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symbolic differentiation to find analytical expressions that are time-consuming to calculate manually?

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SA-W 2023-6-6

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编辑： Torsten 2023-6-6

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Given a function of two variables

f(x,y) =

I want to find the analytial expression of the determinant of the Hessian of f.

are real numbers that I can not specify further, and

are scalar functions that I also can not specify further because they depend on n.

If I compute, for instance, the partial derivatives on a paper, I would write

It is quite time-consuming and messy to do similar calculations manually on a paper.

I know that symbolic differentiation can be used in a context like

syms x
f = x^2
diff(f,x) = 
   2x

but can it also be applied to expressions that are not completely defined like in my context?

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Dyuman Joshi 2023-6-6

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Ran in:

You can try something like this -

However this only works when n is defined as a (natural) number, it won't work if you define n as a symbolic variable.

n=5;

syms N(x,y) [1 n]

syms E [n 1] real

N(x, y) =

E =

f = N*E

f(x, y) =

fx = diff(f,x)

fx(x, y) =

fy = diff(f,y)

fy(x, y) =

Torsten 2023-6-6

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编辑：Torsten 2023-6-6

but can it also be applied to expressions that are not completely defined like in my context?

It can be applied, but I doubt that it will be helpful for your application, especially if n is large. You might want to take a look at

https://uk.mathworks.com/help/symbolic/sym.hessian.html

SA-W 2023-6-6

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@Dyuman Joshi @Torsten

n is at least greater than 30 in my application. I hoped that the sum is kept as it is and not written out if I write, for instance,

. This is indeed not helpful then.

What I really want to do is to compute the determinant of the Hessian and find constraints on the

(e.g.

) such that the determinant is greater than zero. To this end, I wanted to find the anatycal expression.

Is there any other matlab toolbox that might help here? Or, based on your experience, any other tool/software you can recoomend?

Torsten 2023-6-6

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What I really want to do is to compute the determinant of the Hessian and find constraints on the E_i such that the determinant is greater than zero.

This is unrealistic for n at least 30.

SA-W 2023-6-6

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This is unrealistic for n at least 30.

Unrealistic to derive those constraints manually or what do you mean?

Torsten 2023-6-6

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编辑：Torsten 2023-6-6

Yes. And much too complicated. You would have to determine the ranges for x and y in which the determinant of a combination of more than 30 functions N_i are positive. Even for n = 1 this can be a difficult task.

Why do you want to do that ?

SA-W 2023-6-6

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You would have to determine the ranges for x and y in which the determinant of a combination of more than 30 functions N_i are positive.

I think it is a bit easier in my case. I require a positive determinant only at n points in the x-y space. Between those points I have something like a bilinear interpolation. This means that the values of all

are known at those points and what remains to do is to adjust the

. This sounds tractable to me.

Why do you want to do that ?

must be convex such that my objective function can be evaluated. I want to encode the above constraints in A and b (terminology of the optimization solvers). The

are the parameters of my optimization.

Torsten 2023-6-6

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If the E_i can be restricted to be >=0 and the N_i are convex, f will also be convex. Does that help ?

SA-W 2023-6-6

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But the E_i are not convex. In 1d, the N_i resemble hat-functions which are not convex.

So I think I have to incorporate "convexity" via the E_i.

Torsten 2023-6-6

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