chebspec--Chebyshev spectral differentiation matrix
C = gallery('chebspec',n,switch) returns a Chebyshev spectral differentiation matrix of order n. Argument switch is a variable that determines the character of the output matrix. By default, switch = 0.
For switch = 0 ("no boundary conditions"), C is nilpotent (C^n = 0) and has the null vector ones(n,1). The matrix C is similar to a Jordan block of size n with eigenvalue zero.
For switch = 1, C is nonsingular and well-conditioned, and its eigenvalues have negative real parts.
The eigenvector matrix of the Chebyshev spectral differentiation matrix is ill-conditioned.
chebvand--Vandermonde-like matrix for the Chebyshev polynomials
C = gallery('chebvand',p) produces the (primal) Chebyshev Vandermonde matrix based on the vector of points p, which define where the Chebyshev polynomial is calculated.
C = gallery('chebvand',m,p) where m is scalar, produces a rectangular version of the above, with m rows.
If p is a vector, then 
where 
is the Chebyshev polynomial of degree i-1. If p is a scalar, then p equally spaced points on the interval [0,1] are used to calculate C.
chow--Singular Toeplitz lower Hessenberg matrix
A = gallery('chow',n,alpha,delta) returns A such that A = H(alpha) + delta*eye(n), where 
and argument n is the order of the Chow matrix. Default value for scalars alpha and delta are 1 and 0, respectively.
H(alpha) has p = floor(n/2) eigenvalues that are equal to zero. The rest of the eigenvalues are equal to 4*alpha*cos(k*pi/(n+2))^2, k=1:n-p.
circul--Circulant matrix
C = gallery('circul',v) returns the circulant matrix whose first row is the vector v.
A circulant matrix has the property that each row is obtained from the previous one by cyclically permuting the entries one step forward. It is a special Toeplitz matrix in which the diagonals "wrap around."
If v is a scalar, then C = gallery('circul',1:v).
The eigensystem of C (n-by-n) is known explicitly: If t is an nth root of unity, then the inner product of v and 
is an eigenvalue of C and w(n:-1:1) is an eigenvector.
clement--Tridiagonal matrix with zero diagonal entries
A = gallery('clement',n,sym) returns an n-by-n tridiagonal matrix with zeros on its main diagonal and known eigenvalues. It is singular if order n is odd. About 64 percent of the entries of the inverse are zero. The eigenvalues include plus and minus the numbers n-1, n-3, n-5, ..., as well as (for odd n) a final eigenvalue of 1 or 0.
Argument sym determines whether the Clement matrix is symmetric. For sym = 0 (the default) the matrix is nonsymmetric, while for sym = 1, it is symmetric.
compar--Comparison matrices
A = gallery('compar',A,1) returns A with each diagonal element replaced by its absolute value, and each off-diagonal element replaced by minus the absolute value of the largest element in absolute value in its row. However, if A is triangular compar(A,1) is too.
gallery('compar',A) is diag(B) - tril(B,-1) - triu(B,1), where B = abs(A). compar(A) is often denoted by M(A) in the literature.
gallery('compar',A,0) is the same as gallery('compar',A).
condex--Counter-examples to matrix condition number estimators
A = gallery('condex',n,k,theta) returns a "counter-example" matrix to a condition estimator. It has order n and scalar parameter theta (default 100).
The matrix, its natural size, and the estimator to which it applies are specified by k:
k = 1
| 4-by-4
| LINPACK
|
k = 2
| 3-by-3
| LINPACK
|
k = 3
| arbitrary
| LINPACK (rcond) (independent of theta)
|
k = 4
| n >= 4
| LAPACK (RCOND) (default). 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--Matrix whose columns repeat cyclically
A = gallery('cycol',[m n],k) returns an m-by-n matrix with cyclically repeating columns, where one "cycle" consists of randn(m,k). Thus, the rank of matrix A cannot exceed k, and k must be a scalar.
Argument k defaults to round(n/4), and need not evenly divide n.
A = gallery('cycol',n,k), where n is a scalar, is the same as gallery('cycol',[n n],k).
dorr--Diagonally dominant, ill-conditioned, tridiagonal matrix
[c,d,e] = gallery('dorr',n,theta) returns the vectors defining 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. The Dorr matrix itself is the same as gallery('tridiag',c,d,e).
A = gallery('dorr',n,theta) returns the matrix itself, rather than the defining vectors.
dramadah--Matrix of zeros and ones whose inverse has large integer entries
A = gallery('dramadah',n,k) returns an n-by-n matrix of 0's and 1's for which mu(A) = norm(inv(A),'fro') is relatively large, although not necessarily maximal. An anti-Hadamard matrix A is a matrix with elements 0 or 1 for which mu(A) is maximal.
n and k must both be scalars. Argument k determines the character of the output matrix:
k = 1
| Default. A is Toeplitz, with abs(det(A)) = 1, and mu(A) > c(1.75)^n, 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 has maximal determinant among lower Hessenberg (0,1) matrices. det(A) = the nth Fibonacci number. A is Toeplitz. The eigenvalues have an interesting distribution in the complex plane.
|
fiedler--Symmetric matrix
A = gallery('fiedler',c), where c is a length n vector, returns the n-by-n symmetric matrix with elements abs(n(i)-n(j)). For scalar c, A = gallery('fiedler',1:c).
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 [Todd, J., Basic Numerical Mathematics, Vol. 2: Numerical Algebra, Birkhauser, Basel, and Academic Press, New York, 1977, p. 159] and attributed to Fiedler. These indicate that inv(A) is tridiagonal except for nonzero (1,n) and (n,1) elements.
forsythe--Perturbed Jordan block
A = gallery('forsythe',n,alpha,lambda) returns the n-by-n matrix equal to the Jordan block with eigenvalue lambda, excepting that A(n,1) = alpha. The default values of scalars alpha and lambda are sqrt(eps) and 0, respectively.
The characteristic polynomial of A is given by:
frank--Matrix with ill-conditioned eigenvalues
F = gallery('frank',n,k) returns the Frank matrix of order n. It is upper Hessenberg with determinant 1. If k = 1, the elements are reflected about the anti-diagonal (1,n)--(n,1). The eigenvalues of F may be obtained in terms of the zeros of the Hermite polynomials. They are positive and occur in reciprocal pairs; thus if n is odd, 1 is an eigenvalue. F has floor(n/2) ill-conditioned eigenvalues--the smaller ones.
gearmat--Gear matrix
A = gallery('gearmat',n,i,j) returns the n-by-n matrix with ones on the sub- and super-diagonals, sign(i) in the (1,abs(i)) position, sign(j) in the (n,n+1-abs(j)) position, and zeros everywhere else. Arguments i and j default to n and -n, respectively.
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+2*a), ..., sin(w+n*a)], where a and w are given in Gear, C. W., "A Simple Set of Test Matrices for Eigenvalue Programs", Math. Comp., Vol. 23 (1969), pp. 119-125.
grcar--Toeplitz matrix with sensitive eigenvalues
A = gallery('grcar',n,k) returns an n-by-n Toeplitz matrix with -1s on the subdiagonal, 1s on the diagonal, and k superdiagonals of 1s. The default is k = 3. The eigenvalues are sensitive.
hanowa--Matrix whose eigenvalues lie on a vertical line in the complex plane
A = gallery('hanowa',n,d) returns an n-by-n block 2-by-2 matrix of the form:
Argument n is an even integer n=2*m. Matrix A has complex eigenvalues of the form d ± k*i, for 1 <= k <= m. The default value of d is -1.
house--Householder matrix
[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:
k = 0
| sign(s) = -sign(x(1)) (default)
|
k = 1
| sign(s) = sign(x(1))
|
k = 2
| sign(s) = 1 (x must be real)
|
If x is complex, then sign(x) = x./abs(x) when x is nonzero.
If x = 0, or if x = alpha*e1 (alpha >= 0) and either k = 1 or k = 2, then V = 0, beta = 1, and s = x(1). In this case, H is the identity matrix, which is not strictly a Householder matrix.
invhess--Inverse of an upper Hessenberg matrix
A = gallery('invhess',x,y), where x is a length n vector and y is a length n-1 vector, returns the matrix whose lower triangle agrees with that of ones(n,1)*x' and whose strict upper triangle agrees with that of [1 y]*ones(1,n).
The matrix is nonsingular if x(1) ~= 0 and x(i+1) ~= y(i) for all i, and its inverse is an upper Hessenberg matrix. Argument y defaults to -x(1:n-1).
If x is a scalar, invhess(x) is the same as invhess(1:x).
invol--Involutory matrix
A = gallery('invol',n) returns an n-by-n involutory (A*A = eye(n)) and ill-conditioned matrix. It is a diagonally scaled version of hilb(n).
B = (eye(n)-A)/2 and B = (eye(n)+A)/2 are idempotent (B*B = B).
ipjfact--Hankel matrix with factorial elements
[A,d] = gallery('ipjfact',n,k) returns A, an n-by-n Hankel matrix, and d, the determinant of A, which is known explicitly. If k = 0 (the default), then the elements of A are A(i,j) = (i+j)! If k = 1, then the elements of A are A(i,j) = 1/(i+j).
Note that the inverse of A is also known explicitly.
jordbloc--Jordan block
A = gallery('jordbloc',n,lambda) returns the n-by-n Jordan block with eigenvalue lambda. The default value for lambda is 1.
kahan--Upper trapezoidal matrix
A = gallery('kahan',n,theta,pert) returns an upper trapezoidal matrix that has interesting properties regarding estimation of condition and rank.
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 <>, 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 diagonal is perturbed by pert*eps*diag([n:-1:1]). The default pert is 25, which ensures no interchanges for gallery('kahan',n) up to at least n = 90 in IEEE arithmetic.
kms--Kac-Murdock-Szego Toeplitz matrix
A = gallery('kms',n,rho) returns the n-by-n Kac-Murdock-Szego 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.
The KMS matrix A has these properties:
- An LDL' factorization with
L = inv(gallery('triw',n,-rho,1))', and D(i,i) = (1-abs(rho)^2)*eye(n), except D(1,1) = 1. - Positive definite if and only if
0 <>. The inverse inv(A) is tridiagonal.
krylov--Krylov matrix
B = gallery('krylov',A,x,j) returns the Krylov matrix
[x, Ax, A^2x, ..., A^(j-1)x]
where A is an n-by-n matrix and x is a length n vector. The defaults are x = ones(n,1), and j = n.
B = gallery('krylov',n) is the same as gallery('krylov',(randn(n)).
lauchli--Rectangular matrix
A = gallery('lauchli',n,mu) returns the (n+1)-by-n matrix
The Lauchli matrix is a well-known example in least squares and other problems that indicates the dangers of forming A'*A. Argument mu defaults to sqrt(eps).
lehmer--Symmetric positive definite matrix
A = gallery('lehmer',n) returns the symmetric positive definite n-by-n matrix such that A(i,j) = i/j for j >= i.
The Lehmer matrix A has these properties:
A is totally nonnegative. - The inverse
inv(A) is tridiagonal and explicitly known. - The order
n <= cond(A) <= 4*n*n.
leslie--
L = gallery('leslie',a,b) is the n-by-n matrix from the Leslie population model with average birth numbers a(1:n) and survival rates b(1:n-1). It is zero, apart from the first row (which contains the a(i)) and the first subdiagonal (which contains the b(i)). For a valid model, the a(i) are nonnegative and the b(i) are positive and bounded by 1, i.e., 0 <>.
L = gallery('leslie',n) generates the Leslie matrix with a = ones(n,1), b = ones(n-1,1).
lesp--Tridiagonal matrix with real, sensitive eigenvalues
A = gallery('lesp',n) returns an n-by-n matrix whose eigenvalues are real and smoothly distributed in the interval approximately [-2*N-3.5, -4.5].
The sensitivities of the eigenvalues increase exponentially as the eigenvalues grow more negative. The matrix is similar to the symmetric tridiagonal matrix with the same diagonal entries and with off-diagonal entries 1, via a similarity transformation with D = diag(1!,2!,...,n!).
lotkin--Lotkin matrix
A = gallery('lotkin',n) returns the Hilbert matrix with its first row altered to all ones. 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.
minij--Symmetric positive definite matrix
A = gallery('minij',n) returns the n-by-n symmetric positive definite matrix with A(i,j) = min(i,j).
The minij matrix has these properties:
- The inverse
inv(A) is tridiagonal and equal to -1 times the second difference matrix, except its (n,n) element is 1. - Givens' matrix,
2*A-ones(size(A)), has tridiagonal inverse and eigenvalues 0.5*sec((2*r-1)*pi/(4*n))^2, where r=1:n. (n+1)*ones(size(A))-A has elements that are max(i,j) and a tridiagonal inverse.
moler--Symmetric positive definite matrix
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 alpha = -1, A(i,j) = min(i,j)-2, and A(i,i) = i. One of the eigenvalues of A is small.
neumann--Singular matrix from the discrete Neumann problem (sparse)
C = 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 is a perfect square integer 
or a two-element vector. C is sparse and has a one-dimensional null space with null vector ones(n,1).
orthog--Orthogonal and nearly orthogonal matrices
Q = gallery('orthog',n,k) returns the kth type of matrix of order n, where k > 0 selects exactly orthogonal matrices, and k <> selects diagonal scalings of orthogonal matrices. Available types are:
k = 1
| Q(i,j) = sqrt(2/(n+1)) * sin(i*j*pi/(n+1))
Symmetric eigenvector matrix for second difference matrix. This is the default.
|
k = 2
| Q(i,j) = 2/(sqrt(2*n+1)) * sin(2*i*j*pi/(2*n+1))
Symmetric.
|
k = 3
| Q(r,s) = exp(2*pi*i*(r-1)*(s-1)/n) / sqrt(n)
Unitary, the Fourier matrix. Q^4 is the identity. This is essentially the same matrix as fft(eye(n))/sqrt(n)!
|
k = 4
| Helmert matrix: a permutation of a lower Hessenberg matrix, whose first row is ones(1:n)/sqrt(n).
|
k = 5
| Q(i,j) = sin(2*pi*(i-1)*(j-1)/n) + cos(2*pi*(i-1)*(j-1)/n)
Symmetric matrix arising in the Hartley transform.
|
K = 6
| Q(i,j) = sqrt(2/n)*cos((i-1/2)*(j-1/2)*pi/n)
Symmetric matrix arising as a discrete cosine transform.
|
k = -1
| Q(i,j) = cos((i-1)*(j-1)*pi/(n-1))
Chebyshev Vandermonde-like matrix, based on extrema of T(n-1).
|
k = -2
| Q(i,j) = cos((i-1)*(j-1/2)*pi/n))
Chebyshev Vandermonde-like matrix, based on zeros of T(n).
|
parter--Toeplitz matrix with singular values near pi
C = gallery('parter',n) returns the matrix C such that C(i,j) = 1/(i-j+0.5).
C is a Cauchy matrix and a Toeplitz matrix. Most of the singular values of C are very close to pi.
pei--Pei matrix
A = gallery('pei',n,alpha), where alpha is a scalar, returns the symmetric matrix alpha*eye(n) + ones(n). The default for alpha is 1. The matrix is singular for alpha equal to either 0 or -n.
poisson--Block tridiagonal matrix from Poisson's equation (sparse)
A = gallery('poisson',n) returns the block tridiagonal (sparse) matrix of order n^2 resulting from discretizing Poisson's equation with the 5-point operator on an n-by-n mesh.
prolate--Symmetric, ill-conditioned Toeplitz matrix
A = gallery('prolate',n,w) returns the n-by-n prolate matrix with parameter w. It is a symmetric Toeplitz matrix.
If 0 <> then A is positive definite
- The eigenvalues of
A are distinct, lie in (0,1), and tend to cluster around 0 and 1. - The default value of
w is 0.25.
randcolu -- Random matrix with normalized cols and specified singular values
A = gallery('randcolu',n) is a random n-by-n matrix with columns of unit 2-norm, with random singular values whose squares are from a uniform distribution.
A'*A is a correlation matrix of the form produced by gallery('randcorr',n).
gallery('randcolu',x) where x is an n-vector (n > 1), produces a random n-by-n matrix having singular values given by the vector x. The vector x must have nonnegative elements whose sum of squares is n.
gallery('randcolu',x,m) where m >= n, produces an m-by-n matrix.
gallery('randcolu',x,m,k) provides a further option:
k = 0
| diag(x) is initially subjected to a random two-sided orthogonal transformation, and then a sequence of Givens rotations is applied (default).
|
k = 1
| The initial transformation is omitted. This is much faster, but the resulting matrix may have zero entries.
|
randcorr -- Random correlation matrix with specified eigenvalues
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 semidefinite matrix with 1s on the diagonal
gallery('randcorr',x) produces a random correlation matrix having eigenvalues given by the vector x, where length(x) > 1. The vector x must have nonnegative elements summing to length(x).
gallery('randcorr',x,k) provides a further option:
k = 0
| The diagonal matrix of eigenvalues is initially subjected to a random orthogonal similarity transformation, and then a sequence of Givens rotations is applied (default).
|
k = 1
| The initial transformation is omitted. This is much faster, but the resulting matrix may have some zero entries.
|
For more information, see:
[1] Bendel, R. B. and M. R. Mickey, "Population Correlation Matrices for Sampling Experiments," Commun. Statist. Simulation Comput., B7, 1978, pp. 163-182.
[2] Davies, P. I. and N. J. Higham, "Numerically Stable Generation of Correlation Matrices and Their Factors," BIT, Vol. 40, 2000, pp. 640-651.
randhess--Random, orthogonal upper Hessenberg matrix
H = gallery('randhess',n) returns an n-by-n real, random, orthogonal upper Hessenberg matrix.
H = gallery('randhess',x) if x is an arbitrary, real, length n vector with n > 1, constructs H nonrandomly using the elements of x as parameters.
Matrix H is constructed via a product of n-1 Givens rotations.
rando--Random matrix composed of elements -1, 0 or 1
A = gallery('rando',n,k) returns a random n-by-n matrix with elements from one of the following discrete distributions:
k = 1
| A(i,j) = 0 or 1 with equal probability (default).
|
k = 2
| A(i,j) = -1 or 1 with equal probability.
|
k = 3
| A(i,j) = -1, 0 or 1 with equal probability.
|
Argument n may be a two-element vector, in which case the matrix is n(1)-by-n(2).
randsvd--Random matrix with preassigned singular values
A = gallery('randsvd',n,kappa,mode,kl,ku) returns a banded (multidiagonal) random matrix of order n with cond(A) = kappa and singular values from the distribution mode. If n is a two-element vector, A is n(1)-by-n(2).
Arguments kl and ku specify the number of lower and upper off-diagonals, respectively, in A. If they are omitted, a full matrix is produced. If only kl is present, ku defaults to kl.
Distribution mode can be:
1
| One large singular value.
|
2
| One small singular value.
|
3
| Geometrically distributed singular values (default).
|
4
| Arithmetically distributed singular values.
|
5
| Random singular values with uniformly distributed logarithm.
|
<>
| If mode is -1, -2, -3, -4, or -5, then randsvd treats mode as abs(mode), except that in the original matrix of singular values the order of the diagonal entries is reversed: small to large instead of large to small.
|
Condition number kappa defaults to sqrt(1/eps). In the special case where kappa <>, A is a random, full, symmetric, positive definite matrix with cond(A) = -kappa and eigenvalues distributed according to mode. Arguments kl and ku, if present, are ignored.
A = gallery('randsvd',n,kappa,mode,kl,ku,method) specifies how the computations are carried out. method = 0 is the default, while method = 1 uses an alternative method that is much faster for large dimensions, even though it uses more flops.
redheff--Redheffer's matrix of 1s and 0s
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.
The Redheffer matrix has these properties:
(n-floor(log2(n)))-1 eigenvalues equal to 1 - A real eigenvalue (the spectral radius) approximately
sqrt(n) - A negative eigenvalue approximately
-sqrt(n) - The remaining eigenvalues are provably "small."
- The Riemann hypothesis is true if and only if
for every epsilon > 0.
Barrett and Jarvis conjecture that "the small eigenvalues all lie inside the unit circle abs(Z) = 1," and a proof of this conjecture, together with a proof that some eigenvalue tends to zero as n tends to infinity, would yield a new proof of the prime number theorem.
riemann--Matrix associated with the Riemann hypothesis
A = gallery('riemann',n) returns an n-by-n matrix for which the Riemann hypothesis is true if and only if
for every 
.
The Riemann matrix is defined by:
where B(i,j) = i-1 if i divides j, and B(i,j) = -1 otherwise.
The Riemann matrix has these properties:
- Each eigenvalue
e(i) satisfies abs(e(i)) <= m-1/m, where m = n+1. i <= e(i) <= i+1 with at most m-sqrt(m) exceptions. - All integers in the interval
(m/3, m/2] are eigenvalues.
ris--Symmetric Hankel matrix
A = gallery('ris',n) returns a symmetric n-by-n Hankel matrix with elements
The eigenvalues of A cluster around 
and 
. This matrix was invented by F.N. Ris.
rosser--Classic symmetric eigenvalue test matrix
A = rosser returns the Rosser matrix. This matrix was a challenge for many matrix eigenvalue algorithms. But the QR algorithm, as perfected by Wilkinson and implemented in MATLAB, has no trouble with it. The matrix is 8-by-8 with integer elements. It has:
- A double eigenvalue
- Three nearly equal eigenvalues
- Dominant eigenvalues of opposite sign
- A zero eigenvalue
- A small, nonzero eigenvalue
smoke--Complex matrix with a 'smoke ring' pseudospectrum
A = gallery('smoke',n) returns an n-by-n matrix with 1's on the superdiagonal, 1 in the (n,1) position, and powers of roots of unity along the diagonal.
A = gallery('smoke',n,1) returns the same except that element A(n,1) is zero.
The eigenvalues of gallery('smoke',n,1) are the nth roots of unity; those of gallery('smoke',n) are the nth roots of unity times 2^(1/n).
toeppd--Symmetric positive definite Toeplitz matrix
A = gallery('toeppd',n,m,w,theta) returns an n-by-n symmetric, positive semi-definite (SPD) Toeplitz matrix composed of the sum of m rank 2 (or, for certain theta, rank 1) SPD Toeplitz matrices. Specifically,
where T(theta(k)) has (i,j) element cos(2*pi*theta(k)*(i-j)).
By default: m = n, w = rand(m,1), and theta = rand(m,1).
toeppen--Pentadiagonal Toeplitz matrix (sparse)
P = gallery('toeppen',n,a,b,c,d,e) returns the n-by-n sparse, pentadiagonal Toeplitz matrix with the diagonals: P(3,1) = a, P(2,1) = b, P(1,1) = c, P(1,2) = d, and P(1,3) = e, where a, b, c, d, and e are scalars.
By default, (a,b,c,d,e) = (1,-10,0,10,1), yielding a matrix of Rutishauser. This matrix has eigenvalues lying approximately on the line segment 2*cos(2*t) + 20*i*sin(t).
tridiag--Tridiagonal matrix (sparse)
A = gallery('tridiag',c,d,e) returns the tridiagonal matrix with subdiagonal c, diagonal d, and superdiagonal e. Vectors c and e must have length(d)-1.
A = gallery('tridiag',n,c,d,e), where c, d, and e are all scalars, yields the Toeplitz tridiagonal matrix of order n with subdiagonal elements c, diagonal elements d, and superdiagonal elements e. This matrix has eigenvalues
where k = 1:n. (see [1].)
A = gallery('tridiag',n) is the same as A = gallery('tridiag',n,-1,2,-1), which is a symmetric positive definite M-matrix (the negative of the second difference matrix).
triw--Upper triangular matrix discussed by Wilkinson and others
A = gallery('triw',n,alpha,k) returns the upper triangular matrix with ones on the diagonal and alphas on the first k >= 0 superdiagonals.
Order n may be a 2-element vector, in which case the matrix is n(1)-by-n(2) and upper trapezoidal.
Ostrowski ["On the Spectrum of a One-parametric Family of Matrices, J. Reine Angew. Math., 1954] shows that
and, for large abs(alpha), cond(gallery('triw',n,alpha)) is approximately abs(alpha)^n*sin(pi/(4*n-2)).
Adding -2^(2-n) to the (n,1) element makes triw(n) singular, as does adding
-2^(1-n) to all the elements in the first column.
vander--Vandermonde matrix
A = gallery('vander',c) returns the Vandermonde matrix whose second to last column is c. The jth column of a Vandermonde matrix is given by A(:,j) = C^(n-j).
wathen--Finite element matrix (sparse, random entries)
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 in this routine.
A = gallery('wathen',nx,ny,1) returns a diagonally scaled matrix such that
where D = diag(diag(A)) for any positive integers nx and ny and any densities rho(nx,ny).
wilk--Various matrices devised or discussed by Wilkinson
[A,b] = gallery('wilk',n) returns a different matrix or linear system depending on the value of n.
n = 3
| Upper triangular system Ux=b illustrating inaccurate solution.
|
n = 4
| Lower triangular system Lx=b, ill-conditioned.
|
n = 5
| hilb(6)(1:5,2:6)*1.8144. A symmetric positive definite matrix.
|
n = 21
| W21+, a tridiagonal matrix. Eigenvalue problem.
|
