dev.nm.analysis.differentiation.multivariate

Class Hessian

java.lang.Object

dev.nm.algebra.linear.matrix.doubles.matrixtype.dense.triangle.SymmetricMatrix

dev.nm.analysis.differentiation.multivariate.Hessian

All Implemented Interfaces:

Matrix, MatrixAccess, MatrixRing, MatrixTable, Densifiable, AbelianGroup<Matrix>, Monoid<Matrix>, Ring<Matrix>, Table, DeepCopyable

public class Hessian
extends SymmetricMatrix

The Hessian matrix is the square matrix of the second-order partial derivatives of a multivariate function. Mathematically, the Hessian of a scalar function is an \(n \times n\) matrix, where n is the domain dimension of f. For a scalar function f, we have \[ H(f) = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\ \\ \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix} \]

This implementation computes the Hessian matrix numerically using the finite difference method. We assume that the function f is continuous so the Hessian matrix is square and symmetric.

See Also:

Wikipedia: Hessian matrix

Constructor Summary

Constructors

Constructor and Description
Hessian(RealScalarFunction f, Vector x) Construct the Hessian matrix for a multivariate function f at point x.

Method Summary

Methods inherited from class dev.nm.algebra.linear.matrix.doubles.matrixtype.dense.triangle.SymmetricMatrix

add, deepCopy, equals, get, getColumn, getRow, hashCode, minus, multiply, multiply, nCols, nRows, ONE, opposite, scaled, set, t, toDense, toString, ZERO

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Methods inherited from interface dev.nm.algebra.linear.matrix.doubles.Matrix

toCSV

Constructor Detail

Hessian

public Hessian(RealScalarFunction f,
               Vector x)

Construct the Hessian matrix for a multivariate function f at point x.

Parameters:

f - a multivariate function

x - the point to evaluate the Hessian of f at