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);
}
public static VectorR Transform(MatrixR m, VectorR v)
{
//VectorR result = new VectorR(v.GetSize());
VectorR result = new VectorR(m.GetRows());
/*if (!m.IsSquared())
* {
* throw new ArgumentOutOfRangeException(
* "Dimension", m.GetRows(), "The matrix must be squared!");
* }*/
if (m.GetCols() != 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] += m[i, j] * v[j];
}
}
return(result);
}
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 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 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 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));
}
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 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 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 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);
}
}
}
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);
}
public static MatrixR Tridiagonalize(MatrixR A)
{
int n = A.GetCols();
MatrixR A1 = new MatrixR(n, n);
A1 = A.Clone();
double h, g, unorm;
for (int i = 0; i < n - 2; i++)
{
VectorR u = new VectorR(n - i - 1);
for (int j = i + 1; j < n; j++)
{
u[j - i - 1] = A[i, j];
}
unorm = u.GetNorm();
if (u[0] < 0.0)
{
unorm = -unorm;
}
u[0] += unorm;
for (int j = i + 1; j < n; j++)
{
A[j, i] = u[j - i - 1];
}
h = VectorR.DotProduct(u, u) * 0.5;
VectorR v = new VectorR(n - i - 1);
MatrixR a1 = new MatrixR(n - i - 1, n - i - 1);
for (int j = i + 1; j < n; j++)
{
for (int k = i + 1; k < n; k++)
{
a1[j - i - 1, k - i - 1] = A[j, k];
}
}
v = MatrixR.Transform(a1, u) / h;
g = VectorR.DotProduct(u, v) / (2.0 * h);
v -= g * u;
for (int j = i + 1; j < n; j++)
{
for (int k = i + 1; k < n; k++)
{
A[j, k] = A[j, k] - v[j - i - 1] * u[k - i - 1] - u[j - i - 1] * v[k - i - 1];
}
}
A[i, i + 1] = -unorm;
}
Alpha = new double[n];
Beta = new double[n - 1];
Alpha[0] = A[0, 0];
for (int i = 1; i < n; i++)
{
Alpha[i] = A[i, i];
Beta[i - 1] = A[i - 1, i];
}
MatrixR V = new MatrixR(n, n);
V = V.Identity();
for (int i = 0; i < n - 2; i++)
{
VectorR u = new VectorR(n - i - 1);
for (int j = i + 1; j < n; j++)
{
u[j - i - 1] = A.GetColVector(i)[j];
}
h = VectorR.DotProduct(u, u) * 0.5;
VectorR v = new VectorR(n - 1);
MatrixR v1 = new MatrixR(n - 1, n - i - 1);
for (int j = 1; j < n; j++)
{
for (int k = i + 1; k < n; k++)
{
v1[j - 1, k - i - 1] = V[j, k];
}
}
v = MatrixR.Transform(v1, u) / h;
for (int j = 1; j < n; j++)
{
for (int k = i + 1; k < n; k++)
{
V[j, k] -= v[j - 1] * u[k - i - 1];
}
}
}
return(V);
}
public MatrixR(MatrixR m)
{
Rows = m.GetRows();
Cols = m.GetCols();
matrix = m.matrix;
}
