These matrices are a collection of interesting matrices that appear in matlab's 'gallery' collection. This collection was originally defined and implemented by Prof. Nicholas Higham of Manchester University and is more fully discussed in "The Test Matrix Toolbox for Matlab (Version 3.0)", N.J. Higham, Numerical Analysis Report No. 276, September 1995 and available here (Now appears to be a dead link)
The pdf of the 1991 TOMS (Algorithm 694) is available from ACM here
By 'interesting' we mean that these matrices either present some challenges to numerical algorithms or have some a set of interesting properties. The documentation of the individual functions contains much more info, as well as references.
Also included are a set of matrix utility functions that are needed for generating some of members of the collection as well as a few functions from Prof. Higham’s matrixcomp package. One of the more interesting routines here is mdsmax, a direct search optimization algorithm.
The rogues package depends on numpy and scipy, both of which must be installed. Additionally, there are a few routines that deal with plotting, and these use matplotlib. While ipython is not strictly necessary, it is a very convenient environment for numpy / scipy / matplotlib. Finally, the unit tests now utilize pytest, and that must be installed in your environment. Then just make certain you are in the rogues top level directory and run pytest, ie
cd rogues
pytest
The included matrix generation functions are:
- cauchy Cauchy matrix
- chebspec Chebyshev spectral differentiation matrix
- chebvand Vandermonde-like matrix for the Chebyshev polynomials
- chow Chow matrix - a singular Toeplitz lower Hessenberg matrix
- clement Clement matrix - tridiagonal with zero diagonal entries
- comp Comparison matrices
- compan Companion matrix
- condex Counterexamples to matrix condition number estimators
- cycol Matrix whose columns repeat cyclically
- dingdong Dingdong matrix - a symmetric Hankel matrix
- dorr Dorr matrix - diagonally dominant, ill conditioned, tridiagonal.
- dramadah A (0,1) matrix whose inverse has large integer entries
- fiedler Fiedler matrix - symmetric
- forsythe Forsythe matrix - a perturbed Jordan block
- frank Frank matrix - ill conditioned eigenvalues.
- gearm Gear matrix
- gfpp Matrix giving maximal growth factor for GW with partial pivoting
- grcar Grcar matrix - a Toeplitz matrix with sensitive eigenvalues.
- hadamard Hadamard matrix
- hankel Hankel matrix
- hanowa A matrix whose eigenvalues lie on a vertical line in C
- hilb Hilbert matrix
- invhess Inverse of an upper Hessenberg matrix
- invol An involutory matrix
- ipjfact A Hankel matrix with factorial elements
- jordbloc Jordan block matrix
- kahan Kahan matrix - upper trapezoidal
- kms Kar-Murdock-Szego Toeplitz matrix
- krylov Krylov matrix
- lauchli Lauchli matrix - rectangular
- lehmer Lehmer matrix - symmetric positive definite
- lesp A tridiagonal matrix with real, sensitve eigenvalues
- lotkin Lotkin matrix
- minij Symmetric positive definite matrix min(i,j)
- moler Moler matrix symmetric positive definite
- neumann Singular matrix from the descrete Neumann problem (sparse)
- ohess Random, orthogonal upper Hessenberg matrix
- parter Parter matrix - a Toeplitz matrix with singular values near pi
- pascal Pascal matrix
- pdtoep Symmetric positive definite Toeplitz matrix
- pei Pei matrix
- pentoep Tentadiagonal Toeplitz matrix (sparse)
- poisson Block tridiagonal matrix from Poisson’s equation (sparse)
- prolate Prolate matrix - symmetric, ill-conditioned Toeplitz matrix
- qmult Pre-multiply by random orthogonal matrix
- rando Random matrix with elements -1, 0, or 1
- randsvd Random matrix with pre-assigned singular values
- redheff A (0,1) matrix of Redheffer associated with the Riemann hypothesis
- riemann A matrix associated with the Riemann hypothesis
- smoke Smoke matrix - complex, with a ‘smoke ring’ pseudospectrum
- triw Upper triangular matrix discussed by Wilkinson and others
- wathen Wathen matrix - a finite element matrix (sparse, random entries)
- wilk Various specific matrices devised /discussed by Wilkenson
- wilkinson Wilkinson matrix of size n, where n must be odd
Some of generally useful matrix utility functions:
- augment Agumented system matrix
- bandred Band reduction by two-sided unitary transformations
- cgs Classical Gram-Schmidt QR factorization
- cond Matrix condition number in 1,2,Frobenius, or infinity norm
- condeig Condition numbers for eigenvalues of a matrix
- cpltaxes Determine suitable axis for plot of complex vector
- dual Dual vector with respect to Holder p-norm
- fv Evaluate and plot the field of values largest leading submatrices
- ge Gaussian elimination without pivoting
- gersh Plots Gershgorin disks for a square matrix
- hankel Given first row, returns a Toeplitz type matrix
- house Householder matrix
- mdsmax Multidimensional search method for direct search optimization
- mgs Modified Gram-Schmidt QR factorization
- pow2 Vector whose i-th element is 2 ** x[i], where x[] is input
- ps Dot plot of a pseudospectrum
- pscont Plots contours and color plots of pseudospectra
- repmat Simple re-implementation of matlab's repmat function
- rq Rayleigh quotient
- skewpart Skew-symmetric (skew-Hermitian) part
- sparsify Randomly sets matrix elements to zero
- sub Principal submatrix
- symmpart Symmetric (Hermitian) part
- toeplitz Returns toeplitz matrix given first row of the matrix
- treshape Reshape vector to or from (unit) triangular matrix
- tridiag Sparse tridiagonl matrix given the diagonals
- vand Vandermonde matrix
- vecperm Vector permutation matrix
More information is available on any of these functions by typing help
Don't use the distribute_setuptools.py stuff for python 2.7.15
Fixed type errors in cycol.py, dorr.py, hanowa.py, ohess.py, and treshape.py that have
developed with newer versions of Python.
Tested on Ubuntu 18.04.1 with Python 2.7.15 & Numpy 1.14.3; Python 3.6.5 & Numpy 1.14.3;
and Python 3.7.0 and Numpy 1.15.0
Added visualization routines fv, gersh, and pscont. Fixed issues with np.r_.
Relaxed tolerances for two problematic unit tests. Only use distribute with
Python 2 series. Cleaned up pep8 warnings.
Tested on Ubuntu 14.04 with Python 3.5.1, Numpy 1.10.2, Scipy 0.16.1, IPython 4.0.1, Matplotlib 1.5.0
Tested on Ubuntu 14.04 with Python 3.4.3, Numpy 1.9.3, Scipy 0.16.0, IPython 4.0.0, Matplotlib 1.4.3
Tested on Ubuntu 14.04 with Python 2.7.10, Numpy 1.9.2, Scipy 0.15.1, IPython 3.2.1, Matplotlib 1.3.1
Tested on Windows 10 with Python 3.5.1, Numpy 1.10.1, Scipy 0.16.0, IPython 4.0.0, Matplotlib 1.4.3
Simplified tests and moved to pytest
Removed deprecation warnings
Tested on Ubuntu 24.04, Python 3.8.19 with Numpy 1.24.4, and Python 3.12.5 with Numpy 2.1.0
Ported to Python 3. Added distribute_setup.py to fix installation problems.
Tested on Ubuntu 12.04 with Python 2.7.3, Numpy 1.6.1, Scipy 0.10.0, IPython 0.12
Tested on Ubuntu 12.04 with Python 3.2.3, Numpy 1.6.2, Scipy 0.11.0, IPython 0.13.1
Tested on Windows 7 with Python 2.7.3, Numpy 1.6.2, Scipy 0.11.0, IPython 0.13.1
Unit tests now included with distribution.