**Kahan compensated summation algorithm and Neumaier variant summation algorithm, 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] 12Dec2018 ---- <> ---- **Introduction** [gold] Here is some TCL starter code for Kahan compensated summation algorithm and Neumaier variant summation algorithm. The Kahan summation algorithm is used to counter round-off error with some effectiveness. Note: One won't see these (single precision) errors in normal TCL 8.6 use. The TCL script below uses to force single precision (sp) on expr and math operator statements. Expr from TCL 8.6 uses double precision inside its calculations and throws round off errors beyond normal sp use. However, this example of the kahan summation algorithm seems to be working correctly and results can be compared to the gimmicked summation procs below. Kahan, Neumaier, Babuska, and others developed balanced or compensated summation algorithms. Some balanced algorithms subtract the compensations from the summing terms as the terms accumulate. Other balanced algorithms model total compensation with a first, second, or nth order curve and subtract the total compensation from the total terms at the end. Second order balanced algorithms can require 50 percent more time, but could be worth it for more accuracy ( eg. function tables). ---- **Pseudocode Section** ====== # using pseudocode for Kahan summation algorithm trying to "cure" or slow growth of tail digits from round off errors, # kahan summation algorithm in pseudocode a(i) = sequence of floating point numbers, a(i)>=3 c = keeper result sum of floating point numbers loop following y = a(i) -c t = sum + y c = (t-sum)-y sum = t 10000.0, 3.14159, 2.71828= (a + b) + c => single precision fp errors => 10005.8. Kahan function a, b, c results in 10005.9 ref normal TCL precision 12 or 17, expr is double precision inside calculations and not show these errors in sp. have to reproduce rd. error in single precision have to add fp format statements correct for inputs of large angles and accumularted round off errors check_answer (a + b) + c =? a + ( b + c ) (yes/no) set answers and printout with resulting values ====== ---- ***References:*** * search keywords for "faster" trig approximations. * sin cos tan trig series efficient approximation "nested polynomial" multiplication * "Horner's method" "Horner's form" recursive telescoping * Payne, Mary H.; Hanek, Robert N., Radian reduction for Large Angles, * implemented on Sparc 3, probably C++ code and bit instructions * Argument reduction for huge arguments: Good to the Last Bit, * K. C. Ng, SunPro Works , March 24, 1992 * SIAM J. Sci. Comput. Volume 31, Issue 1, pp. 189-224 (2008) * Accurate floating-point summation, part 1: faithful rounding * Siegfried M. Rump, Takeshi Ogita, and Shin Ichi Oishi * Design and Implementation of a High Precision Arithmetic * with Rigorous Error Bounds, * Alexander Wittig, MSUHEP-081126, December 2008 * A generalized Kahan-Babuska Summation Algorithm, * Andreas Klein, April 21, 2005 * implement a better summation algorithm, · Issue #199, · JuliaLang * Kahan's compensated summation algorithm, * William Kahan, Further remarks on reducing truncation errors. * Comm. ACM, 8:40, 1965 * Kahan and Babuska summation algorithm, Neumaier variant * A. Neumaier, Rundungsfehleranalyse einiger Verfahren * zur Summation endlicher Summen, in German * Math. Mechanik, 54:39–51, 1974. ---- *** Console TCL script for kahan summation algorithm*** ====== # console program for kahan summation algorithm # pretty print from autoindent and ased editor # kahan summation algorithm into TCL # written on Windows XP on TCL # working under TCL version 8.6 # gold on TCL WIKI , 8jan2017, kahan_baby3 package require Tk package require math::numtheory package require math::geometry package require math::constants package require math::bigfloat namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::geometry math::constants math::bigfloat} set tclprecision 17 wm title . "kahan summation algorithm" console show proc summer_w_rd_error { lister } { set sum 0 foreach number $lister { set sum [+ $sum [ format %7.1f $number ] ] } # statement to recreate single precision r. errors set sum [ format %7.1f $sum ] return $sum} proc summer_w_rd_expr { lister } { set sum 0 foreach number $lister { set number [ format %7.1f $number ] set sum [ expr { $sum + $number } ] } # statement to recreate single precision r. errors set sum [ format %7.1f $sum ] return $sum} proc kahan_summation {lister} { set a {10000.0 3.14159 2.71828 } set b {9879879.88 6585497.99 875870989.54 765864865479.32 } set c { 1. 2. 3. } set compensation_keeper2 0 set sum 0 set counter 0 set yaada2 0 set term2 0 foreach number $lister { set yaada2 [- $number $compensation_keeper2 ] set term2 [+ $sum $yaada2 ] set compensation_keeper2 [- [- $term2 $sum ] $yaada2 ] set sum $term2 incr $counter} # statement to recreate single precision r. errors set sum [ format %7.1f $sum ] return $sum } set kipper { 10000.0 3.14159 2.71828 } puts " normal expr summation, dp inside expr,format %7.1f [format %7.1f [ expr { 10000.0 + 3.14159 + 2.71828 } ] ]" puts " modify summer_w_rd_error, format %7.1f [ summer_w_rd_error $kipper ] " puts " modify summer_w_rd_expr, format %7.1f [ summer_w_rd_expr $kipper ] " puts " trial kahan summation, format %7.1f [ kahan_summation $kipper ] " ====== ---- output ====== # normal expr summation, dp inside expr,format %7.1f 10005.9 # modify summer_w_rd_error, format %7.1f 10005.8 # modify summer_w_rd_expr, format %7.1f 10005.8 # trial kahan summation, format %7.1f 10005.9 ====== ---- *** Console TCL script for neumaier summation algorithm*** ====== # console program for neumaier summation algorithm # pretty print from autoindent and ased editor # neumaier algorithm summation algorithm into TCL # written on Windows XP on TCL # working under TCL version 8.6 # gold on TCL WIKI , 8jan2017,neumaier_baby5 package require Tk package require math::numtheory package require math::geometry package require math::constants package require math::bigfloat namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::geometry math::constants math::bigfloat} set tclprecision 17 wm title . " neumaier summation algorithm" console show #10000.0, 3.14159, 2.71828= (a + b) + c # (a + b) + c =? a + (b + c) # => single precision fp errors => 10005.8. # neumaier algorithm a, b, c results in 10005.9 proc neumaier_summation {lister} { set a {10000.0 3.14159 2.71828 } set b {9879879.88 6585497.99 875870989.54 765864865479.32 } set c { 1. 2. 3. } set k { 1.0 +1.E100 1.0 -1.E100 } set keeper 0 set sum 0 set counter 0 set term2 0 set keeperx 0 foreach number $lister { set term2 [+ $sum $number ] if { [abs $sum ] >= [abs $number ] } { set keeper [+ [- $sum $term2 ] $number ] } else { set keeper [+ [- $number $term2 ] $sum ] } set sum $term2 set keeper2 $keeper set keeperx [+ $keeperx $keeper2 ] incr $counter} # statement to recreate single precision r. errors set sum [ format %7.1f $sum ] set keeper [ format %7.1f $keeper ] # puts " sum $sum keeper $keeper sum+keeper [+ $sum $keeper ] $sum $keeperx [+ $sum $keeperx ]" return [ format %7.1f [+ $sum $keeperx ]] } set kipper { 10000.0 3.14159 2.71828 } set kipper { 1.0 +1.E16 1.0 -1.E16 } set kipper { 1. 2. 3. } set kipper { 10000.0 3.14159 2.71828 } puts " normal expr summation, dp inside expr,format %7.1f sum { 10000.0 3.14159 2.71828 } [format %7.1f [ expr { 10000.0 + 3.14159 + 2.71828 } ] ]" puts " normal expr summation, dp inside expr,format %7.1f sum { +1.0 + +1.E15 + +1.0 + -1.E15 } [format %7.1f [ expr { +1.0 + +1.E15 + +1.0 + -1.E15 } ] ]" puts " trial neumaier summation, format %7.1f sum { 10000.0 3.14159 2.71828 } [ neumaier_summation $kipper ] " set kipper { 1.0 +1.E16 1.0 -1.E16 } puts " trial neumaier summation, format %7.1f sum { 1.0 +1.E16 1.0 -1.E16 } [ neumaier_summation $kipper ] " set kipper { 1. 2. 3. } puts " trial neumaier summation, format %7.1f sum { 1. 2. 3. } = [ neumaier_summation $kipper ] " # normal expr summation, dp inside expr,format %7.1f sum { 10000.0 3.14159 2.71828 } 10005.9 # normal expr summation, dp inside expr,format %7.1f sum { +1.0 + +1.E15 + +1.0 + -1.E15 } 2.0 # trial neumaier summation, format %7.1f sum { 10000.0 3.14159 2.71828 } 10005.9 # trial neumaier summation, format %7.1f sum { 1.0 +1.E16 1.0 -1.E16 } 2.0 # trial neumaier summation, format %7.1f sum { 1. 2. 3. } = 6.0 ====== ---- [gold] This page is copyrighted under the TCL/TK license terms, [http://tcl.tk/software/tcltk/license.html%|%this license]. **Comments Section** <> Please place any comments here, Thanks. <> Numerical Analysis | Toys | Calculator | Mathematics| Example| Toys and Games | Games | Application | GUI ---- <> Development | Concept| Algorithm | Spreadsheet