public static bool IsNearlyEqual(Complex x, Complex y, EvaluationSettings s)
{
double m = ComplexMath.Abs(x) + ComplexMath.Abs(y);
double e = s.AbsolutePrecision + m * s.RelativePrecision;
return(ComplexMath.Abs(x - y) <= e);
}
public void ComplexAbsSquared()
{
foreach (Complex z in TestUtilities.GenerateComplexValues(1.0E-4, 1.0E4, 10))
{
double abs = ComplexMath.Abs(z);
Assert.IsTrue(TestUtilities.IsNearlyEqual(abs * abs, z * z.Conjugate));
}
}
public void ComplexGammaInequality()
{
foreach (Complex z in TestUtilities.GenerateComplexValues(1.0E-2, 1.0E2, 10))
{
Console.WriteLine(z);
Assert.IsTrue(ComplexMath.Abs(AdvancedComplexMath.Gamma(z)) <= Math.Abs(AdvancedMath.Gamma(z.Re)));
}
}
public void ComplexAbsSpecialValues()
{
Assert.IsTrue(ComplexMath.Abs(Complex.Zero) == 0.0);
Assert.IsTrue(ComplexMath.Abs(Complex.One) == 1.0);
Assert.IsTrue(ComplexMath.Abs(-Complex.One) == 1.0);
Assert.IsTrue(ComplexMath.Abs(Complex.I) == 1.0);
Assert.IsTrue(ComplexMath.Abs(-Complex.I) == 1.0);
}
public void ComplexAbsExtremeValues()
{
Assert.IsTrue(ComplexMath.Abs(Double.MaxValue) == Double.MaxValue);
Assert.IsTrue(ComplexMath.Abs(-Double.MaxValue) == Double.MaxValue);
Assert.IsTrue(ComplexMath.Abs(Complex.I * Double.MaxValue) == Double.MaxValue);
Assert.IsTrue(ComplexMath.Abs(Double.PositiveInfinity) == Double.PositiveInfinity);
Assert.IsTrue(ComplexMath.Abs(Double.NegativeInfinity) == Double.PositiveInfinity);
Assert.IsTrue(Complex.IsNaN(ComplexMath.Abs(Double.NaN)));
}
public void ComplexSqrt()
{
foreach (Complex z in TestUtilities.GenerateComplexValues(1.0E-4, 1.0E4, 32))
{
Complex sz = ComplexMath.Sqrt(z);
Assert.IsTrue(TestUtilities.IsNearlyEqual(ComplexMath.Sqr(sz), z, TestUtilities.RelativeTarget));
Assert.IsTrue(TestUtilities.IsNearlyEqual(ComplexMath.Arg(z) / 2.0, ComplexMath.Arg(sz), TestUtilities.RelativeTarget));
Assert.IsTrue(TestUtilities.IsNearlyEqual(Math.Sqrt(ComplexMath.Abs(z)), ComplexMath.Abs(sz), TestUtilities.RelativeTarget));
}
}
public static bool IsNearlyEqual(Complex u, Complex v, double e)
{
if (ComplexMath.Abs(u - v) <= e * (ComplexMath.Abs(u) + ComplexMath.Abs(v)))
{
return(true);
}
else
{
return(false);
}
}
private IntegrationResult RambleIntegral(int d, int s, IntegrationSettings settings)
{
return(MultiFunctionMath.Integrate((IReadOnlyList <double> x) => {
Complex z = 0.0;
for (int k = 0; k < d; k++)
{
z += ComplexMath.Exp(2.0 * Math.PI * Complex.I * x[k]);
}
return (MoreMath.Pow(ComplexMath.Abs(z), s));
}, UnitCube(d), settings));
}
public void ComplexGammaInequalities()
{
foreach (Complex z in TestUtilities.GenerateComplexValues(1.0E-2, 1.0E2, 16))
{
Assert.IsTrue(ComplexMath.Abs(AdvancedComplexMath.Gamma(z)) <= Math.Abs(AdvancedMath.Gamma(z.Re)));
if (z.Re >= 0.5)
{
Assert.IsTrue(ComplexMath.Abs(AdvancedComplexMath.Gamma(z)) >= Math.Abs(AdvancedMath.Gamma(z.Re)) / Math.Sqrt(Math.Cosh(Math.PI * z.Im)));
}
}
}
public void ComplexRealAgreement()
{
foreach (double x in TestUtilities.GenerateRealValues(0.01, 1000.0, 8))
{
Assert.IsTrue(ComplexMath.Exp(x) == Math.Exp(x));
Assert.IsTrue(ComplexMath.Log(x) == Math.Log(x));
Assert.IsTrue(ComplexMath.Sqrt(x) == Math.Sqrt(x));
Assert.IsTrue(ComplexMath.Abs(x) == Math.Abs(x));
Assert.IsTrue(ComplexMath.Arg(x) == 0.0);
}
}
public static bool IsSumNearlyEqual(IEnumerable <Complex> zs, Complex zz)
{
Complex sum = 0.0;
double error = 0.0;
foreach (Complex z in zs)
{
sum += z;
error += TargetPrecision * ComplexMath.Abs(z);
}
error += TargetPrecision * ComplexMath.Abs(zz);
return(ComplexMath.Abs(sum - zz) <= error);
}
public void Bug5504()
{
// this eigenvalue non-convergence is solved by an ad hoc shift
// these "cyclic matrices" are good examples of situations that are difficult for the QR algorithm
for (int d = 2; d <= 8; d++)
{
Console.WriteLine(d);
SquareMatrix C = CyclicMatrix(d);
Complex[] lambdas = C.Eigenvalues();
foreach (Complex lambda in lambdas)
{
Console.WriteLine("{0} ({1} {2})", lambda, ComplexMath.Abs(lambda), ComplexMath.Arg(lambda));
Assert.IsTrue(TestUtilities.IsNearlyEqual(ComplexMath.Abs(lambda), 1.0));
}
}
}
public static bool IsSumNearlyEqual(Complex z1, Complex z2, Complex zz)
{
Complex z = z1 + z2;
double u1 = ComplexMath.Abs(z1) * TargetPrecision;
double u2 = ComplexMath.Abs(z2) * TargetPrecision;
double u = u1 + u2;
double uu = ComplexMath.Abs(zz) * TargetPrecision;
if (2.0 * ComplexMath.Abs(z - zz) <= (u + uu))
{
return(true);
}
else
{
return(false);
}
}
public void DifficultEigenvalue()
{
// This is from a paper describing difficult eigenvalue problems.
// https://www.mathworks.com/company/newsletters/news_notes/pdf/sum95cleve.pdf
SquareMatrix A = new SquareMatrix(4);
A[0, 0] = 0.0; A[0, 1] = 2.0; A[0, 2] = 0.0; A[0, 3] = -1.0;
A[1, 0] = 1.0; A[1, 1] = 0.0; A[1, 2] = 0.0; A[1, 3] = 0.0;
A[2, 0] = 0.0; A[2, 1] = 1.0; A[2, 2] = 0.0; A[2, 3] = 0.0;
A[3, 0] = 0.0; A[3, 1] = 0.0; A[3, 2] = 1.0; A[3, 3] = 0.0;
Complex[] zs = A.Eigenvalues();
foreach (Complex z in zs)
{
Console.WriteLine("{0} ({1} {2})", z, ComplexMath.Abs(z), ComplexMath.Arg(z));
}
}
public static bool IsNearlyEigenpair(AnySquareMatrix A, Complex[] v, Complex a)
{
int d = A.Dimension;
// compute products
Complex[] Av = new Complex[d];
for (int i = 0; i < d; i++)
{
Av[i] = 0.0;
for (int j = 0; j < d; j++)
{
Av[i] += A[i, j] * v[j];
}
}
Complex[] av = new Complex[d];
for (int i = 0; i < d; i++)
{
av[i] = a * v[i];
}
// compute tolorance
double N = MatrixNorm(A);
double n = 0.0;
for (int i = 0; i < d; i++)
{
n += ComplexMath.Abs(v[i]);
}
double ep = TargetPrecision * (N * n / d + ComplexMath.Abs(a) * n);
// compare elements within tollerance
for (int i = 0; i < d; i++)
{
if (ComplexMath.Abs(Av[i] - av[i]) > ep)
{
return(false);
}
}
return(true);
}
public static bool IsNearlyEigenpair(AnySquareMatrix A, ComplexColumnVector v, Complex a)
{
int d = A.Dimension;
// compute products
/*
* Complex[] Av = new Complex[d];
* for (int i=0; i<d; i++) {
* Av[i] = 0.0;
* for (int j=0; j<d; j++) {
* Av[i] += A[i,j]*v[j];
* }
* }
*/
ComplexColumnVector Av = A * v;
ComplexColumnVector av = a * v;
// compute tolerance
double N = A.MaxNorm();
double n = 0.0;
for (int i = 0; i < d; i++)
{
n += ComplexMath.Abs(v[i]);
}
double ep = TargetPrecision * (N * n / d + ComplexMath.Abs(a) * n);
// compare elements within tolerance
for (int i = 0; i < d; i++)
{
if (ComplexMath.Abs(Av[i] - av[i]) > ep)
{
return(false);
}
}
return(true);
}
public void FourierSpecialCases()
{
for (int n = 2; n <= 10; n++)
{
FourierTransformer ft = new FourierTransformer(n);
Assert.IsTrue(ft.Length == n);
// transform of uniform is zero frequency only
Complex[] x1 = new Complex[n];
for (int i = 0; i < n; i++)
{
x1[i] = 1.0;
}
Complex[] y1 = ft.Transform(x1);
Assert.IsTrue(TestUtilities.IsNearlyEqual(y1[0], n));
for (int i = 1; i < n; i++)
{
Assert.IsTrue(ComplexMath.Abs(y1[i]) < TestUtilities.TargetPrecision);
}
Complex[] z1 = ft.InverseTransform(y1);
for (int i = 0; i < n; i++)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(z1[i], 1.0));
}
// transform of pulse at 1 are nth roots of unity, read clockwise
Complex[] x2 = new Complex[n];
x2[1] = 1.0;
Complex[] y2 = ft.Transform(x2);
for (int i = 0; i < n; i++)
{
double t = -2.0 * Math.PI / n * i;
Assert.IsTrue(TestUtilities.IsNearlyEqual(y2[i], new Complex(Math.Cos(t), Math.Sin(t))));
}
}
}