The following functions return famous matrix forms.
Create interesting matrices for testing.
Create a Cauchy matrix.
Create a Chebyshev spectral differentiation matrix.
Create a Vandermonde-like matrix for the Chebyshev polynomials.
Create a Chow matrix – a singular Toeplitz lower Hessenberg matrix.
Create a tridiagonal matrix with zero diagonal entries.
Create a comparison matrix.
Create a `counterexample' matrix to a condition estimator.
Create a matrix whose columns repeat cyclically.
Create a diagonally dominant, ill conditioned, tridiagonal matrix.
Create a (0, 1) matrix whose inverse has large integer entries.
Create a Forsythe matrix (a perturbed Jordan block).
Create a Frank matrix (ill conditioned eigenvalues).
Create a greatest common divisor matrix.
c is an n-by-n matrix whose values correspond to the greatest common divisor of its coordinate values, i.e., c(i,j) correspond
gcd (i, j)
.
Create a Gear matrix.
Create a Toeplitz matrix with sensitive eigenvalues.
Create a matrix whose eigenvalues lie on a vertical line in the complex plane.
Create a householder matrix.
Create a matrix with random integers in the range [1, imax]. If imin is given then the integers are in the range [imin, imax].
The second input is a matrix of dimensions describing the size of the output. The dimensions can also be input as comma-separated arguments.
The input j is an integer index in the range [0, 2^32-1]. The values of the output matrix are always exactly the same (reproducibility) for a given size input and j index.
The final optional argument determines the class of the resulting matrix. Possible values for class:
"uint8"
,"uint16"
,"uint32"
,"int8"
,"int16"
, int32","single"
,"double"
. The default is"double"
.
Create the inverse of an upper Hessenberg matrix.
Create an Hankel matrix with factorial elements.
Create a Jordan block.
Create a Kahan matrix (upper trapezoidal).
Create a Kac-Murdock-Szego Toeplitz matrix.
Create a Krylov matrix.
Create a Lauchli matrix (rectangular).
Create a tridiagonal matrix with real, sensitive eigenvalues.
Create a Moler matrix (symmetric positive definite).
Create a singular matrix from the discrete Neumann problem (sparse).
Create a matrix with random samples from the standard normal distribution (mean = 0, std = 1).
The first input is a matrix of dimensions describing the size of the output. The dimensions can also be input as comma-separated arguments.
The input j is an integer index in the range [0, 2^32-1]. The values of the output matrix are always exactly the same (reproducibility) for a given size input and j index.
The final optional argument determines the class of the resulting matrix. Possible values for class:
"single"
,"double"
. The default is"double"
.
Create orthogonal and nearly orthogonal matrices.
Create a Parter matrix (a Toeplitz matrix with singular values near pi).
Create a Pei matrix.
Create a block tridiagonal matrix from Poisson's equation (sparse).
Create a prolate matrix (symmetric, ill-conditioned Toeplitz matrix).
Create a random matrix with elements -1, 0 or 1.
Create a random matrix with pre-assigned singular values.
Create a zero and ones matrix of Redheffer associated with the Riemann hypothesis.
Create a complex matrix, with a `smoke ring' pseudospectrum.
Create a symmetric positive definite Toeplitz matrix.
Create a pentadiagonal Toeplitz matrix (sparse).
Create a tridiagonal matrix (sparse).
Create an upper triangular matrix discussed by Kahan, Golub and Wilkinson.
Create a matrix with random samples from the standard uniform distribution (range [0,1]).
The first input is a matrix of dimensions describing the size of the output. The dimensions can also be input as comma-separated arguments.
The input j is an integer index in the range [0, 2^32-1]. The values of the output matrix are always exactly the same (reproducibility) for a given size input and j index.
The final optional argument determines the class of the resulting matrix. Possible values for class:
"single"
,"double"
. The default is"double"
.
Create the Wathen matrix.
Create various specific matrices devised/discussed by Wilkinson.
Construct a Hadamard matrix (Hn) of size n-by-n. The size n must be of the form 2^k * p in which p is one of 1, 12, 20 or 28. The returned matrix is normalized, meaning
Hn(:,1) == 1
andHn(1,:) == 1
.Some of the properties of Hadamard matrices are:
kron (Hm, Hn)
is a Hadamard matrix of size m-by-n.Hn * Hn' =
n* eye (
n)
.- The rows of Hn are orthogonal.
det (
A) <= abs (det (Hn))
for all A withabs (
A(i, j)) <= 1
.- Multiplying any row or column by -1 and the matrix will remain a Hadamard matrix.
Return the Hankel matrix constructed from the first column c, and (optionally) the last row r. If the last element of c is not the same as the first element of r, the last element of c is used. If the second argument is omitted, it is assumed to be a vector of zeros with the same size as c.
A Hankel matrix formed from an m-vector c, and an n-vector r, has the elements
H(i,j) = c(i+j-1), i+j-1 <= m; H(i,j) = r(i+j-m), otherwise
Return the Hilbert matrix of order n. The i,j element of a Hilbert matrix is defined as
H(i, j) = 1 / (i + j - 1)Hilbert matrices are close to being singular which make them difficult to invert with numerical routines. Comparing the condition number of a random matrix 5x5 matrix with that of a Hilbert matrix of order 5 reveals just how difficult the problem is.
cond (rand (5)) ⇒ 14.392 cond (hilb (5)) ⇒ 4.7661e+05See also: invhilb.
Return the inverse of the Hilbert matrix of order n. This can be computed exactly using
(i+j) /n+i-1\ /n+j-1\ /i+j-2\ 2 A(i,j) = -1 (i+j-1)( )( ) ( ) \ n-j / \ n-i / \ i-2 / = p(i) p(j) / (i+j-1)where
k /k+n-1\ /n\ p(k) = -1 ( ) ( ) \ k-1 / \k/The validity of this formula can easily be checked by expanding the binomial coefficients in both formulas as factorials. It can be derived more directly via the theory of Cauchy matrices. See J. W. Demmel, Applied Numerical Linear Algebra, p. 92.
Compare this with the numerical calculation of
inverse (hilb (n))
, which suffers from the ill-conditioning of the Hilbert matrix, and the finite precision of your computer's floating point arithmetic.See also: hilb.
Create an n-by-n magic square. A magic square is an arrangement of the integers
1:n^2
such that the row sums, column sums, and diagonal sums are all equal to the same value.Note: n must be greater than 2 for the magic square to exist.
Return the Pascal matrix of order n if t
= 0
. t defaults to 0. Return the pseudo-lower triangular Cholesky factor of the Pascal matrix if t= 1
(The sign of some columns may be negative). This matrix is its own inverse, that ispascal (
n, 1) ^ 2 == eye (
n)
. If t= -1
, return the true Cholesky factor with strictly positive values on the diagonal. If t= 2
, return a transposed and permuted version ofpascal (
n, 1)
, which is the cube root of the identity matrix. That is,pascal (
n, 2) ^ 3 == eye (
n)
.See also: chol.
Return the Rosser matrix. This is a difficult test case used to evaluate eigenvalue algorithms.
Return the Toeplitz matrix constructed from the first column c, and (optionally) the first row r. If the first element of r is not the same as the first element of c, the first element of c is used. If the second argument is omitted, the first row is taken to be the same as the first column.
A square Toeplitz matrix has the form:
c(0) r(1) r(2) ... r(n) c(1) c(0) r(1) ... r(n-1) c(2) c(1) c(0) ... r(n-2) . . . . . . . . . . . . . . . c(n) c(n-1) c(n-2) ... c(0)See also: hankel.
Return the Vandermonde matrix whose next to last column is c. If n is specified, it determines the number of columns; otherwise, n is taken to be equal to the length of c.
A Vandermonde matrix has the form:
c(1)^(n-1) ... c(1)^2 c(1) 1 c(2)^(n-1) ... c(2)^2 c(2) 1 . . . . . . . . . . . . . . . c(n)^(n-1) ... c(n)^2 c(n) 1See also: polyfit.