Frobenius Norm
| This documentation was generated with the assistance of AI. Please report any inaccuracies. |
FrobeniusNormComputer computes the Frobenius norm of a matrix or array: the square root of the
sum of the squares of all its elements.
For an array, this is exactly its Euclidean (L2) length, treating the array as a column vector.
The Frobenius norm is NormComputer.DEFAULT_NORM_TYPE, so NormComputer.create() (with no
arguments) returns a FrobeniusNormComputer, and Utils.normF is built directly on it.
Utils.isOrthonormal also relies on the Frobenius norm: in addition to being orthogonal, an
orthonormal matrix must have Frobenius norm equal to 1.
For an array, the Frobenius norm is the p = 2 case of the general p-norm family; its unit ball
is the familiar circle below, in contrast with the diamond of the one norm
(p = 1) and the square of the infinity norm (p tending to infinity):
Usage
The getNorm instance methods implement the abstract NormComputer contract; static norm
methods are also available for direct use without instantiating the class:
import com.irurueta.algebra.FrobeniusNormComputer;
import com.irurueta.algebra.Matrix;
import com.irurueta.algebra.NormComputer;
import com.irurueta.algebra.NormType;
Matrix a = new Matrix(2, 2);
// ... fill a with data ...
// static usage
double norm = FrobeniusNormComputer.norm(a);
double[] array = {3.0, 4.0};
double arrayNorm = FrobeniusNormComputer.norm(array); // 5.0
// through the NormComputer factory
NormComputer computer = NormComputer.create(NormType.FROBENIUS_NORM);
double sameNorm = computer.getNorm(a);A norm(double[], Matrix jacobian) overload also computes the norm’s 1xN Jacobian with respect
to the input array, useful when the norm is used inside a larger optimization.
Utils.normF(Matrix) and Utils.normF(double[]) are convenience wrappers around this class.
Error handling
The Jacobian overload throws WrongSizeException if the provided Jacobian matrix is not 1xN,
where N is the length of the input array.
Reference
Frobenius norm is one of the standard matrix norms described in Wikipedia, "Matrix norm".