the n*n*(n-1) series. The solution from the tablet was length 6, front 4, depth 6, and volume 144 cubits.
Some fragmented Babylonian tables known as n*n*(n+1) tables were used in solving some cubic equations, ref Joran Friberg. The equations were of the form n*n*(b*n+1) = c. The eTCL calculator could generate the expected tables of n*n*(n+1).
Other Babylonian tables known as n*(n + 1)*(n + 2) and n*n*(n – 1) tables have been identified, but no abundant use has been cited from the known Babylonian math problems. Although not clear, tables of the n*(n + 1) might have existed. From modern theory, n · (n + 1)/2 = sum of integers (1,2,3,4...) and n*(n + 1)*(n + 2) /6 = sum of squares (1,4,9....). Possibly, the Seleucid math problem used an n*(n + 1)*(n + 2) table. Possibly, the tables for n*(n + 1)*(n + 2) and n*n*(n – 1) could have been used for cubic equations.
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).
table 1 | printed in | tcl wiki format |
---|---|---|
quantity | value | comment, if any |
1: | testcase_number | |
1.0 : | length cubits | |
0.666 : | front cubits | |
0.832 : | depth cubits | |
1.0 : | depth 2nd term cubits | |
144.0 : | volume cubic cubits: | |
180 : | volume limit table look up: | |
6 : | table look up solution: | |
143.808 : | check; product length*front*depth =? vol : | |
6.0 : | length cubits | |
3.996 : | front cubits | |
5.997 : | depth cubits |
table 2 | printed in | tcl wiki format |
---|---|---|
quantity | value | comment, if any |
2: | testcase_number | |
1.0 : | length cubits | |
0.666 : | front cubits | |
0.8329 : | depth cubits | |
1.0 : | depth 2nd term cubits | |
200.0 : | volume cubic cubits: | |
294 : | volume limit table look up: | |
7 : | table look up solution: | |
222.922 : | check; product length*front*depth =? vol : | |
7.0 : | length cubits | |
4.6619 : | front cubits | |
6.8309 : | depth cubits |
table 3 | printed in | tcl wiki format |
---|---|---|
quantity | value | comment, if any |
3: | testcase_number | |
1.0 : | length cubits | |
0.666 : | front cubits | |
0.8329 : | depth cubits | |
1.0 : | depth 2nd term cubits | |
300.0 : | volume cubic cubits: | |
448 : | volume limit table look up: | |
8 : | table look up solution: | |
326.670 : | check; product length*front*depth =? vol : | |
8.0 : | length cubits | |
5.328 : | front cubits | |
7.6639 : | depth cubits |
# pretty print from autoindent and ased editor # Babylonian Cubic Equation Algorithm calculator # written on Windows XP on eTCL # working under TCL version 8.5.6 and eTCL 1.0.1 # gold on TCL WIKI, 15jan2017 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 {{} { length f(a) cubits : a*1 } } lappend names { front f(a) cubits : a*(2/3)} lappend names { depth f(a) cubits : a*12+1 : } lappend names { depth f(a), 2nd term cubits : a*12+1 : } lappend names { volume cubic cubits:} lappend names {answers: length : } lappend names {front : } lappend names {depth :} 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$i grid .frame.label$i .frame.entry$i -sticky ew -pady 2 -padx 1 } proc about {} { set msg "Calculator for Babylonian Cubic Equation Algorithm from TCL WIKI, written on eTCL " tk_messageBox -title "About" -message $msg } proc table_look_up {limit } { global look_up_function counter set counter 1 while { $counter < 50. } { set look_up_function [* $counter $counter [- $counter 1] ] if { [* $counter $counter [- $counter 1] ] > [* $limit] } {return $counter ; break} incr counter } } proc calculate { } { global answer2 global side1 side2 side3 side4 side5 global side6 side7 side8 global look_up_function counter check_product_lfd 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. ] # a*1* a*(2/3)*(a*(5/6)+1)= vol set a 6. set room_volume $side5 set a [ table_look_up $room_volume ] set room_volume [expr ($a*$side1)*($a*$side2)*($a*$side3+$side4) ] set length [expr $a*$side1 ] set front [expr $a*$side2 ] set depth [expr $a*$side3+$side4 ] set check_product_lfd [* $length $front $depth 1. ] set side6 $length set side7 $front set side8 $depth } proc fillup {aa bb cc dd ee ff gg hh} { .frame.entry1 insert 0 "$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 look_up_function counter check_product_lfd 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 :|length cubits| |&" puts "&| $side2 :|front cubits | |& " puts "&| $side3 :|depth cubits | |& " puts "&| $side4 :|depth 2nd term cubits | |&" puts "&| $side5 :|volume cubic cubits: | |&" puts "&| $look_up_function :|volume limit table look up: | |&" puts "&| $counter :|table look up solution: | |&" puts "&| $check_product_lfd :|check; product length*front*depth =? vol : | |&" puts "&| $side6 :|length cubits | |&" puts "&| $side7 :|front cubits | |&" puts "&| $side8 :|depth cubits | |&" } frame .buttons -bg aquamarine4 ::ttk::button .calculator -text "Solve" -command { calculate } ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 1.0 0.666 0.833 1. 144.0 6. 4. 6.} ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 1.0 0.666 0.833 1.0 200.0 7. 4.6 6.8 } ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 1.0 0.666 0.833 1.0 300.0 8. 5.3 7.6 } ::ttk::button .clearallx -text clear -command {clearx } ::ttk::button .about -text about -command {about} ::ttk::button .cons -text report -command { reportx } ::ttk::button .exit -text exit -command {exit} pack .calculator -in .buttons -side top -padx 10 -pady 5 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 Cubic Equation Algorithm Calculator"
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 |&"
# autoindent from ased editor # console program for babylonian algorithm for roots. # combined tablet formulas and Newton's method # written on Windows XP on eTCL # working under TCL version 8.5.6 and eTCL 1.0.1 # TCL WIKI , 12dec2016 console show package require math::numtheory namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory } set tcl_precision 17 proc square_root_function { number_for_root } { set counter 0 set epsilon .0001 while { $counter < 50. } { if { [* $counter $counter 1. ] > [* $number_for_root 1.] } {break} incr counter } set square_root_estimate $counter while {1} { set keeper $square_root_estimate set starter $square_root_estimate set remainder [* $starter $starter 1. ] set remainder [- $number_for_root [* $starter $starter 1. ] ] set square_root_estimate [+ $starter [/ $remainder [* 2. $starter ]]] if {abs($keeper - $square_root_estimate) < $epsilon} break } return $square_root_estimate } puts " square root of 2 is >> [square_root_function 2 ] "
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