/// <summary>
/// Returns the Pearson correlation coefficient of two sets of values.
/// </summary>
/// <param name="a">An array of complex numbers.</param>
/// <param name="b">An array of complex numbers.</param>
/// <returns>The Pearson correlation coefficient of the elements of a and b.</returns>
/// <exception cref="System.ArgumentException">
/// The length of the array a does not match the length of the array b.
/// </exception>
public static Complex Correlation(IList <Complex> a, IList <Complex> b)
{
if (a.Count != b.Count)
{
throw new ArgumentException("The lengths of the two arrays do not match.");
}
Complex mean_a = Mean(a);
Complex mean_b = Mean(b);
Complex sum = Complex.Zero;
double sum1 = 0.0;
double sum2 = 0.0;
for (int i = 0; i < a.Count; i++)
{
Complex c1 = a[i] - mean_a;
Complex c2 = b[i] - mean_b;
sum += Complex.ConjugateMultiply(c2, c1);
sum1 += Complex.AbsSquared(c1);
sum2 += Complex.AbsSquared(c2);
}
return(sum / Math.Sqrt(sum1 * sum2));
}
/// <summary>
/// Returns the population variance of the elements of an array.
/// </summary>
/// <param name="v">An array of complex numbers.</param>
/// <returns>The population variance of the elements of v.</returns>
public static double PopulationVariance(IEnumerable <Complex> v)
{
int count = 0;
double sum = 0.0;
Complex mean = Mean(v);
foreach (Complex item in v)
{
sum += Complex.AbsSquared(item - mean);
count++;
}
return(sum / count);
}
/// <summary>
/// Returns the sample variance of the elements of an array.
/// </summary>
/// <param name="v">An array of complex numbers.</param>
/// <returns>The sample variance of the elements of v.</returns>
public static double SampleVariance(IEnumerable <Complex> v)
{
int count = 0;
double sum = 0.0;
Complex mean = Mean(v);
foreach (Complex item in v)
{
sum += Complex.AbsSquared(item - mean);
count++;
}
if (count == 1)
{
return(0.0);
}
return(sum / (count - 1));
}
public static Complex InfiniteProduct(Func <Complex, Complex> term, int m, double relativeTolerance)
{
double tolsq = relativeTolerance * relativeTolerance;
Complex product = Complex.One;
Complex termValue;
for (int i = m; i < _maxIters; i++)
{
termValue = term(i);
product *= termValue;
if (Complex.AbsSquared(termValue / product) <= tolsq)
{
return(product);
}
}
throw new NotConvergenceException();
}
public static Complex InfiniteSummation(Func <Complex, Complex> term, int m, double relativeTolerance)
{
double tolsq = relativeTolerance * relativeTolerance;
Complex sum = Complex.Zero;
Complex termValue;
for (int i = m; i <= _maxIters; i++)
{
termValue = term(i);
sum += termValue;
if (Complex.AbsSquared(termValue / sum) <= tolsq)
{
return(sum);
}
}
throw new NotConvergenceException();
}
/// <summary>
/// Reduces a complex hermitian matrix to a real symmetric tridiagonal matrix using unitary similarity transformations.
/// </summary>
/// <param name="matrixA">Source matrix to reduce</param>
/// <param name="d">Output: Arrays for internal storage of real parts of eigenvalues</param>
/// <param name="e">Output: Arrays for internal storage of imaginary parts of eigenvalues</param>
/// <param name="tau">Output: Arrays that contains further information about the transformations.</param>
/// <param name="order">Order of initial matrix</param>
/// <remarks>This is derived from the Algol procedures HTRIDI by
/// Smith, Boyle, Dongarra, Garbow, Ikebe, Klema, Moler, and Wilkinson, Handbook for
/// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
/// Fortran subroutine in EISPACK.</remarks>
internal static void SymmetricTridiagonalize(Complex[] matrixA, double[] d, double[] e, Complex[] tau, int order)
{
double hh;
tau[order - 1] = Complex.One;
for (var i = 0; i < order; i++)
{
d[i] = matrixA[i * order + i].Re;
}
// Householder reduction to tridiagonal form.
for (var i = order - 1; i > 0; i--)
{
// Scale to avoid under/overflow.
var scale = 0.0;
var h = 0.0;
for (var k = 0; k < i; k++)
{
scale = scale + Math.Abs(matrixA[k * order + i].Re) + Math.Abs(matrixA[k * order + i].Im);
}
if (scale == 0.0)
{
tau[i - 1] = Complex.One;
e[i] = 0.0;
}
else
{
for (var k = 0; k < i; k++)
{
matrixA[k * order + i] /= scale;
h += Complex.AbsSquared(matrixA[k * order + i]);
}
Complex g = Math.Sqrt(h);
e[i] = scale * g.Re;
Complex temp;
var im1Oi = (i - 1) * order + i;
var f = matrixA[im1Oi];
if (f.Modulus != 0)
{
temp = -(Complex.Conjugate(matrixA[im1Oi]) * Complex.Conjugate(tau[i])) / f.Modulus;
h += f.Modulus * g.Re;
g = 1.0 + (g / f.Modulus);
matrixA[im1Oi] *= g;
}
else
{
temp = -Complex.Conjugate(tau[i]);
matrixA[im1Oi] = g;
}
if ((f.Modulus == 0) || (i != 1))
{
f = Complex.Zero;
for (var j = 0; j < i; j++)
{
var tmp = Complex.Zero;
var jO = j * order;
// Form element of A*U.
for (var k = 0; k <= j; k++)
{
tmp += matrixA[k * order + j] * Complex.Conjugate(matrixA[k * order + i]);
}
for (var k = j + 1; k <= i - 1; k++)
{
tmp += Complex.Conjugate(matrixA[jO + k]) * Complex.Conjugate(matrixA[k * order + i]);
}
// Form element of P
tau[j] = tmp / h;
f += (tmp / h) * matrixA[jO + i];
}
hh = f.Re / (h + h);
// Form the reduced A.
for (var j = 0; j < i; j++)
{
f = Complex.Conjugate(matrixA[j * order + i]);
g = tau[j] - (hh * f);
tau[j] = Complex.Conjugate(g);
for (var k = 0; k <= j; k++)
{
matrixA[k * order + j] -= (f * tau[k]) + (g * matrixA[k * order + i]);
}
}
}
for (var k = 0; k < i; k++)
{
matrixA[k * order + i] *= scale;
}
tau[i - 1] = Complex.Conjugate(temp);
}
hh = d[i];
d[i] = matrixA[i * order + i].Re;
matrixA[i * order + i] = new Complex(hh, scale * Math.Sqrt(h));
}
hh = d[0];
d[0] = matrixA[0].Re;
matrixA[0] = hh;
e[0] = 0.0;
}
/// <summary>
/// Nonsymmetric reduction to Hessenberg form.
/// </summary>
/// <param name="dataEv">Data array of matrix V (eigenvectors)</param>
/// <param name="matrixH">Array for internal storage of nonsymmetric Hessenberg form.</param>
/// <param name="order">Order of initial matrix</param>
/// <remarks>This is derived from the Algol procedures orthes and ortran,
/// by Martin and Wilkinson, Handbook for Auto. Comp.,
/// Vol.ii-Linear Algebra, and the corresponding
/// Fortran subroutines in EISPACK.</remarks>
internal static void NonsymmetricReduceToHessenberg(Complex[] dataEv, Complex[] matrixH, int order)
{
var ort = new Complex[order];
for (var m = 1; m < order - 1; m++)
{
// Scale column.
var scale = 0.0;
var mm1O = (m - 1) * order;
for (var i = m; i < order; i++)
{
scale += Math.Abs(matrixH[mm1O + i].Re) + Math.Abs(matrixH[mm1O + i].Im);
}
if (scale != 0.0)
{
// Compute Householder transformation.
var h = 0.0;
for (var i = order - 1; i >= m; i--)
{
ort[i] = matrixH[mm1O + i] / scale;
h += Complex.AbsSquared(ort[i]);
}
var g = Math.Sqrt(h);
if (ort[m].Modulus != 0)
{
h = h + (ort[m].Modulus * g);
g /= ort[m].Modulus;
ort[m] = (1.0 + g) * ort[m];
}
else
{
ort[m] = g;
matrixH[mm1O + m] = scale;
}
// Apply Householder similarity transformation
// H = (I-u*u'/h)*H*(I-u*u')/h)
for (var j = m; j < order; j++)
{
var f = Complex.Zero;
var jO = j * order;
for (var i = order - 1; i >= m; i--)
{
f += Complex.Conjugate(ort[i]) * matrixH[jO + i];
}
f = f / h;
for (var i = m; i < order; i++)
{
matrixH[jO + i] -= f * ort[i];
}
}
for (var i = 0; i < order; i++)
{
var f = Complex.Zero;
for (var j = order - 1; j >= m; j--)
{
f += ort[j] * matrixH[j * order + i];
}
f = f / h;
for (var j = m; j < order; j++)
{
matrixH[j * order + i] -= f * Complex.Conjugate(ort[j]);
}
}
ort[m] = scale * ort[m];
matrixH[mm1O + m] *= -g;
}
}
// Accumulate transformations (Algol's ortran).
for (var i = 0; i < order; i++)
{
for (var j = 0; j < order; j++)
{
dataEv[(j * order) + i] = i == j ? Complex.One : Complex.Zero;
}
}
for (var m = order - 2; m >= 1; m--)
{
var mm1O = (m - 1) * order;
var mm1Om = mm1O + m;
if (matrixH[mm1Om] != Complex.Zero && ort[m] != Complex.Zero)
{
var norm = (matrixH[mm1Om].Re * ort[m].Re) + (matrixH[mm1Om].Im * ort[m].Im);
for (var i = m + 1; i < order; i++)
{
ort[i] = matrixH[mm1O + i];
}
for (var j = m; j < order; j++)
{
var g = Complex.Zero;
for (var i = m; i < order; i++)
{
g += Complex.Conjugate(ort[i]) * dataEv[(j * order) + i];
}
// Double division avoids possible underflow
g /= norm;
for (var i = m; i < order; i++)
{
dataEv[(j * order) + i] += g * ort[i];
}
}
}
}
// Create real subdiagonal elements.
for (var i = 1; i < order; i++)
{
var im1 = i - 1;
var im1O = im1 * order;
var im1Oi = im1O + i;
var iO = i * order;
if (matrixH[im1Oi].Im != 0.0)
{
var y = matrixH[im1Oi] / matrixH[im1Oi].Modulus;
matrixH[im1Oi] = matrixH[im1Oi].Modulus;
for (var j = i; j < order; j++)
{
matrixH[j * order + i] *= Complex.Conjugate(y);
}
for (var j = 0; j <= Math.Min(i + 1, order - 1); j++)
{
matrixH[iO + j] *= y;
}
for (var j = 0; j < order; j++)
{
dataEv[(i * order) + j] *= y;
}
}
}
}