/**
* <p>
* Creates a pivot matrix that exchanges the rows in a matrix:
* <br>
* A' = P*A<br>
* </p>
* <p>
* For example, if element 0 in 'pivots' is 2 then the first row in A' will be the 3rd row in A.
* </p>
*
* @param ret If null then a new matrix is declared otherwise the results are written to it. Is modified.
* @param pivots Specifies the new order of rows in a matrix.
* @param numPivots How many elements in pivots are being used.
* @param transposed If the transpose of the matrix is returned.
* @return A pivot matrix.
*/
public static ZMatrixRMaj pivotMatrix(ZMatrixRMaj ret, int[] pivots, int numPivots, bool transposed)
{
if (ret == null)
{
ret = new ZMatrixRMaj(numPivots, numPivots);
}
else
{
if (ret.numCols != numPivots || ret.numRows != numPivots)
{
throw new ArgumentException("Unexpected matrix dimension");
}
CommonOps_ZDRM.fill(ret, 0, 0);
}
if (transposed)
{
for (int i = 0; i < numPivots; i++)
{
ret.set(pivots[i], i, 1, 0);
}
}
else
{
for (int i = 0; i < numPivots; i++)
{
ret.set(i, pivots[i], 1, 0);
}
}
return(ret);
}
/**
* Computes the householder vector used in QR decomposition.
*
* u = x / max(x)
* u(0) = u(0) + |u|
* u = u / u(0)
*
* @param x Input vector. Unmodified.
* @return The found householder reflector vector
*/
public static ZMatrixRMaj householderVector(ZMatrixRMaj x)
{
ZMatrixRMaj u = (ZMatrixRMaj)x.copy();
double max = CommonOps_ZDRM.elementMaxAbs(u);
CommonOps_ZDRM.elementDivide(u, max, 0, u);
double nx = NormOps_ZDRM.normF(u);
Complex_F64 c = new Complex_F64();
u.get(0, 0, c);
double realTau, imagTau;
if (c.getMagnitude() == 0)
{
realTau = nx;
imagTau = 0;
}
else
{
realTau = c.real / c.getMagnitude() * nx;
imagTau = c.imaginary / c.getMagnitude() * nx;
}
u.set(0, 0, c.real + realTau, c.imaginary + imagTau);
CommonOps_ZDRM.elementDivide(u, u.getReal(0, 0), u.getImag(0, 0), u);
return(u);
}
/**
* <p>
* Creates a reflector from the provided vector and gamma.<br>
* <br>
* Q = I - γ u u<sup>H</sup><br>
* </p>
*
* @param u A vector. Not modified.
* @param gamma To produce a reflector gamma needs to be equal to 2/||u||.
* @return An orthogonal reflector.
*/
public static ZMatrixRMaj createReflector(ZMatrixRMaj u, double gamma)
{
if (!MatrixFeatures_ZDRM.isVector(u))
{
throw new ArgumentException("u must be a vector");
}
ZMatrixRMaj Q = CommonOps_ZDRM.identity(u.getNumElements());
CommonOps_ZDRM.multAddTransB(-gamma, 0, u, u, Q);
return(Q);
}
/**
* <p>
* Creates a reflector from the provided vector.<br>
* <br>
* Q = I - γ u u<sup>T</sup><br>
* γ = 2/||u||<sup>2</sup>
* </p>
*
* @param u A vector. Not modified.
* @return An orthogonal reflector.
*/
public static ZMatrixRMaj createReflector(ZMatrixRMaj u)
{
if (!MatrixFeatures_ZDRM.isVector(u))
{
throw new ArgumentException("u must be a vector");
}
double norm = NormOps_ZDRM.normF(u);
double gamma = -2.0 / (norm * norm);
ZMatrixRMaj Q = CommonOps_ZDRM.identity(u.getNumElements());
CommonOps_ZDRM.multAddTransB(gamma, 0, u, u, Q);
return(Q);
}
/**
* Creates a random symmetric positive definite matrix.
*
* @param width The width of the square matrix it returns.
* @param rand Random number generator used to make the matrix.
* @return The random symmetric positive definite matrix.
*/
public static ZMatrixRMaj hermitianPosDef(int width, IMersenneTwister rand)
{
// This is not formally proven to work. It just seems to work.
ZMatrixRMaj a = RandomMatrices_ZDRM.rectangle(width, 1, rand);
ZMatrixRMaj b = new ZMatrixRMaj(1, width);
ZMatrixRMaj c = new ZMatrixRMaj(width, width);
CommonOps_ZDRM.transposeConjugate(a, b);
CommonOps_ZDRM.mult(a, b, c);
for (int i = 0; i < width; i++)
{
c.data[2 * (i * width + i)] += 1;
}
return(c);
}
/**
* Q = I - gamma*u*u<sup>H</sup>
*/
public static ZMatrixRMaj householder(ZMatrixRMaj u, double gamma)
{
int N = u.getDataLength() / 2;
// u*u^H
ZMatrixRMaj uut = new ZMatrixRMaj(N, N);
VectorVectorMult_ZDRM.outerProdH(u, u, uut);
// foo = -gamma*u*u^H
CommonOps_ZDRM.elementMultiply(uut, -gamma, 0, uut);
// I + foo
for (int i = 0; i < N; i++)
{
int index = (i * uut.numCols + i) * 2;
uut.data[index] = 1 + uut.data[index];
}
return(uut);
}
/**
* <p>
* Unitary matrices have the following properties:<br><br>
* Q*Q<sup>H</sup> = I
* </p>
* <p>
* This is the complex equivalent of orthogonal matrix.
* </p>
* @param Q The matrix being tested. Not modified.
* @param tol Tolerance.
* @return True if it passes the test.
*/
public static bool isUnitary(ZMatrixRMaj Q, double tol)
{
if (Q.numRows < Q.numCols)
{
throw new ArgumentException("The number of rows must be more than or equal to the number of columns");
}
Complex_F64 prod = new Complex_F64();
ZMatrixRMaj[] u = CommonOps_ZDRM.columnsToVector(Q, null);
for (int i = 0; i < u.Length; i++)
{
ZMatrixRMaj a = u[i];
VectorVectorMult_ZDRM.innerProdH(a, a, prod);
if (Math.Abs(prod.real - 1) > tol)
{
return(false);
}
if (Math.Abs(prod.imaginary) > tol)
{
return(false);
}
for (int j = i + 1; j < u.Length; j++)
{
VectorVectorMult_ZDRM.innerProdH(a, u[j], prod);
if (!(prod.getMagnitude2() <= tol * tol))
{
return(false);
}
}
}
return(true);
}
/**
* <p>
* Computes the Frobenius matrix norm:<br>
* <br>
* normF = Sqrt{ ∑<sub>i=1:m</sub> ∑<sub>j=1:n</sub> { a<sub>ij</sub><sup>2</sup>} }
* </p>
* <p>
* This is equivalent to the element wise p=2 norm.
* </p>
*
* @param a The matrix whose norm is computed. Not modified.
* @return The norm's value.
*/
public static double normF(ZMatrixRMaj a)
{
double total = 0;
double scale = CommonOps_ZDRM.elementMaxAbs(a);
if (scale == 0.0)
{
return(0.0);
}
int size = a.getDataLength();
for (int i = 0; i < size; i += 2)
{
double real = a.data[i] / scale;
double imag = a.data[i + 1] / scale;
total += real * real + imag * imag;
}
return(scale * Math.Sqrt(total));
}