public void TestInterpolationMethod_CubicSpline_BoundarySecondDerivativeFixed()
{
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.CreateCubicSpline(t, x, SplineBoundaryCondition.SecondDerivative, -5.0, SplineBoundaryCondition.SecondDerivative, -1.0);
Assert.That(method, Is.TypeOf(typeof(CubicSplineInterpolation)), "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=-2.4},Spline([[-2,1],[-1,2],[0,-1],[1,0],[2,1]], x, degree=3, endpoints=Matrix(2,13,{(1,3)=1,(1,13)=-5,(2,10)=1,(2,13)=-1}))),20);"
Assert.That(method.Interpolate(-2.4), NumericIs.AlmostEqualTo(-.8999999999999999993, 1e-15), "A -2.4");
Assert.That(method.Interpolate(-0.9), NumericIs.AlmostEqualTo(1.7590357142857142857, 1e-15), "A -0.9");
Assert.That(method.Interpolate(-0.5), NumericIs.AlmostEqualTo(.41517857142857142854, 1e-15), "A -0.5");
Assert.That(method.Interpolate(-0.1), NumericIs.AlmostEqualTo(-.82010714285714285714, 1e-15), "A -0.1");
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(-1.1026071428571428572, 1e-15), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(-1.0211428571428571429, 1e-15), "A 0.4");
Assert.That(method.Interpolate(1.2), NumericIs.AlmostEqualTo(.31771428571428571421, 1e-15), "A 1.2");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo((double)39, 1e-14), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo((double)(-37), 1e-14), "A -10.0");
}
public void TestInterpolationMethod_LinearSpline()
{
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.CreateLinearSpline(t, x);
Assert.IsInstanceOfType(typeof(LinearSplineInterpolation), method, "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.AreEqual(x[i], method.Interpolate(t[i]), "A Exact Point " + i.ToString());
}
// Maple: "f := x -> piecewise(x<-1,3+x,x<0,-1-3*x,x<1,-1+x,-1+x);"
// Maple: "f(x)"
NumericAssert.AreAlmostEqual(.6, method.Interpolate(-2.4), 1e-15, "A -2.4");
NumericAssert.AreAlmostEqual(1.7, method.Interpolate(-0.9), 1e-15, "A -0.9");
NumericAssert.AreAlmostEqual(.5, method.Interpolate(-0.5), 1e-15, "A -0.5");
NumericAssert.AreAlmostEqual(-.7, method.Interpolate(-0.1), 1e-15, "A -0.1");
NumericAssert.AreAlmostEqual(-.9, method.Interpolate(0.1), 1e-15, "A 0.1");
NumericAssert.AreAlmostEqual(-.6, method.Interpolate(0.4), 1e-15, "A 0.4");
NumericAssert.AreAlmostEqual(.2, method.Interpolate(1.2), 1e-15, "A 1.2");
NumericAssert.AreAlmostEqual(9.0, method.Interpolate(10.0), 1e-15, "A 10.0");
NumericAssert.AreAlmostEqual(-7.0, method.Interpolate(-10.0), 1e-15, "A -10.0");
}
public void TestInterpolationMethod_AkimaSpline()
{
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.CreateAkimaCubicSpline(t, x);
Assert.IsInstanceOfType(typeof(AkimaSplineInterpolation), method, "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.AreEqual(x[i], method.Interpolate(t[i]), "A Exact Point " + i.ToString());
}
// TODO: Verify the expected values (that they are really the expected ones)
NumericAssert.AreAlmostEqual(-0.52, method.Interpolate(-2.4), 1e-15, "A -2.4");
NumericAssert.AreAlmostEqual(1.826, method.Interpolate(-0.9), 1e-15, "A -0.9");
NumericAssert.AreAlmostEqual(0.25, method.Interpolate(-0.5), 1e-15, "A -0.5");
NumericAssert.AreAlmostEqual(-1.006, method.Interpolate(-0.1), 1e-15, "A -0.1");
NumericAssert.AreAlmostEqual(-0.9, method.Interpolate(0.1), 1e-15, "A 0.1");
NumericAssert.AreAlmostEqual(-0.6, method.Interpolate(0.4), 1e-15, "A 0.4");
NumericAssert.AreAlmostEqual(0.2, method.Interpolate(1.2), 1e-15, "A 1.2");
NumericAssert.AreAlmostEqual(9, method.Interpolate(10.0), 1e-14, "A 10.0");
NumericAssert.AreAlmostEqual(-151, method.Interpolate(-10.0), 1e-14, "A -10.0");
}
public void TestInterpolationMethod_CubicSpline_BoundaryNatural()
{
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.CreateNaturalCubicSpline(t, x);
Assert.IsInstanceOfType(typeof(CubicSplineInterpolation), method, "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.AreEqual(x[i], method.Interpolate(t[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=-2.4},Spline([[-2,1],[-1,2],[0,-1],[1,0],[2,1]], x, degree=3, endpoints='natural')),20);"
NumericAssert.AreAlmostEqual(.144000000000000000, method.Interpolate(-2.4), 1e-15, "A -2.4");
NumericAssert.AreAlmostEqual(1.7906428571428571429, method.Interpolate(-0.9), 1e-15, "A -0.9");
NumericAssert.AreAlmostEqual(.47321428571428571431, method.Interpolate(-0.5), 1e-15, "A -0.5");
NumericAssert.AreAlmostEqual(-.80992857142857142857, method.Interpolate(-0.1), 1e-15, "A -0.1");
NumericAssert.AreAlmostEqual(-1.1089285714285714286, method.Interpolate(0.1), 1e-15, "A 0.1");
NumericAssert.AreAlmostEqual(-1.0285714285714285714, method.Interpolate(0.4), 1e-15, "A 0.4");
NumericAssert.AreAlmostEqual(.30285714285714285716, method.Interpolate(1.2), 1e-15, "A 1.2");
NumericAssert.AreAlmostEqual(189, method.Interpolate(10.0), 1e-15, "A 10.0");
NumericAssert.AreAlmostEqual(677, method.Interpolate(-10.0), 1e-15, "A -10.0");
}
public void TestInterpolationMethod_CubicSpline_BoundaryFirstDerivativeFixed()
{
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.CreateCubicSpline(t, x, SplineBoundaryCondition.FirstDerivative, 1.0, SplineBoundaryCondition.FirstDerivative, -1.0);
Assert.That(method, Is.TypeOf(typeof(CubicSplineInterpolation)), "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=-2.4},Spline([[-2,1],[-1,2],[0,-1],[1,0],[2,1]], x, degree=3, endpoints=[1,-1])),20);"
Assert.That(method.Interpolate(-2.4), NumericIs.AlmostEqualTo(1.120000000000000001, 1e-15), "A -2.4");
Assert.That(method.Interpolate(-0.9), NumericIs.AlmostEqualTo(1.8243928571428571428, 1e-15), "A -0.9");
Assert.That(method.Interpolate(-0.5), NumericIs.AlmostEqualTo(.54910714285714285715, 1e-15), "A -0.5");
Assert.That(method.Interpolate(-0.1), NumericIs.AlmostEqualTo(-.78903571428571428572, 1e-15), "A -0.1");
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(-1.1304642857142857143, 1e-15), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(-1.1040000000000000000, 1e-15), "A 0.4");
Assert.That(method.Interpolate(1.2), NumericIs.AlmostEqualTo(.4148571428571428571, 1e-15), "A 1.2");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo(-608.14285714285714286, 1e-15), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo(1330.1428571428571429, 1e-15), "A -10.0");
}
public void TestInterpolationMethod_Chebyshev2BarycentricPolynomial()
{
double[] x = new double[] { 0.0, 3.0, 2.5, 1.0, 3.0 };
IInterpolationMethod method = Interpolation.CreateOnChebyshevSecondKindPoints(0.0, 4.0, x);
Assert.That(method, Is.TypeOf(typeof(ChebyshevSecondKindPolynomialInterpolation)), "Type");
double[] t = Interpolation.GenerateChebyshevSecondKindSamplePoints(0.0, 4.0, 5);
for (int i = 0; i < 4; i++)
{
// verify the generated chebyshev2 points
double tt = 2.0 + (2.0 * Math.Cos(Math.PI * i * 0.25));
Assert.That(tt, NumericIs.AlmostEqualTo(t[i]), "Point " + i.ToString());
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(tt), NumericIs.AlmostEqualTo(x[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=0.1},PolynomialInterpolation(evalf([[2*cos(0*Pi/4)+2,0],[2*cos(1*Pi/4)+2,3],[2*cos(2*Pi/4)+2,2.5],[2*cos(3*Pi/4)+2,1],[2*cos(4*Pi/4)+2,3]]), x)),20);"
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(2.4826419375703841423, 1e-14), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(1.3814129880730972522, 1e-14), "A 0.4");
Assert.That(method.Interpolate(1.1), NumericIs.AlmostEqualTo(.8808232156067110292, 1e-15), "A 1.1");
Assert.That(method.Interpolate(3.2), NumericIs.AlmostEqualTo(3.478116015902536997, 1e-15), "A 3.2");
Assert.That(method.Interpolate(4.5), NumericIs.AlmostEqualTo(-5.035612822087164912, 1e-15), "A 4.5");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo(-369.20562748477140583, 1e-13), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo(1199.4696961966999204, 1e-12), "A -10.0");
}
public void TestInterpolationMethod_Chebyshev1BarycentricPolynomial()
{
double[] x = new double[] { 0.0, 3.0, 2.5, 1.0, 3.0 };
IInterpolationMethod method = Interpolation.CreateOnChebyshevFirstKindPoints(0.0, 4.0, x);
Assert.That(method, Is.TypeOf(typeof(ChebyshevFirstKindPolynomialInterpolation)), "Type");
double[] t = Interpolation.GenerateChebyshevFirstKindSamplePoints(0.0, 4.0, 5);
for (int i = 0; i < 4; i++)
{
// verify the generated chebyshev1 points
double tt = 2.0 + (2.0 * Math.Cos(Math.PI * 0.1 * ((2 * i) + 1)));
Assert.That(tt, NumericIs.AlmostEqualTo(t[i]), "Point " + i.ToString());
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(tt), NumericIs.AlmostEqualTo(x[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=0.1},PolynomialInterpolation(evalf([[2*cos(Pi/10)+2,0],[2*cos(3*Pi/10)+2,3],[2*cos(5*Pi/10)+2,2.5],[2*cos(7*Pi/10)+2,1],[2*cos(9*Pi/10)+2,3]]), x)),20);"
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(2.9882560375702001608, 1e-15), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(1.7097090371118968872, 1e-15), "A 0.4");
Assert.That(method.Interpolate(1.1), NumericIs.AlmostEqualTo(1.0462830804302586508, 1e-15), "A 1.1");
Assert.That(method.Interpolate(3.2), NumericIs.AlmostEqualTo(2.951922899377369724, 1e-15), "A 3.2");
Assert.That(method.Interpolate(4.5), NumericIs.AlmostEqualTo(-5.394317844683536750, 1e-15), "A 4.5");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo(-228.01438153088988107, 1e-13), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo(1979.2646653044133954, 1e-12), "A -10.0");
}
public void TestInterpolationMethod_NevillePolynomial()
{
double[] t = new double[] { 0.0, 1.0, 3.0, 4.0 };
double[] x = new double[] { 0.0, 3.0, 1.0, 3.0 };
IInterpolationMethod method = Interpolation.CreatePolynomial(t, x);
Assert.IsInstanceOfType(typeof(PolynomialInterpolation), method, "Type");
double dx, d2x;
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.AreEqual(x[i], method.Interpolate(t[i]), "A Exact Point " + i.ToString());
Assert.AreEqual(x[i], method.Differentiate(t[i], out dx, out d2x), "B Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=0.1},PolynomialInterpolation([[0,0],[1,3],[3,1],[4,3]], x)),20);"
NumericAssert.AreAlmostEqual(.57225000000000000000, method.Interpolate(0.1), 1e-15, "A 0.1");
NumericAssert.AreAlmostEqual(.57225000000000000000, method.Differentiate(0.1, out dx, out d2x), 1e-15, "B 0.1");
NumericAssert.AreAlmostEqual(1.8840000000000000000, method.Interpolate(0.4), 1e-15, "A 0.4");
NumericAssert.AreAlmostEqual(1.8840000000000000000, method.Differentiate(0.4, out dx, out d2x), 1e-15, "B 0.4");
NumericAssert.AreAlmostEqual(3.0314166666666666667, method.Interpolate(1.1), 1e-15, "A 1.1");
NumericAssert.AreAlmostEqual(3.0314166666666666667, method.Differentiate(1.1, out dx, out d2x), 1e-15, "B 1.1");
NumericAssert.AreAlmostEqual(1.034666666666666667, method.Interpolate(3.2), 1e-15, "A 3.2");
NumericAssert.AreAlmostEqual(1.034666666666666667, method.Differentiate(3.2, out dx, out d2x), 1e-15, "B 3.2");
NumericAssert.AreAlmostEqual(6.281250000000000000, method.Interpolate(4.5), 1e-15, "A 4.5");
NumericAssert.AreAlmostEqual(6.281250000000000000, method.Differentiate(4.5, out dx, out d2x), 1e-15, "B 4.5");
NumericAssert.AreAlmostEqual(277.50000000000000000, method.Interpolate(10.0), 1e-15, "A 10.0");
NumericAssert.AreAlmostEqual(277.50000000000000000, method.Differentiate(10.0, out dx, out d2x), 1e-15, "B 10.0");
NumericAssert.AreAlmostEqual(-1010.8333333333333333, method.Interpolate(-10.0), 1e-15, "A -10.0");
NumericAssert.AreAlmostEqual(-1010.8333333333333333, method.Differentiate(-10.0, out dx, out d2x), 1e-15, "B -10.0");
}
void Update()
{
if (currentResolution != resolution || optimizationPoints == null)
{
CreateOptimizationPoints();
}
FunctionDelegate f = functionDelegates[(int)function];
float t = Time.timeSinceLevelLoad;
//optimization steps
Matrix a = QuadraticFormMatrix(t);
Matrix currentPoint = new Matrix(new double[][] {
new double[] { xStart },
new double[] { zStart }
});
Matrix currentGradient;
Matrix currentHessianInv;
Matrix lastPoint = currentPoint.Clone();
double[] ts = new double[iterationCount + 1];
double[] xs = new double[iterationCount + 1];
double[] zs = new double[iterationCount + 1];
xs[0] = currentPoint.GetArray()[0][0];
zs[0] = currentPoint.GetArray()[1][0];
for (int i = 0; i < iterationCount; i++)
{
ts[i] = i;
currentGradient = a * lastPoint;
//currentHessianInv = a.Inverse();
//currentPoint = lastPoint - ((double) learningRate) * currentHessianInv * currentGradient;
currentPoint = lastPoint - ((double)learningRate) * currentGradient;
xs[i + 1] = currentPoint.GetArray()[0][0];
zs[i + 1] = currentPoint.GetArray()[1][0];
lastPoint = currentPoint.Clone();
}
ts[iterationCount] = iterationCount;
IInterpolationMethod xInterp = Interpolation.Create(ts, xs);
IInterpolationMethod zInterp = Interpolation.Create(ts, zs);
float increment = ((float)iterationCount) / ((float)(resolution - 1));
for (int i = 0; i < resolution; i++)
{
float curInc = increment * i;
Vector3 p = new Vector3((float)xInterp.Interpolate(curInc), 0.0f, (float)zInterp.Interpolate(curInc));
p.y = f(p, t);
optimizationPoints[i].position = p;
Color c = optimizationPoints [i].color;
c.g = ((float)i) / ((float)resolution);
c.b = ((float)curInc) / ((float)iterationCount);
//c.g = 1.0f / Mathf.Exp((float) i / 15.0f);
optimizationPoints[i].color = c;
}
particleSystem.SetParticles(optimizationPoints, optimizationPoints.Length);
}
public void TestInterpolationMethod_CubicSpline_BoundaryNatural()
{
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.CreateNaturalCubicSpline(t, x);
Assert.That(method, Is.TypeOf(typeof(CubicSplineInterpolation)), "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=-2.4},Spline([[-2,1],[-1,2],[0,-1],[1,0],[2,1]], x, degree=3, endpoints='natural')),20);"
Assert.That(method.Interpolate(-2.4), NumericIs.AlmostEqualTo(.144000000000000000, 1e-15), "A -2.4");
Assert.That(method.Interpolate(-0.9), NumericIs.AlmostEqualTo(1.7906428571428571429, 1e-15), "A -0.9");
Assert.That(method.Interpolate(-0.5), NumericIs.AlmostEqualTo(.47321428571428571431, 1e-15), "A -0.5");
Assert.That(method.Interpolate(-0.1), NumericIs.AlmostEqualTo(-.80992857142857142857, 1e-15), "A -0.1");
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(-1.1089285714285714286, 1e-15), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(-1.0285714285714285714, 1e-15), "A 0.4");
Assert.That(method.Interpolate(1.2), NumericIs.AlmostEqualTo(.30285714285714285716, 1e-15), "A 1.2");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo((double)189, 1e-15), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo((double)677, 1e-15), "A -10.0");
// Test Linear Case
for (int k = 2; k < 6; k++)
{
double[] linx, liny, linxtest, linytest;
BuildLinearCase(2, k + 1, out linx, out liny, out linxtest, out linytest);
IInterpolationMethod linearMethod = Interpolation.CreateNaturalCubicSpline(linx, liny);
for (int i = 0; i < linxtest.Length; i++)
{
Assert.That(linearMethod.Interpolate(linxtest[i]), NumericIs.AlmostEqualTo(linytest[i], 1e-12), String.Format("Linear k={0} i={1}", k, i));
}
}
}
public void TestInterpolationMethod_LinearSpline()
{
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.CreateLinearSpline(t, x);
Assert.That(method, Is.TypeOf(typeof(LinearSplineInterpolation)), "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "A Exact Point " + i.ToString());
}
// Maple: "f := x -> piecewise(x<-1,3+x,x<0,-1-3*x,x<1,-1+x,-1+x);"
// Maple: "f(x)"
Assert.That(method.Interpolate(-2.4), NumericIs.AlmostEqualTo(.6, 1e-15), "A -2.4");
Assert.That(method.Interpolate(-0.9), NumericIs.AlmostEqualTo(1.7, 1e-15), "A -0.9");
Assert.That(method.Interpolate(-0.5), NumericIs.AlmostEqualTo(.5, 1e-15), "A -0.5");
Assert.That(method.Interpolate(-0.1), NumericIs.AlmostEqualTo(-.7, 1e-15), "A -0.1");
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(-.9, 1e-15), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(-.6, 1e-15), "A 0.4");
Assert.That(method.Interpolate(1.2), NumericIs.AlmostEqualTo(.2, 1e-15), "A 1.2");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo(9.0, 1e-15), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo(-7.0, 1e-15), "A -10.0");
// Test Linear Case
for (int k = 2; k < 6; k++)
{
double[] linx, liny, linxtest, linytest;
BuildLinearCase(2, k + 1, out linx, out liny, out linxtest, out linytest);
IInterpolationMethod linearMethod = Interpolation.CreateLinearSpline(linx, liny);
for (int i = 0; i < linxtest.Length; i++)
{
Assert.That(linearMethod.Interpolate(linxtest[i]), NumericIs.AlmostEqualTo(linytest[i], 1e-12), String.Format("Linear k={0} i={1}", k, i));
}
}
}
public void TestInterpolationMethod_AkimaSpline()
{
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.CreateAkimaCubicSpline(t, x);
Assert.That(method, Is.TypeOf(typeof(AkimaSplineInterpolation)), "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "A Exact Point " + i.ToString());
}
// TODO: Verify the expected values (that they are really the expected ones)
Assert.That(method.Interpolate(-2.4), NumericIs.AlmostEqualTo(-0.52, 1e-15), "A -2.4");
Assert.That(method.Interpolate(-0.9), NumericIs.AlmostEqualTo(1.826, 1e-15), "A -0.9");
Assert.That(method.Interpolate(-0.5), NumericIs.AlmostEqualTo(0.25, 1e-15), "A -0.5");
Assert.That(method.Interpolate(-0.1), NumericIs.AlmostEqualTo(-1.006, 1e-15), "A -0.1");
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(-0.9, 1e-15), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(-0.6, 1e-15), "A 0.4");
Assert.That(method.Interpolate(1.2), NumericIs.AlmostEqualTo(0.2, 1e-15), "A 1.2");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo((double)9, 1e-14), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo((double)(-151), 1e-14), "A -10.0");
// Test Linear Case
for (int k = 2; k < 6; k++)
{
double[] linx, liny, linxtest, linytest;
BuildLinearCase(2, k + 4, out linx, out liny, out linxtest, out linytest);
IInterpolationMethod linearMethod = Interpolation.CreateAkimaCubicSpline(linx, liny);
for (int i = 0; i < linxtest.Length; i++)
{
Assert.That(linearMethod.Interpolate(linxtest[i]), NumericIs.AlmostEqualTo(linytest[i], 1e-12), String.Format("Linear k={0} i={1}", k, i));
}
}
}
public void TestInterpolationMethod_NevillePolynomial()
{
double[] t = new double[] { 0.0, 1.0, 3.0, 4.0 };
double[] x = new double[] { 0.0, 3.0, 1.0, 3.0 };
IInterpolationMethod method = Interpolation.CreatePolynomial(t, x);
Assert.That(method, Is.TypeOf(typeof(PolynomialInterpolation)), "Type");
double dx, d2x;
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "A Exact Point " + i.ToString());
Assert.That(method.Differentiate(t[i], out dx, out d2x), Is.EqualTo(x[i]), "B Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=0.1},PolynomialInterpolation([[0,0],[1,3],[3,1],[4,3]], x)),20);"
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(.57225000000000000000, 1e-15), "A 0.1");
Assert.That(method.Differentiate(0.1, out dx, out d2x), NumericIs.AlmostEqualTo(.57225000000000000000, 1e-15), "B 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(1.8840000000000000000, 1e-15), "A 0.4");
Assert.That(method.Differentiate(0.4, out dx, out d2x), NumericIs.AlmostEqualTo(1.8840000000000000000, 1e-15), "B 0.4");
Assert.That(method.Interpolate(1.1), NumericIs.AlmostEqualTo(3.0314166666666666667, 1e-15), "A 1.1");
Assert.That(method.Differentiate(1.1, out dx, out d2x), NumericIs.AlmostEqualTo(3.0314166666666666667, 1e-15), "B 1.1");
Assert.That(method.Interpolate(3.2), NumericIs.AlmostEqualTo(1.034666666666666667, 1e-15), "A 3.2");
Assert.That(method.Differentiate(3.2, out dx, out d2x), NumericIs.AlmostEqualTo(1.034666666666666667, 1e-15), "B 3.2");
Assert.That(method.Interpolate(4.5), NumericIs.AlmostEqualTo(6.281250000000000000, 1e-15), "A 4.5");
Assert.That(method.Differentiate(4.5, out dx, out d2x), NumericIs.AlmostEqualTo(6.281250000000000000, 1e-15), "B 4.5");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo(277.50000000000000000, 1e-15), "A 10.0");
Assert.That(method.Differentiate(10.0, out dx, out d2x), NumericIs.AlmostEqualTo(277.50000000000000000, 1e-15), "B 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo(-1010.8333333333333333, 1e-15), "A -10.0");
Assert.That(method.Differentiate(-10.0, out dx, out d2x), NumericIs.AlmostEqualTo(-1010.8333333333333333, 1e-15), "B -10.0");
// Test Linear Case
for (int k = 2; k < 7; k++)
{
double[] linx, liny, linxtest, linytest;
BuildLinearCase(2, k, out linx, out liny, out linxtest, out linytest);
IInterpolationMethod linearMethod = Interpolation.CreatePolynomial(linx, liny);
for (int i = 0; i < linxtest.Length; i++)
{
Assert.That(linearMethod.Interpolate(linxtest[i]), NumericIs.AlmostEqualTo(linytest[i], 1e-12), String.Format("Linear k={0} i={1}", k, i));
}
}
}
public void IRID209_InterpolationWithThreeSamples()
{
double[] values = new double[] { 6.0, 12.0, 16.0 };
double[] points = new double[] { 1.0, 2.0, 3.0 };
IInterpolationMethod method2 = Interpolation.CreatePolynomial(points, values);
double b = method2.Interpolate(2.5);
Assert.That(b, Is.Not.NaN, "polynomial (neville)");
IInterpolationMethod method = Interpolation.Create(points, values);
double a = method.Interpolate(2.5);
Assert.That(a, Is.Not.NaN, "rational pole-free");
}
public void TestInterpolationMethod_RationalWithPoles()
{
double[] t = new double[] { 0, 1, 3, 4, 5 };
double[] x = new double[] { 0, 3, 1000, -1000, 3 };
RationalInterpolation method = new RationalInterpolation();
method.Init(t, x);
for (int i = 0; i < t.Length; i++)
{
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=0.1},RationalInterpolation([[0,0],[1,3],[3,1000],[4,-1000], [5,3]], x)),20);"
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(.19389203383553566255, 1e-14), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(.88132900698869875369, 1e-14), "A 0.4");
Assert.That(method.Interpolate(1.1), NumericIs.AlmostEqualTo(3.5057665681580626913, 1e-15), "A 1.1");
Assert.That(method.Interpolate(3.01), NumericIs.AlmostEqualTo(1548.7666642693586902, 1e-13), "A 3.01");
Assert.That(method.Interpolate(3.02), NumericIs.AlmostEqualTo(3362.2564334253633516, 1e-13), "A 3.02");
Assert.That(method.Interpolate(3.03), NumericIs.AlmostEqualTo(-22332.603641443806014, 1e-12), "A 3.03");
Assert.That(method.Interpolate(3.1), NumericIs.AlmostEqualTo(-440.30323769822443789, 1e-14), "A 3.1");
Assert.That(method.Interpolate(3.2), NumericIs.AlmostEqualTo(-202.42421196280566349, 1e-14), "A 3.2");
Assert.That(method.Interpolate(4.5), NumericIs.AlmostEqualTo(21.208249625210155439, 1e-14), "A 4.5");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo(-4.8936986959784751517, 1e-13), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo(-3.6017584308603731307, 1e-13), "A -10.0");
// Test Linear Case
for (int k = 2; k < 6; k++)
{
double[] linx, liny, linxtest, linytest;
BuildLinearCase(2, k, out linx, out liny, out linxtest, out linytest);
IInterpolationMethod linearMethod = Interpolation.CreateRational(linx, liny);
for (int i = 0; i < linxtest.Length; i++)
{
// very weak test, but rational with poles is incredibly bad in the linear case
Assert.That(linearMethod.Interpolate(linxtest[i]), Is.Not.NaN, String.Format("Linear k={0} i={1}", k, i));
}
}
}
public void TestInterpolationMethod_EquidistantBarycentricPolynomial()
{
double[] x = new double[] { 0.0, 3.0, 2.5, 1.0, 3.0 };
IInterpolationMethod method = Interpolation.CreateOnEquidistantPoints(0.0, 4.0, x);
Assert.IsInstanceOfType(typeof(EquidistantPolynomialInterpolation), method, "Type");
for (int i = 0; i < 4; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.AreEqual(x[i], method.Interpolate(i), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=0.1},PolynomialInterpolation([[0,0],[1,3],[2,2.5],[3,1],[4,3]], x)),20);"
NumericAssert.AreAlmostEqual(.48742500000000000000, method.Interpolate(0.1), 1e-15, "A 0.1");
NumericAssert.AreAlmostEqual(1.6968000000000000000, method.Interpolate(0.4), 1e-15, "A 0.4");
NumericAssert.AreAlmostEqual(3.0819250000000000000, method.Interpolate(1.1), 1e-15, "A 1.1");
NumericAssert.AreAlmostEqual(.940800000000000001, method.Interpolate(3.2), 1e-15, "A 3.2");
NumericAssert.AreAlmostEqual(7.265625000000000001, method.Interpolate(4.5), 1e-15, "A 4.5");
NumericAssert.AreAlmostEqual(592.50000000000000000, method.Interpolate(10.0), 1e-13, "A 10.0");
NumericAssert.AreAlmostEqual(657.50000000000000000, method.Interpolate(-10.0), 1e-12, "A -10.0");
}
public void TestInterpolationMethod_EquidistantBarycentricPolynomial()
{
double[] x = new double[] { 0.0, 3.0, 2.5, 1.0, 3.0 };
IInterpolationMethod method = Interpolation.CreateOnEquidistantPoints(0.0, 4.0, x);
Assert.That(method, Is.TypeOf(typeof(EquidistantPolynomialInterpolation)), "Type");
for (int i = 0; i < 4; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(i), Is.EqualTo(x[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "evalf(subs({x=0.1},PolynomialInterpolation([[0,0],[1,3],[2,2.5],[3,1],[4,3]], x)),20);"
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(.48742500000000000000, 1e-15), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(1.6968000000000000000, 1e-15), "A 0.4");
Assert.That(method.Interpolate(1.1), NumericIs.AlmostEqualTo(3.0819250000000000000, 1e-15), "A 1.1");
Assert.That(method.Interpolate(3.2), NumericIs.AlmostEqualTo(.940800000000000001, 1e-15), "A 3.2");
Assert.That(method.Interpolate(4.5), NumericIs.AlmostEqualTo(7.265625000000000001, 1e-15), "A 4.5");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo(592.50000000000000000, 1e-13), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo(657.50000000000000000, 1e-12), "A -10.0");
// Test Linear Case
for (int k = 2; k < 7; k++)
{
double[] linx, liny, linxtest, linytest;
BuildLinearCase(2, k, out linx, out liny, out linxtest, out linytest);
IInterpolationMethod linearMethod = Interpolation.CreateOnEquidistantPoints(2, Math.Max(k, 3), liny);
for (int i = 0; i < linxtest.Length; i++)
{
Assert.That(linearMethod.Interpolate(linxtest[i]), NumericIs.AlmostEqualTo(linytest[i], 1e-12), String.Format("Linear k={0} i={1}", k, i));
}
}
}
public void TestInterpolationMethod_RationalPoleFreeBarycentric()
{
// *************************************************************************************************
// 1st: polynomial case (equidistant polynomial generates the same values; rational would have pole)
// *************************************************************************************************
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.Create(t, x);
Assert.IsInstanceOfType(typeof(RationalPoleFreeInterpolation), method, "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.AreEqual(x[i], method.Interpolate(t[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "PolynomialInterpolation([[-2,1],[-1,2],[0,-1],[1,0],[2,1]], x);"
NumericAssert.AreAlmostEqual(-4.5968, method.Interpolate(-2.4), 1e-15, "A -2.4");
NumericAssert.AreAlmostEqual(1.65395, method.Interpolate(-0.9), 1e-15, "A -0.9");
NumericAssert.AreAlmostEqual(0.21875, method.Interpolate(-0.5), 1e-15, "A -0.5");
NumericAssert.AreAlmostEqual(-0.84205, method.Interpolate(-0.1), 1e-15, "A -0.1");
NumericAssert.AreAlmostEqual(-1.10805, method.Interpolate(0.1), 1e-15, "A 0.1");
NumericAssert.AreAlmostEqual(-1.1248, method.Interpolate(0.4), 1e-15, "A 0.4");
NumericAssert.AreAlmostEqual(0.5392, method.Interpolate(1.2), 1e-15, "A 1.2");
NumericAssert.AreAlmostEqual(-4431, method.Interpolate(10.0), 1e-12, "A 10.0");
NumericAssert.AreAlmostEqual(-5071, method.Interpolate(-10.0), 1e-12, "A -10.0");
// *****************************************************************************
// 2nd: x(t) = 1/(1+t^2), t=-5..5 (polynomial can' t interpolate that function!)
// *****************************************************************************
t = new double[40];
x = new double[40];
double step = 10.0 / 39.0;
for (int i = 0; i < t.Length; i++)
{
double tt = -5 + i * step;
t[i] = tt;
x[i] = 1.0 / (1.0 + tt * tt);
}
RationalPoleFreeInterpolation methodTyped = (RationalPoleFreeInterpolation)method;
methodTyped.Init(t, x); // re-initialize for another set of points/values.
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.AreEqual(x[i], method.Interpolate(t[i]), "B Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "tt := [seq(-5+(i-1)*10/39,i=1..40)]: xx := [seq(1/(1+tt[i]*tt[i]),i=1..40)]:"
// Maple: "RationalInterpolation(tt, xx, x);"
}
//private void CreateOptimizationPoints () {
// currentResolution = resolution;
// optimizationPoints = new ParticleSystem.Particle[resolution];
// float increment = 1f / (resolution - 1);
// int i = 0;
// for (int t = 0; t < resolution; t++) {
// Vector3 p = new Vector3(0f, 0f, 0f);
// optimizationPoints[i].position = p;
// optimizationPoints[i].color = new Color(p.x + increment * resolution / 2, 0f, p.z + increment * resolution / 2);
// optimizationPoints[i++].size = 0.15f;
// }
//}
void Update()
{
if (currentResolution != resolution || points == null || currentGridOption != gridOption)
{
CreateGridPoints();
}
//if (currentResolution != resolution || optimizationPoints == null) {
// CreateOptimizationPoints();
//}
FunctionDelegate f = functionDelegates[(int)function];
float t = Time.timeSinceLevelLoad;
//function graph steps
for (int i = 0; i < points.Length; i++)
{
Vector3 p = points [i].position;
p.y = f(p, t);
points[i].position = p;
Color c = points [i].color;
c.g = p.y;
points[i].color = c;
}
//optimization steps
Matrix a = QuadraticFormMatrix(t);
Matrix currentPoint = new Matrix(new double[][] {
new double[] { xStart },
new double[] { zStart }
});
Matrix lastPoint = currentPoint.Clone();
double[] ts = new double[iterationCount + 1];
double[] xs = new double[iterationCount + 1];
double[] zs = new double[iterationCount + 1];
xs[0] = currentPoint.GetArray()[0][0];
zs[0] = currentPoint.GetArray()[1][0];
for (int i = 0; i < iterationCount; i++)
{
ts[i] = i;
currentPoint = lastPoint - ((double)learningRate) * a * lastPoint;
xs[i + 1] = currentPoint.GetArray()[0][0];
zs[i + 1] = currentPoint.GetArray()[1][0];
lastPoint = currentPoint.Clone();
}
ts[iterationCount] = iterationCount;
IInterpolationMethod xInterp = Interpolation.Create(ts, xs);
IInterpolationMethod zInterp = Interpolation.Create(ts, zs);
float increment = ((float)iterationCount) / ((float)(resolution - 1));
for (int i = 0; i < resolution; i++)
{
float curInc = increment * i;
Vector3 p = new Vector3((float)xInterp.Interpolate(curInc), 0.0f, (float)zInterp.Interpolate(curInc));
p.y = f(p, t);
points[i].position = p;
Color c = points [i].color;
c.g = p.y;
points[i].color = c;
}
particleSystem.SetParticles(points, points.Length);
//particleSystem.SetParticles(optimizationPoints, optimizationPoints.Length);
//Matrix m = new Matrix(new double[][] {
// new double[] { 10.0, -18.0 },
// new double[] { 6.0, -11.0 } });
// alternative way to create the matrix:
// double[][] data = Matrix.CreateMatrixData(2, 2);
// data[0][0] = 10.0;
// data[1][0] = 6.0;
// data[0][1] = -18.0;
// data[1][1] = -11.0;
// Matrix m = new Matrix(data);
//EigenvalueDecomposition eigen = m.EigenvalueDecomposition;
//Complex[] eigenValues = eigen.EigenValues;
// eigenvalues: 1, -2
//Matrix eigenVectors = eigen.EigenVectors;
// eigenvectors: [0.894...,0.447...] and [6.708...,4.473...]
// alternative way to access the eigenvalues witout the Complex type:
// double[] eigenValuesReal = eigen.RealEigenvalues; // real part
// double[] eigenValuesImag = eigen.ImagEigenvalues; // imaginary part
}
public void TestInterpolationMethod_RationalPoleFreeBarycentric()
{
/**************************************************************************************************
* 1st: polynomial case (equidistant polynomial generates the same values; rational would have pole)
**************************************************************************************************/
double[] t = new double[] { -2.0, -1.0, 0.0, 1.0, 2.0 };
double[] x = new double[] { 1.0, 2.0, -1.0, 0.0, 1.0 };
IInterpolationMethod method = Interpolation.Create(t, x);
Assert.That(method, Is.TypeOf(typeof(RationalPoleFreeInterpolation)), "Type");
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "A Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "PolynomialInterpolation([[-2,1],[-1,2],[0,-1],[1,0],[2,1]], x);"
Assert.That(method.Interpolate(-2.4), NumericIs.AlmostEqualTo(-4.5968, 1e-15), "A -2.4");
Assert.That(method.Interpolate(-0.9), NumericIs.AlmostEqualTo(1.65395, 1e-15), "A -0.9");
Assert.That(method.Interpolate(-0.5), NumericIs.AlmostEqualTo(0.21875, 1e-15), "A -0.5");
Assert.That(method.Interpolate(-0.1), NumericIs.AlmostEqualTo(-0.84205, 1e-15), "A -0.1");
Assert.That(method.Interpolate(0.1), NumericIs.AlmostEqualTo(-1.10805, 1e-15), "A 0.1");
Assert.That(method.Interpolate(0.4), NumericIs.AlmostEqualTo(-1.1248, 1e-15), "A 0.4");
Assert.That(method.Interpolate(1.2), NumericIs.AlmostEqualTo(0.5392, 1e-15), "A 1.2");
Assert.That(method.Interpolate(10.0), NumericIs.AlmostEqualTo((double)(-4431), 1e-12), "A 10.0");
Assert.That(method.Interpolate(-10.0), NumericIs.AlmostEqualTo((double)(-5071), 1e-12), "A -10.0");
/******************************************************************************
* 2nd: x(t) = 1/(1+t^2), t=-5..5 (polynomial can' t interpolate that function!)
******************************************************************************/
t = new double[40];
x = new double[40];
const double step = 10.0 / 39.0;
for (int i = 0; i < t.Length; i++)
{
double tt = -5 + (i * step);
t[i] = tt;
x[i] = 1.0 / (1.0 + (tt * tt));
}
RationalPoleFreeInterpolation methodTyped = (RationalPoleFreeInterpolation)method;
methodTyped.Init(t, x); // re-initialize for another set of points/values.
for (int i = 0; i < t.Length; i++)
{
// verify the interpolated values exactly at the sample points.
Assert.That(method.Interpolate(t[i]), Is.EqualTo(x[i]), "B Exact Point " + i.ToString());
}
// Maple: "with(CurveFitting);"
// Maple: "tt := [seq(-5+(i-1)*10/39,i=1..40)]: xx := [seq(1/(1+tt[i]*tt[i]),i=1..40)]:"
// Maple: "RationalInterpolation(tt, xx, x);"
// Test Linear Case
for (int k = 2; k < 7; k++)
{
double[] linx, liny, linxtest, linytest;
BuildLinearCase(2, k, out linx, out liny, out linxtest, out linytest);
IInterpolationMethod linearMethod = Interpolation.Create(linx, liny);
for (int i = 0; i < linxtest.Length; i++)
{
Assert.That(linearMethod.Interpolate(linxtest[i]), NumericIs.AlmostEqualTo(linytest[i], 1e-12), String.Format("Linear k={0} i={1}", k, i));
}
}
}