DOC: add missing details to linalg.lstsq docstring #18062
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numpy/linalg/linalg.py
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| ``a @ x = b``, where `a` is an `m`-by-`n` matrix and `b` is a `m` element | ||
| vector. When the system is overdetermined (``m >= n`` and ``rank(a) = n``), | ||
| a solution `x` that minimizes the Euclidean 2-norm ``||b-ax||`` is found. | ||
| Else, when the system is underdetermined (``m < n`` and ``rank(A) = m``) |
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Maybe "If there are multiple minimizing solutions, the one with the smallest norm is returned". Note that the norm of the solution depends on column scaling (units) and choice of variables, so the smallest norm solution is seldom meaningful when the solution is not unique.
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Sounds better - I've updated my commit to reflect the suggestion, thanks!
The reason I stumbled on this smallest norm matter was that I was specifically looking for that solution in a rank-deficient problem. I was intrigued I got exactly what I wanted for free, so I did some digging and found why :)
It turns out that lstsq also minimizes the 2-norm of x when a is rank-deficient. I found that by searching the documentation for the LAPACK library, which is the current implementation of lstsq (as of Dec 2020). Ref: https://www.netlib.org/lapack/lug/node27.html
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numpy/linalg/linalg.py
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| ``a @ x = b``. The equation may be under-, well-, or over-determined | ||
| (i.e., the number of linearly independent rows of `a` can be less than, | ||
| equal to, or greater than its number of linearly independent columns). | ||
| If `a` is square and of full rank, then `x` (but for round-off error) | ||
| is the "exact" solution of the equation. Else, `x` minimizes the | ||
| Euclidean 2-norm :math:`|| b - a x ||`. | ||
| Euclidean 2-norm :math:`||b-ax||`. If there are multiple minimizing |
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Keep spaces around - for readability.
[ci skip]
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Thanks @krnilo . |
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Worth pointing out that this is the behavior of the pseudoinverse? |
It turns out that
linalg.lstsqalso minimizes the 2-norm ofxwhenais rank-deficient. I found that by searching the documentation for the LAPACK library, which is the current implementation oflstsq.This PR rewrites the docstring for
linalg.lstsqsuch that it contains details for all cases.