TEST_MATRIX
Higham's Test Matrices

TEST_MATRIX is a MATLAB library which defines a set of test matrices.

A wide range of matrix dimensions, forms and properties are available. These matrices may be useful in testing an algorithm for correctness on a variety of problems.

Many of the matrices can be rectangular, with the user specifying the number of rows and columns. Almost all the matrices can be made of arbitrary size, with the user specifying the dimension.

Many different matrix zero structures are available, including diagonal, bidiagonal, tridiagonal, pentadiagonal, banded, upper and lower triangular, and Hessenberg.

Many of the matrices come from a MATLAB M file collection developed by Nicholas Higham, Department of Mathematics, University of Manchester, and maintained in the "testmatrix" file somewhere at the MATLAB web site.

An earlier version of the collection is available, again as MATLAB M files, in ACM TOMS Algorithm 694, in the TOMS directory of the NETLIB web site.

Related Data and Programs:

LINPACK, a MATLAB library which provides some linear algebra operations for certain standard storage formats.

LINPLUS, a MATLAB library which provides some simple linear algebra operations for a number of storage formats.

TEST_MAT, a MATLAB library which defines a number of test matrices.

Reference:

1. TS Chow,
   A class of Hessenberg matrices with known eigenvalues and inverses,
   SIAM Review,
   Volume 11, Number 3, July 1969, pages 391-395.
2. Robert Gregory, David Karney,
   A Collection of Matrices for Testing Computational Algorithms,
   Wiley, 1969,
   ISBN: 0882756494,
   LC: QA263 G862.
3. Nicholas Higham,
   Algorithm 694: A Collection of Test Matrices in MATLAB,
   ACM Transactions on Mathematical Software,
   Volume 17, Number 3, September 1991, pages 289-305.
4. Morris Newman, John Todd,
   The evaluation of matrix inversion programs,
   Journal of the Society for Industrial and Applied Mathematics,
   Volume 6, Number 4, 1958, pages 466-476.
5. Andrew Wathen,
   Realistic eigenvalue bounds for the Galerkin mass matrix,
   IMA Journal of Numerical Analysis,
   Volume 7, 1987, pages 449-457.
6. Joan Westlake,
   A Handbook of Numerical Matrix Inversion and Solution of Linear Equations,
   John Wiley, 1968.
7. James Wilkinson,
   Rounding Errors in Algebraic Processes,
   Prentice Hall, 1963,
   ISBN: 0-486-67999-3.
8. James Wilkinson,
   The Algebraic Eigenvalue Problem,
   Oxford University Press, 1988,
   ISBN: 0198534183.
9. James Wilkinson, Christian Reinsch,
   Handbook for Automatic Computation,
   Volume II, Linear Algebra, Part 2,
   Springer, 1971,
   ISBN: 0387054146.

Source Code:

• adsmax.m, alternating directions direct search method.
• augment.m, augmented system matrix.
• bandred.m, band reduction by two-sided unitary transformation.
• cauchy.m, the Cauchy matrix.
• cgs.m, classical Gram-Scmdidt QR orthogonalization.
• chebspec.m, Chebyshev spectral differentiation matrix.
• chebvand.m, Vandermonde-like matrix for the Chebyshev polynomials.
• cholp.m, Cholesky factorization with pivoting;
• chop.m, round matrix elements.
• chow.m, the Chow matrix.
• circul.m, a circulant matrix.
• clement.m, the Clement matrix.
• cod.m, complete orthogonal decomposition.
• comp.m, comparison matrices.
• compan.m, companion matrix.
• cond.m, compute matrix condition number.
• condex.m, Linpack condition number estimate counterexamples.
• contents.m, contents.
• cpltaxes.m, determines suitable axis for plot of complex vector.
• cycol.m, matrix with cyclically repeating columns.
• diagpiv.m, diagonal pivoting factorization with partial pivoting.
• dingdong.m, the Dingdong matrix.
• dorr.m, the Dorr matrix.
• dramadah.m, the Dramadah matrix (inverse Hadamard).
• dual.m, dual vector in Hoelder P-norm.
• eigsens.m, eigenvalue condition numbers.
• fdemo.m, demonstration function for direct search maximizers.
• fiedler.m, the Fiedler matrix.
• forsythe.m, the Forsythe matrix.
• frank.m, the Frank matrix.
• fv.m, field of values.
• gallery.m, a gallery of test matrices.
• ge.m, Gaussian elimination without pivoting.
• gearm.m, the Gear matrix.
• gecp.m, Gaussian elimination with complete pivoting.
• gersh.m, Gershgorin eigenvalue estimates.
• gfpp.m, matrix with maximal pivot growth factor for Gaussian elimination.
• gj.m, Gauss-Jordan elimination.
• grcar.m, the Grcar matrix.
• hadamard.m, the Hadamard matrix.
• hilb.m, the Hilbert matrix.
• house.m, the Householder matrix.
• invhess.m, inverse of an upper Hessenberg matrix.
• invol.m, an involutory matrix.
• ipjfact.m, a Hankel matrix with factorial elements.
• jordbloc.m, Jordan block matrix.
• kahan.m, the Kahan matrix.
• kms.m, the KMS matrix.
• krylov.m, a Krylov matrix.
• lauchli.m, the Lauchli matrix.
• lehmer.m, the Lehmer matrix.
• lesp.m, the Lesp matrix.
• lotkin.m, the Lotkin matrix.
• makejcf.m, a matrix with given Jordan canonical form.
• matrix.m, access matrix toolbox matrices by number.
• matsignt.m, matrix sign function for a triangular matrix.
• mdsmax.m, multidirectional search method for direct search optimization.
• mgs.m, modified Gram-Schmidt QR orthogonalization.
• minij.m, the min(i,j) matrix.
• moler.m, the Moler matrix.
• neumann.m, the Neumann matrix.
• nmsmax.m, the Nelder-Mead simplex method for direct search optimization.
• ohess.m, random orthogonal upper Hessenberg matrix.
• orthog.m, some orthogonal and nearly orthogonal matrices.
• parter.m, the Parter matrix.
• pascal.m, the Pascal matrix.
• pdtoep.m, a positive definite Toeplitz matrix.
• pei.m, the Pei matrix.
• pentoep.m, a pentadiagonal Toeplitz matrix.
• pnorm.m, estimate of matrix P-norm.
• poisson.m, the Poisson matrix.
• poldec.m, polar decomposition of a matrix.
• prolate.m, the prolate matrix.
• ps.m, dot plot of a pseudospectrum.
• pscont.m, contour plots of pseudospectrum.
• qmult.m, premultiply by a random orthogonal matrix.
• rando.m, random matrix with elements of -1, 0 and 1.
• randsvd.m, random matrix with given singular values.
• redheff.m, the Redheffer matrix.
• riemann.m, the Riemann matrix.
• rq.m, the Rayleigh quotient.
• rschur.m, an upper quasi-triangular matrix.
• see.m, pictures of a matrix and its pseudoinverse.
• seqa.m, additive sequence.
• seqcheb.m, sequence of points related to Chebyshev polynomials.
• seqm.m, multiplicative sequence.
• show.m, display signs of matrix elements.
• signm.m, matrix sign decomposition.
• skewpart.m, skew-symmetric part of a matrix.
• smoke.m, the Smoke matrix.
• sparsify.m, randomly set matrix elements to 0.
• sub.m, principal submatrix.
• symmpart.m, symmetric part of matrix.
• timestamp.m, prints the current HMSDMY date as a timestamp.
• tmtdemo.m, demonstration of test matrix toolbox.
• trap2tri.m, unitary reduction of trapezoidal matrix to triangular form.
• tridiag_dense.m, tridiagonal matrix, dense form.
• tridiag_sparse.m, tridiagonal matrix, sparse form.
• triw.m, upper triangular Wilkinson matrix.
• vand.m, Vandermonde matrix.
• wathen.m, the Wathen matrix.
• wilk.m, the Wilkinson matrix.

Examples and Tests:

• test_matrix_test.m, calls all the tests.
• test_matrix_test.out, the output from all the tests.
• tridiag_dense_test.m, runs a test of TRIDIAG_DENSE.
• tridiag_sparse_test.m, runs a test of TRIDIAG_SPARSE.

You can go up one level to the MATLAB source codes.