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|
# rational_funcs.tcl --
# Implement procedures to deal with rational functions
#
package require math::polynomials
namespace eval ::math::rationalfunctions {
variable count 0 ;# Count the number of specific commands
namespace eval v {}
namespace export rationalFunction ratioCmd evalRatio \
coeffsNumerator coeffsDenominator \
derivRatio \
addRatio subRatio multRatio \
divRatio
namespace import ::math::polynomials::*
}
# rationalFunction --
# Return a rational function definition
#
# Arguments:
# num The coefficients of the numerator
# den The coefficients of the denominator
# Result:
# Rational function definition
#
proc ::math::rationalfunctions::rationalFunction {num den} {
foreach coeffs [list $num $den] {
foreach coeff $coeffs {
if { ! [string is double -strict $coeff] } {
return -code error "Coefficients must be real numbers"
}
}
}
#
# The leading coefficient must be non-zero
#
return [list RATIONAL_FUNCTION [polynomial $num] [polynomial $den]]
}
# ratioCmd --
# Return a procedure that implements a rational function evaluation
#
# Arguments:
# num The coefficients of the numerator
# den The coefficients of the denominator
# Result:
# New procedure
#
proc ::math::rationalfunctions::ratioCmd {num {den {}}} {
variable count
if { [llength $den] == 0 } {
if { [lindex $num 0] == "RATIONAL_FUNCTION" } {
set den [lindex $num 2]
set num [lindex $num 1]
}
}
set degree1 [expr {[llength $num]-1}]
set degree2 [expr {[llength $num]-1}]
set body "expr \{([join $num +\$x*(][string repeat ) $degree1])/\
(double([join $den +\$x*(][string repeat ) $degree2])\}"
incr count
set name "::math::rationalfunctions::v::RATIO$count"
proc $name {x} $body
return $name
}
# evalRatio --
# Evaluate a rational function at a given coordinate
#
# Arguments:
# ratio Rational function definition
# x Coordinate
# Result:
# Value at x
#
proc ::math::rationalfunctions::evalRatio {ratio x} {
if { [lindex $ratio 0] != "RATIONAL_FUNCTION" } {
return -code error "Not a rational function"
}
if { ! [string is double $x] } {
return -code error "Coordinate must be a real number"
}
set num 0.0
foreach c [lindex [lindex $ratio 1] 1] {
set num [expr {$num*$x+$c}]
}
set den 0.0
foreach c [lindex [lindex $ratio 2] 1] {
set den [expr {$den*$x+$c}]
}
return [expr {$num/double($den)}]
}
# coeffsNumerator --
# Return the coefficients of the numerator
#
# Arguments:
# ratio Rational function definition
# Result:
# The coefficients in ascending order
#
proc ::math::rationalfunctions::coeffsNumerator {ratio} {
if { [lindex $ratio 0] != "RATIONAL_FUNCTION" } {
return -code error "Not a rational function"
}
set polyn [lindex $ratio 1]
return [allCoeffsPolyn $polyn]
}
# coeffsDenominator --
# Return the coefficients of the denominator
#
# Arguments:
# ratio Rational function definition
# Result:
# The coefficients in ascending order
#
proc ::math::rationalfunctions::coeffsDenominator {ratio} {
if { [lindex $ratio 0] != "RATIONAL_FUNCTION" } {
return -code error "Not a rational function"
}
set polyn [lindex $ratio 2]
return [allCoeffsPolyn $polyn]
}
# derivRatio --
# Return the derivative of the rational function
#
# Arguments:
# polyn Polynomial definition
# Result:
# The new polynomial
#
proc ::math::rationalfunctions::derivRatio {ratio} {
if { [lindex $ratio 0] != "RATIONAL_FUNCTION" } {
return -code error "Not a rational function"
}
set num_polyn [lindex $ratio 1]
set den_polyn [lindex $ratio 2]
set num_deriv [derivPolyn $num_polyn]
set den_deriv [derivPolyn $den_polyn]
set num [subPolyn [multPolyn $num_deriv $den_polyn] \
[multPolyn $den_deriv $num_polyn] ]
set den [multPolyn $den_polyn $den_polyn]
return [list RATIONAL_FUNCTION $num $den]
}
# addRatio --
# Add two rational functions and return the result
#
# Arguments:
# ratio1 First rational function or a scalar
# ratio2 Second rational function or a scalar
# Result:
# The sum of the two functions
# Note:
# TODO: Check for the same denominator
#
proc ::math::rationalfunctions::addRatio {ratio1 ratio2} {
if { [llength $ratio1] == 1 && [string is double -strict $ratio1] } {
set polyn1 [rationalFunction $ratio1 1.0]
}
if { [llength $ratio2] == 1 && [string is double -strict $ratio2] } {
set ratio2 [rationalFunction $ratio1 1.0]
}
if { [lindex $ratio1 0] != "RATIONAL_FUNCTION" ||
[lindex $ratio2 0] != "RATIONAL_FUNCTION" } {
return -code error "Both arguments must be rational functions or a real number"
}
set num1 [lindex $ratio1 1]
set den1 [lindex $ratio1 2]
set num2 [lindex $ratio2 1]
set den2 [lindex $ratio2 2]
set newnum [addPolyn [multPolyn $num1 $den2] \
[multPolyn $num2 $den1] ]
set newden [multPolyn $den1 $den2]
return [list RATIONAL_FUNCTION $newnum $newden]
}
# subRatio --
# Subtract two rational functions and return the result
#
# Arguments:
# ratio1 First rational function or a scalar
# ratio2 Second rational function or a scalar
# Result:
# The difference of the two functions
# Note:
# TODO: Check for the same denominator
#
proc ::math::rationalfunctions::subRatio {ratio1 ratio2} {
if { [llength $ratio1] == 1 && [string is double -strict $ratio1] } {
set polyn1 [rationalFunction $ratio1 1.0]
}
if { [llength $ratio2] == 1 && [string is double -strict $ratio2] } {
set ratio2 [rationalFunction $ratio1 1.0]
}
if { [lindex $ratio1 0] != "RATIONAL_FUNCTION" ||
[lindex $ratio2 0] != "RATIONAL_FUNCTION" } {
return -code error "Both arguments must be rational functions or a real number"
}
set num1 [lindex $ratio1 1]
set den1 [lindex $ratio1 2]
set num2 [lindex $ratio2 1]
set den2 [lindex $ratio2 2]
set newnum [subPolyn [multPolyn $num1 $den2] \
[multPolyn $num2 $den1] ]
set newden [multPolyn $den1 $den2]
return [list RATIONAL_FUNCTION $newnum $newden]
}
# multRatio --
# Multiply two rational functions and return the result
#
# Arguments:
# ratio1 First rational function or a scalar
# ratio2 Second rational function or a scalar
# Result:
# The product of the two functions
# Note:
# TODO: Check for the same denominator
#
proc ::math::rationalfunctions::multRatio {ratio1 ratio2} {
if { [llength $ratio1] == 1 && [string is double -strict $ratio1] } {
set polyn1 [rationalFunction $ratio1 1.0]
}
if { [llength $ratio2] == 1 && [string is double -strict $ratio2] } {
set ratio2 [rationalFunction $ratio1 1.0]
}
if { [lindex $ratio1 0] != "RATIONAL_FUNCTION" ||
[lindex $ratio2 0] != "RATIONAL_FUNCTION" } {
return -code error "Both arguments must be rational functions or a real number"
}
set num1 [lindex $ratio1 1]
set den1 [lindex $ratio1 2]
set num2 [lindex $ratio2 1]
set den2 [lindex $ratio2 2]
set newnum [multPolyn $num1 $num2]
set newden [multPolyn $den1 $den2]
return [list RATIONAL_FUNCTION $newnum $newden]
}
# divRatio --
# Divide two rational functions and return the result
#
# Arguments:
# ratio1 First rational function or a scalar
# ratio2 Second rational function or a scalar
# Result:
# The quotient of the two functions
# Note:
# TODO: Check for the same denominator
#
proc ::math::rationalfunctions::divRatio {ratio1 ratio2} {
if { [llength $ratio1] == 1 && [string is double -strict $ratio1] } {
set polyn1 [rationalFunction $ratio1 1.0]
}
if { [llength $ratio2] == 1 && [string is double -strict $ratio2] } {
set ratio2 [rationalFunction $ratio1 1.0]
}
if { [lindex $ratio1 0] != "RATIONAL_FUNCTION" ||
[lindex $ratio2 0] != "RATIONAL_FUNCTION" } {
return -code error "Both arguments must be rational functions or a real number"
}
set num1 [lindex $ratio1 1]
set den1 [lindex $ratio1 2]
set num2 [lindex $ratio2 1]
set den2 [lindex $ratio2 2]
set newnum [multPolyn $num1 $den2]
set newden [multPolyn $num2 $den1]
return [list RATIONAL_FUNCTION $newnum $newden]
}
#
# Announce our presence
#
package provide math::rationalfunctions 1.0.2
# some tests --
#
if { 0 } {
if {![package vsatisfies [package provide Tcl] 9]} {
set prec $::tcl_precision
if {![package vsatisfies [package provide Tcl] 8.5 9]} {
set ::tcl_precision 17
} else {
set ::tcl_precision 0
}
}
set f1 [::math::rationalfunctions::rationalFunction {1 2 3} {1 4}]
set f2 [::math::rationalfunctions::rationalFunction {1 2 3 0} {1 4}]
set f3 [::math::rationalfunctions::rationalFunction {0 0 0 0} {1}]
set f4 [::math::rationalfunctions::rationalFunction {5 7} {1}]
set cmdf1 [::math::rationalfunctions::ratioCmd {1 2 3} {1 4}]
foreach x {0 1 2 3 4 5} {
puts "[::math::rationalfunctions::evalRatio $f1 $x] -- \
[expr {(1.0+2.0*$x+3.0*$x*$x)/double(1.0+4.0*$x)}] -- \
[$cmdf1 $x] -- [::math::rationalfunctions::evalRatio $f3 $x]"
}
puts "All coefficients = [::math::rationalfunctions::coeffsNumerator $f2]"
puts " [::math::rationalfunctions::coeffsDenominator $f2]"
puts "Derivative = [::math::rationalfunctions::derivRatio $f1]"
puts "Add: [::math::rationalfunctions::addRatio $f1 $f4]"
puts "Add: [::math::rationalfunctions::addRatio $f4 $f1]"
puts "Subtract: [::math::rationalfunctions::subRatio $f1 $f4]"
puts "Multiply: [::math::rationalfunctions::multRatio $f1 $f4]"
set f1 [::math::rationalfunctions::rationalFunction {1 2 3} 1]
set f2 [::math::rationalfunctions::rationalFunction {0 1} 1]
puts "Divide: [::math::rationalfunctions::divRatio $f1 $f2]"
set f1 [::math::rationalfunctions::rationalFunction {1 2 3} 1]
set f2 [::math::rationalfunctions::rationalFunction {1 1} {1 2}]
puts "Divide: [::math::rationalfunctions::divRatio $f1 $f2]"
set f1 [::math::rationalfunctions::rationalFunction {1 2 3} 1]
set f2 [::math::rationalfunctions::rationalFunction {0 1} {0 0 1}]
set f3 [::math::rationalfunctions::divRatio $f2 $f1]
set coeffs [::math::rationalfunctions::coeffsNumerator $f3]
puts "Coefficients: $coeffs"
set f3 [::math::rationalfunctions::divRatio $f1 $f2]
set coeffs [::math::rationalfunctions::coeffsNumerator $f3]
puts "Coefficients: $coeffs"
set f1 [::math::rationalfunctions::rationalFunction {1 2 3} {1 2}]
set f2 [::math::rationalfunctions::rationalFunction {0} {1}]
set f3 [::math::rationalfunctions::divRatio $f2 $f1]
set coeffs [::math::rationalfunctions::coeffsNumerator $f3]
puts "Coefficients: $coeffs"
puts "Eval null function: [::math::rationalfunctions::evalRatio $f2 1]"
if {![package vsatisfies [package provide Tcl] 9]} {
set ::tcl_precision $prec
}
}
|
