public VectorR GaussSeidel(MatrixR A, VectorR b, int MaxIterations, double tolerance)
{
int n = b.GetSize();
VectorR x = new VectorR(n);
for (int nIteration = 0; nIteration < MaxIterations; nIteration++)
{
VectorR xOld = x.Clone();
for (int i = 0; i < n; i++)
{
double db = b[i];
double da = A[i, i];
if (Math.Abs(da) < epsilon)
{
throw new ArgumentException("Diagonal element is too small!");
}
for (int j = 0; j < i; j++)
{
db -= A[i, j] * x[j];
}
for (int j = i + 1; j < n; j++)
{
db -= A[i, j] * xOld[j];
}
x[i] = db / da;
}
VectorR dx = x - xOld;
if (dx.GetNorm() < tolerance)
{
//MessageBox.Show(nIteration.ToString());
return(x);
}
}
return(x);
}
private double LUSubstitute(MatrixR m, VectorR v) // m = A, v = b
{
int n = v.GetSize();
double det = 1.0;
for (int i = 0; i < n; i++) // Ly = b
{
double d = v[i];
for (int j = 0; j < i; j++)
{
d -= m[i, j] * v[j];
}
double dd = m[i, i];
if (Math.Abs(d) < epsilon)
{
throw new ArgumentException("Diagonal element is too small!");
}
d /= dd;
v[i] = d; //v = y
det *= m[i, i];
}
for (int i = n - 1; i >= 0; i--)
{
double d = v[i];
for (int j = i + 1; j < n; j++)
{
d -= m[i, j] * v[j];
}
v[i] = d; // v=x
}
return(det);
}
public static VectorR Transform(VectorR v, MatrixR m)
{
VectorR result = new VectorR(v.GetSize());
if (!m.IsSquared())
{
throw new ArgumentOutOfRangeException(
"Dimension", m.GetRows(), "The matrix must be squared!");
}
if (m.GetRows() != v.GetSize())
{
throw new ArgumentOutOfRangeException(
"Size", v.GetSize(), "The size of the vector must be equal"
+ "to the number of rows of the matrix!");
}
for (int i = 0; i < m.GetRows(); i++)
{
result[i] = 0.0;
for (int j = 0; j < m.GetCols(); j++)
{
result[i] += v[j] * m[j, i];
}
}
return(result);
}
public static void Power(MatrixR A, double tolerance, out VectorR x, out double lambda)
{
int n = A.GetCols();
x = new VectorR(n);
lambda = 0.0;
double delta = 0.0;
Random random = new Random();
for (int i = 0; i < n; i++)
{
x[i] = random.NextDouble();
}
do
{
VectorR temp = x;
x = MatrixR.Transform(A, x);
x.Normalize();
if (VectorR.DotProduct(temp, x) < 0)
{
x = -x;
}
VectorR dx = temp - x;
delta = dx.GetNorm();
}while (delta > tolerance);
lambda = VectorR.DotProduct(x, MatrixR.Transform(A, x));
}
public static void Inverse(MatrixR A, double s, double tolerance, out VectorR x, out double lambda)
{
int n = A.GetCols();
x = new VectorR(n);
lambda = 0.0;
double delta = 0.0;
MatrixR identity = new MatrixR(n, n);
A = A - s * (identity.Identity());
LinearSystem ls = new LinearSystem();
A = ls.LUInverse(A);
Random random = new Random();
for (int i = 0; i < n; i++)
{
x[i] = random.NextDouble();
}
do
{
VectorR temp = x;
x = MatrixR.Transform(A, x);
x.Normalize();
if (VectorR.DotProduct(temp, x) < 0)
{
x = -x;
}
VectorR dx = temp - x;
delta = dx.GetNorm();
}while (delta > tolerance);
lambda = s + 1.0 / (VectorR.DotProduct(x, MatrixR.Transform(A, x)));
}
public static void Rayleigh(MatrixR A, double tolerance, out VectorR x, out double lambda)
{
int n = A.GetCols();
double delta = 0.0;
Random random = new Random();
x = new VectorR(n);
for (int i = 0; i < n; i++)
{
x[i] = random.NextDouble();
}
x.Normalize();
VectorR x0 = MatrixR.Transform(A, x);
x0.Normalize();
lambda = VectorR.DotProduct(x, x0);
double temp = lambda;
do
{
temp = lambda;
x0 = x;
x0.Normalize();
x = MatrixR.Transform(A, x0);
lambda = VectorR.DotProduct(x, x0);
delta = Math.Abs((temp - lambda) / lambda);
}while (delta > tolerance);
x.Normalize();
}
public MatrixR GetTranspose()
{
MatrixR m = this;
m.Transpose();
return(m);
}
public static void RayleighQuotient(MatrixR A, double tolerance, int flag, out VectorR x, out double lambda)
{
int n = A.GetCols();
double delta = 0.0;
Random random = new Random();
x = new VectorR(n);
if (flag != 2)
{
for (int i = 0; i < n; i++)
{
x[i] = random.NextDouble();
}
x.Normalize();
lambda = VectorR.DotProduct(x, MatrixR.Transform(A, x));
}
else
{
lambda = 0.0;
Rayleigh(A, 1e-2, out x, out lambda);
}
double temp = lambda;
MatrixR identity = new MatrixR(n, n);
LinearSystem ls = new LinearSystem();
do
{
temp = lambda;
double d = ls.LUCrout(A - lambda * identity.Identity(), x);
x.Normalize();
lambda = VectorR.DotProduct(x, MatrixR.Transform(A, x));
delta = Math.Abs((temp - lambda) / lambda);
}while (delta > tolerance);
}
public static MatrixR Minor(MatrixR m, int row, int col)
{
MatrixR mm = new MatrixR(m.GetRows() - 1, m.GetCols() - 1);
int ii = 0, jj = 0;
for (int i = 0; i < m.GetRows(); i++)
{
if (i == row)
{
continue;
}
jj = 0;
for (int j = 0; j < m.GetCols(); j++)
{
if (j == col)
{
continue;
}
mm[ii, jj] = m[i, j];
jj++;
}
ii++;
}
return(mm);
}
private void LUDecompose(MatrixR m)
{
int n = m.GetRows();
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
double d = m[i, j];
for (int k = 0; k < Math.Min(i, j); k++)
{
d -= m[i, k] * m[k, j];
}
if (j > i)
{
double dd = m[i, i];
if (Math.Abs(d) < epsilon)
{
throw new ArgumentException("Diagonal element is too small!");
}
d /= dd;
}
m[i, j] = d;
}
}
}
public MatrixR Clone()
{
// returns a deep copy of the matrix
MatrixR m = new MatrixR(matrix);
m.matrix = (double[, ])matrix.Clone();
return(m);
}
private static void Transformation(MatrixR A, MatrixR R, int I, int J, out MatrixR A1, out MatrixR R1)
{
int n = A.GetCols();
double t = 0.0;
double da = A[J, J] - A[I, I];
if (Math.Abs(A[I, J]) < Math.Abs(da) * 1e-30)
{
t = A[I, J] / da;
}
else
{
double phi = da / (2.0 * A[I, J]);
t = 1.0 / (Math.Abs(phi) + Math.Sqrt(1.0 + phi * phi));
if (phi < 0.0)
{
t = -t;
}
}
double c = 1.0 / Math.Sqrt(Math.Abs(t * t + 1.0));
double s = t * c;
double tau = s / (1.0 + c);
double temp = A[I, J];
A[I, J] = 0.0;
A[I, I] -= t * temp;
A[J, J] += t * temp;
for (int i = 0; i < I; i++)
{
temp = A[i, I];
A[i, I] = temp - s * (A[i, J] + tau * temp);
A[i, J] += s * (temp - tau * A[i, J]);
}
for (int i = I + 1; i < J; i++)
{
temp = A[I, i];
A[I, i] = temp - s * (A[i, J] + tau * A[I, i]);
A[i, J] += s * (temp - tau * A[i, J]);
}
for (int i = J + 1; i < n; i++)
{
temp = A[I, i];
A[I, i] = temp - s * (A[J, i] + tau * temp);
A[J, i] += s * (temp - tau * A[J, i]);
}
for (int i = 0; i < n; i++)
{
temp = R[i, I];
R[i, I] = temp - s * (R[i, J] + tau * R[i, I]);
R[i, J] += s * (temp - tau * R[i, J]);
}
A1 = A;
R1 = R;
}
public static MatrixR Inverse(MatrixR m)
{
if (Determinant(m) == 0)
{
throw new DivideByZeroException(
"Cannot inverse a matrix with a zero determinant!");
}
return(Adjoint(m) / Determinant(m));
}
public static bool CompareDimension(MatrixR m1, MatrixR m2)
{
if (m1.GetRows() == m2.GetRows() && m1.GetCols() == m2.GetCols())
{
return(true);
}
else
{
return(false);
}
}
// Eigenvalues and eigenvectors of a tridiagonal matrix:
public static void SetAlphaBeta(MatrixR T)
{
int n = T.GetCols();
Alpha = new double[n];
Beta = new double[n - 1];
Alpha[0] = T[0, 0];
for (int i = 1; i < n; i++)
{
Alpha[i] = T[i, i];
Beta[i - 1] = T[i - 1, i];
}
}
public static MatrixR operator /(double d, MatrixR m)
{
MatrixR result = new MatrixR(m.GetRows(), m.GetCols());
for (int i = 0; i < m.GetRows(); i++)
{
for (int j = 0; j < m.GetCols(); j++)
{
result[i, j] = d / m[i, j];
}
}
return(result);
}
public void Transpose()
{
MatrixR m = new MatrixR(Cols, Rows);
for (int i = 0; i < Rows; i++)
{
for (int j = 0; j < Cols; j++)
{
m[j, i] = matrix[i, j];
}
}
this = m;
}
public static MatrixR SetTridiagonalMatrix()
{
int n = Alpha.GetLength(0);
MatrixR t = new MatrixR(n, n);
t[0, 0] = Alpha[0];
for (int i = 1; i < n; i++)
{
t[i, i] = Alpha[i];
t[i - 1, i] = Beta[i - 1];
t[i, i - 1] = Beta[i - 1];
}
return(t);
}
public static void Jacobi(MatrixR A, double tolerance, out MatrixR x, out VectorR lambda)
{
MatrixR AA = A.Clone();
int n = A.GetCols();
int maxTransform = 5 * n * n;
MatrixR matrix = new MatrixR(n, n);
MatrixR R = matrix.Identity();
MatrixR R1 = R;
MatrixR A1 = A;
lambda = new VectorR(n);
x = R;
double maxTerm = 0.0;
int I, J;
do
{
maxTerm = MaxTerm(A, out I, out J);
Transformation(A, R, I, J, out A1, out R1);
A = A1;
R = R1;
}while (maxTerm > tolerance);
x = R;
for (int i = 0; i < n; i++)
{
lambda[i] = A[i, i];
}
for (int i = 0; i < n - 1; i++)
{
int index = i;
double d = lambda[i];
for (int j = i + 1; j < n; j++)
{
if (lambda[j] > d)
{
index = j;
d = lambda[j];
}
}
if (index != i)
{
lambda = lambda.GetSwap(i, index);
x = x.GetColSwap(i, index);
}
}
}
public MatrixR Identity()
{
MatrixR m = new MatrixR(Rows, Cols);
for (int i = 0; i < Rows; i++)
{
for (int j = 0; j < Cols; j++)
{
if (i == j)
{
m[i, j] = 1;
}
}
}
return(m);
}
public VectorR GaussJordan(MatrixR A, VectorR b)
{
Triangulate(A, b);
int n = b.GetSize();
VectorR x = new VectorR(n);
for (int i = n - 1; i >= 0; i--)
{
double d = A[i, i];
if (Math.Abs(d) < epsilon)
{
throw new ArgumentException("Diagonal element is too small!");
}
x[i] = (b[i] - VectorR.DotProduct(A.GetRowVector(i), x)) / d;
}
return(x);
}
private static MatrixR Jacobian(MFunction f, VectorR x)
{
double h = 0.0001;
int n = x.GetSize();
MatrixR jacobian = new MatrixR(n, n);
VectorR x1 = x.Clone();
for (int j = 0; j < n; j++)
{
x1[j] = x[j] + h;
for (int i = 0; i < n; i++)
{
jacobian[i, j] = (f(x1)[i] - f(x)[i]) / h;
}
}
return(jacobian);
}
public static VectorR NewtonMultiEquations(MFunction f, VectorR x0, double tolerance)
{
LinearSystem ls = new LinearSystem();
VectorR dx = new VectorR(x0.GetSize());
do
{
MatrixR A = Jacobian(f, x0);
if (Math.Sqrt(VectorR.DotProduct(f(x0), f(x0)) / x0.GetSize()) < tolerance)
{
return(x0);
}
dx = ls.GaussJordan(A, -f(x0));
x0 = x0 + dx;
}while (Math.Sqrt(VectorR.DotProduct(dx, dx)) > tolerance);
return(x0);
}
public MatrixR LUInverse(MatrixR m)
{
int n = m.GetRows();
MatrixR u = m.Identity();
LUDecompose(m);
VectorR uv = new VectorR(n);
for (int i = 0; i < n; i++)
{
uv = u.GetRowVector(i);
LUSubstitute(m, uv);
u.ReplaceRow(uv, i);
}
MatrixR inv = u.GetTranspose();
return(inv);
}
public static MatrixR operator -(MatrixR m1, MatrixR m2)
{
if (!MatrixR.CompareDimension(m1, m2))
{
throw new ArgumentOutOfRangeException(
"Dimension", m1, "The dimensions of two matrices must be the same!");
}
MatrixR result = new MatrixR(m1.GetRows(), m1.GetCols());
for (int i = 0; i < m1.GetRows(); i++)
{
for (int j = 0; j < m1.GetCols(); j++)
{
result[i, j] = m1[i, j] - m2[i, j];
}
}
return(result);
}
public static MatrixR Adjoint(MatrixR m)
{
if (!m.IsSquared())
{
throw new ArgumentOutOfRangeException(
"Dimension", m.GetRows(), "The matrix must be squared!");
}
MatrixR ma = new MatrixR(m.GetRows(), m.GetCols());
for (int i = 0; i < m.GetRows(); i++)
{
for (int j = 0; j < m.GetCols(); j++)
{
ma[i, j] = Math.Pow(-1, i + j) * (Determinant(Minor(m, i, j)));
}
}
return(ma.GetTranspose());
}
public static MatrixR Transform(VectorR v1, VectorR v2)
{
/*if (v1.GetSize() != v2.GetSize())
* {
* throw new ArgumentOutOfRangeException(
* "v1", v1.GetSize(), "The vectors must have the same size!");
* }*/
MatrixR result = new MatrixR(v1.GetSize(), v2.GetSize());
for (int i = 0; i < v1.GetSize(); i++)
{
for (int j = 0; j < v2.GetSize(); j++)
{
result[i, j] = v1[i] * v2[j];
}
}
return(result);
}
public static double Determinant(MatrixR m)
{
double result = 0.0;
if (!m.IsSquared())
{
throw new ArgumentOutOfRangeException(
"Dimension", m.GetRows(), "The matrix must be squared!");
}
if (m.GetRows() == 1)
{
result = m[0, 0];
}
else
{
for (int i = 0; i < m.GetRows(); i++)
{
result += Math.Pow(-1, i) * m[0, i] * Determinant(MatrixR.Minor(m, 0, i));
}
}
return(result);
}
private static double MaxTerm(MatrixR A, out int I, out int J)
{
int n = A.GetCols();
double result = 0.0;
I = 0;
J = 1;
for (int i = 0; i < n - 1; i++)
{
for (int j = i + 1; j < n; j++)
{
if (Math.Abs(A[i, j]) > result)
{
result = Math.Abs(A[i, j]);
I = i;
J = j;
}
}
}
return(result);
}
private double pivot(MatrixR A, VectorR b, int q)
{
int n = b.GetSize();
int i = q;
double d = 0.0;
for (int j = q; j < n; j++)
{
double dd = Math.Abs(A[j, q]);
if (dd > d)
{
d = dd;
i = j;
}
}
if (i > q)
{
A.GetRowSwap(q, i);
b.GetSwap(q, i);
}
return(A[q, q]);
}