com.numericalmethod.suanshu.analysis.differentiation.multivariate
Class Hessian

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.matrixtype.MatrixMathImpl<T>
      com.numericalmethod.suanshu.matrix.doubles.matrixtype.MatrixStorageImpl<SymmetricMatrix>
          com.numericalmethod.suanshu.matrix.doubles.matrixtype.dense.triangle.SymmetricMatrix
              com.numericalmethod.suanshu.analysis.differentiation.multivariate.Hessian

All Implemented Interfaces:

DeepCopyable, AbelianGroup<Matrix>, Monoid<Matrix>, Ring<Matrix>, Matrix, MatrixAccessor, MatrixRing, Densifiable, MatrixDimension

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 x n matrix, where n is the domain dimension of f.

      | ∂f         ∂f   |
      |----, ..., ----  |
      |∂x21       ∂x1∂xn |
 H =  |                 |
      |                 |
      | ∂f         ∂f   |
      |----, ..., ----  |
      |∂xn∂x1     ∂x2∂xn |
 

The Hessian matrix is computed numerically using the finite difference method.

We assume that the function f is continuous so the Hessian matrix is square and symmetric.

See Also:

FiniteDifference, Wikipedia: Hessian matrix

Field Summary

Fields inherited from class com.numericalmethod.suanshu.matrix.doubles.matrixtype.dense.triangle.SymmetricMatrix

dim

Constructor Summary

Hessian(RealScalarFunction f, double... x)
Construct a Hessian matrix for a multivariate function f at point x.

Method Summary

double

discriminant() Compute the determinant of the Hessian at x, which is called the discriminant.

Methods inherited from class com.numericalmethod.suanshu.matrix.doubles.matrixtype.dense.triangle.SymmetricMatrix

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

Methods inherited from class com.numericalmethod.suanshu.matrix.doubles.matrixtype.MatrixStorageImpl

getMatrixData, setMatrixData

Methods inherited from class com.numericalmethod.suanshu.matrix.doubles.matrixtype.MatrixMathImpl

add, call, getSample, minus, multiply, multiply, nCols, nRows, opposite, setColumn, setColumn, setRow, setRow

Methods inherited from class java.lang.Object

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

Constructor Detail

Hessian

public Hessian(RealScalarFunction f,
               double... x)

Construct a 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

Method Detail

discriminant

public double discriminant()

Compute the determinant of the Hessian at x, which is called the discriminant.

Returns:

the discriminant