TC Quadratics

TurboCAD Quadratics first created 28/09/05 - last modified xx/xx/xx Page Author: Ty Harness

Work Unfinished.

Aims:

Use TurboCAD to plot a quadratic equation

Help understand the nature of polynomial of the second degree

Use Least Squares curve fitting to find coefficients from xy points

Apply LU decomposition and Doolittle's method to solve the quadratic coefficients

Fit parabolic curves through data points

Reference Material
[XX] Kreyszig, E., Advanced Engineering Mathematics 8th Ed., Wiley(1999), 895 ~ 916.

The below screen shots of TC: Red circles are the data points and the black line is the polynomial where the coefficients (shown in the text) are calculated using the Method of Least Squares.

Figure 1 - Sample picture description here

Figure XX - Sample picture description here

Private Type Ty2Dcoord
 x As Double
 y As Double

End Type

Private Type TyCoordList
Coords() As Ty2Dcoord
End Type

Private Type TySIGMA
TySIGMAX As Double
TySIGMAY As Double
TySIGMAXY As Double

TySIGMAXX As Double
TySIGMAXXX As Double
TySIGMAXXXX As Double
TySIGMAXXY As Double

End Type


Private Type TyMatrix33
mat(1 To 3, 1 To 3) As Double

End Type

Private Type TyMatrix13
mat(1 To 3) As Double

End Type

Private Type TyXYrange
xmin As Double
ymin As Double

xmax As Double
ymax As Double
End Type

Sub LeastSquaresQuadratic()


Dim rad As Double
Dim cl As TyCoordList
Dim range As TyXYrange

Dim b As TyMatrix13
'/////////// Prob get the points from user selection or a file
'n = 4
'ReDim cl.Coords(n - 1)
'cl.Coords(0).x = -1.3: cl.Coords(0).y = 0.103
'cl.Coords(1).x = -0.1: cl.Coords(1).y = 1.099
'cl.Coords(2).x = 0.2: cl.Coords(2).y = 0.808

'cl.Coords(3).x = 1.3: cl.Coords(3).y = 1.897
'///////////////////////////////////////////////////////////////////

'n = 4
'ReDim cl.Coords(n - 1)
'cl.Coords(0).x = -2: cl.Coords(0).y = 5
'cl.Coords(1).x = -1: cl.Coords(1).y = 2
'cl.Coords(2).x = 1: cl.Coords(2).y = 2
'cl.Coords(3).x = 2: cl.Coords(3).y = 5

'reverse test calculate quadratic from known coefficients

n = 10
ReDim cl.Coords(n - 1)

x = -1.5
For i = 0 To n - 1
y = -0.5 * x ^ 2 + 2 * x + 2
cl.Coords(i).x = x
cl.Coords(i).y = y
x = x + 0.3
Next i

'//////////////////////////////////////////////////////////////////


rad = Abs(cl.Coords(0).x - cl.Coords(1).x) * 0.1
plotcoordinatepoints cl, rad  'plot the orginal points
range = findrange(cl)
b = FindQuadraticCoefficients(cl)

ActiveDrawing.Graphics.AddText "p(x) = " & CStr(b.mat(1)) & " + " & CStr(b.mat(2)) & "x + " _
& CStr(b.mat(3)) & "x^2", 2, 2, 0, rad * 10

plotquadtatic b.mat(1), b.mat(2), b.mat(3), range.xmin, range.xmax 'plot out the polynominal

DrawXYaxes range

End Sub

Private Function findrange(cl As TyCoordList) As TyXYrange


Dim i As Integer

xmin = cl.Coords(0).x
ymin = cl.Coords(0).y
xmax = cl.Coords(0).x + 0.000001
ymax = cl.Coords(0).y + 0.000001

For i = LBound(cl.Coords) + 1 To UBound(cl.Coords)
 If cl.Coords(i).x < xmin Then xmin = cl.Coords(i).x
 If cl.Coords(i).x > xmax Then xmax = cl.Coords(i).x
 If cl.Coords(i).y < ymin Then ymin = cl.Coords(i).y
 If cl.Coords(i).y > ymax Then ymax = cl.Coords(i).y

Next i

findrange.xmin = xmin
findrange.xmax = xmax
findrange.ymin = ymin
findrange.ymax = ymax


End Function


Private Sub plotquadtatic(b0 As Double, b1 As Double, b2 As Double, xmin As Double, xmax As Double)

'plot out a quadratic equations
'p(x) = b0 + b1*x + b2*x^2

Dim v As Vertex
Set gr = ActiveDrawing.Graphics.Add



x = xmin
For i = 0 To 100
y = b0 + b1 * x + b2 * x ^ 2

'ActiveDrawing.Graphics.AddCircleCenterAndPoint x, y, 0, x + 0.01, y, 0
Set v = gr.Vertices.Add(x, y, 0)

x = x + (xmax - xmin) / 100
Next i

Set v = Nothing
Set gr = Nothing


End Sub

Private Sub plotcoordinatepoints(cl As TyCoordList, circlerad As Double)

Dim c As Graphic
Dim i As Integer

For i = LBound(cl.Coords) To UBound(cl.Coords)
 Set c = ActiveDrawing.Graphics.AddCircleCenterAndPoint(cl.Coords(i).x, cl.Coords(i).y, 0, cl.Coords(i).x + circlerad, cl.Coords(i).y, 0)
 c.Properties("Info").Value = CStr(i)
 c.Properties("PenColor").Value = RGB(255, 0, 0)

Next i

Set c = Nothing
End Sub


Private Function FindQuadraticCoefficients(cl As TyCoordList) As TyMatrix13



Dim sum As TySIGMA
Dim n As Integer
Dim a As TyMatrix33

Dim c As TyMatrix13
Dim b As TyMatrix13


sum = TySIGMA1(cl)
'MsgBox sum.TySIGMAX & " " & sum.TySIGMAY & " " & sum.TySIGMAXY & " " & sum.TySIGMAXX


n = UBound(cl.Coords) - LBound(cl.Coords)
'MsgBox n + 1
a.mat(1, 1) = n + 1:          a.mat(2, 1) = sum.TySIGMAX:   a.mat(3, 1) = sum.TySIGMAXX:       c.mat(1) = sum.TySIGMAY
a.mat(1, 2) = sum.TySIGMAX:   a.mat(2, 2) = sum.TySIGMAXX:  a.mat(3, 2) = sum.TySIGMAXXX:      c.mat(2) = sum.TySIGMAXY
a.mat(1, 3) = sum.TySIGMAXX:  a.mat(2, 3) = sum.TySIGMAXXX: a.mat(3, 3) = sum.TySIGMAXXXX:     c.mat(3) = sum.TySIGMAXXY
'MsgBox a.mat(1, 1) & " " & a.mat(2, 1) & " " & a.mat(3, 1) & " " & c.mat(1)

'MsgBox a.mat(1, 2) & " " & a.mat(2, 2) & " " & a.mat(3, 2) & " " & c.mat(2)
'MsgBox a.mat(1, 3) & " " & a.mat(2, 3) & " " & a.mat(3, 3) & " " & c.mat(3)

b = DooLittle(a, c)
'MsgBox b.mat(1) & " " & b.mat(2) & " " & b.mat(3)

FindQuadraticCoefficients.mat(1) = b.mat(1)
FindQuadraticCoefficients.mat(2) = b.mat(2)
FindQuadraticCoefficients.mat(3) = b.mat(3)



End Function


Private Function TySIGMA1(coordinates As TyCoordList) As TySIGMA

Dim i As Integer
Dim SX As Double
SX = 0

For i = LBound(coordinates.Coords) To UBound(coordinates.Coords)
SX = SX + coordinates.Coords(i).x
SY = SY + coordinates.Coords(i).y
SXY = SXY + coordinates.Coords(i).x * coordinates.Coords(i).y
SXX = SXX + coordinates.Coords(i).x * coordinates.Coords(i).x

SXXX = SXXX + coordinates.Coords(i).x * coordinates.Coords(i).x * coordinates.Coords(i).x
SXXXX = SXXXX + coordinates.Coords(i).x * coordinates.Coords(i).x * coordinates.Coords(i).x * coordinates.Coords(i).x
SXXY = SXXY + coordinates.Coords(i).x * coordinates.Coords(i).x * coordinates.Coords(i).y


Next i

TySIGMA1.TySIGMAX = SX
TySIGMA1.TySIGMAY = SY
TySIGMA1.TySIGMAXY = SXY
TySIGMA1.TySIGMAXX = SXX

TySIGMA1.TySIGMAXXX = SXXX
TySIGMA1.TySIGMAXXXX = SXXXX
TySIGMA1.TySIGMAXXY = SXXY

End Function

Sub DooLittletest()
Dim a As TyMatrix33

Dim b As TyMatrix13
Dim x As TyMatrix13

a.mat(1, 1) = 3: a.mat(1, 2) = 5: a.mat(1, 3) = 2: b.mat(1) = 8
a.mat(2, 1) = 0: a.mat(2, 2) = 8: a.mat(2, 3) = 2: b.mat(2) = -7
a.mat(3, 1) = 6: a.mat(3, 2) = 2: a.mat(3, 3) = 8: b.mat(3) = 26

x = DooLittle(a, b)

MsgBox x.mat(1) & "  " & x.mat(2) & "  " & x.mat(3)

End Sub



Private Function DooLittle(a As TyMatrix33, b As TyMatrix13) As TyMatrix13

'A = LU


u11 = a.mat(1, 1)
u12 = a.mat(1, 2)
u13 = a.mat(1, 3)

m21 = a.mat(2, 1) / u11
u22 = a.mat(2, 2) - m21 * u12
u23 = a.mat(2, 3) - m21 * u13
m31 = a.mat(3, 1) / u11
m32 = (a.mat(3, 2) - m31 * u12) / u22
u33 = a.mat(3, 3) - m31 * u13 - m32 * u23
'MsgBox "u11 =" & u11 & " u12=" & u12 & " u13=" & u13 & " m21=" & m21 & " u22=" & u22 & " u23=" & u23 & " " & m31 & " " & m32 & " " & u33

'Ly = b


y1 = b.mat(1)
y2 = b.mat(2) - m21 * y1
y3 = b.mat(3) - m32 * y2 - m31 * y1
'MsgBox y1 & "  " & y2 & "  " & y3



x3 = y3 / u33
x2 = (y2 - u23 * x3) / u22
x1 = (y1 - u13 * x3 - u12 * x2) / u11
'MsgBox x1 & "  " & x2 & "  " & x3


 DooLittle.mat(1) = x1
 DooLittle.mat(2) = x2
 DooLittle.mat(3) = x3



End Function


Private Sub DrawXYaxes(range As TyXYrange)


Dim th As Double

th = (range.ymax - range.xmin) / 50

If range.ymin > 0 Then range.ymin = 0

ActiveDrawing.Graphics.AddLineSingle range.xmin, 0, 0, range.xmax, 0, 0
ActiveDrawing.Graphics.AddLineSingle 0, range.ymin, 0, 0, range.ymax, 0

ActiveDrawing.Graphics.AddText "0", 0, 0, 0, th
ActiveDrawing.Graphics.AddText "X", range.xmax, 0, 0, th
ActiveDrawing.Graphics.AddText "Y", 0, range.ymax, 0, th




End Sub

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