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Test and fix Singular Value Decomposition (double) at different scales #1146

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97 changes: 97 additions & 0 deletions src/Numerics.Tests/LinearAlgebraTests/Double/Factorization/SvdTests.cs
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Original file line number Diff line number Diff line change
Expand Up @@ -115,6 +115,103 @@ public void CanFactorizeRandomMatrix(int row, int column)
}
}

/// <summary>
/// Can handle a matrix at all scales (small, medium and large elements).
/// </summary>
/// <param name="scale">Scale to test.</param>
[TestCase(1)]
[TestCase(1E40)]
[TestCase(1E-40)]
public void CanFactorizeMatrixAtDifferentScales(double scale)
{
/* Values computed with Mathematica in this way:
* matrix = {{4, 7, 4}, {10, 8, 8}, {5, 3, 4}, {5, 4, 5}}
* {u, w, v} = SingularValueDecomposition[matrix];
* N[u, 20]
* N[w, 20]
* N[Transpose[v], 20]
*/

var matrixX = CreateMatrix.DenseOfRowArrays<double>(
new double[] { 4, 7, 4 },
new double[] { 10, 8, 8 },
new double[] { 5, 3, 4 },
new double[] { 5, 4, 5 }
);

var expectedU = CreateMatrix.DenseOfRowArrays<double>(
new double[] { 0.42215668141346500811, -0.87675136383986560620, 0.047995632277482876247, -0.22536015980024164867 },
new double[] { 0.73993840997833451205, 0.22051356023428738286, -0.46959889408365937068, 0.42818430362045913247 },
new double[] { 0.34231752064357875227, 0.38153678983509828442, -0.062299290485614282346, -0.85636860724091826495 },
new double[] { 0.39635035112528586728, 0.19264084136045230895, 0.87937028398887675954, 0.18028812784019331894 }
);

var expectedW = CreateMatrix.DenseOfRowArrays<double>(
new double[] { 20.381761793431507357, 0, 0 },
new double[] { 0, 3.0100938773437216581, 0 },
new double[] { 0, 0, 0.72327107324536682962 },
new double[] { 0, 0, 0 }
);


var expectedVT = CreateMatrix.DenseOfRowArrays<double>(
new double[] { 0.62709741747647217739, 0.56359004351894387921, 0.53769423638407554691 },
new double[] { 0.52125228212010301763, -0.81657829629776043208, 0.24798375833919346295 },
new double[] { -0.57883062063001098569, -0.12476437336740771587, 0.80584673714008074090 }
);

var matrixXScaled = matrixX * scale;
var result = matrixXScaled.Svd(computeVectors: true);

var reconstructedX = result.U * result.W * result.VT;
for (var i = 0; i < reconstructedX.RowCount; ++i)
{
for (var j = 0; j < reconstructedX.ColumnCount; ++j)
{
Assert.AreEqual(matrixXScaled[i, j], reconstructedX[i, j], 1E-14 * scale);
}
}

var signU = new DiagonalMatrix(4); // The sign of the columns of U after SVD is not unique, thus we need to adopt it
for (var i = 0; i < expectedU.ColumnCount; ++i)
{
signU[i, i] = expectedU[0, i] * result.U[0, i] < 0 ? -1 : 1;
}

expectedU *= signU;
for (var i = 0; i < expectedU.RowCount; ++i)
{
for (var j = 0; j < expectedU.ColumnCount; ++j)
{
Assert.AreEqual(expectedU[i, j], result.U[i, j], 1E-14); // U should be independent of scaling
}
}

expectedW *= scale; // W is dependent on scaling
for (var i = 0; i < expectedW.RowCount; ++i)
{
for (var j = 0; j < expectedW.ColumnCount; ++j)
{
Assert.AreEqual(expectedW[i, j], result.W[i, j], 1E-14 * scale); // W should be scaled by scaling
}
}

var signVT = new DiagonalMatrix(3); // The sign of the columns of VT after SVD is not unique, thus we need to adopt it
for (int i = 0; i < expectedVT.ColumnCount; ++i)
{
signVT[i, i] = expectedVT[0, i] * result.VT[0, i] < 0 ? -1 : 1;
}

expectedVT *= signVT;
for (var i = 0; i < expectedVT.RowCount; ++i)
{
for (var j = 0; j < expectedVT.ColumnCount; ++j)
{
Assert.AreEqual(expectedVT[i, j], result.VT[i, j], 1E-14); // VT should be independent of scaling
}
}
}

/// <summary>
/// Can check rank of a non-square matrix.
/// </summary>
Expand Down
16 changes: 14 additions & 2 deletions src/Numerics/Providers/LinearAlgebra/ManagedLinearAlgebraProvider.Double.cs
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Original file line number Diff line number Diff line change
Expand Up @@ -2008,7 +2008,7 @@ public void SingularValueDecomposition(bool computeVectors, double[] a, int rows
{
test = Math.Abs(stemp[l]) + Math.Abs(stemp[l + 1]);
ztest = test + Math.Abs(e[l]);
if (ztest.AlmostEqualRelative(test, 15))
if (AlmostEqualRelative(ztest, test, 4E-15))
{
e[l] = 0.0;
break;
Expand Down Expand Up @@ -2037,7 +2037,7 @@ public void SingularValueDecomposition(bool computeVectors, double[] a, int rows
}

ztest = test + Math.Abs(stemp[ls]);
if (ztest.AlmostEqualRelative(test, 15))
if (AlmostEqualRelative(ztest, test, 4E-15))
{
stemp[ls] = 0.0;
break;
Expand Down Expand Up @@ -3575,5 +3575,17 @@ static Complex Cdiv(double xreal, double ximag, double yreal, double yimag)

return new Complex((ximag + (xreal*(yreal/yimag)))/(yimag + (yreal*(yreal/yimag))), (-xreal + (ximag*(yreal/yimag)))/(yimag + (yreal*(yreal/yimag))));
}

/// <summary>
/// Determines whether <paramref name="x1"/> and <paramref name="x2"/> are equal within a given relative tolerance.
/// </summary>
/// <param name="x1">The value <c>x1</c>.</param>
/// <param name="x2">The value <c>x2</c>.</param>
/// <param name="relativeTolerance">The relative tolerance value.</param>
/// <returns><see langword="true"/> if the values are equal within tolerance; otherwise, <see langword="false"/>.</returns>
static bool AlmostEqualRelative(double x1, double x2, double relativeTolerance)
{
return (x1 == x2) || Math.Abs(x1 - x2) < relativeTolerance * Math.Max(Math.Abs(x1), Math.Abs(x2));
}
}
}