8000 Add initial linear algebra function specifications by kgryte · Pull Request #20 · data-apis/array-api · GitHub
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Add initial linear algebra function specifications #20

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Merged
merged 21 commits into from
Sep 8, 2020
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Add initial linear algebra function specifications #20

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bc73de3
Add cross signature
kgryte Aug 24, 2020
7d9be38
Add det specification
kgryte Aug 24, 2020
88090b4
Add diagonal specification
kgryte Aug 24, 2020
02f80be
Add inv specification
kgryte Aug 24, 2020
41697d9
Add norm specification
kgryte Aug 24, 2020
20164af
Add outer product specification
kgryte Aug 24, 2020
f046571
Add outer specification
kgryte Aug 24, 2020
e0b1e18
Add trace specification
kgryte Aug 24, 2020
95194aa
Add transpose
kgryte Aug 24, 2020
5095031
Update index
kgryte Aug 24, 2020
6a7fde8
Fix type annotation
kgryte Aug 24, 2020
83593dc
Update norm behavior for multi-dimensional arrays
kgryte Aug 24, 2020
16fc53c
Support all of NumPy's norms
kgryte Aug 25, 2020
4745713
Switch order
kgryte Aug 25, 2020
< 8000 code class="float-right">46f0bfc
Split into separate tables
kgryte Aug 25, 2020
8f65e94
Escape markup
kgryte Aug 25, 2020
2ab06d5
Escape markup
kgryte Aug 25, 2020
fdc07e1
Add matrix_transpose interface
kgryte Aug 25, 2020
cd076e5
Rename parameters
kgryte Aug 25, 2020
1bff1b5
Merge branch 'master' of https://github.com/pydata-apis/array-api int…
kgryte Aug 26, 2020
4015620
Remove matrix_transpose signature
kgryte Aug 31, 2020
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Split into separate tables
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@kgryte
kgryte committed Aug 25, 2020
commit 46f0bfc1bf1ca91852fdd204d74d4bdf6b6c7d9e
47 changes: 30 additions & 17 deletions spec/API_specification/linear_algebra_functions.md
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Expand Up @@ -133,23 +133,36 @@ Computes the matrix or vector norm of `a`.

- **ord**: _Optional\[ int, float, Literal\[ inf, -inf, 'fro', 'nuc' ] ]_

- order of the norm. The following norms must be supported:

| ord | matrix | vector |
| ------------ | ------------------------------- | -------------------------- |
| None | 'fro' | L2-norm (Euclidean) |
| 'fro' | 'fro' | - |
| 'nuc' | 'nuc' | - |
| 1 | max(sum(abs(x), axis=0)) | L1-norm |
| -1 | min(sum(abs(x), axis=0)) | sum(1./abs(x))**(-1) |
| 2 | largest singular value | L2-norm (Euclidean) |
| -2 | smallest singular value | 1./sqrt(sum(1./abs(x)**2)) |
| inf | max(sum(abs(x), axis=1)) | infinity norm |
| -inf | min(sum(abs(x), axis=1)) | min(abs(x)) |
| (+int,+float) | - 959D | p-norm (for p >= 1) |
| (-int,-float) | - | sum(abs(x)**ord)**(1./ord) |

where `fro` corresponds to the **Frobenius norm**, `nuc` corresponds to the **nuclear norm**, and `-` indicates that the norm is **not** supported. For matrices,
- order of the norm. The following mathematical norms must be supported:

| ord | matrix | vector |
| ---------------- | ------------------------------- | -------------------------- |
| 'fro' | 'fro' | - |
| 'nuc' | 'nuc' | - |
| 1 | max(sum(abs(x), axis=0)) | L1-norm (Manhattan) |
| 2 | largest singular value | L2-norm (Euclidean) |
| inf | max(sum(abs(x), axis=1)) | infinity norm |
| (int,float >= 1) | - | p-norm |

The following non-mathematical "norms" must be supported:

| ord | matrix | vector |
| ---------------- | ------------------------------- | -------------------------- |
| 0 | - | sum(a != 0) |
| -1 | min(sum(abs(x), axis=0)) | sum(1./abs(a))**(-1) |
| -2 | smallest singular value | 1./sqrt(sum(1./abs(a)**2)) |
| -inf | min(sum(abs(x), axis=1)) | min(abs(a)) |
| (int,float < 1) | - | sum(abs(a)**ord)**(1./ord) |

When `ord` is `None`, the following norms must be the default norms:

| ord | matrix | vector |
| ---------------- | ------------------------------- | -------------------------- |
| None | 'fro' | L2-norm (Euclidean) |

where `fro` corresponds to the **Frobenius norm**, `nuc` corresponds to the **nuclear norm**, and `-` indicates that the norm is **not** supported.

For matrices,

- if `ord=1`, the norm corresponds to the induced matrix norm where `p=1` (i.e., the maximum absolute value column sum).
- if `ord=2`, the norm corresponds to the induced matrix norm where `p=inf` (i.e., the maximum absolute value row sum).
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