public MatrixValue Function(ScalarValue j1, ScalarValue j2)
{
if (isNotHalf(j1))
throw new ArgumentException("0, +-0.5, +-1, +-1.5, ...", j1.Re.ToString());
if (isNotHalf(j2))
throw new ArgumentException("0, +-0.5, +-1, +-1.5, ...", j2.Re.ToString());
var l = 1;
var M = new MatrixValue();
for (var m1 = -j1.Re; m1 <= j1.Re; m1 += 1.0)
{
for (var m2 = -j2.Re; m2 <= j2.Re; m2 += 1.0)
{
var m = m1 + m2;
var ja = j1.Re + j2.Re;
for (var j = Math.Abs(m); j <= ja; j += 1.0)
{
var v = CGCoefficients(j1.Re, j2.Re, j, m1, m2);
M[l, 1] = new ScalarValue(m);
M[l, 2] = new ScalarValue(m1);
M[l, 3] = new ScalarValue(m2);
M[l, 4] = new ScalarValue(j);
M[l, 5] = new ScalarValue(v);
l++;
}
}
}
return M;
}
public MatrixValue Function(MatrixValue M)
{
if (M.DimensionX != 2 && M.DimensionY != 2)
throw new YAMPMatrixDimensionException(2, M.DimensionX, M.DimensionY, M.DimensionX);
var isTransposed = M.DimensionY != 2;
if (isTransposed)
{
M = M.Transpose();
}
var m = new MatrixValue(2, M.DimensionX);
for (var i = 1; i <= M.DimensionX; i++)
{
var x = M[1, i].Re;
var y = M[2, i].Re;
var phi = Math.Atan2(y, x);
var r = x == 0.0 ? y : (y == 0.0 ? x : x / Math.Cos(phi));
m[1, i] = new ScalarValue(r * Math.Cos(phi));
m[2, i] = new ScalarValue(r * Math.Sin(phi));
}
return isTransposed ? m.Transpose() : m;
}
public MatrixValue Function(MatrixValue Y, MatrixValue x)
{
var X = new Double[x.Length];
for (var i = 0; i < x.Length; i++)
{
X[i] = x[i + 1].Re;
}
if (!Y.IsVector)
{
var M = new MatrixValue();
for (var i = 1; i <= Y.DimensionX; i++)
{
var N = YMath.Histogram(Y.GetColumnVector(i), X);
for (var j = 1; j <= N.Length; j++)
{
M[j, i] = N[j];
}
}
return M;
}
return YMath.Histogram(Y, X);
}
public MatrixValue Function(MatrixValue M)
{
if (M.DimensionX != 3 && M.DimensionY != 3)
throw new YAMPMatrixDimensionException(3, M.DimensionX, M.DimensionY, M.DimensionX);
var isTransposed = M.DimensionY != 3;
if (isTransposed)
{
M = M.Transpose();
}
var m = new MatrixValue(3, M.DimensionX);
for (var i = 1; i <= M.DimensionX; i++)
{
var x = M[1, i].Re;
var y = M[2, i].Re;
var z = M[3, i].Re;
var r = Math.Sqrt(x * x + y * y + z * z);
var phi = Math.Atan2(y, x);
var theta = Math.Acos(z / r);
m[1, i] = new ScalarValue(r);
m[2, i] = new ScalarValue(theta);
m[3, i] = new ScalarValue(phi);
}
return isTransposed ? m.Transpose() : m;
}
/// <summary>
/// Creates a new instance.
/// </summary>
/// <param name="samples">The Nx2 matrix containing the sample data.</param>
public SplineInterpolation(MatrixValue samples)
: base(samples)
{
a = new double[Np];
h = new double[Np];
for (int i = 2; i <= Np; i++)
h[i - 1] = samples[i, 1].Re - samples[i - 1, 1].Re;
if (Np > 2)
{
double[] sub = new double[Np - 1];
double[] diag = new double[Np - 1];
double[] sup = new double[Np - 1];
for (int i = 2; i < Np; i++)
{
int j = i - 1;
diag[j] = (h[j] + h[j + 1]) / 3;
sup[j] = h[j + 1] / 6;
sub[j] = h[j] / 6;
a[j] = (samples[i + 1, 2].Re - samples[i, 2].Re) / h[j + 1] - (samples[i, 2].Re - samples[i - 1, 2].Re) / h[j];
}
SolveTridiag(sub, diag, sup, ref a, Np - 2);
}
}
public override Value Perform(Value argument)
{
if (argument is ScalarValue)
{
return argument;
}
if (argument is MatrixValue)
{
var m = argument as MatrixValue;
if (m.DimensionX == 1)
{
return GetVectorMin(m.GetColumnVector(1));
}
else if (m.DimensionY == 1)
{
return GetVectorMin(m.GetRowVector(1));
}
else
{
var M = new MatrixValue(1, m.DimensionX);
for (var i = 1; i <= m.DimensionX; i++)
{
M[1, i] = GetVectorMin(m.GetColumnVector(i));
}
return M;
}
}
throw new YAMPOperationInvalidException("min", argument);
}
/// <summary>
/// Creates the right QR decomposition (Givens or Householder) depending on the given matrix.
/// </summary>
/// <param name="A">The matrix to decompose.</param>
/// <returns>The right QR decomposition implementation.</returns>
public static QRDecomposition Create(MatrixValue A)
{
if (A.IsComplex)
return new GivensDecomposition(A);
return new HouseholderDecomposition(A);
}
static MatrixValue Generate(Int32 n, Int32 m)
{
var distribution = new DiscreteUniformDistribution(Rng);
var l = n * m;
var M = new MatrixValue(n, m);
var numbers = new List<Int32>();
distribution.Alpha = 0;
for (var i = 1; i <= l; i++)
{
numbers.Add(i);
}
for (var j = 1; j <= n; j++)
{
for (var i = 1; i <= m; i++)
{
distribution.Beta = numbers.Count - 1;
var index = distribution.Next();
index = Math.Max(Math.Min(0, index), numbers.Count - 1);
M[j, i] = new ScalarValue(numbers[index]);
numbers.RemoveAt(index);
}
}
return M;
}
public MatrixValue Function(MatrixValue x, MatrixValue y, ScalarValue n)
{
if (x.Length != y.Length)
throw new YAMPDifferentLengthsException(x.Length, y.Length);
var nn = n.GetIntegerOrThrowException("n", Name);
var m = nn + 1;
if (m < 2)
throw new YAMPArgumentRangeException("n", 0.0);
var M = new MatrixValue(x.Length, m);
var b = new MatrixValue(x.Length, 1);
for (var j = 1; j <= M.Rows; j++)
{
var el = ScalarValue.One;
var z = x[j];
for (var i = 1; i <= M.Columns; i++)
{
M[j, i] = el;
el *= z;
}
b[j, 1] = y[j];
}
var qr = QRDecomposition.Create(M);
return qr.Solve(b);
}
Plot2DValue Plot(IFunction f, Double minx, Double maxx, Double precision)
{
var cp = new Plot2DValue();
var N = (Int32)((maxx - minx) / precision) + 1;
var M = new MatrixValue(N, 2);
var x = new ScalarValue(minx);
for (var i = 0; i < N; i++)
{
var row = i + 1;
var y = f.Perform(Context, x);
M[row, 1] = x.Clone();
if (y is ScalarValue)
{
M[row, 2] = (ScalarValue)y;
}
else if (y is MatrixValue)
{
var Y = (MatrixValue)y;
for (var j = 1; j <= Y.Length; j++)
{
M[row, j + 1] = Y[j];
}
}
x.Re += precision;
}
cp.AddPoints(M);
return cp;
}
public MatrixValue Function(ArgumentsValue values)
{
var m = new MatrixValue();
for (var i = 1; i <= values.Length; i++)
{
var sy = m.DimensionY;
var sx = m.DimensionX;
if (values[i] is ScalarValue)
{
var s = (ScalarValue)values[i];
m[sy + 1, sx + 1] = s.Clone();
}
else if (values[i] is MatrixValue)
{
var n = (MatrixValue)values[i];
for (var l = 1; l <= n.DimensionX; l++)
{
for (var k = 1; k <= n.DimensionY; k++)
{
m[sy + k, sx + l] = n[k, l].Clone();
}
}
}
else
{
throw new YAMPArgumentInvalidException(Name, values[i].Header, i);
}
}
return m;
}
public BarPlotValue Function(MatrixValue Y, ScalarValue nbins)
{
var nn = nbins.GetIntegerOrThrowException("nbins", Name);
var bp = new BarPlotValue();
if (Y.IsVector)
{
bp.AddPoints(YMath.Histogram(Y, nn));
}
else
{
var M = new MatrixValue();
for (var i = 1; i <= Y.DimensionX; i++)
{
var N = YMath.Histogram(Y.GetColumnVector(i), nn);
for (var j = 1; j <= N.Length; j++)
{
M[j, i] = N[j];
}
}
bp.AddPoints(M);
}
return bp;
}
public ArgumentsValue Function(MatrixValue X, MatrixValue Y)
{
if (X.Length != Y.Length)
throw new YAMPDifferentLengthsException(X.Length, Y.Length);
var x1 = new ScalarValue();
var y1 = new ScalarValue();
var xy = new ScalarValue();
var x2 = new ScalarValue();
var slope = new ScalarValue();
var offset = new ScalarValue();
for (var i = 1; i <= X.Length; i++)
{
x1 = x1 + X[i];
y1 = y1 + Y[i];
xy = xy + X[i] * Y[i];
x2 = x2 + X[i] * X[i];
}
var J = ((Double)X.Length * x2) - (x1 * x1);
if (J.Re != 0.0)
{
slope = (((Double)X.Length * xy) - (x1 * y1)) / J.Re;
offset = ((y1 * x2) - (x1 * xy)) / J.Re;
}
return new ArgumentsValue(slope, offset);
}
/// <summary>
/// Performs the function - maps each entry of a matrix to a matrix.
/// </summary>
/// <param name="argument">Either a Scalar or a Matrix.</param>
/// <returns>The scalar or matrix.</returns>
public override Value Perform(Value argument)
{
if (argument is ScalarValue)
{
return GetValue((ScalarValue)argument);
}
else if (argument is MatrixValue)
{
var A = (MatrixValue)argument;
var M = new MatrixValue(A.DimensionY, A.DimensionX);
for (var j = 1; j <= A.DimensionY; j++)
{
for (var i = 1; i <= A.DimensionX; i++)
{
M[j, i] = GetValue(A[j, i]);
}
}
return M;
}
else if (argument is ArgumentsValue)
{
throw new YAMPArgumentNumberException(Name, ((ArgumentsValue)argument).Length, 1);
}
throw new YAMPOperationInvalidException(Name, argument);
}
public MatrixValue Function(MatrixValue M)
{
if (M.DimensionX != 3 && M.DimensionY != 3)
throw new YAMPMatrixDimensionException(3, M.DimensionX, M.DimensionY, M.DimensionX);
var isTransposed = M.DimensionY != 3;
if (isTransposed)
{
M = M.Transpose();
}
var m = new MatrixValue(3, M.DimensionX);
for (var i = 1; i <= M.DimensionX; i++)
{
var r = M[1, i].Re;
var theta = M[2, i].Re;
var phi = M[3, i].Re;
var rt = r * Math.Sin(theta);
m[1, i] = new ScalarValue(rt * Math.Cos(phi));
m[2, i] = new ScalarValue(rt * Math.Sin(phi));
m[3, i] = new ScalarValue(r * Math.Cos(theta));
}
return isTransposed ? m.Transpose() : m;
}
/// <summary>
/// QR Decomposition, computed by Householder reflections.
/// </summary>
/// <param name="A">Rectangular matrix</param>
/// <returns>Structure to access R and the Householder vectors and compute Q.</returns>
protected QRDecomposition(MatrixValue A)
{
// Initialize.
FullRank = true;
m = A.DimensionY;
n = A.DimensionX;
}
public ArgumentsValue Function(ScalarValue n)
{
var nn = n.GetIntegerOrThrowException("n", Name);
int dim = nn + 1;
if (dim < 2)
throw new YAMPArgumentRangeException("n", 1.0);
var X = new MatrixValue(dim, dim); // x = sin(phi) * cos(theta)
var Y = new MatrixValue(dim, dim); // y = sin(phi) * sin(theta)
var Z = new MatrixValue(dim, dim); // z = cos(phi)
var stheta = Table(0.0, 2.0 * Math.PI, dim, Math.Sin);
var ctheta = Table(0.0, 2.0 * Math.PI, dim, Math.Cos);
var sphi = Table(0.0, Math.PI, dim, Math.Sin);
var cphi = Table(0.0, Math.PI, dim, Math.Cos);
for (var j = 0; j < dim; j++)
{
var col = j + 1;
for (var i = 0; i < dim; i++)
{
var row = i + 1;
X[row, col] = new ScalarValue(sphi[j] * ctheta[i]);
Y[row, col] = new ScalarValue(sphi[j] * stheta[i]);
Z[row, col] = new ScalarValue(cphi[j]);
}
}
return new ArgumentsValue(X, Y, Z);
}
public BarPlotValue Function(MatrixValue Y, MatrixValue x)
{
var bp = new BarPlotValue();
var X = new Double[x.Length];
for (var i = 0; i < x.Length; i++)
{
X[i] = x[i + 1].Re;
}
if (Y.IsVector)
{
bp.AddPoints(YMath.Histogram(Y, X));
}
else
{
var M = new MatrixValue();
for (var i = 1; i <= Y.DimensionX; i++)
{
var N = YMath.Histogram(Y.GetColumnVector(i), X);
for (var j = 1; j <= N.Length; j++)
{
M[j, i] = N[j];
}
}
bp.AddPoints(M);
}
return bp;
}
public MatrixValue Function(MatrixValue original, ScalarValue x)
{
var spline = new SplineInterpolation(original);
var M = new MatrixValue(1, 2);
M[1, 1].Re = x.Re;
M[1, 2].Re = spline.ComputeValue(x.Re);
return M;
}
public SurfacePlotValue Function(MatrixValue X, MatrixValue Y, MatrixValue Z)
{
var splot = new SurfacePlotValue();
splot.IsSurf = true;
splot.IsMesh = false;
splot.AddPoints(X, Y, Z);
return splot;
}
/// <summary>
/// Check for symmetry, then construct the eigenvalue decomposition
/// </summary>
/// <param name="Arg">Square matrix</param>
/// <returns>Structure to access D and V.</returns>
public Eigenvalues(MatrixValue Arg)
{
var A = Arg.GetRealMatrix();
n = Arg.DimensionX;
V = new double[n][];
for (int i = 0; i < n; i++)
V[i] = new double[n];
d = new double[n];
e = new double[n];
issymmetric = true;
for (int j = 0; (j < n) && issymmetric; j++)
{
for (int i = 0; (i < n) && issymmetric; i++)
issymmetric = (A[i][j] == A[j][i]);
}
if (issymmetric)
{
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
V[i][j] = A[i][j];
}
// Tridiagonalize.
tred2();
// Diagonalize.
tql2();
}
else
{
H = new double[n][];
for (int i2 = 0; i2 < n; i2++)
H[i2] = new double[n];
ort = new double[n];
for (int j = 0; j < n; j++)
{
for (int i = 0; i < n; i++)
H[i][j] = A[i][j];
}
// Reduce to Hessenberg form.
orthes();
// Reduce Hessenberg to real Schur form.
hqr2();
}
}
public ArgumentsValue Function(MatrixValue M)
{
if (M.DimensionX != 2 && M.DimensionY != 2)
throw new YAMPOperationInvalidException("lsq", "because exactly two rows or columns are required.");
if (M.DimensionX > M.DimensionY)
return Function(M.GetSubMatrix(0, 1, 0, M.DimensionX), M.GetSubMatrix(1, 2, 0, M.DimensionX));
return Function(M.GetSubMatrix(0, M.DimensionY, 0, 1), M.GetSubMatrix(0, M.DimensionY, 1, 2));
}
public override MatrixValue Function(MatrixValue m)
{
var M = new MatrixValue(1, m.DimensionX);
for (var i = 1; i <= m.DimensionX; i++)
{
M[1, i] = GetVectorMax(m.GetColumnVector(i));
}
return M;
}
public MatrixValue Function(MatrixValue M)
{
var m = new MatrixValue(M.Length, M.Length);
for (var i = 1; i <= M.Length; i++)
{
m[i, i] = M[i].Clone();
}
return m;
}
public MatrixValue Function(ScalarValue s, MatrixValue Z)
{
var n = s.GetIntegerOrThrowException("s", Name);
var M = new MatrixValue(Z.DimensionY, Z.DimensionX);
for (var j = 1; j <= Z.DimensionY; j++)
for (var i = 1; i <= Z.DimensionX; i++)
M[j, i] = GetValue(n, Z[j, i]);
return M;
}
public MatrixValue Function(MatrixValue M)
{
var ev = new Eigenvalues(M as MatrixValue);
var m = new MatrixValue(ev.RealEigenvalues.Length, 1);
for (var i = 1; i <= ev.RealEigenvalues.Length; i++)
{
m[i, 1] = new ScalarValue(ev.RealEigenvalues[i - 1], ev.ImagEigenvalues[i - 1]);
}
return m;
}
public ScalarValue Function(MatrixValue M)
{
var deviation = ScalarValue.Zero;
var mean = M.Sum() / M.Length;
for (var i = 1; i <= M.Length; i++)
{
deviation += (M[i] - mean).Square();
}
return new ScalarValue(Math.Sqrt(deviation.Abs() / M.Length));
}
public MatrixValue Function(MatrixValue Z, ScalarValue w)
{
var M = new MatrixValue(Z.DimensionY, Z.DimensionX);
for (var i = 1; i <= Z.DimensionX; i++)
{
for (var j = 1; j <= Z.DimensionY; j++)
{
M[j, i] = Function(Z[j, i], w);
}
}
return M;
}
public MatrixValue Function(ScalarValue n, MatrixValue Z)
{
var nn = n.GetIntegerOrThrowException("n", Name);
if (nn < 0)
throw new Exception("Hermite polynomial of order n < 0 does not make sense.");
var M = new MatrixValue(Z.DimensionY, Z.DimensionX);
for (var i = 1; i <= Z.Length; i++)
M[i] = HermitePolynomial(nn, Z[i]);
return M;
}
public MatrixValue Function(ScalarValue n, MatrixValue X)
{
var nn = n.GetIntegerOrThrowException("n", Name);
if (nn < 0)
throw new Exception("Chebyshev polynomial of order n < 0 does not make sense.");
var M = new MatrixValue(X.DimensionY, X.DimensionX);
var f = GetPolynom(nn);
for (var i = 1; i <= X.Length; i++)
M[i] = new ScalarValue(f(X[i].Re));
return M;
}
static void MatrixBenchmark()
{
Console.WriteLine("Running matrix benchmarks ...");
//var n = 1000;
//var m = 1000;
for (var n = 20; n <= 500; n += 20)
{
var m = n;
var A = new YAMP.MatrixValue(n, m);
var B = new YAMP.MatrixValue(m, n);
A.Randomize();
B.Randomize();
A[n, m] = YAMP.ScalarValue.One;
B[m, n] = YAMP.ScalarValue.One;
var sw = Stopwatch.StartNew();
var C = A * B;
sw.Stop();
#region Outputs
//---
// Output for usual multiplication
//---
//Time for n = 20, m = 20 : 22 ms
//Time for n = 40, m = 40 : 222 ms
//Time for n = 60, m = 60 : 1328 ms
//Time for n = 80, m = 80 : 3107 ms
//Time for n = 100, m = 100 : 8244 ms
// stop ...
//---
// Output for BLAS L3 multiplication (1st order approx.)
//---
//Time for n = 20, m = 20 : 7 ms
//Time for n = 40, m = 40 : 8 ms
//Time for n = 60, m = 60 : 28 ms
//Time for n = 80, m = 80 : 51 ms
//Time for n = 100, m = 100 : 135 ms
//Time for n = 120, m = 120 : 273 ms
//Time for n = 140, m = 140 : 281 ms
//Time for n = 160, m = 160 : 387 ms
//Time for n = 180, m = 180 : 585 ms
//Time for n = 200, m = 200 : 845 ms
//Time for n = 220, m = 220 : 1196 ms
//Time for n = 240, m = 240 : 1709 ms
//Time for n = 260, m = 260 : 2318 ms
//Time for n = 280, m = 280 : 2451 ms
//Time for n = 300, m = 300 : 2771 ms
// and so on !
//---
/// Output for copying arrays only (required for perf. BLAS L3)
//---
//Time for n = 20, m = 20 : 7 ms
//Time for n = 40, m = 40 : 15 ms
//Time for n = 60, m = 60 : 20 ms
//Time for n = 80, m = 80 : 37 ms
//Time for n = 100, m = 100 : 78 ms
//Time for n = 120, m = 120 : 158 ms
//Time for n = 140, m = 140 : 207 ms
//Time for n = 160, m = 160 : 276 ms
//Time for n = 180, m = 180 : 411 ms
//Time for n = 200, m = 200 : 628 ms
//Time for n = 220, m = 220 : 897 ms
//Time for n = 240, m = 240 : 1271 ms
//Time for n = 260, m = 260 : 1703 ms
//Time for n = 280, m = 280 : 1956 ms
//Time for n = 300, m = 300 : 1974 ms
// and so on !
#endregion
Console.WriteLine("Time for n = {0}, m = {1} : {2} ms", n, m, sw.ElapsedMilliseconds);
}
Console.WriteLine("Finished !");
}
