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|
# rootfind.tcl --
# Root-finding procedures:
# - Bisection
# - Secant
# - Brent
# - Chandrupatla
#
# TODO: f(root), number of steps, converged or not
#
# TOMS748? Seems quite complicated
# Brent in stead of secant - more robust, Chandrupatla is supposed to be faster
#
namespace eval ::math::calculus {}
# root_bisection --
# Find a root of a function of one variable via bisection
#
# Arguments:
# f Procedure implementing the function
# a Left point of the interval
# b Right point of the interval
# tol Tolerance (optional)
#
# Returns:
# The approximation of the root
#
# Note:
# The interval [a,b] must enclose an odd number of roots,
# that is: f(a)*f(b) < 0
#
proc ::math::calculus::root_bisection {f a b {tol 1.0e-7}} {
if { $tol <= 0.0 } {
return -code error "The tolerance must be a small positive value"
}
set fa [$f $a]
set fb [$f $b]
if { ($fa < 0.0 && $fb < 0.0) || ($fa > 0.0 && $fb > 0.0) } {
return -code error "The given interval does not enclose an odd number of roots: f($a) = $fa, f($b) = $fb"
}
set steps 0
set reduction [expr {min( 2.0e16, abs($b-$a)/$tol )}]
set maxsteps [expr {int( log($reduction)/log(2.0) )}] ;# Maximum number of halvings that makes sense
while { $steps < $maxsteps } {
incr steps
set c [expr {($a + $b)/ 2.0}]
set fc [$f $c]
if { ($fc <= 0.0 && $fa <= 0.0) || ($fc >= 0.0 && $fa >= 0.0) } {
set a $c
set fa $fc
} else {
set b $c
set fb $fc
}
# Special case ...
if { $fc == 0.0 } {
return $c
}
}
return $c
}
# root_secant --
# Find a root of a function of one variable via the secant method
#
# Arguments:
# f Procedure implementing the function
# a Left point of the interval
# b Right point of the interval
# tol Tolerance (optional)
#
# Returns:
# The approximation of the root
#
# Note:
# The method is not guranteed to converge, but if it does, it does so
# quicker than linear. The maximum number of steps is derived from the
# idea that the interval will be roughly halved in length.
#
proc ::math::calculus::root_secant {f a b {tol 1.0e-7}} {
if { $tol <= 0.0 } {
return -code error "The tolerance must be a small positive value"
}
set fa [$f $a]
set fb [$f $b]
if { ($fa < 0.0 && $fb < 0.0) || ($fa > 0.0 && $fb > 0.0) } {
return -code error "The given interval does not enclose an odd number of roots: f($a) = $fa, f($b) = $fb"
}
set steps 0
set reduction [expr {min( 2.0e16, abs($b-$a)/$tol )}]
set maxsteps [expr {int( log($reduction)/log(2.0) )}] ;# Maximum number of halvings that makes sense
while { $steps < $maxsteps } {
incr steps
set c [expr {($a * $fb - $b * $fa) / double($fb - $fa)}]
set fc [$f $c]
if { abs($c - $b) <= $tol } {
break
}
set a $b
set fa $fb
set b $c
set fb $fc
}
return $c
}
# root_chandrupatla --
# Find a root of a function of one variable via the method by Chandrupatla (variation on Brent's method)
#
# Arguments:
# f Procedure implementing the function
# x0 Left point of the interval
# x1 Right point of the interval
# tol Tolerance (optional)
#
# Returns:
# The approximation of the root
#
# Note:
# The method brackets a root, like the bisection method.
#
# Adopted from www.embeddedrelated.com/showarticle/855.php
#
proc ::math::calculus::root_chandrupatla {f x0 x1 {tol 1.0e-7}} {
if { $tol <= 0.0 } {
return -code error "The tolerance must be a small positive value"
}
set b $x0
set a $x1
set c $x1
set fa [$f $a]
set fb [$f $b]
set fc $fa
if { ($fa < 0.0 && $fb < 0.0) || ($fa > 0.0 && $fb > 0.0) } {
return -code error "The given interval does not enclose an odd number of roots: f($a) = $fa, f($b) = $fb"
}
set steps 0
set t 0.5
set eps_m $tol
set eps_a [expr {2.0 * $tol}] ;# TODO
while { 1 } {
incr steps
#
# Get a new estimate of the root via interpolation
#
set xt [expr {$a + $t * ($b - $a)}]
set ft [$f $xt]
#
# Update the three points we keep track of
#
if { ($ft < 0.0 && $fa < 0.0) || ($ft > 0.0 && $fa > 0.0) } {
set c $a
set fc $fa
} else {
set c $b
set fc $fb
set b $a
set fb $fa
}
set a $xt
set fa $ft
#
# Determine the point with the smallest function value
#
if { abs($fa) < abs($fb) } {
set xm $a
set fm $fa
} else {
set xm $b
set fm $fb
}
if { $fm == 0.0 } {
return $xm
}
#
# Critical values xi and phi (decisions on how to proceed)
#
set phtol [expr {2.0*$eps_m * abs($xm) + $eps_a}]
set tlim [expr {$phtol / abs($b-$c)}]
if { $tlim > 0.5 } {
return $xm
}
set xi [expr {($a - $b) / ($c - $b)}]
set phi [expr {($fa - $fb) / ($fc - $fb)}]
set do_iqi [expr {$phi**2 < $xi && (1.0 - $phi)**2 < 1.0 - $xi}]
if { $do_iqi } {
#
# Inverse quadratic interpolation
#
set t [expr {$fa / ($fb-$fa) * $fc / ($fb-$fc) +
($c-$a) / ($b-$a) * $fa / ($fc-$a) * $fb / ($fc-$fb)}]
} else {
#
# Bisection
#
set t 0.5
}
#
# Limit t between (tlim,1-tlim)
#
set t [expr {min( 1.0 -$tlim, max($tlim, $t) )}]
}
}
# root_brent --
# Find a root of a function of one variable via Brent's method
#
# Arguments:
# f Procedure implementing the function
# x0 Left point of the interval
# x1 Right point of the interval
# tol Tolerance (optional)
#
# Returns:
# The approximation of the root
#
# Note:
# The method brackets a root, like the bisection method but combines it with the secant method and
# inverse quadratic interpolation.
#
# Adopted from Wikipedia
#
proc ::math::calculus::root_brent {f x0 x1 {tol 1.0e-7}} {
if { $tol <= 0.0 } {
return -code error "The tolerance must be a small positive value"
}
set b $x0
set a $x1
set c $x1
#
# Minimal distance between points at which to calculate the function
#
set delta [expr {1.0e-8 * (abs($x0) + abs($x1)) + $tol/3.0}]
set fa [$f $a]
set fb [$f $b]
if { ($fa < 0.0 && $fb < 0.0) || ($fa > 0.0 && $fb > 0.0) } {
return -code error "The given interval does not enclose an odd number of roots: f($a) = $fa, f($b) = $fb"
}
if { abs($fa) < abs($fb) } {
set tmp $a ; set tmpf $fa
set a $b ; set fa $fb
set b $tmp ; set fb $tmpf
}
set c $a
set fc $fa
set flag 1
set d 0.0 ;# Dummy for the first iteration step
set s $b ;# Make sure s is defined
set fs $fb
#set step 0
while { abs($b - $a) > $tol && $fb != 0.0 && $fs != 0.0 } {
#incr step
if { $fa != $fc && $fb != $fc } {
set s [expr { $a * $fb * $fc / (($fa-$fb)*($fa-$fc)) +
$b * $fa * $fc / (($fb-$fa)*($fb-$fc)) +
$c * $fa * $fb / (($fc-$fa)*($fc-$fb)) }]
} else {
set s [expr {$b - $fb * ($b-$a) / ($fb -$fa)}]
}
# Check the conditions for the next step
if { ( ((3.0*$a+$b) / 4.0 - $s) * ($s - $b) < 0.0 ) ||
( $flag && abs($s-$b) >= abs($b-$c)/2.0 ) ||
( !$flag && abs($s-$b) >= abs($c-$d)/2.0 ) ||
( $flag && abs($b-$c) < abs($delta) ) ||
( !$flag && abs($c-$d) < abs($delta) ) } {
set s [expr {($a + $b) / 2.0}]
set flag 1
} else {
set flag 0
}
set fs [$f $s]
set d $c
set c $b
if { $fa * $fs < 0.0 } {
set b $s
set fb $fs
} else {
set a $s
set fa $fs
}
if { abs($a) < abs($fb) } {
set tmp $a ; set tmpf $fa
set a $b ; set fa $fb
set b $tmp ; set fb $tmpf
}
}
return $s
}
|
