| Type: | Package |
| Title: | Generate Test Matrices for Numerical Experiments |
| Version: | 1.0.0 |
| Maintainer: | Thomas Hsiao <thomas.hsiao@emory.edu> |
| Description: | Generates a variety of structured test matrices commonly used in numerical linear algebra and computational experiments. Includes well-known matrices for benchmarking and testing the performance, stability, and accuracy of linear algebra algorithms. Inspired by 'MATLAB' 'gallery' functions. |
| License: | MIT + file LICENSE |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.3.1 |
| Imports: | Matrix, pracma (≥ 2.2.5) |
| Suggests: | testthat (≥ 2.1.0), knitr, rmarkdown |
| NeedsCompilation: | no |
| Packaged: | 2024-09-25 23:29:40 UTC; thsiao3 |
| Author: | Thomas Hsiao 
[aut, cre] |
| Repository: | CRAN |
| Date/Publication: | 2024-09-26 13:40:02 UTC |
Create binomial matrix
Description
Binomial matrix: an N-by-N multiple of an involutory matrix with integer entries such that $A^2 = 2^(N-1)*I_N$ Thus B = A*2^((1-N)/2) is involutory, that is B^2 = EYE(N)
Usage
binomial_matrix(n)
Arguments
n |
row dimension |
Value
a binomial matrix, which is a multiple of involutory matrix
Create Cauchy matrix
Description
Arguments x and y are vectors of length n.
C[i,j] = 1 / (x[i] + y[j])
Usage
cauchy_matrix(x, y = NULL)
Arguments
x |
vector of length n |
y |
vector of length n |
Value
a Cauchy matrix
Create Chebyshev spectral differentiation matrix
Description
Chebyshev spectral differentiation matrix of order n. k determines
the character of the output matrix. For either form, the eigenvector matrix is ill-conditioned.
Usage
chebspec(n, k = NULL)
Arguments
n |
order of the matrix. |
k |
|
Value
Chebyshev spectral differentiation matrix
Creating Vandermonde-like matrix for the Chebyshev polynomials
Description
Produces the (primal) Chebyshev Vandermonde matrix based on
the points p. C[i,j] = T_{i-1}p[j], where T_{i-1} is the Chebyshev
polynomial of degree i-1
Usage
chebvand(p, m = NULL)
Arguments
p |
points to evaluate. If a scalar, then |
m |
number of rows of the matrix. |
Value
Vandermonde-like matrix for the Chebyshev polynomials
Creating singular Toeplitz lower Hessenberg matrix
Description
returns matrix A = H(alpha) + delta * EYE, such that
H[i,j] = alpha^(i-j+1).
Usage
chow(n, alpha = 1, delta = 0)
Arguments
n |
order of the matrix |
alpha |
defaults to 1 |
delta |
defaults to 0 |
Value
Singular Toeplitz lower Hessenberg matrix
Create circulant matrix
Description
Each row is obtained from the previous by cyclically permuting the entries one step forward. A special Toeplitz matrix in which diagonals "wrap around"
Usage
circul(v)
Arguments
v |
first row of the matrix. If |
Value
a circulant matrix whose first row is the vector v
Create Clement tridiagonal matrix with zero diagonal entries
Description
Returns an n-by-n tridiagonal matrix with zeros on the
main diagonal. For k=0, A is nonsymmetric. For k=1, A is
symmetric
Usage
clement(n, k = 0)
Arguments
n |
order of matrix |
k |
0 indicates symmetric matrix, 1 asymmetric |
Value
Clement tridiagonal matrix with zero diagonal entries
Create comparison matrix A
Description
For k=0, if i==j, $A[i,j]=abs(B[i,j])$ and
A[i,j]=-abs(B[i,j]) otherwise. For k=1, A replaces each
diagonal element of B with its absolute value, and replaces each
off-diagonal with the negative of the largest absolute value off-diagonal
in the same row.
Usage
compar(B, k = 0)
Arguments
B |
input matrix |
k |
decides what matrix to return |
Value
Comparison matrix
Create matrix A whose columns repeat cyclically
Description
Returns an n-by-n matrix with cyclically repeating columns
where one cycle consists of the columns defined by randn(n,k). Thus, the
rank of matrix A cannot exceed k, and k must be scalar.
Usage
cycol(n, k, m = NULL)
Arguments
n |
number of columns of matrix |
k |
upper limit of rank |
m |
number of rows of matrix |
Value
Matrix whose columns repeat cyclically
Create Dorr matrix
Description
Returns a n-by-n row diagonally dominant,
tridiagonal matrix that is ill-conditioned for small nonnegative values
of theta. The default value of theta is 0.01.
Usage
dorr(n, theta = 0.01)
Arguments
n |
order of matrix |
theta |
determines conditionality. Ill-conditioned when theta is nonnegative. |
Value
Dorr matrix of class 'dgcMatrix'.
Create anti-Hadamard matrix A
Description
Returns a n-by-n nonsingular matrix of 0's and 1's.
With large determinant or inverse. If k=1, A is Toeplitz and abs(det(A))=1.
If k=2, A is upper triangular and Toeplitz. If k=3, A has maximal
determinant among (0,1) lower Hessenberg matrices. Also is Toeplitz.
Also known as an anti-Hadamard matrix.
Usage
dramadah(n, k = 1)
Arguments
n |
order of matrix |
k |
decides type of matrix returned. |
Value
Matrix of zeros and ones
Create Fiedler matrix
Description
Fiedler matrix that has a dominant positive eigenvalue and all others are negative
Usage
fiedler(c)
Arguments
c |
N-vector. If |
Value
a symmetric dense matrix A with a dominant positive eigenvalue and all others are negative.
Create Forsythe matrix or perturbed Jordan block
Description
Returns a n-by-n matrix equal to the Jordan block with
eigenvalue lambda, except that A[n,1]=alpha.
Usage
forsythe(n, alpha = .Machine$double.eps, lambda = 0)
Arguments
n |
order of matrix |
alpha |
value of perturbation at |
lambda |
eigenvalue of Jordan block |
Value
Forsythe matrix or perturbed Jordan block
Frank matrix of order N
Description
Frank matrix of order N. It is upper Hessenberg with
determinant 1.
Usage
frank(n, k = 0)
Arguments
n |
order of the matrix |
k |
If k is 1, the elements are reflected about the anti-diagonal. |
Value
Frank matrix with ill-conditioned eigenvalues.
Create Toeplitz matrix with sensitive eigenvalues
Description
Eigenvalues are sensitive.
Usage
grcar(n, k = NULL)
Arguments
n |
dimension of the square matrix |
k |
number of superdiagonals of ones |
Value
n-by-n Toeplitz matrix with -1 on subdiagonal, 1 on diagonal, and k superdiagionals of 1s.
Hanowa matrix
Description
Matrix whose eigenvalues lie on vertical plane in complex plane.
Returns a 2-by-2 block matrix with four n/2 by n/2 blocks. n
must be an even integer.
[d*eye(m) -diag(1:m),
diag(1:m) d*eye(m)]
Usage
hanowa(n, d = NULL)
Arguments
n |
order of matrix |
d |
value of main diagonal |
Value
Matrix whose eigenvalues lie on a vertical line in the complex plane.
Involutory matrix (a matrix that is its own inverse)
Description
a n-byn involutory matrix and ill-conditioned.
It is a diagonally scaled version of a Hilbert matrix.
Usage
invol(n)
Arguments
n |
order of matrix |
Value
Involutory matrix (a matrix that is its own inverse).
Create Jordan block matrix
Description
Returns a n-by-n Jordan block with eigenvalue
lambda. The default is 1.
Usage
jordbloc(n, lambda = 1)
Arguments
n |
order of matrix |
lambda |
eigenvalue of Jordan block |
Value
Jordan block matrix
Create Lauchli Matrix
Description
the (N + 1) x (N) matrix [ones(1,n); mu*eye(n)]. Well-known example in least squares of the danger of forming t(A)
Usage
lauchli(n, mu = NULL, sparse = FALSE)
Arguments
n |
number of columns |
mu |
constant applied to identity |
sparse |
whether matrix should be sparse |
Value
Lauchli rectangular matrix.
Create Lehmer matrix
Description
the symmetric positive-definite matrix such that A[i,j] = i/j, for j >= i
Usage
lehmer(n)
Arguments
n |
order of matrix |
Value
Lehmer symmetric positive definite matrix.
Create Leslie population model matrix
Description
N by N matrix from Leslie population model with average birth and survival rates.
Usage
leslie(a, b = NULL, sparse = FALSE)
Arguments
a |
average birth numbers (first row) |
b |
survival rates (subdiagonal) |
sparse |
whether to return a sparse matrix |
Value
N by N Leslie population model matrix
Symmetric positive definite matrix MIN(i,j)
Description
The N-by-N SPD matrix with A[i,j]=min(i,j)
Usage
minij(n)
Arguments
n |
order of the matrix |
Value
Symmetric positive definite matrix with entries A[i,j]=min(i,j)
Create sparse diagonal matrix
Description
Creates a sparse representation of multiple diagonal matrix
Usage
spdiags(A, d, m, n)
Arguments
A |
matrix where columns correspond to the desired diagonals |
d |
indices of the diagonals to be filled in. 0 is main diagonal. -1 is first subdiagonal and +1 is first superdiagonal. |
m |
row dim |
n |
col dim |
Value
sparse diagonal matrix of class 'dgcMatrix'
Create sparse tridiagonal matrix
Description
Create a sparse tridiagonal matrix of dgcMatrix class.
Usage
tridiag(n, x = NULL, y = NULL, z = NULL)
Arguments
n |
dimension of the square matrix |
x |
subdiagonal (-1) |
y |
diagonal (0) |
z |
superdiagonal (+1) |
Value
Sparse tridiagonal matrix of class 'dgcMatrix'