Undergrad Function value at different points

[Apashanka]

If for a field say $f=xyz$ ,it's value is defined at 10 points in the 3-D Cartesian co-ordinate system...now using these 10 values of f and the corresponding coordinates is it possible to find the value of f at any $(x,y,z)$ of choice??

 

[pasmith]

Do you mean 10 points in total, or 10 points along each axis (ie. 1000 points in total)?

Either you are trying to fit a function to given values at the grid points, in which case you to get values elsewhere you must interpolate using, for example, lagrange polynomials (MATLAB solution here), or your function is defined by an expression in closed form in which case you already know its value everywhere.

 

[Apashanka]

Do you mean 10 points in total, or 10 points along each axis (ie. 1000 points in total)?

Either you are trying to fit a function to given values at the grid points, in which case you to get values elsewhere you must interpolate using, for example, lagrange polynomials (MATLAB solution here), or your function is defined by an expression in closed form in which case you already know its value everywhere.

(10) points in total

 

[Vanadium 50]

It was the second paragraph of pasmith's that was most important.

 

[Apashanka]

It's ok if $f(x,y,z)$ is known for 5 values of $(x,y,z)$ then how will the formula of $f(x,y,z)$ at any $x,y,z$ change??for 1-D it is simple

 

[Stephen Tashi]

If for a field say $f=xyz$ ,it's value is defined at 10 points in the 3-D Cartesian co-ordinate system...now using these 10 values of f and the corresponding coordinates is it possible to find the value of f at any $(x,y,z)$ of choice??

The value of $f$ at 10 points does not determine a unique value of $f$ at other points. You can invent many different functions that agree with $f$ on the 10 points but disagree with each other elsewhere. The general topic of inventing a function that has specified values on a finite number of points can be studied under the topics of "fitting a function to data" or "interpolation".

In your example, a straightforward method is to use multidimensional Lagrange interpolation. I think Answer 1 to https://math.stackexchange.com/ques...omial-in-3-d-variable-of-interest-is-a-vector describes this method.

The wikipedia gives a list of multivariate interpolation methods suited to different applications. https://en.wikipedia.org/wiki/Multivariate_interpolation

 

[DaveE]

No. I would think about this in 1-D first. How many points does it take to determine a quadratic function ( f(x) = ax² +bx +c )? Would those points define a unique cubic equation?
More info here: https://en.wikipedia.org/wiki/Curve_fitting

 

[PeterDonis]

How many points does it take to determine a quadratic function ( f(x) = ax2 +bx +c )? Would those points define a unique cubic equation?

The OP just said "function", not "quadratic function" or "cubic function". With no information about what kind of function it is, no finite set of points can tell you the function.

 

[DaveE]

The OP just said "function", not "quadratic function" or "cubic function". With no information about what kind of function it is, no finite set of points can tell you the function.

Yes, I saw that. That's why my first sentence was "No."
I was suggesting he look at a simpler problem to highlight the limitations of curve fitting. If you can't do it for simple polynomials (even the 11th order ones), it won't work in the more general case.