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Ultima2

Attempt at solving roots of polynomial

3 posts in this topic

#1 ·  Posted (edited)

Here is my attempt at writing a script that will solve any roots. It uses the Aberth method

http://en.wikipedia.org/wiki/Aberth_method And it can even solve complex roots. It only goes up to the third degree for right now but I will improve it.

Here is the code

#include <GUIConstantsEx.au3>
#include "complex.au3"

$gui = GUIcreate("Root Solver", 900, 500)
$exit = GUICtrlCreateButton("EXIT", 100, 100, 50, 50)
$input = GUICtrlCreateinput("",20, 50, 220, 30)
$ok = GUICtrlCreatebutton("OK", 30, 100, 50, 50)
$list = GUICtrlCreateList("", 280, 50, 380, 400)
$list2 = GUICtrlCreateList("",680, 50, 180, 400)
GUICtrlCreateLabel("Iteration Trail",680, 25)
GUICtrlCreateLabel("Put in standard form, put 0x^2 if there is no x^2",15, 22)

GUIsetstate(@SW_SHOW)

While 1
    $msg = GUIgetmsg()
    Select
    Case $msg = $exit
        Exit
    Case $msg = $GUI_EVENT_CLOSE 
        Exit
    Case $msg = $ok
        GUICtrlSetData($list, "")
        GUICtrlSetData($list2, "")
        polynomial()
    Endselect
WEnd

Func polynomial()
    $userinput = GUICtrlRead($input)
    $array = Stringsplit($userinput, "x")
    Select 
    Case StringinStr($array[2], "^2") = 1 
    second()
    Case StringinStr($array[2], "^3") = 1
    third()
    EndSelect
EndFunc

Func second()
    $userinput = GUICtrlRead($input)
    $array = Stringsplit($userinput, "x")
    For $i = 1 to $array[0]
        $array[$i] = Stringstripws($array[$i], 8)
    next

    If Stringinstr($array[1], "-") = 1 Then
        $array[1] = (-1)*(StringTrimleft($array[1], 1))
        If StringTrimLeft($array[1], 1) = "0" Then
            $array[1] = -1
        EndIf
    EndIf
    
    If StringLen($array[1]) = 0 Then
        $array[1] = 1
    EndIf
    
    If StringInStr($array[2], "-") > 0 Then
        $b = (-1)*(StringTrimLeft($array[2], 3))
        If StringTrimleft($b, 1) = "0" Then
            $b = -1
        EndIf
    ElseIf StringinStr($array[2], "+") > 0 Then
        $b = StringTrimLeft($array[2], 3)
    EndIf
    
    If Stringlen($b) = 0 Then
        $b = 1
    EndIf
    
    If StringinStr($array[3], "+") = 1 Then
        $c = StringtrimLeft($array[3], 1) + 0.0
    ElseIf StringinStr($array[3], "-") = 1 Then
        $c = (-1)*(StringTrimLeft($array[3], 1))
    EndIf
    
    If StringTrimLeft($array[3], 1) = 0 Then
        $c = $array[3]
    EndIf
    
    $a = complex($array[1], 0)
    $b = complex($b, 0)
    $c = complex($c, 0)
    $two = complex(2,0)
    $l = complex(4, 0)
    $i = complex(-1,0)
    $g = complex_pow($b, 2)
    $l = complex_mul(complex_mul($a, $c), $l)
    $g = complex_pow(complex_minus($g, $l), 0.5)
    $b = complex_mul($b, $i)
    $x1 = complex_div(complex_add($b, $g), complex_mul($a, $two))
    $x2 = complex_div(complex_minus($b, $g), complex_mul($a, $two))
    GUICtrlSetData($list, "x = "&$x1)
    GUICtrlSetData($list, "x = "&$x2)
EndFunc

;uses Aberth Method, it is much better than Newton or Durand-Kerner
Func third()
    $userinput = GUICtrlRead($input)
    $array = Stringsplit($userinput, "x")
    For $i = 1 to $array[0]
        $array[$i] = Stringstripws($array[$i], 8)
    next
    
    For $i = 2 to $array[0] - 1
    If StringInStr($array[$i], "-") > 0 Then
        $array[$i] = (-1)*(StringTrimLeft($array[$i], 3))
        If StringTrimleft($array[$i], 1) = "0" Then
            $array[$i] = -1
        EndIf
    ElseIf StringinStr($array[$i], "+") > 0 Then
        $array[$i] = StringTrimLeft($array[$i], 3)
    EndIf
    Next

    For $i = 2 to $array[0] - 1
    If Stringlen($array[$i]) = 0 Then
        $array[$i] = 1
    EndIf
    Next
    
    For $i = 1 to $array[0]
    If StringLen($array[$i]) = 0 Then
        $array[$i] = 1
    EndIf
    Next
    
    For $i = 1 to $array[0] step 3
    If StringInStr($array[$i], "-") > 0 Then
        $array[$i] = (-1)*(StringTrimLeft($array[$i], 1))
        If StringTrimLeft($array[$i], 1) = "0" Then
            $array[$i] = -1
        EndIf
    EndIf
    Next
    
    $z = complex(Random(), Random()); Must get better seeds
    $z1 = complex(Random(), Random())
    $z2 = complex(Random(), Random())
;Cannot change the coefficients to negative
    $a = complex($array[1], 0)
    $b = complex($array[2], 0)
    $c = complex($array[3], 0)
    $d = complex($array[4], 0)
    $one = complex(1, 0)
    $l = complex(-1, 0)
    $i = 1
    While abs(complex_extract_re(complex_add(complex_mul($a, complex_pow($z, 3)),complex_mul($b, complex_pow($z, 2)),complex_mul($c, $z), $d))) > 1e-10
        $v = complex_add(complex_div($one, complex_minus($z, $z1)), complex_div($one, complex_minus($z, $z2)))
        $m = complex_div(complex_add(complex_mul($a, complex_pow($z, 3)),complex_mul($b, complex_pow($z, 2)),complex_mul($c, $z), $d) , complex_add(complex_mul(complex(3, 0),$a, complex_pow($z, 2)), complex_mul(complex(2, 0),$b,$z), $c))
        $u = complex_mul($l, complex_div($m, complex_minus($one, complex_mul($m, $v))))
        $z = complex_add($z, $u);
        $v = complex_add(complex_div($one, complex_minus($z1, $z)), complex_div($one, complex_minus($z1, $z2)))
        $m = complex_div(complex_add(complex_mul($a, complex_pow($z1, 3)),complex_mul($b, complex_pow($z1, 2)),complex_mul($c, $z1), $d) , complex_add(complex_mul(complex(3, 0),$a, complex_pow($z1, 2)), complex_mul(complex(2, 0),$b,$z1), $c))
        $u = complex_mul($l, complex_div($m, complex_minus($one, complex_mul($m, $v))))
        $z1 = complex_add($z1, $u);
        $v = complex_add(complex_div($one, complex_minus($z2, $z1)), complex_div($one, complex_minus($z2, $z)))
        $m = complex_div(complex_add(complex_mul($a, complex_pow($z2, 3)),complex_mul($b, complex_pow($z2, 2)),complex_mul($c, $z2), $d) , complex_add(complex_mul(complex(3, 0),$a, complex_pow($z2, 2)), complex_mul(complex(2, 0),$b,$z2), $c))
        $u = complex_mul($l, complex_div($m, complex_minus($one, complex_mul($m, $v))))
        $z2 = complex_add($z2, $u)
        GUICtrlSetData($list2, "Loop "&$i&" = "&$z&", "&$z1&", "&$z2)
        $i = $i + 1
        If abs(complex_extract_re(complex_add(complex_mul($a, complex_pow($z, 3)),complex_mul($b, complex_pow($z, 2)),complex_mul($c, $z), $d))) < 1e-10 or abs(complex_extract_re(complex_add(complex_mul($a, complex_pow($z1, 3)),complex_mul($b, complex_pow($z1, 2)),complex_mul($c, $z1), $d))) < 1e-10 or abs(complex_extract_re(complex_add(complex_mul($a, complex_pow($z2, 3)),complex_mul($b, complex_pow($z2, 2)),complex_mul($c, $z2), $d))) < 1e-10 Then
            ExitLoop
        EndIf
    Wend
        
    GUICtrlSetData($list, "x1 = "&$z)
    GUICtrlSetData($list, "x2 = "&$z1)
    GUICtrlSetData($list, "x3 = "&$z2)
EndFunc

Here is the code for the complex number include file. It is my own UDF that I created because I couldnt find any complex numbers UDFs.

#include <Math.au3>
$pi = 3.141592653589793
$toRad = $pi/180

Global const $ni = complex(0,0)
Global const $gi = complex(1,0)

Func complex($re, $im)
    $complex = ""&$re&","&$im&"i"
    return $complex
EndFunc

;returns the magnitude and the argument(theta). This will be used a lot in conjunction with other
;functions
Func complex_polar($x)
    $x = StringSplit($x, ","); Splits the complex to a, bi
    $x[2] = StringTrimright($x[2], 1); Strips the i off of the imaginary number
    $x[1] = $x[1] + 0.0; needs to add 0.0 to make it a float for squaring, type conversion
    $x[2] = $x[2] + 0.0
    $mag = sqrt($x[1]^2 + $x[2]^2)
    $theta = (atan(abs($x[2])/abs($x[1])))*(180/$pi)
;this is to select the quadrant for the magnitude and theta. 3 -7i would be in the 4th quadrant
    Select
    Case $x[1] > 0 and $x[2] >= 0
        $theta = $theta
    Case $x[1] < 0 and $x[2] >= 0
        $theta = 180 - $theta
    Case $x[1] < 0 and $x[2] <= 0
        $theta = 180 + $theta
    Case $x[1] > 0 and $x[2] <= 0
        $theta = 360 - $theta
    EndSelect
    If $theta <> $theta Then
        $theta = 0
    EndIf
    $complex = ""&$mag&","&$theta&""
    Return $complex
EndFunc

;when this function is used with polar, you can do a lot of things
Func complex_rect($complex)
    $complex = StringSplit($complex, ",")
    $real = $complex[1]*cos($toRad*($complex[2]))
    $imag = $complex[1]*sin($toRad*($complex[2]))
    $real = Round($real, 12); rounds the number, can change if you want more precision
    $imag = Round($imag, 12)
    If abs($real) < 1e-10 Then; more ifs and thens for more precision handling >_>
        $real = 0
    EndIf
    If abs($imag) < 1e-10 Then
        $imag = 0
    EndIf
    $complex = ""&$real&","&$imag&"i"
    Return $complex
EndFunc

;this is an example of using the function polar and rect together.
;this can raise to a complex number to any power, even to float like 2.3
Func complex_pow($complex, $pow)
    $complex = complex_polar($complex)
    $complex = StringSplit($complex, ",")
    $complex[1] = $complex[1]^$pow
    $complex[2] = $complex[2]*$pow
    $complex = ""&$complex[1]&","&$complex[2]&""
    $complex = complex_rect($complex)
    Return $complex
EndFunc

Func complex_conj($complex1)
    $complex1 = StringSplit($complex1, ",")
    $complex1[2] = StringTrimRight($complex1[2], 1)
    $complex1[2] = (-1)*$complex1[2] + 0.0
    $real = $complex1[1]
    $imag = $complex1[2]
    $complex = ""&$real&","&$imag&"i"
    Return $complex
EndFunc

Func complex_add($complex1, $complex2, $complex3 = $ni, $complex4 = $ni, $complex5 = $ni)
    $complex1 = StringSplit($complex1, ",")
    $complex1[2] = StringTrimRight($complex1[2], 1)
    $complex2 = StringSplit($complex2, ",")
    $complex2[2] = StringTrimRight($complex2[2], 1)
    $complex3 = StringSplit($complex3, ",")
    $complex3[2] = StringTrimRight($complex3[2], 1)
    $complex4 = StringSplit($complex4, ",")
    $complex4[2] = StringTrimRight($complex4[2], 1)
    $complex5 = StringSplit($complex5, ",")
    $complex5[2] = StringTrimRight($complex5[2], 1)
    $real = $complex1[1] + $complex2[1] + $complex3[1] + $complex4[1] + $complex5[1]
    $imag = $complex1[2] + $complex2[2] + $complex3[2] + $complex4[2] + $complex5[2]
    $complex = ""&$real&","&$imag&"i"
    Return $complex
EndFunc

Func complex_minus($complex1, $complex2, $complex3 = $ni, $complex4 = $ni, $complex5 = $ni)
    $complex1 = StringSplit($complex1, ",")
    $complex1[2] = StringTrimRight($complex1[2], 1)
    $complex2 = StringSplit($complex2, ",")
    $complex2[2] = StringTrimRight($complex2[2], 1)
    $complex3 = StringSplit($complex3, ",")
    $complex3[2] = StringTrimRight($complex3[2], 1)
    $complex4 = StringSplit($complex4, ",")
    $complex4[2] = StringTrimRight($complex4[2], 1)
    $complex5 = StringSplit($complex5, ",")
    $complex5[2] = StringTrimRight($complex5[2], 1)
    $real = $complex1[1] - $complex2[1] - $complex3[1] - $complex4[1] - $complex5[1]
    $imag = $complex1[2] - $complex2[2] - $complex3[2] - $complex4[2] - $complex5[2]
    $complex = ""&$real&","&$imag&"i"
    Return $complex
EndFunc

Func complex_mul($complex1, $complex2, $complex3 = $gi, $complex4 = $gi, $complex5 = $gi)
    $complex1 = complex_polar($complex1)
    $complex2 = complex_polar($complex2)
    $complex3 = complex_polar($complex3)
    $complex4 = complex_polar($complex4)
    $complex5 = complex_polar($complex5)
    $complex1 = StringSplit($complex1, ",")
    $complex2 = StringSplit($complex2, ",")
    $complex3 = StringSplit($complex3, ",")
    $complex4 = StringSplit($complex4, ",")
    $complex5 = StringSplit($complex5, ",")
    $mag = $complex1[1]*$complex2[1]*$complex3[1]*$complex4[1]*$complex5[1]
    $theta = $complex1[2] + $complex2[2] + $complex3[2] + $complex4[2] + $complex5[2]
    $complex = complex_rect(""&$mag&","&$theta&"")
    Return $complex
EndFunc

Func complex_div($complex1, $complex2)
    $complex3 = complex_conj($complex2)
    $complex1 = complex_mul($complex1, $complex3)
    $complex2 = complex_mul($complex2, $complex3)
    $complex1 = StringSplit($complex1, ",")
    $complex2 = StringSplit($complex2, ",")
    $complex1[2] = StringTrimRight($complex1[2], 1)
    $real = $complex1[1]/$complex2[1]
    $imag = $complex1[2]/$complex2[1]
    $complex = ""&$real&","&$imag&"i"
    Return $complex
EndFunc

Func complex_abs($complex1)
    $complex1 = StringSplit($complex1, ",")
    $complex1[2] = StringTrimRight($complex1[2], 1)
    $complex1[1] = $complex1[1] + 0.0
    $complex1[2] = $complex1[2] + 0.0
    $mag = sqrt($complex1[1]^2 + $complex1[2]^2)
    Return $mag
EndFunc

Func complex_extract_re($complex1)
    $complex1 = StringSplit($complex1, ",")
    $complex1[1] = $complex1[1] + 0.0
    Return $complex1[1]
EndFunc

Func complex_extract_im($complex1)
    $complex1 = StringSplit($complex1, ",")
    $complex1[2] = StringTrimRight($complex1[2], 1)
    $complex1[2] = $complex1[2] + 0.0
    Return $complex1[2]
EndFunc

Dont be too harsh, it is my first attempt at writing complex numbers UDF. You need to write x^3 -2 as x^3 +0x^2 +0x -2. Btw can anyone show me the optimizing process or equation for choosing the seeds for the Aberth method? Right now the seeds are chosen randomly. Thanks

Edited by Ultima2

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#2 ·  Posted (edited)

So as a math major who's taken numerical analysis I enjoyed seeing this-- not much non-trivial math appears on these forums at all. That said, I feel like you'd have an easier time of this if you either (1) Used a language that inherently understood imaginary numbers (i.e. matlab) or (2) Used an object oriented language where you could create Polynomial and Complex classes. It's not that I think that what your doing is at all bad, just that your using AutoIT for something that it really wasn't designed for. You wouldn't use a tack hammer to drive in a 16d nail, so I wouldn't suggest using AutoIT for doing complicated numerical work.

That said I will PM you the original AMS article this approximation was published in. And I note here that this scheme is cubically convergent... which is pretty damn good. Makes me wonder why we didn't cover it in class.

Edited by Wus

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Thank you, Wus

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