/// <summary>Solves a set of equation systems of type <c>A * X = B</c>.</summary>
/// <param name="value">Right hand side matrix with as many rows as <c>A</c> and any number of columns.</param>
/// <returns>Matrix <c>X</c> so that <c>L * L' * X = B</c>.</returns>
/// <exception cref="T:System.ArgumentException">Matrix dimensions do not match.</exception>
/// <exception cref="T:System.InvalidOperationException">Matrix is not symmetrix and positive definite.</exception>
public Matrix Solve(Matrix value)
{
if (value == null)
{
throw new ArgumentNullException("value");
}
if (value.Rows != L.Rows)
{
throw new ArgumentException("Matrix dimensions do not match.");
}
if (!this.symmetric)
{
throw new InvalidOperationException("Matrix is not symmetric.");
}
if (!this.positiveDefinite)
{
throw new InvalidOperationException("Matrix is not positive definite.");
}
int dimension = L.Rows;
int count = value.Columns;
Matrix B = (Matrix)value.Clone();
double[][] l = L.Array;
// Solve L*Y = B;
for (int k = 0; k < L.Rows; k++)
{
for (int i = k + 1; i < dimension; i++)
{
for (int j = 0; j < count; j++)
{
B[i, j] -= B[k, j] * l[i][k];
}
}
for (int j = 0; j < count; j++)
{
B[k, j] /= l[k][k];
}
}
// Solve L'*X = Y;
for (int k = dimension - 1; k >= 0; k--)
{
for (int j = 0; j < count; j++)
{
B[k, j] /= l[k][k];
}
for (int i = 0; i < k; i++)
{
for (int j = 0; j < count; j++)
{
B[i, j] -= B[k, j] * l[k][i];
}
}
}
return(B);
}
/// <summary>Construct singular value decomposition.</summary>
public SingularValueDecomposition(Matrix value)
{
if (value == null)
{
throw new ArgumentNullException("value");
}
Matrix copy = (Matrix)value.Clone();
double[][] a = copy.Array;
m = value.Rows;
n = value.Columns;
int nu = Math.Min(m, n);
s = new double [Math.Min(m + 1, n)];
U = new Matrix(m, nu);
V = new Matrix(n, n);
double[][] u = U.Array;
double[][] v = V.Array;
double[] e = new double [n];
double[] work = new double [m];
bool wantu = true;
bool wantv = true;
// Reduce A to bidiagonal form, storing the diagonal elements in s and the super-diagonal elements in e.
int nct = Math.Min(m - 1, n);
int nrt = Math.Max(0, Math.Min(n - 2, m));
for (int k = 0; k < Math.Max(nct, nrt); k++)
{
if (k < nct)
{
// Compute the transformation for the k-th column and place the k-th diagonal in s[k].
// Compute 2-norm of k-th column without under/overflow.
s[k] = 0;
for (int i = k; i < m; i++)
{
s[k] = Hypotenuse(s[k], a[i][k]);
}
if (s[k] != 0.0)
{
if (a[k][k] < 0.0)
{
s[k] = -s[k];
}
for (int i = k; i < m; i++)
{
a[i][k] /= s[k];
}
a[k][k] += 1.0;
}
s[k] = -s[k];
}
for (int j = k + 1; j < n; j++)
{
if ((k < nct) & (s[k] != 0.0))
{
// Apply the transformation.
double t = 0;
for (int i = k; i < m; i++)
{
t += a[i][k] * a[i][j];
}
t = -t / a[k][k];
for (int i = k; i < m; i++)
{
a[i][j] += t * a[i][k];
}
}
// Place the k-th row of A into e for the subsequent calculation of the row transformation.
e[j] = a[k][j];
}
if (wantu & (k < nct))
{
// Place the transformation in U for subsequent back
// multiplication.
for (int i = k; i < m; i++)
{
u[i][k] = a[i][k];
}
}
if (k < nrt)
{
// Compute the k-th row transformation and place the k-th super-diagonal in e[k].
// Compute 2-norm without under/overflow.
e[k] = 0;
for (int i = k + 1; i < n; i++)
{
e[k] = Hypotenuse(e[k], e[i]);
}
if (e[k] != 0.0)
{
if (e[k + 1] < 0.0)
{
e[k] = -e[k];
}
for (int i = k + 1; i < n; i++)
{
e[i] /= e[k];
}
e[k + 1] += 1.0;
}
e[k] = -e[k];
if ((k + 1 < m) & (e[k] != 0.0))
{
// Apply the transformation.
for (int i = k + 1; i < m; i++)
{
work[i] = 0.0;
}
for (int j = k + 1; j < n; j++)
{
for (int i = k + 1; i < m; i++)
{
work[i] += e[j] * a[i][j];
}
}
for (int j = k + 1; j < n; j++)
{
double t = -e[j] / e[k + 1];
for (int i = k + 1; i < m; i++)
{
a[i][j] += t * work[i];
}
}
}
if (wantv)
{
// Place the transformation in V for subsequent back multiplication.
for (int i = k + 1; i < n; i++)
{
v[i][k] = e[i];
}
}
}
}
// Set up the final bidiagonal matrix or order p.
int p = Math.Min(n, m + 1);
if (nct < n)
{
s[nct] = a[nct][nct];
}
if (m < p)
{
s[p - 1] = 0.0;
}
if (nrt + 1 < p)
{
e[nrt] = a[nrt][p - 1];
}
e[p - 1] = 0.0;
// If required, generate U.
if (wantu)
{
for (int j = nct; j < nu; j++)
{
for (int i = 0; i < m; i++)
{
u[i][j] = 0.0;
}
u[j][j] = 1.0;
}
for (int k = nct - 1; k >= 0; k--)
{
if (s[k] != 0.0)
{
for (int j = k + 1; j < nu; j++)
{
double t = 0;
for (int i = k; i < m; i++)
{
t += u[i][k] * u[i][j];
}
t = -t / u[k][k];
for (int i = k; i < m; i++)
{
u[i][j] += t * u[i][k];
}
}
for (int i = k; i < m; i++)
{
u[i][k] = -u[i][k];
}
u[k][k] = 1.0 + u[k][k];
for (int i = 0; i < k - 1; i++)
{
u[i][k] = 0.0;
}
}
else
{
for (int i = 0; i < m; i++)
{
u[i][k] = 0.0;
}
u[k][k] = 1.0;
}
}
}
// If required, generate V.
if (wantv)
{
for (int k = n - 1; k >= 0; k--)
{
if ((k < nrt) & (e[k] != 0.0))
{
for (int j = k + 1; j < nu; j++)
{
double t = 0;
for (int i = k + 1; i < n; i++)
{
t += v[i][k] * v[i][j];
}
t = -t / v[k + 1][k];
for (int i = k + 1; i < n; i++)
{
v[i][j] += t * v[i][k];
}
}
}
for (int i = 0; i < n; i++)
{
v[i][k] = 0.0;
}
v[k][k] = 1.0;
}
}
// Main iteration loop for the singular values.
int pp = p - 1;
int iter = 0;
double eps = Math.Pow(2.0, -52.0);
while (p > 0)
{
int k, kase;
// Here is where a test for too many iterations would go.
// This section of the program inspects for
// negligible elements in the s and e arrays. On
// completion the variables kase and k are set as follows.
// kase = 1 if s(p) and e[k-1] are negligible and k<p
// kase = 2 if s(k) is negligible and k<p
// kase = 3 if e[k-1] is negligible, k<p, and s(k), ..., s(p) are not negligible (qr step).
// kase = 4 if e(p-1) is negligible (convergence).
for (k = p - 2; k >= -1; k--)
{
if (k == -1)
{
break;
}
if (Math.Abs(e[k]) <= eps * (Math.Abs(s[k]) + Math.Abs(s[k + 1])))
{
e[k] = 0.0;
break;
}
}
if (k == p - 2)
{
kase = 4;
}
else
{
int ks;
for (ks = p - 1; ks >= k; ks--)
{
if (ks == k)
{
break;
}
double t = (ks != p ? Math.Abs(e[ks]) : 0.0) + (ks != k + 1 ? Math.Abs(e[ks - 1]) : 0.0);
if (Math.Abs(s[ks]) <= eps * t)
{
s[ks] = 0.0;
break;
}
}
if (ks == k)
{
kase = 3;
}
else if (ks == p - 1)
{
kase = 1;
}
else
{
kase = 2;
k = ks;
}
}
k++;
// Perform the task indicated by kase.
switch (kase)
{
// Deflate negligible s(p).
case 1:
{
double f = e[p - 2];
e[p - 2] = 0.0;
for (int j = p - 2; j >= k; j--)
{
double t = Hypotenuse(s[j], f);
double cs = s[j] / t;
double sn = f / t;
s[j] = t;
if (j != k)
{
f = -sn * e[j - 1];
e[j - 1] = cs * e[j - 1];
}
if (wantv)
{
for (int i = 0; i < n; i++)
{
t = cs * v[i][j] + sn * v[i][p - 1];
v[i][p - 1] = -sn * v[i][j] + cs * v[i][p - 1];
v[i][j] = t;
}
}
}
}
break;
// Split at negligible s(k).
case 2:
{
double f = e[k - 1];
e[k - 1] = 0.0;
for (int j = k; j < p; j++)
{
double t = Hypotenuse(s[j], f);
double cs = s[j] / t;
double sn = f / t;
s[j] = t;
f = -sn * e[j];
e[j] = cs * e[j];
if (wantu)
{
for (int i = 0; i < m; i++)
{
t = cs * u[i][j] + sn * u[i][k - 1];
u[i][k - 1] = -sn * u[i][j] + cs * u[i][k - 1];
u[i][j] = t;
}
}
}
}
break;
// Perform one qr step.
case 3:
{
// Calculate the shift.
double scale = Math.Max(Math.Max(Math.Max(Math.Max(Math.Abs(s[p - 1]), Math.Abs(s[p - 2])), Math.Abs(e[p - 2])), Math.Abs(s[k])), Math.Abs(e[k]));
double sp = s[p - 1] / scale;
double spm1 = s[p - 2] / scale;
double epm1 = e[p - 2] / scale;
double sk = s[k] / scale;
double ek = e[k] / scale;
double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
double c = (sp * epm1) * (sp * epm1);
double 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);
}
double f = (sk + sp) * (sk - sp) + shift;
double g = sk * ek;
// Chase zeros.
for (int j = k; j < p - 1; j++)
{
double t = Hypotenuse(f, g);
double cs = f / t;
double sn = g / t;
if (j != k)
{
e[j - 1] = t;
}
f = cs * s[j] + sn * e[j];
e[j] = cs * e[j] - sn * s[j];
g = sn * s[j + 1];
s[j + 1] = cs * s[j + 1];
if (wantv)
{
for (int i = 0; i < n; i++)
{
t = cs * v[i][j] + sn * v[i][j + 1];
v[i][j + 1] = -sn * v[i][j] + cs * v[i][j + 1];
v[i][j] = t;
}
}
t = Hypotenuse(f, g);
cs = f / t;
sn = g / t;
s[j] = t;
f = cs * e[j] + sn * s[j + 1];
s[j + 1] = -sn * e[j] + cs * s[j + 1];
g = sn * e[j + 1];
e[j + 1] = cs * e[j + 1];
if (wantu && (j < m - 1))
{
for (int i = 0; i < m; i++)
{
t = cs * u[i][j] + sn * u[i][j + 1];
u[i][j + 1] = -sn * u[i][j] + cs * u[i][j + 1];
u[i][j] = t;
}
}
}
e[p - 2] = f;
iter = iter + 1;
}
break;
// Convergence.
case 4:
{
// Make the singular values positive.
if (s[k] <= 0.0)
{
s[k] = (s[k] < 0.0 ? -s[k] : 0.0);
if (wantv)
{
for (int i = 0; i <= pp; i++)
{
v[i][k] = -v[i][k];
}
}
}
// Order the singular values.
while (k < pp)
{
if (s[k] >= s[k + 1])
{
break;
}
double t = s[k];
s[k] = s[k + 1];
s[k + 1] = t;
if (wantv && (k < n - 1))
{
for (int i = 0; i < n; i++)
{
t = v[i][k + 1];
v[i][k + 1] = v[i][k];
v[i][k] = t;
}
}
if (wantu && (k < m - 1))
{
for (int i = 0; i < m; i++)
{
t = u[i][k + 1];
u[i][k + 1] = u[i][k];
u[i][k] = t;
}
}
k++;
}
iter = 0;
p--;
}
break;
}
}
}
/// <summary>Least squares solution of <c>A * X = B</c></summary>
/// <param name="value">Right-hand-side matrix with as many rows as <c>A</c> and any number of columns.</param>
/// <returns>A matrix that minimized the two norm of <c>Q * R * X - B</c>.</returns>
/// <exception cref="T:System.ArgumentException">Matrix row dimensions must be the same.</exception>
/// <exception cref="T:System.InvalidOperationException">Matrix is rank deficient.</exception>
public Matrix Solve(Matrix value)
{
if (value == null)
{
throw new ArgumentNullException("value");
}
if (value.Rows != QR.Rows)
{
throw new ArgumentException("Matrix row dimensions must agree.");
}
if (!this.FullRank)
{
throw new InvalidOperationException("Matrix is rank deficient.");
}
// Copy right hand side
int count = value.Columns;
Matrix X = value.Clone();
int m = QR.Rows;
int n = QR.Columns;
double[][] qr = QR.Array;
// Compute Y = transpose(Q)*B
for (int k = 0; k < n; k++)
{
for (int j = 0; j < count; j++)
{
double s = 0.0;
for (int i = k; i < m; i++)
{
s += qr[i][k] * X[i, j];
}
s = -s / qr[k][k];
for (int i = k; i < m; i++)
{
X[i, j] += s * qr[i][k];
}
}
}
// Solve R*X = Y;
for (int k = n - 1; k >= 0; k--)
{
for (int j = 0; j < count; j++)
{
X[k, j] /= Rdiag[k];
}
for (int i = 0; i < k; i++)
{
for (int j = 0; j < count; j++)
{
X[i, j] -= X[k, j] * qr[i][k];
}
}
}
return(X.Submatrix(0, n - 1, 0, count - 1));
}
/// <summary>Construct a LU decomposition.</summary>
public LuDecomposition(Matrix value)
{
if (value == null)
{
throw new ArgumentNullException("value");
}
this.LU = (Matrix)value.Clone();
double[][] lu = LU.Array;
int rows = value.Rows;
int columns = value.Columns;
pivotVector = new int[rows];
for (int i = 0; i < rows; i++)
{
pivotVector[i] = i;
}
pivotSign = 1;
double[] LUrowi;
double[] LUcolj = new double[rows];
// Outer loop.
for (int j = 0; j < columns; j++)
{
// Make a copy of the j-th column to localize references.
for (int i = 0; i < rows; i++)
{
LUcolj[i] = lu[i][j];
}
// Apply previous transformations.
for (int i = 0; i < rows; i++)
{
LUrowi = lu[i];
// Most of the time is spent in the following dot product.
int kmax = Math.Min(i, j);
double s = 0.0;
for (int k = 0; k < kmax; k++)
{
s += LUrowi[k] * LUcolj[k];
}
LUrowi[j] = LUcolj[i] -= s;
}
// Find pivot and exchange if necessary.
int p = j;
for (int i = j + 1; i < rows; i++)
{
if (Math.Abs(LUcolj[i]) > Math.Abs(LUcolj[p]))
{
p = i;
}
}
if (p != j)
{
for (int k = 0; k < columns; k++)
{
double t = lu[p][k];
lu[p][k] = lu[j][k];
lu[j][k] = t;
}
int v = pivotVector[p];
pivotVector[p] = pivotVector[j];
pivotVector[j] = v;
pivotSign = -pivotSign;
}
// Compute multipliers.
if (j < rows & lu[j][j] != 0.0)
{
for (int i = j + 1; i < rows; i++)
{
lu[i][j] /= lu[j][j];
}
}
}
}
