Takao Inou´e, Noboru Endou, and Yasunari Shidama. Differentiation of vector-valued functions on n-dimensional real normed linear spaces. MML, (5.4.1165):7907-7913. MML id.: PDIFF 6.Noboru Endou, and Yasunari
On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functionsAuthor links open overlay panelHans-Peter HeinzShow more Add to Mendeley Share Cite https://doi.org/10.1016/0362-546X(83)90006-8Get rights and content...
Derivative of a vector-valued functionfcan be defined as the limit [latex]\\frac{df}{d\\boldsymbol{u}}=\\lim_{h \to 0}\\frac{f(\\boldsymbol{x}+h \\boldsymbol{u})-f(\\boldsymbol{x})}{h}[/latex] wherever it exists finitely. As mentioned before, this gives us the rate of...
Let \{f_k\} be a sequence of increasing real-valued functions on [a,b] f(x)=\sum_{k=1}^\infty f_k(x) converges on [a,b] , then (1) f is increasing on [a,b] (2) f'(x)=\sum_{k=1}^\infty f'_k(x) a.e. on [a,b] ...
LetL(X,Y)be the set of all linear transformations of the vector space X into vector space Y, let the addition, multiplication (composition), and scalar multiplication be equipped with the spaceL(X,Y). ForA∈L(Rn,Rm), we define the norm‖A‖ofAas ...
You can evaluate gradients using automatic differentiation only for scalar-valued functions. Intermediate calculations can have any number of variables, but the final function value must be scalar. If you need to take derivatives of a vector-valued function, take derivatives of one component at a ...
Vectorized transcendental functions (exp,cos,erf, ..) are not widely available. Intel, AMD, and CUDA provide proprietary implementations, but many compilers don't include them by default. It is desirable to retain both scalar and vector versions of an algorithm, but ensuring their consistency thr...
Jacobian: Compute the Jacobian matrix of a vector valued function of one or more variables. Hessian: Compute the Hessian matrix of all 2nd partial derivatives of a scalar function of one or more variables. Hessdiag: Compute only the diagonal elements of the Hessian matrix ...
where for each i>n, ni is the arity of fi and τi is a map from {1,…,ni} into {1,…,i−1}. In this formulation, we allow the functions fi to be arbitrary differentiable scalar-valued functions. However we could, by introducing additional Wengert variables, ensure that the funct...
(9) in line 43. The shape derivative is a bounded linear functional on a space of vector fields. We introduce a vector-valued finite element spaceVECand define the object representing the shape derivativedJOmega_fas a linear functional onVEC. In line 48, we differentiate with respect to the...