| Title: | Generate test matrices |
|---|---|
| Description: | Functions for generating various test matrices. Inspired by the MATLAB gallery of test matrices. |
| Authors: | Thomas Hsiao [aut, cre] |
| Maintainer: | Thomas Hsiao <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 1.0.0 |
| Built: | 2025-02-24 05:03:39 UTC |
| Source: | https://github.com/txiao95/gallery |
Binomial matrix: an N-by-N multiple of an involuntory 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)
binomial_matrix(n)binomial_matrix(n)
n |
- row dimension |
Arguments x and y are vectors of length n.
C[i,j] = 1 / (x[i] + y[j])
cauchy_matrix(x, y = NULL)cauchy_matrix(x, y = NULL)
x |
vector of length n |
y |
vector of length n |
Chebyshev spectral differentiation matrix of order n. k determines
the character of the output matrix. For either form, the eigenvector matrix is ill-conditioned.
chebspec(n, k = NULL)chebspec(n, k = NULL)
n |
order of the matrix. |
k |
|
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
chebvand(p, m = NULL)chebvand(p, m = NULL)
p |
points to evaluate. If a scalar, then |
m |
number of rows of the matrix. |
returns matrix A = H(alpha) + delta * EYE, such that
H[i,j] = alpha^(i-j+1).
chow(n, alpha = 1, delta = 0)chow(n, alpha = 1, delta = 0)
n |
order of the matrix |
alpha |
defaults to 1 |
delta |
defaults to 0 |
Each row is obtained from the previous by cyclically permuting the entries one step forward. A special Toeplitz matrix in which diagonals "wrap around"
circul(v)circul(v)
v |
first row of the matrix. If |
a circulant matrix whose first row is the vector v
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
clement(n, k = 0)clement(n, k = 0)
n |
order of matrix |
k |
0 indicates symmetric matrix, 1 asymmetric |
A
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.
compar(B, k = 0)compar(B, k = 0)
B |
input matrix |
k |
decides what matrix to return |
A whose columns repeat cyclicallyReturns 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.
cycol(n, k, m = NULL)cycol(n, k, m = NULL)
n |
number of columns of matrix |
k |
upper limit of rank |
m |
number of rows of matrix |
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.
dorr(n, theta = 0.01)dorr(n, theta = 0.01)
n |
order of matrix |
theta |
determines conditionality. Ill-conditioned when theta is nonnegative. |
A
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.
dramadah(n, k = 1)dramadah(n, k = 1)
n |
order of matrix |
k |
decides type of matrix returned. |
Fiedler matrix that has a dominant positive eigenvalue and all others are negative
fiedler(c)fiedler(c)
c |
N-vector. If |
a symmetric dense matrix A with a dominant positive eigenvalue and all others are negative.
Returns a n-by-n matrix equal to the Jordan block with
eigenvalue lambda, except that A[n,1]=alpha.
forsythe(n, alpha = .Machine$double.eps, lambda = 0)forsythe(n, alpha = .Machine$double.eps, lambda = 0)
n |
order of matrix |
alpha |
value of perturbation at |
lambda |
eigenvalue of Jordan block |
N
Frank matrix of order N. It is upper Hessenberg with
determinant 1.
frank(n, k = 0)frank(n, k = 0)
n |
order of the matrix |
k |
If k is 1, the elements are reflected about the anti-diagonal. |
Eigenvalues are sensitive.
grcar(n, k = NULL)grcar(n, k = NULL)
n |
dimension of the square matrix |
k |
number of superdiagonals of ones |
n-by-n Toeplitz matrix with -1 on subdiagonal, 1 on diagonal, and k superdiagionals of 1s.
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)]
hanowa(n, d = NULL)hanowa(n, d = NULL)
n |
order of matrix |
d |
value of main diagonal |
a n-byn involutory matrix and ill-conditioned.
It is a diagonally scaled version of a Hilbert matrix.
invol(n)invol(n)
n |
order of matrix |
Returns a n-by-n JOrdan block with eigenvalue
lambda. The default is 1.
jordbloc(n, lambda = 1)jordbloc(n, lambda = 1)
n |
order of matrix |
lambda |
eigenvalue of Jordan block |
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)
lauchli(n, mu = NULL, sparse = F)lauchli(n, mu = NULL, sparse = F)
n |
number of columns |
mu |
constant applied to identity |
sparse |
whether matrix should be sparse |
Lauchli matrix.
the symmetric positive-definite matrix such that A[i,j] = i/j, for j >= i
lehmer(n)lehmer(n)
n |
order of matrix |
N by N matrix from Leslie population model with average birth and survival rates.
leslie(a, b = NULL, sparse = F)leslie(a, b = NULL, sparse = F)
a |
average birth numbers (first row) |
b |
survival rates (subdiagonal) |
sparse |
whether to return a sparse matrix |
N by N Leslie population model matrix
MIN(i,j)
The N-by-N SPD matrix with A[i,j]=min(i,j)
minij(n)minij(n)
n |
order of the matrix |
Creates a sparse representation of multiple diagonal matrix
spdiags(A, d, m, n)spdiags(A, d, m, n)
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 |
dgcMatrix sparse diagonal
Create a sparse tridiagonal matrix of dgcMatrix class.
tridiag(n, x = NULL, y = NULL, z = NULL)tridiag(n, x = NULL, y = NULL, z = NULL)
n |
dimension of the square matrix |
x |
subdiagonal (-1) |
y |
diagonal (0) |
z |
superdiagonal (+1) |
Sparse tridiagonal matrix