docs: add band matrix storage format documentation in the README of lapack/base

[aayush0325]

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This pull request:

• adds band matrix storage format documentation in the README of lapack/base as discussed

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Coverage Report

Package	Statements	Branches	Functions	Lines
lapack/base	$\color{green}294/294$ $\color{green}+100.00\%$	$\color{green}1/1$ $\color{green}+100.00\%$	$\color{green}0/0$ $\color{green}+100.00\%$	$\color{green}294/294$ $\color{green}+100.00\%$

The above coverage report was generated for the changes in this PR.

[kgryte]

It would be good to actually show matrices which are upper and lower triangular here.

[kgryte]

Overall, this is looking good. It's a more readable version of https://netlib.org/lapack/lug/node124.html. That stated, one thing which needs to be changed is that currently everything is written using one-based indexing. We need to update everything to zero-based indexing.

[kgryte]

One question I have: how do row- and column-major orders factor in here. For example, in dgbbrd, the routine accepts a layout argument.

It may be good to also show how the band storage matrices are stored in linear memory for the two supported layouts. IIRC, this was a point of confusion during development.

[kgryte]

In these comments, you do not need to preserve extra whitespace for aligning array elements. In fact, removing is preferred.

[aayush0325]

One question I have: how do row- and column-major orders factor in here. For example, in dgbbrd, the routine accepts a layout argument.

this is how i understand the row and column major interpretations of these matrices:

for column major:

{
  "order": "column-major",
  "type": "banded",
  "M": 5,
  "N": 5,
  "KL": 2,
  "KU": 1,
  "A": [
    0.0, 0.0, 0.0, 1.1, 2.1, 3.1,
    0.0, 0.0, 1.2, 2.2, 3.2, 4.2,
    0.0, 0.0, 2.3, 3.3, 4.3, 5.3,
    0.0, 0.0, 3.4, 4.4, 5.4, 0.0,
    0.0, 0.0, 4.5, 5.5, 0.0, 0.0
  ],
  "A_mat": [
    [ 1.1, 1.2, 0.0, 0.0, 0.0 ],
    [ 2.1, 2.2, 2.3, 0.0, 0.0 ],
    [ 3.1, 3.2, 3.3, 3.4, 0.0 ],
    [ 0.0, 4.2, 4.3, 4.4, 4.5 ],
    [ 0.0, 0.0, 5.3, 5.4, 5.5 ]
  ],
}

for row major:

{
  "order": "row-major",
  "type": "banded",
  "M": 5,
  "N": 5,
  "KL": 2,
  "KU": 1,
  "A": [
    0.0, 0.0, 0.0, 0.0, 0.0,
    0.0, 0.0, 0.0, 0.0, 0.0,
    0.0, 1.2, 2.3, 3.4, 4.5,
    1.1, 2.2, 3.3, 4.4, 5.5,
    2.1, 3.2, 4.3, 5.4, 0.0,
    3.1, 4.2, 5.3, 0.0, 0.0
  ],
  "A_mat": [
    [ 1.1, 1.2, 0.0, 0.0, 0.0 ],
    [ 2.1, 2.2, 2.3, 0.0, 0.0 ],
    [ 3.1, 3.2, 3.3, 3.4, 0.0 ],
    [ 0.0, 4.2, 4.3, 4.4, 4.5 ],
    [ 0.0, 0.0, 5.3, 5.4, 5.5 ]
  ]
}

i am basically reading this column-by-column for col-major and row-by-row for row-major

[kgryte]

@aayush0325 What you are displaying is specific for LU correct with the extra cols/rows?

Regardless, I suggest making explicit with a couple of brief examples showing how banded matrices are stored according to layouts.