Gradient, Hessian, Jacobian

Explained:

gradient

Hessian

Jacobian


   
 

The gradient and Hessian of a function f: are the vector of its first partial derivatives and matrix of its second partial derivatives:

[1]

The Hessian is symmetric if the second partials are continuous.

The Jacobian J f of a function f: is the matrix of its first partial derivatives.

[2]

Note that the Hessian of a function f: is the Jacobian of its gradient.

 

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Exercises

Consider the function: h(x,y) = x2y. Calculate its gradient at the point (2,5). [solution]

Related Internal Links

Cholesky matrix A lower-triangular matrix that acts as a matrix "square root" for a positive definite matrix.

eigenvalue, eigenvector Concepts from linear algebra.

interpolation Any procedure for fitting a function to a set of points in such a manner that the function intercepts each of the points.

positive definite matrix A real symmetric matrix, all of whose eigenvalues are real and positive.

Taylor series expansion In calculus, a power series obtained as a limit of Taylor polynomials that may approximate or equal the function from which it is constructed.

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