## 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 <format %7.1f \$inputs> 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 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  ```