Since this will mainly be a list of which files have been modified and which are left to do this might not be very interesting if you are not familiar with the structure of the interval package.

### Interval constant (3)

All of these have been modified to support N-dimensional arrays.### Interval constructor (5)

All of these have been modified to support N-dimensional arrays.### Interval function (most with tightest accuracy) (63)

Almost all of these functions worked out of the box! At least after the API functions to the MPFR and crlibm libraries were fixed, they are further down in the list.The only function that did not work immediately were $linspace$. Even though this function could be generalized to N-dimensinal arrays the standard Octave function only works for matrices (I think the Matlab version only allows scalars). This means that adding support for N-dimensional vectors for the interval version is not a priority. I might do it later on but it is not necessary.

### Interval matrix operation (16)

Most of the matrix functions does not make sense for N-dimensional arrays. For example matrix multiplication and matrix inversion only makes sense for matrices. However all of the reduction functions are also here, they include $dot$, $prod$, $sum$, $sumabs$ and $sumsq$.At the moment I have implemented support for N-dimensional arrays for $sum$, $sumabs$ and $prod$. The functions $dot$ and $sumsq$ are not ready, I'm waiting to see what happens with bug #51333 [2] before I continue with that work. Depending on the bug I might also have to modify the behaviour of $sum$, $sumabs$ and $prod$ slightly.

### Interval comparison (19)

All of these have been modified to support N-dimensional arrays.### Set operation (7)

All of these functions have been modified to support N-dimensional arrays except one, $mince$. The function $mince$ is an interval version of $linspace$ and reasoning here is the same as that for $linspace$ above.### Interval reverse operation (12)

Like the interval functions above, all of the functions worked out of the box!### Interval numeric function (11)

Also these functions worked out of the box, with some small modifications to the documentation for some of them.### Interval input and output (9)

Here there are some functions which require some comments, the ones I do not comment about have all gotten support for N-dimensional arrays.$interval\_bitpack$

I think that this function does not make sense to generalize to N-dimensions. It could perhaps take an N-dimensional arrays as input, but it will always return a row vector. I have left it as it is for now at least.

$disp$ and $display$

These are functions that might be subject to change later on. At the moment it prints N-dimensional arrays of intervals in the same way Octave does for normal arrays. It's however not clear how to handle the $\subset$ symbol and we might decide to change it.

### Interval solver or optimizer (5)

The functions $gauss$ and $polyval$ are not generalizable to N-dimensional vectors. I don't think that $fzero$ can be generalized either, for it to work the functions must be real-valued.The function $fsolve$ can perhaps be modified to support N-dimensional vectors. It uses the SIVIA algorithm [3] and I have to dive deeper into how it works to see if it can be done.

For $fminsearch$ nothing needed to be done, it worked for N-dimensional arrays directly.

### Interval contractor arithmetic (2)

Both of these functions are used together with $fsolve$ so they also depend on if SIVIA can be generalized or not.### Verified solver or optimizer (6)

All of these functions work on matrices and cannot be generalized.### Utility function (29)

All of these for which it made sense have been modified to support N-dimensional arrays. Some of them only works for matrices, these are $ctranspose$, $diag$, $transpose$, $tril$ and $triu$. I have left them as they were, though I fixed a bug in $diag$.### API function to the MPFR and crlibm libraries (8)

These are the functions that in general required most work. The ones I have added full support for N-dimensional arrays in are $crlibm\_function$, $mpfr\_function\_d$ and $mpfr\_vector\_sum\_d$. Some of them cannot be generalized, these are $mpfr\_matrix\_mul_d$, $mpfr\_matrix\_sqr\_d$ and $mpfr\_to\_string\_d$. The functions $mpfr\_linspace\_d$ and $mpfr\_vector\_dot\_d$ are related to what I mentioned above for $linspace$ and $dot$.### Summary

So summing up the functions that still require some work are- Functions related to $fsolve$
- The functions $dot$ and $sumsq$
- The functions $linspace$ and $mince$

Apart from this there are also some more things that needs to be considered. The documentation for the package will need to be updated. This includes adding some examples which make use of the new functionality.

The interval package also did not follow the coding style for Octave. All the functions which I have made changes to have been updated with the correct coding style, but many of the functions that worked out of the box still use the old style. It might be that we want to unify the coding standard for all files before the next release.

[1] The $INDEX$ file https://sourceforge.net/u/urathai/octave/ci/default/tree/INDEX

[2] Bug #51333 https://savannah.gnu.org/bugs/index.php?51333

[3] The SIVIA algorithm https://www.youtube.com/watch?v=kxYh2cXHdNQ

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