/// <summary>
/// Returns a deep-copy clone of the vector.
/// </summary>
/// <returns>A deep-copy clone of the vector.</returns>
public Vector <T> Clone()
{
var result = Build.SameAs(this);
Storage.CopyToUnchecked(result.Storage, ExistingData.AssumeZeros);
return(result);
}
/// <summary>
/// Solves a system of linear equations, <b>Ax = b</b>, with A LU factorized.
/// </summary>
/// <param name="input">The right hand side vector, <b>b</b>.</param>
/// <returns>The left hand side <see cref="Vector{T}"/>, <b>x</b>.</returns>
public virtual Vector <T> Solve(Vector <T> input)
{
var x = v_builder.SameAs(input, input.Count);
Solve(input, x);
return(x);
}
/// <summary>
/// Solves a system of linear equations, <b>Ax = b</b>, with A EVD factorized.
/// </summary>
/// <param name="input">The right hand side vector, <b>b</b>.</param>
/// <returns>The left hand side <see cref="Vector{T}"/>, <b>x</b>.</returns>
public virtual Vector <T> Solve(Vector <T> input)
{
var x = v_builder.SameAs(EigenVectors, EigenVectors.ColumnCount);
Solve(input, x);
return(x);
}
/// <summary>
/// Solves a system of linear equations, <b>Ax = b</b>, with A SVD factorized.
/// </summary>
/// <param name="input">The right hand side vector, <b>b</b>.</param>
/// <returns>The left hand side <see cref="Vector{T}"/>, <b>x</b>.</returns>
public virtual Vector <T> Solve(Vector <T> input)
{
if (!VectorsComputed)
{
throw new InvalidOperationException("Resources.SingularVectorsNotComputed");
}
var x = v_builder.SameAs(U, VT.ColumnCount);
Solve(input, x);
return(x);
}
public static Vector <T> SameAs <T>(this VectorBuilder <T> builder, Vector <T> example, Func <T> f) where T : struct, global::System.IEquatable <T>, global::System.IFormattable
{
Contract.Requires(builder != null);
Contract.Requires(example != null);
Contract.Requires(f != null);
var result = builder.SameAs(example);
Contract.Assume(result != null);
result.MapInplace(x => f());
return(result);
}
public static UserSvd Create(Matrix <double> matrix, bool computeVectors)
{
var nm = Math.Min(matrix.RowCount + 1, matrix.ColumnCount);
var matrixCopy = matrix.Clone();
var s = v_builder.SameAs(matrixCopy, nm);
var u = m_builder.SameAs(matrixCopy, matrixCopy.RowCount, matrixCopy.RowCount, fullyMutable: true);
var vt = m_builder.SameAs(matrixCopy, matrixCopy.ColumnCount, matrixCopy.ColumnCount, fullyMutable: true);
const int maxiter = 1000;
var e = new double[matrixCopy.ColumnCount];
var work = new double[matrixCopy.RowCount];
int i, j;
int l, lp1;
double t;
var ncu = matrixCopy.RowCount;
// Reduce matrixCopy to bidiagonal form, storing the diagonal elements
// In s and the super-diagonal elements in e.
var nct = Math.Min(matrixCopy.RowCount - 1, matrixCopy.ColumnCount);
var nrt = Math.Max(0, Math.Min(matrixCopy.ColumnCount - 2, matrixCopy.RowCount));
var lu = Math.Max(nct, nrt);
for (l = 0; l < lu; l++)
{
lp1 = l + 1;
if (l < nct)
{
// Compute the transformation for the l-th column and place the l-th diagonal in VectorS[l].
var xnorm = Dnrm2Column(matrixCopy, matrixCopy.RowCount, l, l);
s[l] = xnorm;
if (s[l] != 0.0)
{
if (matrixCopy.At(l, l) != 0.0)
{
s[l] = Dsign(s[l], matrixCopy.At(l, l));
}
DscalColumn(matrixCopy, matrixCopy.RowCount, l, l, 1.0 / s[l]);
matrixCopy.At(l, l, (1.0 + matrixCopy.At(l, l)));
}
s[l] = -s[l];
}
for (j = lp1; j < matrixCopy.ColumnCount; j++)
{
if (l < nct)
{
if (s[l] != 0.0)
{
// Apply the transformation.
t = -Ddot(matrixCopy, matrixCopy.RowCount, l, j, l) / matrixCopy.At(l, l);
for (var ii = l; ii < matrixCopy.RowCount; ii++)
{
matrixCopy.At(ii, j, matrixCopy.At(ii, j) + (t * matrixCopy.At(ii, l)));
}
}
}
// Place the l-th row of matrixCopy into e for the
// Subsequent calculation of the row transformation.
e[j] = matrixCopy.At(l, j);
}
if (computeVectors && l < nct)
{
// Place the transformation in u for subsequent back multiplication.
for (i = l; i < matrixCopy.RowCount; i++)
{
u.At(i, l, matrixCopy.At(i, l));
}
}
if (l >= nrt)
{
continue;
}
// Compute the l-th row transformation and place the l-th super-diagonal in e(l).
var enorm = Dnrm2Vector(e, lp1);
e[l] = enorm;
if (e[l] != 0.0)
{
if (e[lp1] != 0.0)
{
e[l] = Dsign(e[l], e[lp1]);
}
DscalVector(e, lp1, 1.0 / e[l]);
e[lp1] = 1.0 + e[lp1];
}
e[l] = -e[l];
if (lp1 < matrixCopy.RowCount && e[l] != 0.0)
{
// Apply the transformation.
for (i = lp1; i < matrixCopy.RowCount; i++)
{
work[i] = 0.0;
}
for (j = lp1; j < matrixCopy.ColumnCount; j++)
{
for (var ii = lp1; ii < matrixCopy.RowCount; ii++)
{
work[ii] += e[j] * matrixCopy.At(ii, j);
}
}
for (j = lp1; j < matrixCopy.ColumnCount; j++)
{
var ww = -e[j] / e[lp1];
for (var ii = lp1; ii < matrixCopy.RowCount; ii++)
{
matrixCopy.At(ii, j, matrixCopy.At(ii, j) + (ww * work[ii]));
}
}
}
if (computeVectors)
{
// Place the transformation in v for subsequent back multiplication.
for (i = lp1; i < matrixCopy.ColumnCount; i++)
{
vt.At(i, l, e[i]);
}
}
}
// Set up the final bidiagonal matrixCopy or order m.
var m = Math.Min(matrixCopy.ColumnCount, matrixCopy.RowCount + 1);
var nctp1 = nct + 1;
var nrtp1 = nrt + 1;
if (nct < matrixCopy.ColumnCount)
{
s[nctp1 - 1] = matrixCopy.At((nctp1 - 1), (nctp1 - 1));
}
if (matrixCopy.RowCount < m)
{
s[m - 1] = 0.0;
}
if (nrtp1 < m)
{
e[nrtp1 - 1] = matrixCopy.At((nrtp1 - 1), (m - 1));
}
e[m - 1] = 0.0;
// If required, generate u.
if (computeVectors)
{
for (j = nctp1 - 1; j < ncu; j++)
{
for (i = 0; i < matrixCopy.RowCount; i++)
{
u.At(i, j, 0.0);
}
u.At(j, j, 1.0);
}
for (l = nct - 1; l >= 0; l--)
{
if (s[l] != 0.0)
{
for (j = l + 1; j < ncu; j++)
{
t = -Ddot(u, matrixCopy.RowCount, l, j, l) / u.At(l, l);
for (var ii = l; ii < matrixCopy.RowCount; ii++)
{
u.At(ii, j, u.At(ii, j) + (t * u.At(ii, l)));
}
}
DscalColumn(u, matrixCopy.RowCount, l, l, -1.0);
u.At(l, l, 1.0 + u.At(l, l));
for (i = 0; i < l; i++)
{
u.At(i, l, 0.0);
}
}
else
{
for (i = 0; i < matrixCopy.RowCount; i++)
{
u.At(i, l, 0.0);
}
u.At(l, l, 1.0);
}
}
}
// If it is required, generate v.
if (computeVectors)
{
for (l = matrixCopy.ColumnCount - 1; l >= 0; l--)
{
lp1 = l + 1;
if (l < nrt)
{
if (e[l] != 0.0)
{
for (j = lp1; j < matrixCopy.ColumnCount; j++)
{
t = -Ddot(vt, matrixCopy.ColumnCount, l, j, lp1) / vt.At(lp1, l);
for (var ii = l; ii < matrixCopy.ColumnCount; ii++)
{
vt.At(ii, j, vt.At(ii, j) + (t * vt.At(ii, l)));
}
}
}
}
for (i = 0; i < matrixCopy.ColumnCount; i++)
{
vt.At(i, l, 0.0);
}
vt.At(l, l, 1.0);
}
}
// Transform s and e so that they are double .
for (i = 0; i < m; i++)
{
double r;
if (s[i] != 0.0)
{
t = s[i];
r = s[i] / t;
s[i] = t;
if (i < m - 1)
{
e[i] = e[i] / r;
}
if (computeVectors)
{
DscalColumn(u, matrixCopy.RowCount, i, 0, r);
}
}
// Exit
if (i == m - 1)
{
break;
}
if (e[i] != 0.0)
{
t = e[i];
r = t / e[i];
e[i] = t;
s[i + 1] = s[i + 1] * r;
if (computeVectors)
{
DscalColumn(vt, matrixCopy.ColumnCount, i + 1, 0, r);
}
}
}
// Main iteration loop for the singular values.
var mn = m;
var iter = 0;
bool AlmostEquals(double val1, double val2)
{
if (Math.Abs(val1 - val2) < 1e-8)
{
return(true);
}
return(false);
}
while (m > 0)
{
// Quit if all the singular values have been found. If too many iterations have been performed,
// throw exception that Convergence Failed
if (iter >= maxiter)
{
throw new ArgumentException("NonConvergenceException()");;
}
// This section of the program inspects for negligible elements in the s and e arrays. On
// completion the variables case and l are set as follows.
// Case = 1 if VectorS[m] and e[l-1] are negligible and l < m
// Case = 2 if VectorS[l] is negligible and l < m
// Case = 3 if e[l-1] is negligible, l < m, and VectorS[l, ..., VectorS[m] are not negligible (qr step).
// Case = 4 if e[m-1] is negligible (convergence).
double ztest;
double test;
for (l = m - 2; l >= 0; l--)
{
test = Math.Abs(s[l]) + Math.Abs(s[l + 1]);
ztest = test + Math.Abs(e[l]);
if (AlmostEquals(test, ztest))
{
e[l] = 0.0;
break;
}
}
int kase;
if (l == m - 2)
{
kase = 4;
}
else
{
int ls;
for (ls = m - 1; ls > l; ls--)
{
test = 0.0;
if (ls != m - 1)
{
test = test + Math.Abs(e[ls]);
}
if (ls != l + 1)
{
test = test + Math.Abs(e[ls - 1]);
}
ztest = test + Math.Abs(s[ls]);
if (AlmostEquals(test, ztest))
{
s[ls] = 0.0;
break;
}
}
if (ls == l)
{
kase = 3;
}
else if (ls == m - 1)
{
kase = 1;
}
else
{
kase = 2;
l = ls;
}
}
l = l + 1;
// Perform the task indicated by case.
int k;
double f;
double sn;
double cs;
double t1;
switch (kase)
{
// Deflate negligible VectorS[m].
case 1:
f = e[m - 2];
e[m - 2] = 0.0;
for (var kk = l; kk < m - 1; kk++)
{
k = m - 2 - kk + l;
t1 = s[k];
Drotg(ref t1, ref f, out cs, out sn);
s[k] = t1;
if (k != l)
{
f = -sn * e[k - 1];
e[k - 1] = cs * e[k - 1];
}
if (computeVectors)
{
Drot(vt, matrixCopy.ColumnCount, k, m - 1, cs, sn);
}
}
break;
// Split at negligible VectorS[l].
case 2:
f = e[l - 1];
e[l - 1] = 0.0;
for (k = l; k < m; k++)
{
t1 = s[k];
Drotg(ref t1, ref f, out cs, out sn);
s[k] = t1;
f = -sn * e[k];
e[k] = cs * e[k];
if (computeVectors)
{
Drot(u, matrixCopy.RowCount, k, l - 1, cs, sn);
}
}
break;
// Perform one qr step.
case 3:
// Calculate the shift.
var scale = 0.0;
scale = Math.Max(scale, Math.Abs(s[m - 1]));
scale = Math.Max(scale, Math.Abs(s[m - 2]));
scale = Math.Max(scale, Math.Abs(e[m - 2]));
scale = Math.Max(scale, Math.Abs(s[l]));
scale = Math.Max(scale, Math.Abs(e[l]));
var sm = s[m - 1] / scale;
var smm1 = s[m - 2] / scale;
var emm1 = e[m - 2] / scale;
var sl = s[l] / scale;
var el = e[l] / scale;
var b = (((smm1 + sm) * (smm1 - sm)) + (emm1 * emm1)) / 2.0;
var c = (sm * emm1) * (sm * emm1);
var shift = 0.0;
if (b != 0.0 || c != 0.0)
{
shift = Math.Sqrt((b * b) + c);
if (b < 0.0)
{
shift = -shift;
}
shift = c / (b + shift);
}
f = ((sl + sm) * (sl - sm)) + shift;
var g = sl * el;
// Chase zeros.
for (k = l; k < m - 1; k++)
{
Drotg(ref f, ref g, out cs, out sn);
if (k != l)
{
e[k - 1] = f;
}
f = (cs * s[k]) + (sn * e[k]);
e[k] = (cs * e[k]) - (sn * s[k]);
g = sn * s[k + 1];
s[k + 1] = cs * s[k + 1];
if (computeVectors)
{
Drot(vt, matrixCopy.ColumnCount, k, k + 1, cs, sn);
}
Drotg(ref f, ref g, out cs, out sn);
s[k] = f;
f = (cs * e[k]) + (sn * s[k + 1]);
s[k + 1] = (-sn * e[k]) + (cs * s[k + 1]);
g = sn * e[k + 1];
e[k + 1] = cs * e[k + 1];
if (computeVectors && k < matrixCopy.RowCount)
{
Drot(u, matrixCopy.RowCount, k, k + 1, cs, sn);
}
}
e[m - 2] = f;
iter = iter + 1;
break;
// Convergence.
case 4:
// Make the singular value positive
if (s[l] < 0.0)
{
s[l] = -s[l];
if (computeVectors)
{
DscalColumn(vt, matrixCopy.ColumnCount, l, 0, -1.0);
}
}
// Order the singular value.
while (l != mn - 1)
{
if (s[l] >= s[l + 1])
{
break;
}
t = s[l];
s[l] = s[l + 1];
s[l + 1] = t;
if (computeVectors && l < matrixCopy.ColumnCount)
{
Dswap(vt, matrixCopy.ColumnCount, l, l + 1);
}
if (computeVectors && l < matrixCopy.RowCount)
{
Dswap(u, matrixCopy.RowCount, l, l + 1);
}
l = l + 1;
}
iter = 0;
m = m - 1;
break;
}
}
if (computeVectors)
{
vt = vt.Transpose();
}
// Adjust the size of s if rows < columns. We are using ported copy of linpack's svd code and it uses
// a singular vector of length mRows+1 when mRows < mColumns. The last element is not used and needs to be removed.
// we should port lapack's svd routine to remove this problem.
if (matrixCopy.RowCount < matrixCopy.ColumnCount)
{
nm--;
var tmp = v_builder.SameAs(matrixCopy, nm);
for (i = 0; i < nm; i++)
{
tmp[i] = s[i];
}
s = tmp;
}
return(new UserSvd(s, u, vt, computeVectors));
}