'binomial'
| Description: Binomial matrix, which
is a multiple of involutory matrix Syntax:
Properties:
|
'cauchy'
| Description: Cauchy
matrix Syntax:
A = gallery('cauchy',x,y) returns an
n-by-n matrix with entries
A(i,j) = 1/(x(i)+y(j)). Arguments
x and y are vectors of length
n. If you pass in scalars for x
and y, they are interpreted as vectors
1:x and 1:y.
A = gallery('cauchy',x) returns the same as above
with y = x. In other words, A(i,j) =
1/(x(i)+x(j)).
Properties:
Explicit formulas are known for the inverse and determinant of a
Cauchy matrix. The determinant det(C) is nonzero if
x and y both have distinct
elements.
C is totally positive if 0 < x(1) <
... < x(n) and 0 < y(1) < ... <
y(n).
|
'chebspec'
| Description: Chebyshev spectral
differentiation matrix Syntax:
A = gallery('chebspec',n,k) returns a Chebyshev
spectral differentiation matrix of size
n-by-n. Argument
k, which can take the value 0
(default) or 1, determines the character of the
output matrix.
For k = 0 (no boundary conditions),
A is nilpotent, meaning there exists a positive
integer c such that A^c = 0. The
matrix A has the null vector
ones(n,1). For k = 1, A is nonsingular
and well conditioned, and its eigenvalues have negative real
parts.
Properties:
For k = 0, the matrix A is
similar to a Jordan block of size n with eigenvalue
zero. Two matrices A and B are
called similar if there is an invertible matrix P of
the same size, such that B = inv(P)*A*P. For both k, the eigenvector matrix of the
Chebyshev spectral differentiation matrix is ill conditioned.
|
'chebvand'
| Description: Vandermonde-like matrix
for the Chebyshev polynomials Syntax:
A = gallery('chebvand',x) produces the primal
Chebyshev Vandermonde matrix based on the vector of points
x, which defines where the Chebyshev polynomial is
calculated. Argument x is a vector of length
n, and the size of A is
n-by-n. The entries of
A are A(i,j)
= Ti –
1(x(j)), where Ti –
1 is the Chebyshev polynomial of the first kind of
degree i – 1. If x is a scalar,
then x equally spaced points on the interval
[0,1] are used to calculate
A.
A = gallery('chebvand',m,x) produces a
rectangular version of the above with m rows, where
m is a scalar.
|
'chow'
| Description: Singular Toeplitz lower
Hessenberg matrix Syntax:
A = gallery('chow',n,alpha,delta) returns a Chow
matrix of order n, which is an
n-by-n lower Hessenberg matrix.
The matrix A is defined as
where the matrix elements of
Hα are
Hα(i,j)
= α(i –
j + 1) for (i –
j + 1) ≥ 0, δ is a coefficient,
and I is an
n-by-n identity matrix. A = gallery('chow',n) uses the default values
1 and 0 for
alpha and delta,
respectively.
Properties:
|
'circul'
| Description: Circulant
matrix Syntax:
A = gallery('circul',v) returns an
n-by-n circulant matrix whose
first row is the vector v of length
n. A circulant matrix is a special kind of Toeplitz
matrix where each row is obtained from the previous one by cyclically
moving the entries one place to the right. If v is a
scalar, then A = gallery('circul',1:v).
Properties:
See also:
toeplitz |
'clement'
| Description: Clement tridiagonal
matrix with zero diagonal entries Syntax:
A = gallery('clement',n,k) returns an
n-by-n tridiagonal matrix with
zeros on its main diagonal. For k = 0 (default),
A is nonsymmetric. For k = 1,
A is symmetric.
Properties:
A = gallery('clement',n,1) is diagonally similar
to B = gallery('clement',n,0), where there exists a
diagonal matrix D of the same size such that
B = inv(D)*A*D.
If n is odd, then A is a
singular matrix. The eigenvalues of A are explicitly known, which
include plus and minus the numbers n-1,
n-3, n-5,
..., 1 or
0. The two previous properties also hold for
gallery('tridiag',x,y,z), where y =
zeros(n,1). If the length of the input vector
y or n is odd, then
gallery('tridiag',x,y,z) is singular. The
eigenvalues still come in plus and minus pairs, but they are not known
explicitly. For odd n = 2*m+1 and k = 0,
gallery('clement',n,0) has m+1
singular values that are equal to sqrt((2*m+1)^2 -
(2*t+1).^2) for t = 0:m.
|
'compar'
| Description: Comparison
matrix Syntax:
A = gallery('compar',B,k) returns the comparison
matrix of B.
For k = 0 (default), A(i,j) =
abs(B(i,j)) if i == j and A(i,j)
= -abs(B(i,j)) otherwise. For k = 1, A =
gallery('compar',B,1) replaces each diagonal element of
B with its absolute value, and replaces each
off-diagonal element with the negative of the largest absolute value of
off-diagonal elements within the same row.
Properties:
|
'condex'
| Description: Counterexamples to
matrix condition number estimators Syntax:
A = gallery('condex',n,k,alpha) returns a
counterexample matrix to a condition number estimator. It takes a scalar
parameter alpha (default to 100) and returns a matrix
of size n-by-n.
The matrix, its natural size, and the estimator to which it applies
are specified by k. k = 1
| 4-by-4 | Counterexample to LINPACK's RCOND | k = 2
| 3-by-3 | Counterexample to LINPACK's RCOND | k = 3
| arbitrary | Counterexample to LINPACK's RCOND (independent of
alpha) | k = 4 (default)
| n >= 4
| Counterexample to LAPACK's RCOND (it is the inverse
of this matrix that is a counter example) |
If n is not equal to the natural size of the
matrix, then the matrix is padded out with an identity matrix to order
n.
|
'cycol'
| Description: Matrix whose columns
repeat cyclically Syntax:
A = gallery('cycol',n,k) 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 a scalar. Argument k
is a scalar with default value round(n/4), and it
does not need to evenly divide n.
A = gallery('cycol',[m n],k) returns an
m-by-n matrix with cyclically
repeating columns, where one cycle consists of the columns defined by
randn(m,k).
|
'dorr'
| Description: Diagonally dominant,
ill-conditioned, tridiagonal matrix (sparse matrix) Syntax:
A = gallery('dorr',n,theta) returns the Dorr
matrix, which is an 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.
[v1,v2,v3] = gallery('dorr',n,alpha) returns the
vectors that define the Dorr matrix. The Dorr matrix itself is the same
as gallery('tridiag',v1,v2,v3).
|
'dramadah'
| Description: Matrix of zeros and
ones Syntax:
A = gallery('dramadah',n,k) returns an
n-by-n matrix of
0's and 1's.
n and k must both be scalars.
For k = 0 and 1, mu(A) =
norm(inv(A),'fro') is relatively large, although not
necessarily maximal [2].
Argument k determines the character of the output
matrix as listed below.
k = 1
(default)
| A is Toeplitz and unimodular with
abs(det(A)) = 1 and mu(A) >
c*(1.75)^2n, where c is a
constant. The inverse of A has integer
entries.
| k = 2
| A is upper triangular and
Toeplitz. The inverse of A has integer
entries.
| k = 3
| A is Toeplitz and has maximal
determinant among lower Hessenberg binary matrices (with all
elements either 0 or 1). det(A) is equal to
the nth Fibonacci number. The eigenvalues
have an interesting distribution in the complex
plane.
|
|
'fiedler'
| Description: Fiedler symmetric
matrix Syntax:
A = gallery('fiedler',x), where
x is a length n vector, returns
the n-by-n symmetric matrix with
elements A(i,j) = abs(x(i)-x(j)). For scalar
x, A =
gallery('fiedler',1:x).
Properties:
Matrix A has a dominant positive eigenvalue and
all the other eigenvalues are negative. Explicit formulas for inv(A) and
det(A) are given in [3] and
attributed to Fiedler. These formulas indicate that
inv(A) is tridiagonal except for nonzero
(1,n) and (n,1) elements.
|
'forsythe'
| Description: Forsythe matrix or
perturbed Jordan block Syntax:
A = gallery('forsythe',n,alpha,lambda) returns
the n-by-n matrix equal to the
Jordan block with eigenvalue lambda, except that
A(n,1) = alpha. The default values of scalars
alpha and lambda are
sqrt(eps) and 0,
respectively.
Properties:
|
'frank'
| Description: Frank matrix with
ill-conditioned eigenvalues Syntax:
A = gallery('frank',n,k) returns the Frank matrix
of size n-by-n for the default
value of k = 0. The Frank matrix is an upper
Hessenberg matrix with determinant 1. If k =
1, the elements are reflected about the anti-diagonal,
(1,n), (2,n-1), …,
(n,1).
Properties:
The eigenvalues of A may be obtained in terms of
the zeros of the Hermite polynomials. They are positive and occur in
reciprocal pairs. If n is odd, then 1 is an
eigenvalue. The smallest half of the eigenvalues, or the smallest
floor(n/2) eigenvalues, are ill conditioned.
|
'gcdmat'
| Description: Greatest common divisor
matrix Syntax:
Properties:
|
'gearmat'
| Description: Gear
matrix Syntax:
A = gallery('gearmat',n,i,j) returns the
n-by-n matrix with
1's on the subdiagonal and superdiagonal,
sign(i) in the (1,abs(i))
position, sign(j) in the
(n,n+1-abs(j)) position, and 0's
everywhere else. Arguments i and j
default to n and -n, respectively.
They must be integers in the range of -n to
n.
Properties:
Matrix A is singular, can have double and triple
eigenvalues, and can be defective. All eigenvalues are of the form 2*cos(a) and the eigenvectors are of the form [sin(w+a),
sin(w+2a), …,
sin(w+na)], where a and w
are given in [4].
|
'grcar'
| Description: Toeplitz matrix with
sensitive eigenvalues Syntax:
A = gallery('grcar',n,k) returns an
n-by-n Toeplitz matrix with
-1's on the subdiagonal, 1's on
the main diagonal, and 1's on k
diagonals above the main diagonal. k must be an
integer, and the default value is k = 3.
A has eigenvalues that are sensitive to
perturbation.
|
'hanowa'
| Description: Matrix whose
eigenvalues lie on a vertical line in the complex plane Syntax:
A = gallery('hanowa',n,alpha) returns a
2-by-2 block matrix with four
n/2-by-n/2 blocks of the
form:
[alpha*eye(n/2) -diag(1:n/2)
diag(1:n/2) alpha*eye(n/2)] Argument n must be an even integer. The default
value of alpha is -1.
A has complex eigenvalues of the form
alpha ± k*i, for 1
<= k <= n/2.
|
'house'
| Description: Householder
matrix Syntax:
[v,beta] = gallery('house',x) takes
x, a scalar or an n-element
column vector, and returns v and
beta such that H = eye(n) -
beta*v*v' is a Householder matrix. A Householder matrix
H satisfies the property
H*x = -sign(x(1))*norm(x)*e1 where e1 is the first column of
eye(n). Note that if x is complex, then sign(x)
= exp(i*arg(x)), which is equal to
x./abs(x) when x is nonzero. If
x is zero, then v is zero and
beta = 1. [v,beta,s] = gallery('house',x,k) takes
x, an n-element column vector,
and returns v and beta such that
H*x = s*e1. In this expression,
e1 is the first column of
eye(n), abs(s) = norm(x), and
H = eye(n) - beta*v*v' is a Householder
matrix.
k determines the sign of s as
listed below.
k = 0 (default)
| sign(s) =
-sign(x(1))
| k = 1
| sign(s) =
sign(x(1))
| k = 2
| sign(s) = 1 (x
must be real)
|
If x is zero, or if x =
alpha*e1 (alpha >= 0) and either
k = 1 or k = 2, then
v is zero, beta = 1, and
s = x(1). In this case, H =
eye(n) is the identity matrix, which is not strictly a
Householder matrix.
|
'integerdata'
| Description: Array of randomly
sampled integers from uniform distribution on specified
range Syntax:
A = gallery('integerdata',imax,[m,n,...],k)
returns an m-by-n-by-... array
A whose values are a sample from the uniform
distribution on the integers 1:imax.
k is a random seed and must be an integer value in
the interval [0,2^32-1]. Calling
gallery('integerdata',...) with different values of
k returns different arrays. Repeated calls to
gallery('integerdata',...) with the same
imax, size vector, and k always
return the same array.
In any call to gallery('integerdata',...) you can
substitute individual inputs m,n,... for the size
vector input [m,n,...]. For example,
gallery('integerdata',7,[1,2,3,4],5) is equivalent
to gallery('integerdata',7,1,2,3,4,5). A = gallery('integerdata',[imin
imax],[m,n,...],k) returns an
m-by-n-by-... array
A whose values are a sample from the uniform
distribution on the integers imin:imax.
[A1,A2,...,Am] = gallery('integerdata',[imin
imax],[m,n,...],k) returns multiple
m-by-n-by-... arrays
A1, A2, ...,
Am containing different values.
[A1,A2,...,Am] = gallery('integerdata',[imin
imax],[m,n,...],k,typename) produces arrays with elements of
type typename. typename must be
'double' (default), 'single',
'uint8', 'uint16',
'uint32', 'int8',
'int16', or int32'.
See also:
randi | rng |
'invhess'
| Description: Inverse of an upper
Hessenberg matrix Syntax:
A = gallery('invhess',x,y), where
x is a length n vector and
y is a length n-1 vector,
returns a matrix with elements A(i,j) = x(j) if
i <= j, and A(i,j) = y(i)
otherwise. The lower triangle of A is based on the
vector x, and the strict upper triangle is based on
y. Argument y defaults to
-x(1:n-1).
For a scalar n,
gallery('invhess',n) is the same as
gallery('invhess',1:n).
Properties:
|
'invol'
| Description: Involutory matrix (a
matrix that is its own inverse) Syntax:
Properties:
See also:
hilb |
'ipjfact'
| Description: Hankel matrix with
factorial elements Syntax:
[A,beta] = gallery('ipjfact',n,k) returns an
n-by-n Hankel matrix
A and its determinant beta,
which is known explicitly. The inverse of A is also
known explicitly.
If k = 0 (default), then the elements of
A are A(i,j) = factorial(i+j).
If k = 1, then the elements of A
are A(i,j) = 1/factorial(i+j).
See also:
hankel | toeplitz |
'jordbloc'
| Description: Jordan block
matrix Syntax:
|
'kahan'
| Description: Upper trapezoidal Kahan
matrix Syntax:
A = gallery('kahan',n,theta,pert) returns an
upper trapezoidal matrix that has interesting properties regarding
estimation of condition and rank. For example, the Kahan matrix
illustrates that using a column-permuted QR decomposition can fail to
give a good rank estimation of a matrix.
If n is a two-element vector, then
A is
n(1)-by-n(2); otherwise,
A is n-by-n.
The useful range of theta is 0 < theta
< pi, with a default value of
1.2. To ensure that the QR factorization with column pivoting does not
interchange columns in the presence of rounding errors, the main
diagonal is perturbed by pert*eps*diag([n:-1:1]). The
default value of pert is 1e3,
which ensures no interchanges for gallery('kahan',n)
up to at least n = 100 in IEEE® arithmetic (for default value of theta =
1.2).
|
'kms'
| Description: Kac-Murdock-Szegö
Toeplitz matrix Syntax:
A = gallery('kms',n,rho) returns the
n-by-n Kac-Murdock-Szegö
Toeplitz matrix such that A(i,j) = rho^(abs(i-j)) for
real rho. For complex rho, the
same formula holds except that elements below the diagonal are
conjugated. rho defaults to 0.5.
Properties:
There exists an LDL factorization of A with
L = inv(gallery('triw',n,-rho,1))', and D
= (1-abs(rho)^2)*eye(n), except the first element of
D is D(1,1) = 1. A is positive definite if and only if 0
< abs(rho) < 1.
The inverse inv(A) is tridiagonal.
|
'krylov'
| Description: Krylov
matrix Syntax:
A = gallery('krylov',B,x,k) returns the Krylov
matrix, where B is an
n-by-n matrix,
x is a vector with length n, and
k must be an integer. The Krylov matrix is equal
to
[x, B*x, B^2*x, ..., B^(k-1)*x] for a column vector x. If you do not specify the
arguments x and k, they take the
default values of x = ones(n,1) and k =
n. A = gallery('krylov',n) for a scalar
n is the same as
gallery('krylov',B) with B =
randn(n).
|
'lauchli'
| Description: Lauchli rectangular
matrix Syntax:
A = gallery('lauchli',n,mu) returns the
(n+1)-by-n matrix of the form
[ones(1,n); mu*eye(n)]. Argument
mu defaults to sqrt(eps).
Properties:
|
'lehmer'
| Description: Lehmer symmetric
positive definite matrix Syntax:
A = gallery('lehmer',n) returns the symmetric
positive definite n-by-n matrix
such that A(i,j) = i/j for j >= i
and A(i,j) = j/i otherwise.
Properties:
A is totally nonnegative.
The inverse inv(A) is tridiagonal and explicitly
known. The order n satisfies n <= cond(A)
<= 4*n*n.
|
'leslie'
| Description: Matrix of birth numbers
and survival rates from the Leslie population model Syntax:
A = gallery('leslie',x,y) is the
n-by-n matrix from the Leslie
population model with average birth numbers x(1:n)
and survival rates y(1:n-1). The matrix elements are
mostly zero, apart from the first row that contains
x(i) and the first subdiagonal that contains
y(i). For a valid model, the elements of
x are nonnegative, and the elements of
y are positive and bounded by 1, where 0
< y(i) <= 1.
A = gallery('leslie',n) generates the Leslie
matrix with x = ones(n,1) and y =
ones(n-1,1).
|
'lesp'
| Description: Tridiagonal matrix with
real, sensitive eigenvalues Syntax:
Properties:
The sensitivities of the eigenvalues increase exponentially as the
eigenvalues grow more negative. The matrix is similar to the symmetric tridiagonal matrix
B with the same diagonal entries and with
off-diagonal entries 1 via a similarity transformation A =
inv(D)*B*D, where D =
diag(factorial(1:n)).
|
'lotkin'
| Description: Lotkin
matrix Syntax:
Properties:
The Lotkin matrix A is nonsymmetric, ill
conditioned, and has many negative eigenvalues of small
magnitude. Its inverse has integer entries and is known explicitly [5].
|
'minij'
| Description: Symmetric positive
definite matrix Syntax:
A = gallery('minij',n) returns the
n-by-n symmetric positive
definite matrix with entries A(i,j) =
min(i,j).
Properties:
A has eigenvalues of
0.25*sec(r*pi/(2*n+1)).^2, where r =
1:n.
The inverse inv(A) is tridiagonal and equal to
-1 times the second difference matrix, except its
(n,n) element is 1. The matrix 2*A-ones(size(A)) has tridiagonal
inverse and eigenvalues of
0.5*sec((2*r-1)*pi/(4*n)).^2, where r =
1:n.
|
'moler'
| Description: Moler symmetric
positive definite matrix Syntax:
A = gallery('moler',n,alpha) returns the
symmetric positive definite n-by-n
matrix U'*U, where U =
gallery('triw',n,alpha). For the default value of
alpha = -1, A(i,j) = min(i,j)-2,
and A(i,i) = i.
Properties:
|
'neumann'
| Description: Singular matrix from
the discrete Neumann problem (sparse matrix) Syntax:
A = gallery('neumann',n) returns the sparse
n-by-n singular, row diagonally
dominant matrix resulting from discretizing the Neumann problem with the
usual five-point operator on a regular mesh. Argument
n must be a perfect square integer.
A is sparse and has a one-dimensional null space
with null vector ones(n,1).
A = gallery('neumann',[m n]) returns an identical
matrix, but with size m*n-by-m*n.
Arguments m and n must be positive
integers, where m must be larger than 1.
|
'normaldata'
| Description: Array of randomly
sampled numbers from standard normal (Gaussian)
distribution Syntax:
A = gallery('normaldata',[m,n,...],k) returns an
m-by-n-by-... array
A. The elements of A are a
random sample of numbers from the standard normal distribution.
k is a random seed and must be an integer value in
the interval [0, 2^32-1]. Calling
gallery('normaldata',...) with different values of
k returns different arrays. Repeated calls to
gallery('normaldata',...) with the same size vector
[m,n,...] and the same value of
k always return the same array.
In any call to gallery('normaldata',...) you can
substitute individual inputs m,n,... for the size
vector input [m,n,...]. For example,
gallery('normaldata',[1,2,3,4],5) is equivalent to
gallery('normaldata',1,2,3,4,5). [A1,A2,...,Am] =
gallery('normaldata',[m,n,...],k) returns multiple
m-by-n-by-... arrays
A1, A2, ...,
Am containing different values.
[A1,A2,...,Am] =
gallery('normaldata',[m,n,...],k,typename) produces arrays
with elements of type typename.
typename must be either
'double'(default) or
'single'.
See also:
randn | rng |
'orthog'
| Description: Orthogonal and nearly
orthogonal matrices Syntax:
A = gallery('orthog',n,k) returns the
kth type of matrix of order n,
where k > 0 returns exactly orthogonal matrices,
and k < 0 returns different diagonal scalings of
orthogonal matrices.
k = 1 (default)
| A(i,j) = sqrt(2/(n+1)) *
sin(i*j*pi/(n+1))
Matrix of
eigenvectors of the second difference matrix
gallery('tridiag',n).
A is symmetric and
orthogonal. | k = 2
| A(i,j) = 2/(sqrt(2*n+1)) *
sin(2*i*j*pi/(2*n+1))
Symmetric and
orthogonal matrix. | k = 3
| A(r,s) = exp(2*pi*i*(r-1)*(s-1)/n) /
sqrt(n)
Complex matrix that is
unitary. A^4 is an identity matrix. This is
exactly the same matrix as
fft(eye(n))/sqrt(n). | k = 4
| Standard Helmert matrix: a permutation of a lower
Hessenberg matrix, whose first row is
ones(1,n)/sqrt(n). | k = 5
| A(i,j) = sin(2*pi*(i-1)*(j-1)/n)/sqrt(n) +
cos(2*pi*(i-1)*(j-1)/n)/sqrt(n)
Symmetric
matrix arising in the Hartley transform. | k = 6
| A(i,j) = sqrt(2/n) *
cos((i-1/2)*(j-1/2)*pi/n)
Symmetric
matrix arising as a discrete cosine transform. | k = -1
| A(i,j) =
cos((i-1)*(j-1)*pi/(n-1))
Chebyshev
Vandermonde-like matrix, based on the extrema of Chebyshev
polynomial of the first kind T(n-1). This
matrix is symmetric. Each even column (row) vector of the
matrix is orthogonal to each odd column (row) vector, that is
A(i,:)*A(j,:)' = 0 for any even
i and odd
j. | k = -2
| A(i,j) =
cos((i-1)*(j-1/2)*pi/n)
Chebyshev
Vandermonde-like matrix, based on the zeros of
T(n). This matrix has row vectors that
are orthogonal to each other, that is A(i,:)*A(j,:)'
= 0 for i not equal to
j. The matrix A*A' is
not the identity matrix, but it is a diagonal
matrix. |
|
'parter'
| Description: Parter
matrix Syntax:
Properties:
|
'pei'
| Description: Pei
matrix Syntax:
A = gallery('pei',n,alpha), where
alpha is a scalar, returns the symmetric matrix
alpha*eye(n) + ones(n). The default value for
alpha is 1. The matrix is
singular for alpha equal to either
0 or -n.
|
'poisson'
| Description: Block tridiagonal
matrix from Poisson's equation (sparse matrix) Syntax:
|
'prolate'
| Description: Prolate
matrix Syntax:
A = gallery('prolate',n,alpha) returns the
n-by-n prolate matrix with
parameter alpha. It is a symmetric, ill-conditioned
Toeplitz matrix. If 0 < alpha < 0.5, then
A is positive definite.
Properties:
The eigenvalues of A are distinct, lie in the
interval (0,1), and tend to cluster around
0 and 1. The default value of w is 0.25.
|
'randcolu'
| Description: Random matrix with
normalized columns and specified singular values Syntax:
A = gallery('randcolu',n) generates a random
n-by-n matrix with normalized
columns of unit 2-norm and random singular values from a uniform
distribution.
A'*A is a correlation matrix of the similar form
of that produced by gallery('randcorr',n). The
eigenvalues of the latter are uniformly distributed, while for the
former, the square roots of the eigenvalues are uniformly
distributed.
A = gallery('randcolu',x), where
x is a vector of length n
(n > 1), produces a random
n-by-n matrix with singular
values given by the vector x. The vector
x must have nonnegative elements whose sum of
squares is n. This matrix also has normalized columns
of unit 2-norm.
A = gallery('randcolu',__,m), where m >=
n, produces an m-by-n
matrix.
A = gallery('randcolu',__,m,k) provides
additional options based on k.
k = 0 (default)
| diag(x) is initially subjected to
a random two-sided orthogonal transformation, and then a
sequence of Givens rotations is applied.
| k = 1
| The initial transformation is omitted. This is faster
when generating the test matrix, but the resulting matrix
typically has many zero entries. |
|
'randcorr'
| Description: Random correlation
matrix with specified eigenvalues Syntax:
A = gallery('randcorr',n) is a random
n-by-n correlation matrix with
random eigenvalues from a uniform distribution. A correlation matrix is
a symmetric positive semi-definite matrix with 1's on
the diagonal.
A = gallery('randcorr',x) produces a random
correlation matrix with eigenvalues given by the vector
x, where length(x) > 1. The
vector x must have nonnegative elements whose sum is
length(x).
A = gallery('randcorr',x,k) provides additional
options based on k.
k = 0
(default)
| The diagonal matrix of eigenvalues is initially
subjected to a random orthogonal similarity transformation,
and then a sequence of Givens rotations is applied [6]. | k =
1
| The initial transformation is omitted. This is faster
when generating the test matrix, but the resulting matrix
typically has many zero entries. |
See also:
corrcoef |
'randhess'
| Description: Random, orthogonal
upper Hessenberg matrix Syntax:
A = gallery('randhess',n) returns an
n-by-n real, random, orthogonal
upper Hessenberg matrix.
A = gallery('randhess',x) constructs
A nonrandomly using the elements of
x as parameters. x must be a
real vector with length n, where n >
1.
In both cases, matrix A is constructed from a
product of n-1 Givens rotations.
|
'randjorth'
| Description: Random
J-orthogonal matrix Syntax:
A = gallery('randjorth',n) produces a random
n-by-n
J-orthogonal matrix A that
satisfies the relation A'*J*A = J (such matrices are
also known as hyperbolic). Here, J =
blkdiag(eye(ceil(n/2)),-eye(floor(n/2))), and
cond(A) = sqrt(1/eps).
A = gallery('randjorth',n,m), for positive
integers n and m, produces a
random (n+m)-by-(n+m)
J-orthogonal matrix A. Here,
J = blkdiag(eye(n),-eye(m)), and cond(A) =
sqrt(1/eps).
A =
gallery('randjorth',n,m,alpha,symm,method)
uses the following optional input arguments: alpha — Specify cond(A) =
alpha, where alpha must be equal to
or larger than 1.
symm — Choose to enforce symmetry or
not. A is nonsymmetric if the scalar
symm is 0. A is symmetric if
the scalar symm is 1 or nonzero.
method — Choose to call
qr to perform the underlying orthogonal
transformations or not. qr is not called if the
scalar method is 0. qr is
called if the scalar method is 1 or nonzero. A
call to qr is much faster than the default method
to create the J-orthogonal matrix for large
dimensions.
|
'rando'
| Description: Random matrix composed
of elements –1, 0 or 1 Syntax:
A = gallery('rando',n,k) returns a random
n-by-n matrix. The input
argument k determines the matrix elements from one of
the following discrete distributions.
k = 1 (default)
|
A(i,j) = 0 or
1 with equal probability. | k = 2
|
A(i,j) = -1 or
1 with equal probability. | k = 3
|
A(i,j) = -1,
0, or 1 with equal
probability. |
A = gallery('rando',[n m],k) returns a random
n-by-m matrix.
|
'randsvd'
| Description: Random matrix with
preassigned singular values Syntax:
A = gallery('randsvd',n,kappa,mode,kl,ku) returns
a banded (multidiagonal) random matrix of order n
with condition number cond(A) = abs(kappa), which
must be larger or equal than 1. If n is a two-element
vector, A has the size of
n(1)-by-n(2).
The default value of kappa is
sqrt(1/eps). The distribution mode
mode determines the singular values of the
matrix. Arguments kl and ku specify
the number of lower and upper off-diagonals in A,
respectively. If they are omitted, a full matrix is produced with
kl = n-1 and ku = kl. If only
kl is present, ku defaults to
kl. Available values of mode are listed below. mode = 1
| One large singular value that is equal to
1 (and the rest of the singular values
are equal to 1/abs(kappa)). | mode = 2
| One small singular value that is equal to
1/abs(kappa) (and the rest of the
singular values are equal to
1). | mode = 3 (default)
| Geometrically distributed singular
values. | mode = 4
| Arithmetically distributed singular
values. | mode = 5
| Random singular values with uniformly distributed
logarithm. | mode < 0
| If mode is -1,
-2, -3,
-4, or -5, then
randsvd treats mode as
abs(mode). However, in the original
matrix of singular values, the order of the diagonal entries
is reversed from small to large instead of from large to
small. |
In the special case of kappa <= 1,
A is a symmetric, positive definite matrix with
cond(A) = -kappa and eigenvalues distributed
according to mode. In this case, the arguments
kl and ku are ignored. A = gallery('randsvd',n,kappa,mode,kl,ku,method)
specifies how the computations to generate the test matrix are carried
out. method = 0 is the default, while method
= 1 uses an alternative method to call qr
that is much faster for large dimensions, even though it uses more
floating-point operations per second.
|
'redheff'
| Description: Redheffer matrix of
ones and zeros Syntax:
A = gallery('redheff',n) returns an
n-by-n matrix of
0's and 1's defined by
A(i,j) = 1, if j = 1 or if
i divides j, and A(i,j)
= 0 otherwise.
Properties:
A has n-floor(log2(n))-1
eigenvalues that are equal to 1.
A has a real eigenvalue (and its spectral radius)
that is approximately sqrt(n).
A has a negative eigenvalue that is approximately
-sqrt(n).
The remaining floor(log2(n))-1 eigenvalues have
relatively small moduli, which lie inside the circle , for ε
> 0 and large enough n. The Riemann hypothesis is true if and only if for every ε
> 0. Barrett and Jarvis conjecture that floor(log2(n))
of the small eigenvalues lie inside the unit circle abs(z) =
1. A proof of this conjecture, together with a proof that
some eigenvalues tend to zero as n tends to infinity,
would yield a new proof of the prime-number theorem [7].
|
'riemann'
| Description: Matrix associated with
the Riemann hypothesis Syntax:
A = gallery('riemann',n) returns an
n-by-n matrix for which the
Riemann hypothesis is true if and only if
for every ε > 0 [8]. The Riemann matrix is defined by A =
B(2:n+1,2:n+1), where B(i,j) = i-1 if
i divides j, and B(i,j)
= -1 otherwise.
Properties: Each eigenvalue e(i) satisfies abs(e(i))
<= m-1/m, where m = n+1. The eigenvalues also satisfies i <= e(i) <
i+1, with at most m-sqrt(m)
exceptions. All integers in the interval (m/3,m/2] are
eigenvalues.
|
'ris'
| Description: Ris
matrix Syntax:
|
'sampling'
| Description: Nonsymmetric matrix
with ill-conditioned integer eigenvalues Syntax:
A = gallery('sampling',x), where
x is a vector of length n,
returns an n-by-n nonsymmetric
matrix. The elements of the matrix are A(i,j) =
x(i)/(x(i)-x(j)) for i ~= j and
A(j,j) equals to the sum of the off-diagonal
elements in column j. If x is a
scalar, then A = gallery('sampling',1:x).
Properties:
A has integer eigenvalues
0:n-1 that are ill conditioned.
For the eigenvalues 0 and n-1,
the corresponding eigenvectors are x and
ones(n,1), respectively. A has the property that A(i,j) + A(j,i)
= 1 for i ~= j.
Explicit formulas are available for the left eigenvectors of
A. A special case of this matrix has appeared in sampling theory where
its right eigenvectors, if properly normalized, give the inclusion
probabilities of the conditional Poisson sampling design [9].
|
'smoke'
| Description: Complex matrix with a
"smoke ring" pseudospectrum Syntax:
A = gallery('smoke',n) returns an
n-by-n matrix with
1's on the superdiagonal and 1
in the (n,1) position. The diagonal consists of the
set of all nth roots of unity.
A = gallery('smoke',n,1) returns the same as
above except that the element A(n,1) is zero.
Properties:
The eigenvalues of gallery('smoke',n,1) are the
nth roots of unity. The eigenvalues of gallery('smoke',n) are the
nth roots of unity times
2^(1/n). The pseudospectrum of a matrix A can be
calculated by finding the minimum singular value of the matrix zI-A in the complex plane. Here, z
represents points in the complex plane, and I is an
identity matrix. The pseudospectra of
gallery('smoke',n) and
gallery('smoke',n,1) have "smoke ring" patterns
near their eigenvalues.
|
'toeppd'
| Description: Symmetric positive
definite Toeplitz matrix Syntax:
A = gallery('toeppd',n,m,x,theta) returns an
n-by-n symmetric Toeplitz matrix
composed of the sum of m Toeplitz matrices (with rank
1 or 2). x and theta are vectors
of length m. The matrix A is
positive definite if all elements of x are positive.
Specifically, A is generated by
A = x(1)*T1 + ... + x(k)*Tk + ... + x(m)*Tm where Tk is an
n-by-n matrix that depends on
theta(k). The elements of Tk are
Tk(i,j) = cos(2*pi*theta(k)*(i-j)). By default, m = n, x =
rand(m,1), and theta = rand(m,1).
|
'toeppen'
| Description: Pentadiagonal Toeplitz
matrix (sparse matrix) Syntax:
A = gallery('toeppen',n,a,b,c,d,e) returns the
n-by-n sparse pentadiagonal
Toeplitz matrix with elements: a on the second
diagonal below the main diagonal, b on the
subdiagonal, c on the main diagonal,
d on the superdiagonal, and e on
the second diagonal above the main diagonal, where a,
b, c, d, and
e are scalars.
By default, (a,b,c,d,e) =
(1,-10,0,10,1), yielding a matrix that was
originally introduced by Rutishauser [10].
This matrix has eigenvalues lying approximately on the curve
2*cos(2*t) + 20*i*sin(t) in the complex
plane.
|
'tridiag'
| Description: Tridiagonal matrix
(sparse matrix) Syntax:
A = gallery('tridiag',n) returns the sparse
tridiagonal matrix of size n-by-n
with subdiagonal elements -1, diagonal elements
2, and superdiagonal elements
-1. This matrix has eigenvalues 2 +
2*cos(k*pi/(n+1)), where k = 1:n.
The generated matrix is a symmetric positive definite
M-matrix with real nonnegative eigenvalues. This
matrix is also the negative of the second difference matrix. A = gallery('tridiag',c,d,e) returns the
tridiagonal matrix with subdiagonal c, diagonal
d, and superdiagonal e defined
by the vectors c, d, and
e. The length of vectors c and
e must be length(d)-1.
A = gallery('tridiag',n,c,d,e), where
c, d, and e
are all scalars, yields the Toeplitz tridiagonal matrix of size
n-by-n with subdiagonal elements
c, diagonal elements d, and
superdiagonal elements e. This matrix has eigenvalues
d + 2*sqrt(c*e)*cos(k*pi/(n+1)), where k =
1:n.
|
'triw'
| Description: Upper triangular matrix
discussed by Wilkinson and others Syntax:
A = gallery('triw',n,alpha,k) returns the upper
triangular matrix with 1's on the diagonal and
alpha's on the first k >= 0
superdiagonals. By default, alpha = -1, and
k = n-1.
The order n can be a 2-element vector, in which
case the matrix is n(1)-by-n(2)
and upper trapezoidal.
Properties:
For alpha = 2, the condition number of the matrix
satisfies: cond(gallery('triw',n,2)) = cot(pi/(4*n))^2,For large abs(alpha),
cond(gallery('triw',n,alpha)) is approximately
abs(alpha)^n*sin(pi/(4*n-2)). The matrix A = gallery('triw',n) becomes singular
when you add -2^(2-n) to its (n,1)
element. The matrix also becomes singular when you add
-2^(1-n) to its elements in the first
column.
|
'uniformdata'
| Description: Array of randomly
sampled numbers from standard uniform distribution Syntax:
A = gallery('uniformdata',[m,n,...],k) returns an
m-by-n-by-... array
A. The elements of A are a
random sample of numbers from the standard uniform distribution.
k is a random seed and must be an integer value in
the interval [0, 2^32-1]. Calling
gallery('uniformdata',...) with different values of
k returns different arrays. Repeated calls to
gallery('uniformdata',...) with the same size
vector [m,n,...] and the same value of
k always return the same array.
In any call to gallery('uniformdata',...) you can
substitute individual inputs m,n,... for the size
vector input [m,n,...]. For example,
gallery('uniformdata',[1,2,3,4],5) is equivalent to
gallery('uniformdata',1,2,3,4,5). [A1,A2,...,Am] =
gallery('uniformdata',[m,n,...],k) returns multiple
m-by-n-by-... arrays
A1, A2, ...,
Am containing different values.
[A1,A2,...,Am] =
gallery('uniformdata',[m,n,...],k,typename) produces arrays
with elements of type typename.
typename must be either 'double'
(default) or 'single'.
See also:
rand | rng |
'wathen'
| Description: Wathen matrix (sparse
matrix) Syntax:
A = gallery('wathen',nx,ny) returns a sparse,
random, n-by-n finite element
matrix where n = 3*nx*ny + 2*nx + 2*ny + 1.
Matrix A is precisely the “consistent mass
matrix” for a regular
nx-by-ny grid of 8-node
(serendipity) elements in two dimensions. A is
symmetric, positive definite for any (positive) values of the
“density” rho(nx,ny), which is chosen
randomly. B = gallery('wathen',nx,ny,1) returns a
diagonally scaled matrix based on the previous syntax, where B
= diag(diag(A))\A. The eigenvalues of this matrix
satisfy
for any positive integers nx and
ny and any densities
rho(nx,ny).
|
'wilk'
| Description: Various matrices
devised or discussed by Wilkinson Syntax:
[U,b] = gallery('wilk',3) returns an upper
triangular system U*x = b illustrating an inaccurate
solution.
[L,b] = gallery('wilk',4) returns a lower
triangular system L*x = b, which is ill
conditioned.
A = gallery('wilk',5) returns a symmetric
positive definite matrix A = B(1:5,2:6)*1.8144, where
B = hilb(6).
A = gallery('wilk',21) returns
W21+, a tridiagonal matrix with pairs of nearly
equal eigenvalues. For more details, see [11].
|
