[AM] (15 july 2004) The code below illustrates graphically how approximating functions with Chebyshev polynomials works, it also forms a nice little package of mathematical methods. Implemented by [kbk], put on the Wiki by me.

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 namespace eval ::math::approximate {
     namespace export cheby_fit
 }

 #----------------------------------------------------------------------
 #
 # cheby_eval --
 #
 #	Approximates a function with a truncated Chebyshev series
 #
 # Parameters:
 #	coef - List of coefficients of the Chebyshev polynomials
 #	a, b - Boundaries of the interval over which the Chebyshev
 #	       fit was computed.
 #	x    - Abscissa at which to evaluate the function
 #
 # Results:
 #	Returns an approximation to the function evaluated at x.
 #
 # Side effects:
 #	None.
 #
 #----------------------------------------------------------------------

 proc ::math::approximate::cheby_eval { coef a b x } {
     if { ( $x - $a ) * ( $x - $b ) > 1e-8 } {
 	return -code error "domain error: x not in interval"
     }
     set u [expr { ( $x + $x - $a - $b ) / ( $b - $a ) }]
     set twou [expr { $u + $u }]
     set v2 0.
     set v1 0.
     set j [llength $coef]
     while { $j >= 2 } {
 	incr j -1
 	foreach { v2 v1 } \
 	    [list $v1 [expr { $twou * $v1 - $v2 + [lindex $coef $j] }]] \
 	    break
     }
     return [expr { $u * $v1 - $v2 + [lindex $coef 0] }]
 }

 #----------------------------------------------------------------------
 #
 # cheby_fit --
 #
 #	Fits a function with a truncated Chebyshev series.
 #
 # Parameters:
 #	f - Function to fit, expressed as a command prefix to which
 #	    the abscissa will be appended. The function is expected
 #	    to return the ordinate.
 #	a,b - Interval over which to fit the function
 #	epsilon - Desired absolute error bound of the fit (approximate)
 #	n - Maximum order of the Chebyshev polynomial to include.
 #
 # Results:
 #	Returns a Tcl command prefix that, when evaluated with an
 #	abscissa postpended, will give an approximation to the function.
 #
 # Side effects:
 #	Evaluates the given function n times, so has whatever side
 #	effects it has.  Also may cache a table of cosines.
 #
 #----------------------------------------------------------------------

 proc ::math::approximate::cheby_fit { f a b {epsilon 1e-8} { n 50 } } {
     set halfwidth [expr { ($b - $a) / 2. }]
     set midpoint [expr { ( $b + $a ) / 2. }]
     set cs [cos_table $n]
     set c {}
     for { set k 0 } { $k < $n } { incr k } {
 	lappend c 0.
     }
     set twon [expr { $n + $n }]
     set fourn [expr { $twon + $twon }]
     for { set k 1 } { $k <= $twon } { incr k 2 } {
 	set x [expr { [lindex $cs $k] * $halfwidth + $midpoint }]
 	set cmd $f; lappend cmd $x; set y [uplevel 1 $cmd]
 	set m 0
 	for { set j 0 } { $j < $n } { incr j } {
 	    lset c $j [expr { [lindex $c $j] + $y * [lindex $cs $m] }]
 	    incr m $k
 	    if { $m >= $fourn } {
 		set m [expr { $m - $fourn }]
 	    }
 	}
     }
     set r {}
     foreach d $c {
 	lappend r [expr { $d * 2. / $n }]
     }
     lset r 0 [expr { [lindex $r 0] / 2. }]
     for { set i [expr { $n-1 }] } { $i > 0 } { incr i -1 } {
 	if { abs([lindex $r $i]) > $epsilon } break
     }
     return [list [namespace which -command cheby_eval] [lrange $r 0 $i] $a $b]
 }

 #----------------------------------------------------------------------
 #
 # cos_table --
 #
 #	Builds a cosine table for Chebyshev fitting.
 #
 # Parameters:
 #	n - Number of points.
 #
 # Results:
 #	Returns a list of length 4n.  The ith element of the list
 #	is cos( i * pi / ( 2 * n ) )
 #
 # Side effects:
 #	Caches the cosine table.
 #
 #----------------------------------------------------------------------

 proc ::math::approximate::cos_table { n } {
     variable costable
     if { [info exists costable($n)] } {
 	return $costable($n)
     }
     set theta [expr { 0.78539816339744828 / $n }]
     set s [expr { sin($theta) }]
     set alpha [expr { 2. * $s * $s }]
     set beta [expr { sin( 2. * $theta ) }]
     set c 1.
     set s 0.
     set table {}
     for { set i 0 } { $i + $i <= $n } { incr i } {
 	lappend ctable $c
 	lappend stable $s
 	foreach { c s } \
 	    [list \
 		 [expr { $c - ( $alpha * $c + $beta * $s ) }] \
 		 [expr { $s - ( $alpha * $s - $beta * $c ) }]] break
     }
     incr i [expr { ( $n % 2 ) - 1 }]
     for { incr i -1 } { $i > 0 } { incr i -1 } {
 	lappend ctable [lindex $stable $i]
 	lappend stable [lindex $ctable $i]
     }
     foreach c $stable {
 	set c [expr {- $c}]
 	lappend ctable $c
     }
     foreach c $ctable  {
 	set c [expr { - $c }]
 	lappend ctable $c
     }
     if { $n <= 100 } {
 	set costable($n) $ctable
     }
     return $ctable
 }

 if { ! [info exists ::argv0] || ![string equal [info script] $::argv0] } {
     return
 }

 set tcl_precision 17

 proc f { x } {
     return [expr { pow( sin( 6.28318 * $x - 1.2 ), 5) }]
 }

 package require Tk

 grid [canvas .c -width 600 -height 400 -bg white]
 .c create line 20 20 20 380
 .c create line 20 380 580 380

 proc cx { x } {
     expr { 560. * $x + 20. }
 }
 proc cy { y } {
     expr { 200 - 180. * ( $y  ) }
 }

 for { set xx 0 } { $xx <= 10 } { incr xx } {
     set x [expr { $xx / 10. }]
     .c create text [cx $x] 382 -anchor n -text [format %.1f $x]
 }
 for { set yy -10 } { $yy <= 10 } { incr yy 2 } {
     set y [expr { $yy / 10.  }]
     .c create text 18 [cy $y] -anchor e -text [format %.1f $y]
 }

 set l {.c create line}
 for { set x 0. } { $x < 1.0025 } { set x [expr { $x + 0.005 }] } {
     set cmd f; lappend cmd $x; set y [eval $cmd]
     lappend l [cx $x] [cy $y]
 }
 lappend l -width 5 -fill gray85
 eval $l

 .c create text 590 10 -anchor ne -tags n -font {Helvetica 12}

 after 1000 set done 1; vwait done
 set apx0 [::math::approximate::cheby_fit f 0. 1. 1.e-8 120]
 for { set n 0 } { $n < [llength [lindex $apx0 1]] } { incr n } {
     .c itemconfigure n -text $n
     set apx $apx0
     lset apx 1 [lrange [lindex $apx 1] 0 $n]
     set l {.c create line}
     set maxy2 0.
     for { set x 0. } { $x < 1.0025 } { set x [expr { $x + 0.005 }] } {
 	set cmd $apx; lappend cmd $x; set y [eval $cmd]
 	lappend l [cx $x] [cy $y]
     }
     lappend l -tags apx
     catch { .c delete apx }
     eval $l
     after 2000 set done 1; vwait done
 }

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[[
Category Mathematics]
]]