Applied Numerical Methods Using MATLAB, 2nd Edition

Appendix CDifferentiation W.R.T. A Vector

The first derivative of a scalar‐valued function f(x) w.r.t. the vector x = [x₁ x₂]^T is called the gradient of f(x) and defined as

(C.1)

Based on this definition, we can write the following equation:

(C.2)

(C.3)

Also with an M × N matrix A = [a_mn], we have

(C.4a)

(C.4b)

where

(C.5)

Especially for a symmetric (and square) matrix A (with M = N), we have

(C.6)

The second derivative of a scalar function f (x) w.r.t. the vector x = [x₁ x₂]^T is called the Hessian (matrix) of f(x) and is defined as

(C.7)

Based on this definition, we can write the following equation:

(C.8)

The first derivative of a vector‐valued function f(x) w.r.t. the vector ...