Function EIGEN_JK(ByRef M As Variant) As Variant

** Function computes the eigenvalues and eigenvectors for a real **
** symmetric positive definite matrix using the "JK Method". The **
** first column of the return matrix contains the eigenvalues and **
** the rest of the p+1 columns contain the eigenvectors. **

Dim A() As Variant, Ematrix() As Double
Dim i As Long, j As Long, k As Long, iter As Long, p As Long
Dim den As Double, hold As Double, Sin_ As Double, num As Double
Dim Sin2 As Double, Cos2 As Double, Cos_ As Double, Test As Double
Dim Tan2 As Double, Cot2 As Double, tmp As Double
Const eps As Double = 1E-16

On Error GoTo EndProc

A = M
p = UBound(A, 1)
ReDim Ematrix(1 To p, 1 To p + 1)

For iter = 1 To 15

'Orthogonalize pairs of columns in upper off diag
For j = 1 To p - 1
For k = j + 1 To p

den = 0#
num = 0#
'Perform single plane rotation
For i = 1 To p
num = num + 2 * A(i, j) * A(i, k) ': numerator eq. 11
den = den + (A(i, j) + A(i, k)) * _
(A(i, j) - A(i, k)) ': denominator eq. 11
Next i

'Skip rotation if aij is zero and correct ordering
If Abs(num) < eps And den >= 0 Then Exit For

'Perform Rotation
If Abs(num) <= Abs(den) Then
Tan2 = Abs(num) / Abs(den) ': eq. 11
Cos2 = 1 / Sqr(1 + Tan2 * Tan2) ': eq. 12
Sin2 = Tan2 * Cos2 ': eq. 13
Else
Cot2 = Abs(den) / Abs(num) ': eq. 16
Sin2 = 1 / Sqr(1 + Cot2 * Cot2) ': eq. 17
Cos2 = Cot2 * Sin2 ': eq. 18
End If

Cos_ = Sqr((1 + Cos2) / 2) ': eq. 14/19
Sin_ = Sin2 / (2 * Cos_) ': eq. 15/20

If den < 0 Then
tmp = Cos_
Cos_ = Sin_ ': table 21
Sin_ = tmp
End If

Sin_ = Sgn(num) * Sin_ ': sign table 21

'Rotate
For i = 1 To p
tmp = A(i, j)
A(i, j) = tmp * Cos_ + A(i, k) * Sin_
A(i, k) = -tmp * Sin_ + A(i, k) * Cos_
Next i

Next k
Next j

'Test for convergence
Test = Application.SumSq(A)
If Abs(Test - hold) < eps And iter > 5 Then Exit For
hold = Test
Next iter

If iter = 16 Then MsgBox "JK Iteration has not converged."

'Compute eigenvalues/eigenvectors
For j = 1 To p
'Compute eigenvalues
For k = 1 To p
Ematrix(j, 1) = Ematrix(j, 1) + A(k, j) ^ 2
Next k
Ematrix(j, 1) = Sqr(Ematrix(j, 1))

'Normalize eigenvectors
For i = 1 To p
If Ematrix(j, 1) <= 0 Then
Ematrix(i, j + 1) = 0
Else
Ematrix(i, j + 1) = A(i, j) / Ematrix(j, 1)
End If
Next i
Next j

EIGEN_JK = Ematrix

Exit Function

EndProc:
MsgBox prompt:="Error in function EIGEN_JK!" & vbCr & vbCr & _
"Error: " & Err.Description & ".", Buttons:=48, _
Title:="Run time error!"
End Function