# Tagged Questions

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### Solution of $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$

I'm trying to prove that $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$,given that $A$ is symmetric positive-definite has only one minimum. I've found the derivative is $f'(x)=Ax-b$, and in order to find ...
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### Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
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### Implementation of Total Variation Regularization Algorithm (Lagged Diffusivity Algorithm)

I am trying to compute the derivative of an experimentally-measured quantity as a function of time. The data are fairly noisy, which causes problems. For instance, using finite differences (central ...
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### Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
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### How to verify the gradient of a symbolic function using numerical gradient?

I have a function $f$, which takes as inputs a three arrays and returns an array. I have written a symbolic function $g$ to calculate the gradient of this function and I want to verify that it ...
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### Using Lagrange polynomial to obtain the Second Derivative Midpoint formula

The Second Derivative Midpoint/Central Formula is $$f^{\prime\prime}(x_0)=\frac{f(x_0-h)-2f(x_0)+f(x_0+h)}{h^2}-\frac{h^2}{12}f^{(4)}(\xi)$$ I tried to get this formula using Lagrange polynomial. ...
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### Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
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### optimal way to approximate second derivative

Suppose there is a function $f: \mathbb R\to \mathbb R$ and that we only know $f(0),f(h),f'(h),f(2h)$ for some $h>0$. and we can't know the value of $f$ with $100$% accuracy at any other point. ...
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### (newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
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### Simpson's rule error rate for N-dimension

I'm doing a project that involves numerical method, but I'm not too familiar on calculus. I'm using Simpson's rule to integrate n-dimension gaussian, I was able to get the integration result for ...
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### Numerical Analysis: Given a function and successive derivatives at one point, what's the value of the function at another point?

Example of an exercise I'm trying to solve: Find the value of $f ( 4)$ given that $f (6 )=350 , f ' (6 )=87 , f'' (6 )=30 , f ''' (6 )=4$ and all other higher derivatives of $f (x) at x=6$ are zero. ...
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### Numerical computation of the $n^{\mathrm {th}}$ derivative of a multivariate function

From a multivariate function $f$, depending on $n\geq 1$ variables, which can be computed numerically, but which does not admit simple analytic expression, I would like to approximate numerically the ...
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### Numerically calculate the second “left hand” derivative

The Problem I have a series of measurements for which I have to calculate the first and second derivative numerically in a "live" fashion, i.e. using only previous data. This is easy for the first ...
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### Knowing if the real derivative exists

The numerical derivative is valid only if the real derivative exists. Is it possible to know if the real derivative exists without using symbolic derivative, and using computer operations?
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### How do I calculate numerically a tensor in polar coordinates?

You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid? a) fit a (spline) surface, ...
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### Derivative of a function defined by the divided difference of another function.

Given a function $f$ of class $C$ $^{n+2}$ in an interval $[a,b]$ and $x_{0}=a<x_1<x_2 ... <x_n = b$ a subdivision of $[a,b]$ into $n+1$ points. Given another function $g$ defined in the ...
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### Given a cubic function, and its quadratic derivative- can I recover the cubic from quadratic?

Background: I'm trying to learn how to work with cubic and quadratic bezier splines for various drawing libraries, and working through how to approximate a cubic spline with a quadratic spline. It's ...
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### Percentage variation dependance of a function of two variables

The language is a sort of barrier in this case (even in my native language) so I'll try to make an example here to clarify the question. Given a function $f(a,b)$ I want to answer the question: to ...
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### Method for estimating the nth derivative?

When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
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### Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ...
I've been using this to compute the first order derivative's value of a function $f$ in a given point: $$f'(x) = \frac{f(x+\epsilon) - f(x-\epsilon)}{2\epsilon}$$ For some $\epsilon = 0.0001$ or ...
$$f''(x) \thickapprox\dfrac{1}{2h^2}[f(x+2h) - 2f(x) + f(x - 2h)]$$ I'm supposed to be deriving the above formula and establish an error formula in using them. This is one of a series of problems ...