/// <summary>
/// Performs forward insertion for regular lower triangular matrix
/// and right side b, such that the solution is saved right within b.
/// The matrix is not changed.
/// </summary>
/// <param name="b">Vector of height n, if matrix is n by n.</param>
public void ForwardInsertion(ArrayMatrix b)
{
if (!IsLowerTriangular()) throw new InvalidOperationException("Cannot perform forward insertion for matrix not being lower triangular.");
if ( /*this.Determinant*/DiagProd() == 0) throw new InvalidOperationException("Warning: Matrix is nearly singular.");
var n = RowCount;
if (b.VectorLength() != n) throw new ArgumentException("Parameter must vector of the same height as matrix.");
for (var j = 1; j <= n - 1; j++)
{
b[j] /= this[j, j];
for (var i = 1; i <= n - j; i++) b[j + i] -= b[j]*this[j + i, j];
}
b[n] /= this[n, n];
}
public static ArrayMatrix HorizontalConcat(ArrayMatrix[] A)
{
if (A == null) throw new ArgumentNullException();
else if (A.Length == 1) return A[0];
else
{
var C = HorizontalConcat(A[0], A[1]);
for (var i = 2; i < A.Length; i++) C = HorizontalConcat(C, A[i]);
return C;
}
}
/// <summary>
/// Creates n by n matrix filled with random values in [0,1].
/// </summary>
/// <param name="m"></param>
/// <param name="n"></param>
/// <returns></returns>
public static ArrayMatrix Random(int n)
{
var M = new ArrayMatrix(n);
var r = new Random();
for (var i = 1; i <= n; i++) for (var j = 1; j <= n; j++) M[i, j] = new Complex(r.NextDouble());
return M;
}
/// <summary>
/// Implements the dot product of two vectors.
/// </summary>
/// <param name="v">Row or column vector.</param>
/// <param name="w">Row or column vector.</param>
/// <returns>Dot product.</returns>
public static Complex Dot(ArrayMatrix v, ArrayMatrix w)
{
var m = v.VectorLength();
var n = w.VectorLength();
if (m == 0 || n == 0) throw new ArgumentException("Arguments need to be vectors.");
else if (m != n) throw new ArgumentException("Vectors must be of the same length.");
var buf = Complex.Zero;
for (var i = 1; i <= m; i++) buf += v[i]*w[i];
return buf;
}
/// <summary>
/// Returns the shortest path between two given vertices i and j as
/// int array.
/// </summary>
/// <param name="P">Path matrix as returned from Floyd().</param>
/// <param name="i">One-based index of start vertex.</param>
/// <param name="j">One-based index of end vertex.</param>
/// <returns></returns>
public static ArrayList FloydPath(ArrayMatrix P, int i, int j)
{
if (!P.IsSquare()) throw new ArgumentException("Path matrix must be square.");
else if (!P.IsReal()) throw new ArgumentException("Adjacence matrices are expected to be real.");
var path = new ArrayList();
path.Add(i);
//int borderliner = 0;
//int n = P.Size()[0] + 1; // shortest path cannot have more than n vertices!
while (P[i, j] != 0)
{
i = Convert.ToInt32(P[i, j]);
path.Add(i);
//borderliner++;
//if (borderliner == n)
// throw new FormatException("P was not a Floyd path matrix.");
}
path.Add(j);
return path;
}
/// <summary>
/// Creates m by n chessboard matrix with interchangíng ones and zeros.
///
/// </summary>
/// <param name="n">Number of columns.</param>
/// <param name="even">Indicates, if matrix entry (1,1) equals zero.</param>
/// <returns></returns>
public static ArrayMatrix ChessboardMatrix(int n, bool even)
{
var M = new ArrayMatrix(n);
if (even) for (var i = 1; i <= n; i++) for (var j = 1; j <= n; j++) M[i, j] = KroneckerDelta((i + j)%2, 0);
else for (var i = 1; i <= n; i++) for (var j = 1; j <= n; j++) M[i, j] = KroneckerDelta((i + j)%2, 1);
return M;
}
/// <summary>
/// Generates diagonal matrix
/// </summary>
/// <param name="diag_vector">column vector containing the diag elements</param>
/// <returns></returns>
public static ArrayMatrix Diag(ArrayMatrix diag_vector)
{
var dim = diag_vector.VectorLength();
if (dim == 0) throw new ArgumentException("diag_vector must be 1xN or Nx1");
var M = new ArrayMatrix(dim, dim);
for (var i = 1; i <= dim; i++) M[i, i] = diag_vector[i];
return M;
}
/// <summary>
/// Executes the QR iteration.
/// </summary>
/// <param name="max_iterations"></param>
/// <returns></returns>
public ArrayMatrix QRIterationBasic(int max_iterations)
{
if (!IsReal()) throw new InvalidOperationException("Basic QR iteration is possible only for real matrices.");
var T = Clone();
var QR = new ArrayMatrix[2];
for (var i = 0; i < max_iterations; i++)
{
QR = T.QRGramSchmidt();
T = QR[1]*QR[0];
}
return T;
}
/// <summary>
/// Returns the matrix of the real parts of the entries of this matrix.
/// </summary>
/// <returns></returns>
public ArrayMatrix Re()
{
var M = new ArrayMatrix(RowCount, ColumnCount);
for (var i = 1; i <= RowCount; i++) for (var j = 1; j <= ColumnCount; j++) M[i, j] = new Complex(this[i, j].Real);
return M;
}
/// <summary>
/// Inserts row at specified index.
/// </summary>
/// <param name="row">Vector to insert</param>
/// <param name="i">One-based index at which to insert</param>
public void InsertRow(ArrayMatrix row, int i)
{
var size = row.VectorLength();
if (size == 0) throw new InvalidOperationException("Row must be a vector of length > 0.");
if (i <= 0) throw new ArgumentException("Row index must be positive.");
if (i > RowCount) this[i, size] = Complex.Zero;
else if (size > ColumnCount)
{
this[i, size] = Complex.Zero;
RowCount++;
}
else RowCount++;
Values.Insert(--i, new ArrayList(size));
//Debug.WriteLine(Values.Count.ToString());
for (var k = 1; k <= size; k++) ((ArrayList) Values[i]).Add(row[k]);
// fill w/ zeros if vector row is too short
for (var k = size; k < ColumnCount; k++) ((ArrayList) Values[i]).Add(Complex.Zero);
}
/// <summary>
/// Gram-Schmidtian orthogonalization of an m by n matrix A, such that
/// {Q, R} is returned, where A = QR, Q is m by n and orthogonal, R is
/// n by n and upper triangular matrix.
/// </summary>
/// <returns></returns>
public ArrayMatrix[] QRGramSchmidt()
{
var m = RowCount;
var n = ColumnCount;
var A = Clone();
var Q = new ArrayMatrix(m, n);
var R = new ArrayMatrix(n, n);
// the first column of Q equals the first column of this matrix
for (var i = 1; i <= m; i++) Q[i, 1] = A[i, 1];
R[1, 1] = Complex.One;
for (var k = 1; k <= n; k++)
{
R[k, k] = new Complex(A.Column(k).Norm());
for (var i = 1; i <= m; i++) Q[i, k] = A[i, k]/R[k, k];
for (var j = k + 1; j <= n; j++)
{
R[k, j] = Dot(Q.Column(k), A.Column(j));
for (var i = 1; i <= m; i++) A[i, j] = A[i, j] - Q[i, k]*R[k, j];
}
}
return new ArrayMatrix[] {Q, R};
}
/// <summary>
/// Inserts column at specified index.
/// </summary>
/// <param name="col">Vector to insert</param>
/// <param name="j">One-based index at which to insert</param>
public void InsertColumn(ArrayMatrix col, int j)
{
var size = col.VectorLength();
if (size == 0) throw new InvalidOperationException("Row must be a vector of length > 0.");
if (j <= 0) throw new ArgumentException("Row index must be positive.");
if (j > ColumnCount) this[size, j] = Complex.Zero;
else ColumnCount++;
if (size > RowCount) this[size, j] = Complex.Zero;
j--;
for (var k = 0; k < size; k++) ((ArrayList) Values[k]).Insert(j, col[k + 1]);
// fill w/ zeros if vector col too short
for (var k = size; k < RowCount; k++) ((ArrayList) Values[k]).Insert(j, 0);
}
/// <summary>
/// Inserts a sub matrix M at row i and column j.
/// </summary>
/// <param name="i">One-based row number to insert.</param>
/// <param name="j">One-based column number to insert.</param>
/// <param name="M">Sub matrix to insert.</param>
public void Insert(int i, int j, ArrayMatrix M)
{
for (var m = 1; m <= M.RowCount; m++) for (var n = 1; n <= M.ColumnCount; n++) this[i + m - 1, j + n - 1] = M[m, n];
}
/// <summary>
/// Performs Hessenberg-Householder reduction, where {H, Q}
/// is returned, with H Hessenbergian, Q orthogonal and H = Q'AQ.
/// </summary>
/// <returns></returns>
public ArrayMatrix[] HessenbergHouseholder()
{
//throw new NotImplementedException("Still buggy!");
if (!IsSquare())
throw new InvalidOperationException(
"Cannot perform Hessenberg Householder decomposition of non-square matrix.");
var n = RowCount;
var Q = Identity(n);
var H = Clone();
ArrayMatrix I, N, R, P;
var vbeta = new ArrayMatrix[2];
int m;
// don't try to understand from the code alone.
// this is pure magic to me - mathematics, reborn as code.
for (var k = 1; k <= n - 2; k++)
{
vbeta = HouseholderVector(H.Extract(k + 1, n, k, k));
I = Identity(k);
N = Zeros(k, n - k);
m = vbeta[0].VectorLength();
R = Identity(m) - vbeta[1][1, 1]*vbeta[0]*vbeta[0].Transpose();
H.Insert(k + 1, k, R*H.Extract(k + 1, n, k, n));
H.Insert(1, k + 1, H.Extract(1, n, k + 1, n)*R);
P = BlockMatrix(I, N, N.Transpose(), R);
Q = Q*P;
}
return new ArrayMatrix[] {H, Q};
}
public static ArrayMatrix operator *(Complex x, ArrayMatrix A)
{
var B = new ArrayMatrix(A.RowCount, A.ColumnCount);
for (var i = 1; i <= A.RowCount; i++) for (var j = 1; j <= A.ColumnCount; j++) B[i, j] = A[i, j]*x;
return B;
}
/// <summary>
/// Retrieves row with one-based index i.
/// </summary>
/// <param name="i"></param>
/// <returns>i-th row...</returns>
public ArrayMatrix Row(int i)
{
if (i <= 0 || i > RowCount) throw new ArgumentException("Index exceed matrix dimension.");
//return (new Matrix((Complex[])((ArrayList)Values[i - 1]).ToArray(typeof(Complex)))).Transpose();
var buf = new ArrayMatrix(ColumnCount, 1);
for (var j = 1; j <= ColumnCount; j++) buf[j] = this[i, j];
return buf;
}
/// <summary>
/// Constructs block matrix [A, B; C, D].
/// </summary>
/// <param name="A">Upper left sub matrix.</param>
/// <param name="B">Upper right sub matrix.</param>
/// <param name="C">Lower left sub matrix.</param>
/// <param name="D">Lower right sub matrix.</param>
/// <returns></returns>
public static ArrayMatrix BlockMatrix(ArrayMatrix A, ArrayMatrix B, ArrayMatrix C, ArrayMatrix D)
{
if (A.RowCount != B.RowCount || C.RowCount != D.RowCount
|| A.ColumnCount != C.ColumnCount || B.ColumnCount != D.ColumnCount) throw new ArgumentException("Matrix dimensions must agree.");
var R = new ArrayMatrix(A.RowCount + C.RowCount, A.ColumnCount + B.ColumnCount);
for (var i = 1; i <= R.RowCount; i++)
for (var j = 1; j <= R.ColumnCount; j++)
if (i <= A.RowCount)
if (j <= A.ColumnCount) R[i, j] = A[i, j];
else R[i, j] = B[i, j - A.ColumnCount];
else if (j <= C.ColumnCount) R[i, j] = C[i - A.RowCount, j];
else R[i, j] = D[i - A.RowCount, j - C.ColumnCount];
return R;
}
/// <summary>
/// Splits matrix into its row vectors.
/// </summary>
/// <returns>Array of row vectors.</returns>
public ArrayMatrix[] RowVectorize()
{
var buf = new ArrayMatrix[RowCount];
for (var i = 1; i <= buf.Length; i++) buf[i] = Row(i);
return buf;
}
/// <summary>
/// Performs depth-first search for a graph given by its adjacence matrix.
/// </summary>
/// <param name="adjacence_matrix">A[i,j] = 0 or +infty, if there is no edge from i to j; any non-zero value otherwise.</param>
/// <param name="root">The vertex to begin the search.</param>
/// <returns>Adjacence matrix of the computed spanning tree.</returns>
public static ArrayMatrix DFS(ArrayMatrix adjacence_matrix, int root)
{
if (!adjacence_matrix.IsSquare()) throw new ArgumentException("Adjacence matrices are expected to be square.");
else if (!adjacence_matrix.IsReal()) throw new ArgumentException("Adjacence matrices are expected to be real.");
var n = adjacence_matrix.RowCount;
if (root < 1 || root > n) throw new ArgumentException("Root must be a vertex of the graph, e.i. in {1, ..., n}.");
var spanTree = new ArrayMatrix(n);
var marked = new bool[n + 1];
var todo = new Stack();
todo.Push(root);
marked[root] = true;
// adajacence lists for each vertex
var A = new ArrayList[n + 1];
for (var i = 1; i <= n; i++)
{
A[i] = new ArrayList();
for (var j = 1; j <= n; j++) if (adjacence_matrix[i, j].Real != 0 && adjacence_matrix[i, j].Imag != double.PositiveInfinity) A[i].Add(j);
}
int v, w;
while (todo.Count > 0)
{
v = (int) todo.Peek();
if (A[v].Count > 0)
{
w = (int) A[v][0];
if (!marked[w])
{
marked[w] = true; // mark w
spanTree[v, w].Real = 1; // mark vw
todo.Push(w); // one more to search
}
A[v].RemoveAt(0);
}
else todo.Pop();
}
return spanTree;
}
/// <summary>
/// Solves equation this*x = b via conjugate gradient method.
/// </summary>
/// <param name="b"></param>
/// <returns></returns>
public ArrayMatrix SolveCG(ArrayMatrix b)
{
//throw new NotImplementedException("Still buggy!");
if (!IsSymmetricPositiveDefinite()) throw new InvalidOperationException("CG method only works for spd matrices.");
else if (!IsReal()) throw new InvalidOperationException("CG method only works for real matrices.");
var n = RowCount;
var max_iterations = 150;
var tolerance = 1e-6;
var x = Ones(n, 1); // x will contain the solution
var r = b - this*x; // residual approaches zero as x converges to the solution
var d = r; // dir = direction of descence
var delta = r.Norm(); // delta denotes the current error
delta *= delta;
tolerance *= tolerance;
var h = Zeros(n, 1);
double alpha, gamma;
double old_delta;
if (delta <= tolerance) return x;
else
{
for (var i = 0; i < max_iterations; i++)
{
h = this*d;
gamma = Dot(h, d).Real;
if (Math.Abs(gamma) <= tolerance) return x;
alpha = delta/gamma;
x += alpha*d; // compute new approximation of solution
r -= alpha*h; // compute new residual
old_delta = delta; // buffer delta
delta = r.Norm();
delta *= delta;
if (delta <= tolerance) return x;
d = r + delta/old_delta*d; // compute new direction of descence
}
return x;
}
}
/// <summary>
/// Generates diagonal matrix
/// </summary>
/// <param name="diag_vector">column vector containing the diag elements</param>
/// <returns></returns>
public static ArrayMatrix Diag(ArrayMatrix diag_vector, int offset)
{
var dim = diag_vector.VectorLength();
if (dim == 0) throw new ArgumentException("diag_vector must be 1xN or Nx1.");
//if (Math.Abs(offset) >= dim)
// throw new ArgumentException("Absolute value of offset must be less than length of diag_vector.");
var M = new ArrayMatrix(dim + Math.Abs(offset), dim + Math.Abs(offset));
dim = M.RowCount;
if (offset >= 0) for (var i = 1; i <= dim - offset; i++) M[i, i + offset] = diag_vector[i];
else for (var i = 1; i <= dim + offset; i++) M[i - offset, i] = diag_vector[i];
return M;
}
/// <summary>
/// Swaps each matrix entry A[i, j] with A[j, i].
/// </summary>
/// <returns>A transposed matrix.</returns>
public ArrayMatrix Transpose()
{
var M = new ArrayMatrix(ColumnCount, RowCount);
for (var i = 1; i <= ColumnCount; i++) for (var j = 1; j <= RowCount; j++) M[i, j] = this[j, i];
return M;
}
/// <summary>
/// Computes all shortest distance between any vertices in a given graph.
/// </summary>
/// <param name="adjacence_matrix">Square adjacence matrix. The main diagonal
/// is expected to consist of zeros, any non-existing edges should be marked
/// positive infinity.</param>
/// <returns>Two matrices D and P, where D[u,v] holds the distance of the shortest
/// path between u and v, and P[u,v] holds the shortcut vertex on the way from
/// u to v.</returns>
public static ArrayMatrix[] Floyd(ArrayMatrix adjacence_matrix)
{
if (!adjacence_matrix.IsSquare()) throw new ArgumentException("Expected square matrix.");
else if (!adjacence_matrix.IsReal()) throw new ArgumentException("Adjacence matrices are expected to be real.");
var n = adjacence_matrix.RowCount;
var D = adjacence_matrix.Clone(); // distance matrix
var P = new ArrayMatrix(n);
double buf;
for (var k = 1; k <= n; k++)
for (var i = 1; i <= n; i++)
for (var j = 1; j <= n; j++)
{
buf = D[i, k].Real + D[k, j].Real;
if (buf < D[i, j].Real)
{
D[i, j].Real = buf;
P[i, j].Real = k;
}
}
return new ArrayMatrix[] {D, P};
}
/// <summary>
/// Computes the Householder vector.
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
static ArrayMatrix[] HouseholderVector(ArrayMatrix x)
{
//throw new NotImplementedException("Supposingly buggy!");
//if (!x.IsReal())
// throw new ArgumentException("Cannot compute housholder vector of non-real vector.");
var n = x.VectorLength();
if (n == 0) throw new InvalidOperationException("Expected vector as argument.");
var y = x/x.Norm();
var buf = y.Extract(2, n, 1, 1);
var s = Dot(buf, buf);
var v = Zeros(n, 1);
v[1] = Complex.One;
v.Insert(2, 1, buf);
double beta = 0;
if (s != 0)
{
var mu = Complex.Sqrt(y[1]*y[1] + s);
if (y[1].Real <= 0) v[1] = y[1] - mu;
else v[1] = -s/(y[1] + mu);
beta = 2*v[1].Real*v[1].Real/(s.Real + v[1].Real*v[1].Real);
v = v/v[1];
}
return new ArrayMatrix[] {v, new ArrayMatrix(beta)};
}
public static ArrayMatrix HorizontalConcat(ArrayMatrix A, ArrayMatrix B)
{
var C = A.Row(1);
for (var i = 2; i <= A.RowCount; i++) C.InsertRow(A.Row(i), i);
for (var i = 1; i <= B.RowCount; i++) C.InsertRow(B.Row(i), C.RowCount + 1);
return C;
}
/// <summary>
/// Givens product. Internal use for QRGivens.
/// </summary>
/// <param name="c"></param>
/// <param name="s"></param>
/// <param name="n"></param>
/// <returns></returns>
ArrayMatrix GivProd(ArrayMatrix c, ArrayMatrix s, int n)
{
var n1 = n - 1;
var n2 = n - 2;
var Q = Eye(n);
Q[n1, n1] = c[n1];
Q[n, n] = c[n1];
Q[n1, n] = s[n1];
Q[n, n1] = -s[n1];
for (var k = n2; k >= 1; k--)
{
var k1 = k + 1;
Q[k, k] = c[k];
Q[k1, k] = -s[k];
var q = Q.Extract(k1, k1, k1, n);
Q.Insert(k, k1, s[k]*q);
Q.Insert(k1, k1, c[k]*q);
}
return Q;
}
/// <summary>
/// Creates n by n matrix filled with ones.
/// </summary>
/// <param name="n">Number of columns.</param>
/// <returns>n by n matrix filled with ones.</returns>
public static ArrayMatrix Ones(int n)
{
var M = new ArrayMatrix(n);
for (var i = 0; i < n; i++) for (var j = 0; j < n; j++) ((ArrayList) M.Values[i])[j] = Complex.One;
return M;
}
public static ArrayMatrix operator *(ArrayMatrix A, ArrayMatrix B)
{
if (A.ColumnCount != B.RowCount) throw new ArgumentException("Inner matrix dimensions must agree.");
var C = new ArrayMatrix(A.RowCount, B.ColumnCount);
for (var i = 1; i <= A.RowCount; i++) for (var j = 1; j <= B.ColumnCount; j++) C[i, j] = Dot(A.Row(i), B.Column(j));
return C;
}
/// <summary>
/// Creates m by n matrix filled with random values in {lo,...,hi-1}.
/// </summary>
///<param name="lo">Inclusive lower bound.</param>
/// <param name="hi">Exclusive upper bound</param>
/// <param name="m">Number of rows.</param>
/// <param name="n">Number of columns.</param>
/// <returns></returns>
public static ArrayMatrix Random(int m, int n, int lo, int hi)
{
var M = new ArrayMatrix(m, n);
var r = new Random();
for (var i = 1; i <= m; i++) for (var j = 1; j <= n; j++) M[i, j] = new Complex((double) r.Next(lo, hi));
return M;
}
/// <summary>
/// Extracts upper trapeze matrix of this matrix.
/// </summary>
/// <returns></returns>
public ArrayMatrix ExtractUpperTrapeze()
{
var buf = new ArrayMatrix(RowCount, ColumnCount);
for (var i = 1; i <= RowCount; i++) for (var j = i; j <= ColumnCount; j++) buf[i, j] = this[i, j];
return buf;
}
