private DoubleVector Pivot(IReadOnlyList <double> B)
{
var ret = new DoubleVector(B.Count);
var retArray = ret.GetInternalData();
for (int i = 0; i < pivots.Length; i++)
{
retArray[i] = B[pivots[i]];
}
return(ret);
}
public void SolveGETest()
{
const int N = 50;
var a = new SparseDoubleMatrix(N, N);
for (int i = 0; i < N; i++)
{
a[i, i] = 1;
}
// Apply random rotations around each pair of axes. This will keep det(A) ~ 1
var rand = new Random();
for (int i = 0; i < N; i++)
{
for (int j = i + 1; j < N; j++)
{
double angle = rand.NextDouble() * 2 * Math.PI;
var r = new SparseDoubleMatrix(N, N);
for (int k = 0; k < N; k++)
{
r[k, k] = 1;
}
r[i, i] = r[j, j] = Math.Cos(angle);
r[i, j] = Math.Sin(angle);
r[j, i] = -Math.Sin(angle);
a = a * r;
}
}
var ainit = a.Clone();
// Generate random vector
var b = new DoubleVector(N);
for (int i = 0; i < N; i++)
{
b[i] = rand.NextDouble();
}
var binit = b.Clone();
// Solve system
var solver = new GaussianEliminationSolver();
var sw = new Stopwatch();
sw.Start();
var x = solver.SolveDestructive(a, b.GetInternalData());
sw.Stop();
Trace.WriteLine("Gaussian elimination took: " + sw.ElapsedTicks);
// Put solution into system
var b2 = ainit * x;
// Verify result is the same
Assert.IsTrue(VectorMath.LInfinityNorm(binit, b2) < 1e-6);
}
///<summary>Solves a system on linear equations, AX=B, where A is the factored matrixed.</summary>
///<param name="B">RHS side of the system.</param>
///<returns>the solution vector, X.</returns>
///<exception cref="ArgumentNullException">B is null.</exception>
///<exception cref="NotPositiveDefiniteException">A is not positive definite.</exception>
///<exception cref="ArgumentException">The number of rows of A and the length of B must be the same.</exception>
public DoubleVector Solve(IReadOnlyList <double> B)
{
if (B == null)
{
throw new System.ArgumentNullException("B cannot be null.");
}
Compute();
if (!ispd)
{
throw new NotPositiveDefiniteException();
}
else
{
if (B.Count != order)
{
throw new System.ArgumentException("The length of B must be the same as the order of the matrix.");
}
#if MANAGED
// Copy right hand side.
var X = new DoubleVector(B);
var xarray = X.GetInternalData();
// Solve L*Y = B;
for (int i = 0; i < order; i++)
{
double sum = B[i];
for (int k = i - 1; k >= 0; k--)
{
sum -= l.data[i][k] * xarray[k];
}
xarray[i] = sum / l.data[i][i];
}
// Solve L'*X = Y;
for (int i = order - 1; i >= 0; i--)
{
double sum = xarray[i];
for (int k = i + 1; k < order; k++)
{
sum -= l.data[k][i] * xarray[k];
}
xarray[i] = sum / l.data[i][i];
}
return(X);
#else
double[] rhs = DoubleMatrix.ToLinearArray(B);
Lapack.Potrs.Compute(Lapack.UpLo.Lower, order, 1, l.data, order, rhs, B.Length);
DoubleVector ret = new DoubleVector(order, B.Length);
ret.data = rhs;
return(ret);
#endif
}
}
public void GetInternalData()
{
double[] testvector = new double[2]{0,1};
DoubleVector test = new DoubleVector(testvector);
double[] internaldata = test.GetInternalData();
Assert.AreEqual(internaldata.Length,testvector.Length);
Assert.AreEqual(internaldata[0],testvector[0]);
Assert.AreEqual(internaldata[1],testvector[1]);
}
/// <summary>
///
/// </summary>
/// <param name="x"></param>
/// <param name="bw"></param>
/// <param name="bwSel"></param>
/// <param name="adjust"></param>
/// <param name="kernel"></param>
/// <param name="weights"></param>
/// <param name="width"></param>
/// <param name="widthSel"></param>
/// <param name="n"></param>
/// <param name="from"></param>
/// <param name="to"></param>
/// <param name="cut"></param>
/// <remarks>Adapted from the R-project (www.r-project.org), Version 2.72, file density.R</remarks>
public static ProbabilityDensityResult ProbabilityDensity(
this IROVector x,
double bw,
string bwSel,
double adjust,
ConvolutionKernel kernel,
IROVector weights,
double width,
string widthSel,
int n,
double from,
double to,
double cut // default: 3
)
{
double wsum;
if (null == weights)
{
weights = VectorMath.GetConstantVector(1.0 / x.Length, x.Length);
wsum = 1;
}
else
{
wsum = weights.Sum();
}
double totMass = 1;
int n_user = n;
n = Math.Max(n, 512);
if (n > 512)
n = BinaryMath.NextPowerOfTwoGreaterOrEqualThan(n);
if (bw.IsNaN() && !(width.IsNaN() && null == widthSel))
{
if (!width.IsNaN())
{
// S has width equal to the length of the support of the kernel
// except for the gaussian where it is 4 * sd.
// R has bw a multiple of the sd.
double fac = 1;
switch (kernel)
{
case ConvolutionKernel.Gaussian:
fac = 4;
break;
case ConvolutionKernel.Rectangular:
fac = 2 * Math.Sqrt(3);
break;
case ConvolutionKernel.Triangular:
fac = 2 * Math.Sqrt(6);
break;
case ConvolutionKernel.Epanechnikov:
fac = 2 * Math.Sqrt(5);
break;
case ConvolutionKernel.Biweight:
fac = 2 * Math.Sqrt(7);
break;
case ConvolutionKernel.Cosine:
fac = 2 / Math.Sqrt(1 / 3 - 2 / (Math.PI * Math.PI));
break;
case ConvolutionKernel.Optcosine:
fac = 2 / Math.Sqrt(1 - 8 / (Math.PI * Math.PI));
break;
default:
throw new ArgumentException("Unknown convolution kernel");
}
bw = width / fac;
}
else
{
bwSel = widthSel;
}
}
if (null != bwSel)
{
if (x.Length < 2)
throw new ArgumentException("need at least 2 points to select a bandwidth automatically");
switch (bwSel.ToLowerInvariant())
{
case "nrd0":
//nrd0 = bw.nrd0(x),
break;
case "nrd":
//nrd = bw.nrd(x),
break;
case "ucv":
//ucv = bw.ucv(x),
break;
case "bcv":
//bcv = bw.bcv(x),
break;
case "sj":
//sj = , "sj-ste" = bw.SJ(x, method="ste"),
break;
case "sj-dpi":
//"sj-dpi" = bw.SJ(x, method="dpi"),
break;
default:
throw new ArgumentException("Unknown bandwith selection rule: " + bwSel.ToString());
}
}
if (!RMath.IsFinite(bw))
throw new ArithmeticException("Bandwidth is not finite");
bw = adjust * bw;
if (!(bw > 0))
throw new ArithmeticException("Bandwith is not positive");
if (from.IsNaN())
from = x.GetMinimum() - cut * bw;
if (to.IsNaN())
to = x.GetMaximum() + cut * bw;
if (!RMath.IsFinite(from))
throw new ArithmeticException("non-finite 'from'");
if (!to.IsFinite())
throw new ArithmeticException("non-finite 'to'");
double lo = from - 4 * bw;
double up = to + 4 * bw;
var y = new DoubleVector(2 * n);
MassDistribution(x, weights, lo, up, y, n);
y.Multiply(totMass);
var kords = new DoubleVector(2 * n);
kords.FillWithLinearSequenceGivenByStartEnd(0, 2 * (up - lo));
for (int i = n + 1, j = n - 1; j >= 0; i++, j--)
kords[i] = -kords[j];
switch (kernel)
{
case ConvolutionKernel.Gaussian:
kords.Apply(new Probability.NormalDistribution(0, bw).PDF);
break;
case ConvolutionKernel.Rectangular:
double a = bw * Math.Sqrt(3);
kords.Apply(delegate(double xx) { return Math.Abs(xx) < a ? 0.5 / a : 0; });
break;
case ConvolutionKernel.Triangular:
a = bw * Math.Sqrt(6);
kords.Apply(delegate(double xx) { return Math.Abs(xx) < a ? (1 - Math.Abs(xx) / a) / a : 0; });
break;
case ConvolutionKernel.Epanechnikov:
a = bw * Math.Sqrt(5);
kords.Apply(delegate(double xx) { return Math.Abs(xx) < a ? 0.75 * (1 - RMath.Pow2(Math.Abs(xx) / a)) / a : 0; });
break;
case ConvolutionKernel.Biweight:
a = bw * Math.Sqrt(7);
kords.Apply(delegate(double xx) { return Math.Abs(xx) < a ? 15.0 / 16.0 * RMath.Pow2(1 - RMath.Pow2(Math.Abs(xx) / a)) / a : 0; });
break;
case ConvolutionKernel.Cosine:
a = bw / Math.Sqrt(1.0 / 3 - 2 / RMath.Pow2(Math.PI));
kords.Apply(delegate(double xx) { return Math.Abs(xx) < a ? (1 + Math.Cos(Math.PI * xx / a)) / (2 * a) : 0; });
break;
case ConvolutionKernel.Optcosine:
a = bw / Math.Sqrt(1 - 8 / RMath.Pow2(Math.PI));
kords.Apply(delegate(double xx) { return Math.Abs(xx) < a ? Math.PI / 4 * Math.Cos(Math.PI * xx / (2 * a)) / a : 0; });
break;
default:
throw new ArgumentException("Unknown convolution kernel");
}
var result = new DoubleVector(2 * n);
Fourier.FastHartleyTransform.CyclicRealConvolution(y.GetInternalData(), kords.GetInternalData(), result.GetInternalData(), 2 * n, null);
y.Multiply(1.0 / (2 * n));
VectorMath.Max(y, 0, y);
var xords = VectorMath.CreateEquidistantSequenceByStartEndLength(lo, up, n);
var xu = VectorMath.CreateEquidistantSequenceByStartEndLength(from, to, n_user);
double[] res2 = new double[xu.Length];
Interpolation.LinearInterpolation.Interpolate(xords, result, n, xu, xu.Length, 0, out res2);
return new ProbabilityDensityResult() { X = xu, Y = VectorMath.ToROVector(res2), Bandwidth = bw };
}
