/**
* <p>
* The condition p = 2 number of a matrix is used to measure the sensitivity of the linear
* system <b>Ax=b</b>. A value near one indicates that it is a well conditioned matrix.<br>
* <br>
* κ<sub>2</sub> = ||A||<sub>2</sub>||A<sup>-1</sup>||<sub>2</sub>
* </p>
* <p>
* This is also known as the spectral condition number.
* </p>
*
* @param A The matrix.
* @return The condition number.
*/
public static double conditionP2(DMatrixRMaj A)
{
SingularValueDecomposition_F64 <DMatrixRMaj> svd =
DecompositionFactory_DDRM.svd(A.numRows, A.numCols, false, false, true);
svd.decompose(A);
double[] singularValues = svd.getSingularValues();
int n = SingularOps_DDRM.rank(svd, UtilEjml.TEST_F64);
if (n == 0)
{
return(0);
}
double smallest = double.MaxValue;
double largest = double.MinValue;
foreach (double s in singularValues)
{
if (s < smallest)
{
smallest = s;
}
if (s > largest)
{
largest = s;
}
}
return(largest / smallest);
}
/**
* Computes the nullity of a matrix using the specified tolerance.
*
* @param A Matrix whose rank is to be calculated. Not modified.
* @param threshold The numerical threshold used to determine a singular value.
* @return The matrix's nullity.
*/
public static int nullity(DMatrixRMaj A, double threshold)
{
SingularValueDecomposition_F64 <DMatrixRMaj> svd =
DecompositionFactory_DDRM.svd(A.numRows, A.numCols, false, false, true);
if (svd.inputModified())
{
A = (DMatrixRMaj)A.copy();
}
if (!svd.decompose(A))
{
throw new InvalidOperationException("Decomposition failed");
}
return(SingularOps_DDRM.nullity(svd, threshold));
}
