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aphirst

Gold Member

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As part of my work, I'm making use of the familiar properties of function minima/maxima in a way which I can't seem to find in the literature. I was hoping that by describing it here, someone else might recognise it and be able to point me to a citation. I think it's highly unlikely that I'm the first to do this, since it's so straightforward.

Say you have a scalar function: $$f(\mathbf{x},q)$$ where: $$\mathbf{x} = \begin{pmatrix}x_1 \\ x_2 \\ \vdots \end{pmatrix}$$ and where ##q## is an auxiliary parameter - perhaps ##f(\mathbf{x})## represents some physical quantity, and ##q## parametrises the physical system somehow.

Let's assume that ##f(\mathbf{x},q)## is continuous and differentiable in ##\mathbf{x}## and ##q## up to at least ##\mathcal{C}^3##.

Let's say you use a robust numerical procedure to obtain the parameters ##\mathbf{x}_i## which give a local minimum of ##f(\mathbf{x})## from some (irrelevant) starting point, at some ##q##: $$\mathbf{x}_i = \mathrm{argmin} f(\mathbf{x})$$ Let's say we're actually interested in ##\partial_q \mathbf{x}_i##: the rate of change of the

From the definition of a minimum (actually any extremum), at the minimum: $$\nabla_{\mathbf{x}} f = 0$$ By taking an additional ##\partial_q## (and swapping the order of partial derivatives via Schwarz' theorem): $$\mathbf{H}_{\mathbf{x}} \partial_q \mathbf{x}_i + \partial_q \nabla_{\mathbf{x}} f= 0$$ where ##\mathbf{H}_\mathbf{x}## is the Hessian of ##f## w.r.t. ##\mathbf{x}##, and where the expression takes advantage of the total derivative (

Issues with my compact notation aside, I can confirm that this approach is very successful, and lets me obtain quasi-analytic Jacobian matrices for physical quantities w.r.t. other physical parameters, which beforehand seemed to unavoidably necessitate numerical derivatives (e.g. finite difference).

Say you have a scalar function: $$f(\mathbf{x},q)$$ where: $$\mathbf{x} = \begin{pmatrix}x_1 \\ x_2 \\ \vdots \end{pmatrix}$$ and where ##q## is an auxiliary parameter - perhaps ##f(\mathbf{x})## represents some physical quantity, and ##q## parametrises the physical system somehow.

__Subscripts in ##\mathbf{x},u## denote partial derivatives w.r.t. those variables.__Let's assume that ##f(\mathbf{x},q)## is continuous and differentiable in ##\mathbf{x}## and ##q## up to at least ##\mathcal{C}^3##.

Let's say you use a robust numerical procedure to obtain the parameters ##\mathbf{x}_i## which give a local minimum of ##f(\mathbf{x})## from some (irrelevant) starting point, at some ##q##: $$\mathbf{x}_i = \mathrm{argmin} f(\mathbf{x})$$ Let's say we're actually interested in ##\partial_q \mathbf{x}_i##: the rate of change of the

__solved__variables ##\mathbf{x}_i##__with respect to__the parameter ##q## describing the system itself. Of course, as you change ##q## (change the system), you expect different solved values of ##\mathbf{x}_i##.From the definition of a minimum (actually any extremum), at the minimum: $$\nabla_{\mathbf{x}} f = 0$$ By taking an additional ##\partial_q## (and swapping the order of partial derivatives via Schwarz' theorem): $$\mathbf{H}_{\mathbf{x}} \partial_q \mathbf{x}_i + \partial_q \nabla_{\mathbf{x}} f= 0$$ where ##\mathbf{H}_\mathbf{x}## is the Hessian of ##f## w.r.t. ##\mathbf{x}##, and where the expression takes advantage of the total derivative (

__is that the correct term?__): $$\frac{\partial g_i(\mathbf{x}_i,q)}{\partial q} = \frac{\partial g}{\partial x_1} \frac{\partial x_{i,1}}{\partial q} + \frac{\partial g}{\partial x_2} \frac{\partial x_{i,2}}{\partial q}+ ... + \frac{\partial g}{\partial q}$$ From here, obtaining ##\partial_q \mathbf{x}_i## involves simply solving the linear equation system involving the Hessian matrix.Issues with my compact notation aside, I can confirm that this approach is very successful, and lets me obtain quasi-analytic Jacobian matrices for physical quantities w.r.t. other physical parameters, which beforehand seemed to unavoidably necessitate numerical derivatives (e.g. finite difference).

__Is there a name for what I've just done here? Surely I can't have invented it?__
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