/**
* 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);
}
/**
* Computes the N<sup>th</sup> root of a complex number. There are
* N distinct N<sup>th</sup> roots.
*
* @param a Complex number
* @param N The root's magnitude
* @param k Specifies which root. 0 ≤ k < N
* @param result Computed root
*/
public static void root(Complex_F64 a, int N, int k, Complex_F64 result)
{
double r = a.getMagnitude();
double theta = Math.Atan2(a.imaginary, a.real);
r = Math.Pow(r, 1.0 / N);
theta = (theta + 2.0 * k * UtilEjml.PI) / N;
result.real = r * Math.Cos(theta);
result.imaginary = r * Math.Sin(theta);
}
/**
* Computes the square root of the complex number.
*
* @param input Input complex number.
* @param root Output. The square root of the input
*/
public static void sqrt(Complex_F64 input, Complex_F64 root)
{
double r = input.getMagnitude();
double a = input.real;
root.real = Math.Sqrt((r + a) / 2.0);
root.imaginary = Math.Sqrt((r - a) / 2.0);
if (input.imaginary < 0)
{
root.imaginary = -root.imaginary;
}
}
/**
* <p>
* Converts a complex number into polar notation.
* </p>
*
* @param input Standard notation
* @param output Polar notation
*/
public static void convert(Complex_F64 input, ComplexPolar_F64 output)
{
output.r = input.getMagnitude();
output.theta = Math.Atan2(input.imaginary, input.real);
}
