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Answering my own question in case someone else ran into the same issue. A should be a 1D array of size N * (N + 1) / 2. Also, the value each element in array A must be initialize to zero. Otherwise, the result will be wrong. Read the description for the cblas_chpr() function for why this is the case.

The input matrix A is 5x5 but it has rank 2. The first two columns are linearly independent, while the last three columns of A are linear combinations of the first two.

For a QR factorization, this means that there is not unicity for the last three columns of Q. An QR factorization implementation (e.g., LAPACK, numpy, etc. ) can return any three columns that are (1) mutually orthonormal and (2) in A^(perp), and that is a correct answer. There are many correct answers! The solution is not unique.

If you want to check the Q and R returned by LAPACK, (or any QR factorization implementation for the matter,) you can (1) compute Q'*Q and check that you get the 5x5 identity matrix, (please use BLAS HERK function to do so) and (2) compute Q'*A and check that you have the 5x5 upper triangular matrix R (as returned by QR). In your case you should see that the last 3 rows of R are all zeros, which indicates that the last three columns of A are linear combinations of the first two. To compute Q'*A, you can use BLAS GEMM.

I took your input A and your output Q from LAPACK and I checked for you that Q'*Q is identity and Q'*R is upper triangular with last three rows being fully zeros. So the Q output from LAPACK looks good to me. Yes, this Q is different from what is returned by another implementation and this is entirely possible.

In general we check the quality of a QR factorization by checking that: (1) || A - QR || / || A || is small, (2) || I - Q^T Q || is small, and R is upper triangular. (Take any norm that is easy to compute.)

Checking that two codes return the same output is not a good idea. Since there is not unicity of the output Q and R for two reasons: (1) when A is rank deficient, see the example that you give for example; (2) for any j, any column scaling with a complex number of modulus 1 for the j-th column of Q is possible as long as you scale the j-th row of R accordingly. This changes the Q and R factors but there are still valid Q and R factors.

In addition, assume that you force all diagonal elements of R to be real nonnegative, (by rescaling columns of Q and columns of R appropriately,) and that A is full rank, then you would expect unicity of Q and R. However checking that the pair (Q,R) is close to another one is related to the condition number of computing Q and R for a QR factorization and so you can see pairs that are far apart (forward error), even though they have a good backward error quality.

Reconsider your third-party dependency version-selection.

FROM python:3.8.3-buster

is Python 3.8

scipy 1.3.3 (released 2019!!!) doesn't support Python 3.8

down to the very bottom of the stack error:

it seems your pip aversion in mac is too old, you need to upgrade pip first

As others have stated, Python 3.10 is not currently compatible with Matplotlib. You need to install and use Python 3.9 until it is supported. Until then you have a few options:

You can use the Python Launcher for Windows (py.exe) script to choose which Python version to run like so:

py.exe script help: py -h

List all installed versions py -0

Use a specified version py -3.9

e.g. 1 Create a virtual environment using python 3.9 py -3.9 venv .venv

e.g. 2 install matplotlib with pip using python 3.9 py -3.9 -m pip install matplotlib

On Linux you can run whatever Python version you like like so:

python3.9

You can set up a local (your working directory and all sub-directories) virtual environment that will use your specified version like so:

python3.9 -m venv .venv

Which will set the version of python used to 3.9 while in the local directory, and allow you to type python instead of python3.9 each time you need it.

Another relevant and helpful post by Rotareti here.

Please note that I have not described how to activate and use Python Virtual Environments here in detail, for more information on using them read the python docs.

Reference this answer if you're interested in installing a matplotlib release candidate.

In scipy's PyPI page, it looks like scipy doesn't support 3.10 as the meta says

Not totally sure, but I ran into the same issue. After some digging and testing I think I've found out that Matplotlib, and pylab aren't compatible with python3.10 yet. For the time being I've kept python3.9 along with 3.10 so whenever I use matplotlib, numpy, or pylab, I just use version 3.9.