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Hi All,

If you love numerical analysis like I do, then you have most likely come across the various Gaussian quadrature algorithms. For many years I implemented these algorithms by using the roots and weights of a related quadrature polynomial (Legendre, Laguerre, Hermite, etc.) listed in tables for a specific polynomial order. Higher orders gave better quadrature results at the price of using more roots and weights. About two years ago I was able to implement dynamic quadrature function in Matlab. Matlab allowed me to get the roots of the quadrature polynomials very quickly and easily. These routines allowed me to specify the order of the polynomial used and have the routine calculate the roots and weights used to calculate the integral numerically. No more hard-coding tables of roots and weights!

A few days ago I was able to implement similar functions in Excel VBA, even though Excel lacked the ability to find the roots of the quadrature polynomials using a simple function call like in Matlab. The approach I used employed my Scan Range Method that I presented in HHC2012. This method finds the roots of any function within a range of values. I was able to adapt this algorithm to the Gauss-Legendre, Gauss-Laguerre, and Gauss-Hermite quadratures to integrate between [A, B], [0, infinity], and [-infinity, infinity] respectively. The functions I present below use the following parameters to find the polynomial roots:

1. sExpress passes a string that specifies the function to be integrated. For example you can pass the argument "EXP(\$X)-3*\$X^2".

2. VarName is the string that specifies the name of the variable in the parameter sExpress. For example you can pass "\$X" or just "X". Avoid "X" if your function string includes "EXP".

3. DeltaX to specify the increments of X used in scanning for teh roots.

4. Toler to specify the tolerance used in calculating the roots.

5. MaxX used to specify when to stop scanning if not all the anticipated roots are found.

In implementing the code in other programming languages, you can access the integrated function using user-defined functions that you declare elsewhere in your code.

I recommend that you use Wikipedia to look up the various Gaussian quadrature methods and their related polynomials.

Here is the Excel VBA code that should be easily readable and adaptable to any other programming language:

```Option Explicit
Function LegendrePoly(ByVal X As Double, ByVal Order As Integer) As Double
' Implementation of recursive relation for Legendre polynomials.
Dim L0 As Double, L1 As Double, L2 As Double
Dim I As Integer, K As Integer
L0 = 1
If Order = 0 Then
LegendrePoly = L0
Exit Function
End If
L1 = X
If Order = 1 Then
LegendrePoly = L1
Exit Function
End If
K = 1
For I = 2 To Order
L2 = ((2 * K + 1) * X * L1 - K * L0) / (K + 1)
K = K + 1
L0 = L1
L1 = L2
Next I
LegendrePoly = L2
End Function
Function LegendrePolyDeriv(ByVal X As Double, ByVal Order As Integer) As Double
' First derivative of Legendre polynomials.
Dim h As Double
h = 0.001 * (1 + Abs(X))
LegendrePolyDeriv = (LegendrePoly(X + h, Order) - LegendrePoly(X - h, Order)) / 2 / h
End Function
Function GaussLegendreQuad(ByVal sExpress As String, ByVal sVarName As String, _
ByVal A As Double, ByVal B As Double, _
ByVal Order As Integer, _
Optional ByVal DeltaX As Double = 0.1, _
Optional ByVal Toler As Double = 0.00000001, _
Optional ByVal MaxX As Double = 1000) As Double
Const MAX_ITER = 1000
Dim Sum As Double
Dim Xa As Double, Fa As Double, Xb As Double, Fb As Double, Fx As Double
Dim Xrt As Double, Wt As Double, Diff As Double, Xval As Double
Dim N As Integer, Iter As Integer
N = Order
Sum = 0
Xb = 0
Fb = LegendrePoly(Xb, Order)
' found root at x=0 (for odd polynomial orders)?
If Abs(Fb) < 0.000001 Then
N = N - 1
Xrt = Xb
Wt = 2 / (1 - Xrt * Xrt) / (LegendrePolyDeriv(Xrt, Order)) ^ 2
Xval = (B - A) * Xrt / 2 + (A + B) / 2
Fx = Evaluate(Replace(sExpress, sVarName, "(" & CStr(Xval) & ")"))
Sum = Sum + Wt * Fx
End If
' find roots of the Legendre polynomial using the scan range method
Do
DoEvents
Xa = Xb
Fa = Fb
Xb = Xa + DeltaX
Fb = LegendrePoly(Xb, Order)
' root in interval [Xa, Xb]?
If Fa * Fb < 0 Then
' start Newton;s method
Xrt = (Xa + Xb) / 2
Iter = 0
Do
DoEvents
Iter = Iter + 1
Diff = LegendrePoly(Xrt, Order) / LegendrePolyDeriv(Xrt, Order)
Xrt = Xrt - Diff
Loop Until Abs(Diff) <= Toler Or Iter > MAX_ITER
N = N - 2 ' decrement 2 ... one for positive-value root and one for it's negative twin
' calculate weight for postive and negative roots
Wt = 2 / (1 - Xrt * Xrt) / (LegendrePolyDeriv(Xrt, Order)) ^ 2
' calculate transformed x for positive root
Xval = (B - A) * Xrt / 2 + (A + B) / 2
Fx = Evaluate(Replace(sExpress, sVarName, "(" & CStr(Xval) & ")"))
' update the value of the integral
Sum = Sum + Wt * Fx
' calculate transformed x for negative root
Xval = (A - B) * Xrt / 2 + (A + B) / 2
Fx = Evaluate(Replace(sExpress, sVarName, "(" & CStr(Xval) & ")"))
' update the value of the integral
Sum = Sum + Wt * Fx
End If
Loop Until N = 0 Or Xa > MaxX
' found all roots?
If N = 0 Then
GaussLegendreQuad = (B - A) / 2 * Sum
End If
End Function
Function LaguerrePoly(ByVal X As Double, ByVal Order As Integer) As Double
' Implementation of recursive relation for Laguerre polynomials.
Dim L0 As Double, L1 As Double, L2 As Double
Dim I As Integer, K As Integer
L0 = 1
If Order = 0 Then
LaguerrePoly = L0
Exit Function
End If
L1 = 1 - X
If Order = 1 Then
LaguerrePoly = L1
Exit Function
End If
K = 1
For I = 2 To Order
L2 = ((2 * K + 1 - X) * L1 - K * L0) / (K + 1)
K = K + 1
L0 = L1
L1 = L2
Next I
LaguerrePoly = L2
End Function
Function GaussLaguerreQuad(ByVal sExpress As String, ByVal sVarName As String, _
ByVal Order As Integer, _
Optional ByVal DeltaX As Double = 0.1, _
Optional ByVal Toler As Double = 0.00000001, _
Optional ByVal MaxX As Double = 1000) As Double
Const MAX_ITER = 1000
Dim Sum As Double
Dim Xa As Double, Fa As Double, Xb As Double, Fb As Double, Fx As Double
Dim Xrt As Double, Wt As Double, h As Double, Diff As Double
Dim N As Integer, Iter As Integer
N = Order
Sum = 0
Xb = 0
Fb = LaguerrePoly(Xb, Order)
' find the roots of the Laguerre polynomials using the scan range method
Do
DoEvents
Xa = Xb
Fa = Fb
Xb = Xa + DeltaX
Fb = LaguerrePoly(Xb, Order)
' root in interval [Xa, Xb]?
If Fa * Fb < 0 Then
' start Newton's method
Xrt = (Xa + Xb) / 2
Iter = 0
Do
DoEvents
Iter = Iter + 1
h = 0.001 * (1 + Abs(Xrt))
Diff = 2 * h * LaguerrePoly(Xrt, Order) / (LaguerrePoly(Xrt + h, Order) - LaguerrePoly(Xrt - h, Order))
Xrt = Xrt - Diff
Loop Until Abs(Diff) <= Toler Or Iter > MAX_ITER
N = N - 1
' calculaet weight at x root
Wt = Xrt / (Order + 1) ^ 2 / LaguerrePoly(Xrt, Order + 1) ^ 2
' calculate function value
Fx = Evaluate(Replace(sExpress, sVarName, "(" & CStr(Xrt) & ")"))
' update area value
Sum = Sum + Wt * Fx
End If
Loop Until N = 0 Or Xa > MaxX
' foudn all roots?
If N = 0 Then
End If
End Function
Function Factorial(ByVal N As Integer) As Double
Factorial = Application.WorksheetFunction.Fact(N)
End Function
Function HermitePoly(ByVal X As Double, ByVal Order As Integer) As Double
' Implementation of recursive relation for Hermite polynomials.
Dim H0 As Double, H1 As Double, H2 As Double
Dim I As Integer, K As Integer
H0 = 1
If Order = 0 Then
HermitePoly = H0
Exit Function
End If
H1 = 2 * X
If Order = 1 Then
HermitePoly = H1
Exit Function
End If
K = 1
For I = 2 To Order
H2 = 2 * X * H1 - 2 * K * H0
K = K + 1
H0 = H1
H1 = H2
Next I
HermitePoly = H2
End Function
Function GaussHermiteQuad(ByVal sExpress As String, ByVal sVarName As String, _
ByVal Order As Integer, _
Optional ByVal DeltaX As Double = 0.1, _
Optional ByVal Toler As Double = 0.00000001, _
Optional ByVal MaxX As Double = 1000) As Double
Const MAX_ITER = 1000
Dim Sum As Double
Dim Xa As Double, Fa As Double, Xb As Double, Fb As Double, Fx As Double
Dim Xrt As Double, Wt As Double, Diff As Double, h As Double
Dim N As Integer, Iter As Integer
N = Order
Sum = 0
Xb = 0
Fb = HermitePoly(Xb, Order)
' found root at x=0 (for odd polynomial orders)?
If Abs(Fb) < 0.000001 Then
N = N - 1
Xrt = Xb
Wt = 2 ^ (Order - 1) * Factorial(Order) * Sqr(4 * Atn(1)) / Order ^ 2 / (HermitePoly(Xrt, Order - 1)) ^ 2
Fx = Evaluate(Replace(sExpress, sVarName, "0"))
Sum = Sum + Wt * Fx
End If
' find roots of the Hermite polynomial using the scan range method
Do
DoEvents
Xa = Xb
Fa = Fb
Xb = Xa + DeltaX
Fb = HermitePoly(Xb, Order)
' root in interval [Xa, Xb]?
If Fa * Fb < 0 Then
' start Newton;s method
Xrt = (Xa + Xb) / 2
Iter = 0
Do
DoEvents
Iter = Iter + 1
h = 0.001 * (1 + Abs(Xrt))
Diff = 2 * h * HermitePoly(Xrt, Order) / (HermitePoly(Xrt + h, Order) - HermitePoly(Xrt - h, Order))
Xrt = Xrt - Diff
Loop Until Abs(Diff) <= Toler Or Iter > MAX_ITER
N = N - 2 ' decrement 2 ... one for positive-value root and one for it's negative twin
' calculate weight for postive and negative roots
Wt = 2 ^ (Order - 1) * Factorial(Order) * Sqr(4 * Atn(1)) / Order ^ 2 / (HermitePoly(Xrt, Order - 1)) ^ 2
Fx = Evaluate(Replace(sExpress, sVarName, CStr(Xrt)))
' update the value of the integral
Sum = Sum + Wt * Fx
Xrt = -Xrt
Fx = Evaluate(Replace(sExpress, sVarName, "(" & CStr(Xrt) & ")"))
' update the value of the integral
Sum = Sum + Wt * Fx
End If
Loop Until N = 0 Or Xa > MaxX
' found all roots?
If N = 0 Then
End If
End Function
```

Enjoy!

Edited: 29 July 2013, 3:44 p.m.

Namir

Very interesting; thanks for the submittal!

SlideRule

ps: I'm s-l-o-w-l-y working my way thru, thanks again.

I guess one can first translate the code to HP-71B BASIC. I have no illusions that an HP-41C/42 version will run slow. In the case of calculators I suggest having the part that calculators the roots of the polynomial and their associated weights separate. This scheme allows you to reuse the roots/weights for the same polynomial order for different integral ranges (this applies to Gauss-Legendre quadrature for the ranges of [A, B]).

The Gaussian quadrature is an interesting numerical analysis algorithm that requires the calculations of ALL of a polynomial roots. This is where a method like my Scan Range method comes in handy.

Namir

Edited: 30 July 2013, 3:09 p.m.

The 34S should be able to do this relatively well. The orthogonal polynomials are built ins, although actually implemented as key stroke programs.

- Pauli

It will be on my list of machines to use!!!

:-)

Namir