public static Vector CalculateFFT(Vector v)
{
if (Math.Log(v.Size, 2) % 1 > 0) {
v = PadWithZeros(v);
}
return CFFT(v);
}
private void MakeDecomposition()
{
//if (!(M.IsSquare())) {
// throw new MatrixException("The matrix needs to be square to compute the eigenvalues/eigenvectors!");
//}
if (!(M.IsReal() && M.IsSymmetric())) {
throw new MatrixException("The matrix needs to be real and symmetric, to reliably compute the eigenvalues/eigenvectors!");
}
HessenbergDecomposition hessen = new HessenbergDecomposition(M);
Matrix P = hessen.P;
Matrix Ai = M.Copy();
QRDecomposition qr = Ai.QR();
Eigenvalues = new Vector(Ai.Width);
Eigenvectors = qr.Q;
for (int i = 0; i < ITERATION_DEPTH; i++) {
Ai = qr.R * qr.Q;
qr = Ai.QR();
Eigenvectors *= qr.Q;
if (Ai.IsUpperTriangular()) {
break;
}
}
for (int j = 0; j < Ai.Height; j++) {
Eigenvalues[j] = Ai[j, j];
}
}
private static Vector DFT_Slow(Vector x)
{
int N = x.Size;
Vector k = Vector.Arrange(N);
Matrix n = k.Transpose();
Matrix M = MatrixMath.Exp(new Complex(0, 2) * Math.PI * k * n / N);
return (M * x).ToColumnVector();
}
public static Vector CalculateIFFT(Vector v)
{
if (Math.Log(v.Size, 2) % 1 > 0) {
v = PadWithZeros(v);
}
// The IFFT is the same as the FFT applied on the complex conjugate of the vector, divided by the size.
return 1.0 / v.Size * CFFT(v.AppyToAllElements(x => x.Conjugate()).ToColumnVector());
}
public Vector this[Vector v]
{
get {
return this[v, 0];
}
set {
this[v, 0] = value;
}
}
public static Matrix CalculateHouseholderTransform(Vector x)
{
if (x.EuclidianNorm() == 0) {
return MatrixFactory.IdentityMatrix(x.Height);
}
double a = x[0].R == 0 ? x.EuclidianNorm() : Math.Sign(x[0].R) * x.EuclidianNorm();
Vector u = x + a * Vector.Ei(x.Size, 0);
Vector v = u / u.EuclidianNorm();
Complex w = (x.ConjugateTranspose() * v).ToComplexNumber() / (v.ConjugateTranspose() * x).ToComplexNumber();
return (MatrixFactory.IdentityMatrix(x.Height) - (1.0 + w) * v * v.ConjugateTranspose());
}
private static Vector PadWithZeros(Vector x)
{
int N = x.Size;
int pow2Size = (int)Math.Pow(2, Math.Ceiling(Math.Log(N) / Math.Log(2)));
Vector newVec = new Vector(pow2Size);
for (int i = 0; i < N; i++) {
newVec[i] = x[i];
}
for (int i = N; i < pow2Size; i++) {
newVec[i] = 0;
}
return newVec;
}
/// <summary>
/// Create an n x n Hankel matrix based on the specified first and last column.
/// </summary>
/// <param name="firstColumn"></param>
/// <param name="lastColumn"></param>
/// <returns></returns>
public static Matrix Hankel(Vector firstColumn, Vector lastColumn)
{
if (firstColumn.Size != lastColumn.Size) {
throw new IncompatibleMatrixDimensionsException(firstColumn, lastColumn);
}
if (firstColumn[firstColumn.Size - 1] != lastColumn[0]) {
throw new MatrixException("The last and first index of the specified vectors don't match!");
}
Vector concat = firstColumn.ConcatenateRows(lastColumn[Vector.Arrange(1, lastColumn.Size)]).ToColumnVector();
Matrix newMat = new Matrix(firstColumn.Size);
for (int i = 0; i < firstColumn.Size; i++) {
newMat[Vector.Arrange(firstColumn.Size), i] = concat.SubMatrix(i, i + firstColumn.Size, 0, 1).ToColumnVector();
}
return newMat;
}
private void MakeDecomposition()
{
Matrix A = M.Copy();
Q = MatrixFactory.IdentityMatrix(M.Height);
for (int k = 0; k < Math.Min(M.Height - 1, M.Width); k++) {
Vector x = new Vector(A, 0);
Matrix Q_k = MatrixMath.CalculateHouseholderTransform(x);
A = (Q_k * A).SubMatrix(1, A.Height, 1, A.Width);
Matrix Q_temp = MatrixFactory.IdentityMatrix(M.Height);
Q_temp[Vector.Arrange(k, M.Height), Vector.Arrange(k, M.Height)] = Q_k;
Q *= Q_temp.ConjugateTranspose();
}
R = Q.ConjugateTranspose() * M;
}
/// <summary>
/// Given a vector space defined by the columns of the matrix x, an orthonormal projection y is calculated using the Gram-Schmidt process.
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
public static Matrix GramSchmidthProcess(Matrix x)
{
Matrix y = MatrixFactory.Zeros(x.Height, x.Width);
y[Vector.Arrange(y.Height), 0] = x[Vector.Arrange(x.Height), 0];
for (int i = 1; i < x.Width; i++) {
Vector xi = x[Vector.Arrange(x.Height), i];
Vector projection = new Vector(xi.Size);
for (int j = 0; j < i - 1; j++) {
Vector yj = y[Vector.Arrange(y.Height), j];
projection += xi.DotProduct(yj) / yj.DotProduct(yj) * yj;
}
y[Vector.Arrange(y.Height), i] = x[Vector.Arrange(x.Height), i] - projection;
}
return y.NormalizeColumns();
}
private static Vector CFFT(Vector x)
{
int N = x.Size;
if (N <= 32) {
return DFT_Slow(x);
}
Vector evenIndices = Vector.Arrange(0, N, 2);
Vector oddIndices = Vector.Arrange(1, N, 2);
Vector X_even = CFFT(x[evenIndices]);
Vector X_odd = CFFT(x[oddIndices]);
Vector firstHalf = Vector.Arrange(N / 2);
Vector secondHalf = Vector.Arrange(N / 2, N);
Vector factor = MatrixMath.Exp(new Complex(0, 2) * Math.PI * Vector.Arrange(N) / N).ToColumnVector();
return (X_even + factor[firstHalf].ElementMultiply(X_odd)).ConcatenateRows(
X_even + factor[secondHalf].ElementMultiply(X_odd)).ToColumnVector();
}
public Vector Solve(Vector b)
{
if (M.Height != b.Height) {
throw new MatrixException("Wrong number of results in solution vector!");
}
Vector x = new Vector(M.Height);
if (M.IsSquare()) {
// use LU decomposition
LUDecomposition lu = M.LU();
Vector bp = (lu.P * b).ToColumnVector();
// Solves Ly = b using forward substitution
Vector y = ForwardSubstitution(lu.L, bp);
// Solves Ux = y using back substitution
x = BackSubstitution(lu.U, y);
} else {
// use QR decomposition
// TODO: Only works if M is full rank
// overdetermined system
if (M.Height >= M.Width) {
QRDecomposition qr = M.QR();
Matrix R1 = qr.R.SubMatrix(0, M.Width, 0, M.Width);
// Solves R1*x = Q.T*b using back substitution
x = BackSubstitution(R1, (qr.Q.ConjugateTranspose() * b).ToColumnVector());
} else {
// underdetermined system
QRDecomposition qr = M.Transpose().QR(); // note the transpose here
Matrix R1 = qr.R.SubMatrix(0, M.Height, 0, M.Height);
Matrix Q1 = qr.Q.SubMatrix(0, M.Width, 0, M.Height);
// Multiplies Q1 with the solution of R1.T*x = b using forward substitution
x = (Q1 * ForwardSubstitution(R1.ConjugateTranspose(), b)).ToColumnVector();
}
}
return x;
}
/// <summary>
/// Index the matrix at the specified column with vector v.
/// </summary>
/// <param name="v"></param>
/// <param name="column"></param>
/// <returns></returns>
public Vector this[Vector v, int column]
{
get {
Vector newVec = new Vector(v.Size);
for (int i = 0; i < v.Size; i++) {
newVec[i] = Mat[(int)v[i].R, column];
}
return newVec;
}
set {
for (int i = 0; i < value.Size; i++) {
Mat[(int)v[i].R, column] = value[i];
}
}
}
public MedianFilter(Vector data, int windowSize = 5)
{
Data = data.Copy();
WindowSize = windowSize;
}
public IFilter Filter() {
RectangularSmoothingFilter rFilter = new RectangularSmoothingFilter(Data);
Data = rFilter.Filter().Filter().Data;
return rFilter;
}
public TriangularSmoothingFilter(Vector data) {
Data = data.Copy();
}
/// <summary>
/// Matrix A needs to be lower triangular
/// </summary>
/// <param name="A"></param>
/// <param name="b"></param>
/// <returns></returns>
private Vector ForwardSubstitution(Matrix A, Vector b)
{
Vector x = new Vector(A.Height);
for (int i = 0; i < A.Height; i++) {
x[i] = b[i];
for (int j = 0; j < i; j++) {
x[i] -= A[i, j] * x[j];
}
x[i] = x[i] / A[i, i];
}
return x;
}
public Vector GetColumnVector(int column)
{
if (column < 0 || column >= Width) {
throw new MatrixException("Exceeds matrix dimensions!");
}
Vector v = new Vector(Height);
for (int i = 0; i < Height; i++) {
v[i] = this[i, column];
}
return v;
}
public Vector(Vector v)
: base(v)
{
Size = this.Height;
}
public Vector ToColumnVector()
{
if (Width != 1) {
throw new MatrixException("Wrong matrix dimension to convert to column vector!");
}
Vector v = new Vector(Height);
for (int i = 0; i < Height; i++) {
v[i] = Mat[i, 0];
}
return v;
}
public Complex DotProduct(Vector v)
{
if (Size != v.Size) {
throw new IncompatibleMatrixDimensionsException(this, v);
}
Complex cSum = new Complex();
for (int i = 0; i < Size; i++) {
cSum += this[i] * v[i].Conjugate();
}
return cSum;
}
public static Vector ParseFrom(string vecString)
{
vecString = vecString.Replace(" ", "");
string[] rows = vecString.Split(new char[] { ',', ';' });
Vector newVec = new Vector(rows.Length);
try {
for (int i = 0; i < newVec.Size; i++) {
newVec[i] = new Complex(rows[i]);
}
} catch (Exception e) {
Console.WriteLine(e.Message);
}
return newVec;
}
/// <summary>
/// Matrix A needs to be upper triangular
/// </summary>
/// <param name="A"></param>
/// <param name="b"></param>
/// <returns></returns>
private Vector BackSubstitution(Matrix A, Vector b)
{
Vector x = new Vector(A.Height);
for (int i = A.Height - 1; i > -1; i--) {
x[i] = b[i];
for (int j = A.Width - 1; j > i; j--) {
x[i] -= A[i, j] * x[j];
}
x[i] = x[i] / A[i, i];
}
return x;
}
/// <summary>
/// Index the matrix at the specified row with vector v.
/// </summary>
/// <param name="row"></param>
/// <param name="v"></param>
/// <returns></returns>
public Vector this[int row, Vector v]
{
get {
Vector newVec = new Vector(v.Size);
for (int i = 0; i < v.Size; i++) {
newVec[i] = Mat[row, (int)v[i].R];
}
return newVec;
}
set {
for (int i = 0; i < value.Size; i++) {
Mat[row, (int)v[i].R] = value[i];
}
}
}
/// <summary>
/// Index the matrix with all column vector and row vector indices.
/// </summary>
/// <param name="column"></param>
/// <param name="row"></param>
/// <returns></returns>
public Matrix this[Vector column, Vector row]
{
get {
Matrix newMat = new Matrix(column.Size, row.Size);
for (int i = 0; i < column.Size; i++) {
for (int j = 0; j < row.Size; j++) {
newMat[i, j] = Mat[(int)column[i].R, (int)row[j].R];
}
}
return newMat;
}
set {
for (int i = 0; i < value.Height; i++) {
for (int j = 0; j < value.Width; j++) {
Mat[(int)column[i].R, (int)row[j].R] = value[i, j];
}
}
}
}
public Vector CrossProduct(Vector v)
{
if (Size != 3) {
throw new MatrixException("Cross product only exists in 3D");
}
Vector newV = new Vector(3);
newV[0] = this[2] * v[3] - v[2] * this[3];
newV[1] = this[3] * v[1] - v[3] * this[1];
newV[2] = this[1] * v[2] - v[1] * this[2];
return newV;
}
public void SetColumnVector(Vector v, int column)
{
if (column < 0 || column >= Width || Height != v.Size) {
throw new MatrixException("Exceeds matrix dimensions!");
}
for (int i = 0; i < Height; i++) {
this[i, column] = v[i];
}
}
public bool IsOrthogonalTo(Vector v)
{
return DotProduct(v) == 0;
}
public PlotData(Vector x, Vector y, string name) {
this.X = x;
this.Y = y;
this.Name = name;
}
public Vector CircShift(int shift)
{
Vector newV = new Vector(Size);
for (int i = 0; i < Size; i++) {
newV[i] = this[ComplexMath.Mod((i + shift), Size)];
}
return newV;
}
