| Name | pygensvd JSON |
| Version |
0.1.0
JSON |
| download |
| home_page | |
| Summary | Python and NumPy extension module implementing the generalized signular value decomposition (GSVD). |
| upload_time | 2023-06-27 02:31:49 |
| maintainer | |
| docs_url | None |
| author | |
| requires_python | >=3.7 |
| license | GNU License |
| keywords |
gsvd
|
| VCS |
 |
| bugtrack_url |
|
| requirements |
numpy
|
| Travis-CI |
No Travis.
|
| coveralls test coverage |
No coveralls.
|
# `pygensvd.py`
An updated version of the original pygsvd package by Benjamin Naecker. This version
uses cmake for compilation rather than distutils, but only supports openblas as a backend (for now).
A Python wrapper to the LAPACK generalized singular value decomposition.
(C) 2017 Benjamin Naecker bnaecker@fastmail.com
## Overview
The `pygsvd` module exports a single function `gsvd`, which computes the
generalized singular value decomposition (GSVD) of a pair of matrices,
`A` and `B`. The [GSVD](https://en.wikipedia.org/wiki/Generalized_singular_value_decomposition)
is a joint decomposition useful for computing regularized solutions
to ill-posed least-squares problems, as well as dimensionality reduction
and clustering.
The `pygsvd` module is very simple: it just wraps the underlying LAPACK
routine `ggsvd3`, both the double-precision (`dggsvd3`) and complex-double
precision versions (`zggsvd3`).
## Building
Because the `pygsvd` module wraps a LAPACK routine itself, it is provided
as a Python and NumPy extension module. The module must be compiled,
and doing so requires a LAPACK header and a shared library. The module
currently supports both the standard C bindings to LAPACK (called
[LAPACKE](http://www.netlib.org/lapack/lapacke.html)),
and those provided by Intel's Math Kernel Library. Notably it does *not*
support Apple's Accelerate framework, which seems to be outdated and
differs in several subtle and annoying ways.
You can build against either of the supported implementations, by editing
the `setup.cfg` file. Set the `define=` line in the file to be one of
`USE_LAPACK` (the default) or `USE_MKL`.
You must also add the include and library directories for these. The
build process already searches `/usr/local/{include,lib}`, but if these
don't contain the header and library, add the directory containing these
to the `include_dirs=` and `library_dirs=` line. Multiple directories are
separated by a `:`. You can also set these on the command line when building.
For example, to use the LAPACK library, with a header in `/some/dir/`
and the library in `/some/libdir/`, you could run:
$ python3 setup.py build_ext --include-dirs="/some/dir" --library-dirs="/some/libdir"
Then you can install the module either as usual or in develop mode as:
$ python3 setup.py {install,develop}
Or via `pip` as:
$ pip3 install .
## Usage
The GSVD of a pair of NumPy ndarrays `a` and `b` can be computed as:
>>> c, s, x = pygsvd.gsvd(a, b)
This returns the generalized singular values, in arrays `c` and `s`, and the
right generalized singular vectors in `x`. Optionally, the transformation matrices
`u` and` `v` may also be computed. E.g.:
>>> c, s, x, u = pygsvd.gsvd(a, b, extras='u')
also returns the left generalized singular vectors of `a`.
By default, the matrices `u` and `v`, if returned, are of shape `(m, n)` and
`(p, n)`. Using the optional argument `full_matrices` is set to `True`, then
the matrices are square, of shape `(m, m)` and `(p, p)`.
## The generalized singular value decomposition
The GSVD is a joint decomposition of a pair of matrices. Given matrices
`A` with shape `(m, n)` and `B` with shape `(p, n)`, it computes:
A = U*C*X.T
B = V*S*X.T
where `U` and `V` are unitary matrices, with shapes `(m, m)` and `(p, p)`,
and `X` is shaped as `(n, n)`, respectively. `C` and `S` are diagonal (possibly non-square)
matrices containing the generalized singular value pairs.
This decomposition has many uses, including least-squares fitting of ill-posed
problems. For example, letting `B` be the "second derivative" operator one can
solve the equation
min_x ||Ax - b||^2 + \lambda ||Bx||^2
using the GSVD, which achieves a smoother solution as `\lambda` is increased.
Similarly, setting `B` to the identity matrix, this becomes the standard
ridge regression problem. These are both versions of the Tichonov regularization
problem, for which the GSVD provides a useful and efficient solution.
Raw data
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