public static ExtendedDualNumber Sqr(ExtendedDualNumber f)
{
return new ExtendedDualNumber(f, f.value * f.value, 2.0 * f.value, 2.0, 0.0);
}
public static ExtendedDualNumber Sqrt(ExtendedDualNumber f)
{
double g = Math.Sqrt(f.value);
double g1 = 0.5 / g;
double g11 = -0.5 * g1 / f.value;
double g111 = -3.0 * g11 / (2.0 * f.value);
return new ExtendedDualNumber(f, g, g1, g11, g111);
}
public static ExtendedDualNumber Pow(ExtendedDualNumber f1, ExtendedDualNumber f2)
{
// FIXME Optimize this!
double g = Math.Pow(f1.value, f2.value);
double g1 = f2.value * Math.Pow(f1.value, f2.value - 1.0);
double g2 = Math.Log(f1.value) * Math.Pow(f1.value, f2.value);
double g11 = f2.value * (f2.value - 1.0) * Math.Pow(f1.value, f2.value - 2.0);
double g22 = Math.Log(f1.value) * Math.Log(f1.value) * Math.Pow(f1.value, f2.value);
double g12 = Math.Pow(f1.value, f2.value - 1.0) * (1.0 + Math.Log(f1.value) * f2.value);
double g111 = f2.value * (f2.value - 1.0) * (f2.value - 2.0) * Math.Pow(f1.value, f2.value - 3.0);
double g222 = Math.Log(f1.value) * Math.Log(f1.value) * Math.Log(f1.value) * Math.Pow(f1.value, f2.value);
double g112 = (2.0 * f2.value - 1.0 + f2.value * (f2.value - 1.0) * Math.Log(f1.value)) * Math.Pow(f1.value, f2.value - 2.0);
double g122 = 2.0 * Math.Log(f1.value) / f1.value * Math.Pow(f1.value, f2.value) + Math.Log(f1.value) * Math.Log(f1.value) * f2.value * Math.Pow(f1.value, f2.value - 1.0);
return new ExtendedDualNumber(f1, f2, g, g1, g2, g11, g12, g22, g111, g112, g122, g222);
}
public static ExtendedDualNumber Sin(ExtendedDualNumber f)
{
double g = Math.Sin(f.value);
double g1 = Math.Cos(f.value);
double g11 = -g;
double g111 = -g1;
return new ExtendedDualNumber(f, g, g1, g11, g111);
}
public static ExtendedDualNumber Log(ExtendedDualNumber f)
{
double g = Math.Log(f.value);
double g1 = 1.0 / f.value;
double g11 = -g1 / f.value;
double g111 = -2.0 * g11 / f.value;
return new ExtendedDualNumber(f, g, g1, g11, g111);
}
public static ExtendedDualNumber Exp(ExtendedDualNumber f)
{
double g = Math.Exp(f.value);
double g1 = g;
double g11 = g;
double g111 = g;
return new ExtendedDualNumber(f, g, g1, g11, g111);
}
static ExtendedDualNumber()
{
zero = new ExtendedDualNumber(0.0);
}
public ExtendedDualNumber(ExtendedDualNumber f1, ExtendedDualNumber f2, double g, double g1, double g2, double g11, double g12, double g22, double g111, double g112, double g122, double g222)
{
value = g;
if (f1.gradientArray != null || f2.gradientArray != null)
{
if (f1.gradientArray != null && f2.gradientArray != null && (f1.n != f2.n || f1.n0 != f2.n0))
{
throw new ArgumentException("Inconsistent number of derivatives.");
}
// One of the counters may be zero if the corresponding ExtendedDualNumber is a constant.
n = Math.Max(f1.n, f2.n);
n0 = Math.Max(f1.n0, f2.n0);
gradientArray = new double[n];
if (g1 != 0.0 && f1.gradientArray != null)
{
for (int i = 0; i < n; i++)
{
gradientArray[i] += g1 * f1.gradientArray[i];
}
}
if (g2 != 0.0 && f2.gradientArray != null)
{
for (int i = 0; i < n; i++)
{
gradientArray[i] += g2 * f2.gradientArray[i];
}
}
if (f1.hessianArray != null || f2.hessianArray != null || g11 != 0.0 || g12 != 0.0 || g22 != 0.0 || g111 != 0.0 || g112 != 0.0 || g122 != 0.0 || g222 != 0.0)
{
hessianArray = new double[HessianSize(n)];
if (g1 != 0.0 && f1.hessianArray != null)
{
for (int i = 0, l = 0; i < n; i++)
{
for (int j = i; j < n; j++, l++)
{
hessianArray[l] += g1 * f1.hessianArray[l];
}
}
}
if (g2 != 0.0 && f2.hessianArray != null)
{
for (int i = 0, l = 0; i < n; i++)
{
for (int j = i; j < n; j++, l++)
{
hessianArray[l] += g2 * f2.hessianArray[l];
}
}
}
if (g11 != 0.0 && f1.gradientArray != null)
{
for (int i = 0, l = 0; i < n; i++)
{
for (int j = i; j < n; j++, l++)
{
hessianArray[l] += g11 * f1.gradientArray[i] * f1.gradientArray[j];
}
}
}
if (g22 != 0.0 && f2.gradientArray != null)
{
for (int i = 0, l = 0; i < n; i++)
{
for (int j = i; j < n; j++, l++)
{
hessianArray[l] += g22 * f2.gradientArray[i] * f2.gradientArray[j];
}
}
}
if (g12 != 0.0 && f1.gradientArray != null && f2.gradientArray != null)
{
for (int i = 0, l = 0; i < n; i++)
{
for (int j = i; j < n; j++, l++)
{
hessianArray[l] += g12 * (f1.gradientArray[i] * f2.gradientArray[j] + f2.gradientArray[i] * f1.gradientArray[j]);
}
}
}
if (f1.hessianArray != null || f2.hessianArray != null || g111 != 0.0 || g112 != 0.0 || g122 != 0.0 || g222 != 0.0)
{
thirdArray = new double[ThirdReducedSize(n, n0)];
// The counters below are constructed so that:
//
// hessianArray[l1] is the (i,j)th entry of the Hessian,
// hessianArray[l2] is the (i,k)th entry of the Hessian,
// hessianArray[l3] is the (j,k)th entry of the Hessian,
// thirdArray[m] is the (i,j,k)th entry of the third derivatives tensor.
//
// It's made this way to eliminate redundant entries. In case of the Hessian, only
// the upper triangular part is stored.
if (g1 != 0.0 && f1.thirdArray != null)
{
for (int i = 0, m = 0; i < n0; i++)
{
for (int j = i; j < n; j++)
{
for (int k = j; k < n; k++, m++)
{
thirdArray[m] += g1 * f1.thirdArray[m];
}
}
}
}
if (g2 != 0.0 && f2.thirdArray != null)
{
for (int i = 0, m = 0; i < n0; i++)
{
for (int j = i; j < n; j++)
{
for (int k = j; k < n; k++, m++)
{
thirdArray[m] += g2 * f2.thirdArray[m];
}
}
}
}
if (g11 != 0.0 && f1.hessianArray != null)
{
for (int i = 0, l1 = 0, l2 = 0, m = 0; i < n0; i++)
{
for (int j = i, l3 = l2; j < n; j++, l1++, l2 -= n - j)
{
for (int k = j; k < n; k++, l2++, l3++, m++)
{
thirdArray[m] += g11 * (f1.gradientArray[i] * f1.hessianArray[l3] + f1.gradientArray[j] * f1.hessianArray[l2] + f1.gradientArray[k] * f1.hessianArray[l1]);
}
}
}
}
if (g22 != 0.0 && f2.hessianArray != null)
{
for (int i = 0, l1 = 0, l2 = 0, m = 0; i < n0; i++)
{
for (int j = i, l3 = l2; j < n; j++, l1++, l2 -= n - j)
{
for (int k = j; k < n; k++, l2++, l3++, m++)
{
thirdArray[m] += g22 * (f2.gradientArray[i] * f2.hessianArray[l3] + f2.gradientArray[j] * f2.hessianArray[l2] + f2.gradientArray[k] * f2.hessianArray[l1]);
}
}
}
}
if (g12 != 0.0 && f1.gradientArray != null && f2.hessianArray != null)
{
for (int i = 0, l1 = 0, l2 = 0, m = 0; i < n0; i++)
{
for (int j = i, l3 = l2; j < n; j++, l1++, l2 -= n - j)
{
for (int k = j; k < n; k++, l2++, l3++, m++)
{
thirdArray[m] += g12 * (f1.gradientArray[i] * f2.hessianArray[l3] + f1.gradientArray[j] * f2.hessianArray[l2] + f1.gradientArray[k] * f2.hessianArray[l1]);
}
}
}
}
if (g12 != 0.0 && f2.gradientArray != null && f1.hessianArray != null)
{
for (int i = 0, l1 = 0, l2 = 0, m = 0; i < n0; i++)
{
for (int j = i, l3 = l2; j < n; j++, l1++, l2 -= n - j)
{
for (int k = j; k < n; k++, l2++, l3++, m++)
{
thirdArray[m] += g12 * (f2.gradientArray[i] * f1.hessianArray[l3] + f2.gradientArray[j] * f1.hessianArray[l2] + f2.gradientArray[k] * f1.hessianArray[l1]);
}
}
}
}
if (g111 != 0.0 && f1.gradientArray != null)
{
for (int i = 0, m = 0; i < n0; i++)
{
for (int j = i; j < n; j++)
{
for (int k = j; k < n; k++, m++)
{
thirdArray[m] += g111 * f1.gradientArray[i] * f1.gradientArray[j] * f1.gradientArray[k];
}
}
}
}
if (g222 != 0.0 && f2.gradientArray != null)
{
for (int i = 0, m = 0; i < n0; i++)
{
for (int j = i; j < n; j++)
{
for (int k = j; k < n; k++, m++)
{
thirdArray[m] += g222 * f2.gradientArray[i] * f2.gradientArray[j] * f2.gradientArray[k];
}
}
}
}
if (g112 != 0.0 && f1.gradientArray != null && f2.gradientArray != null)
{
for (int i = 0, m = 0; i < n0; i++)
{
for (int j = i; j < n; j++)
{
for (int k = j; k < n; k++, m++)
{
thirdArray[m] += g112 * (f1.gradientArray[i] * f1.gradientArray[j] * f2.gradientArray[k] + f1.gradientArray[i] * f2.gradientArray[j] * f1.gradientArray[k] + f2.gradientArray[i] * f1.gradientArray[j] * f1.gradientArray[k]);
}
}
}
}
if (g122 != 0.0 && f1.gradientArray != null && f2.gradientArray != null)
{
for (int i = 0, m = 0; i < n0; i++)
{
for (int j = i; j < n; j++)
{
for (int k = j; k < n; k++, m++)
{
thirdArray[m] += g122 * (f2.gradientArray[i] * f2.gradientArray[j] * f1.gradientArray[k] + f2.gradientArray[i] * f1.gradientArray[j] * f2.gradientArray[k] + f1.gradientArray[i] * f2.gradientArray[j] * f2.gradientArray[k]);
}
}
}
}
}
}
}
}
public ExtendedDualNumber(ExtendedDualNumber f, double g, double g1, double g11, double g111)
{
value = g;
if (f.gradientArray != null)
{
n = f.n;
n0 = f.n0;
gradientArray = new double[n];
if (g1 != 0.0)
{
for (int i = 0; i < n; i++)
{
gradientArray[i] += g1 * f.gradientArray[i];
}
}
if (f.hessianArray != null || g11 != 0.0 || g111 != 0.0)
{
hessianArray = new double[HessianSize(n)];
if (g1 != 0.0 && f.hessianArray != null)
{
for (int i = 0, l = 0; i < n; i++)
{
for (int j = i; j < n; j++, l++)
{
hessianArray[l] += g1 * f.hessianArray[l];
}
}
}
if (g11 != 0.0)
{
for (int i = 0, l = 0; i < n; i++)
{
for (int j = i; j < n; j++, l++)
{
hessianArray[l] += g11 * f.gradientArray[i] * f.gradientArray[j];
}
}
}
if (f.hessianArray != null || g111 != 0.0)
{
thirdArray = new double[ThirdReducedSize(n, n0)];
// See how the counters work in the constructor below.
if (g1 != 0.0 && f.thirdArray != null)
{
for (int i = 0, m = 0; i < n0; i++)
{
for (int j = i; j < n; j++)
{
for (int k = j; k < n; k++, m++)
{
thirdArray[m] += g1 * f.thirdArray[m];
}
}
}
}
if (g11 != 0.0 && f.hessianArray != null)
{
for (int i = 0, l1 = 0, l2 = 0, m = 0; i < n0; i++)
{
for (int j = i, l3 = l2; j < n; j++, l1++, l2 -= n - j)
{
for (int k = j; k < n; k++, l2++, l3++, m++)
{
thirdArray[m] += g11 * (f.gradientArray[i] * f.hessianArray[l3] + f.gradientArray[j] * f.hessianArray[l2] + f.gradientArray[k] * f.hessianArray[l1]);
}
}
}
}
if (g111 != 0.0)
{
for (int i = 0, m = 0; i < n0; i++)
{
for (int j = i; j < n; j++)
{
for (int k = j; k < n; k++, m++)
{
thirdArray[m] += g111 * f.gradientArray[i] * f.gradientArray[j] * f.gradientArray[k];
}
}
}
}
}
}
}
}
