octave: Famous Matrices
16.4 Famous Matrices
====================
The following functions return famous matrix forms.
-- : gallery (NAME)
-- : gallery (NAME, ARGS)
Create interesting matrices for testing.
-- : C = gallery ("cauchy", X)
-- : C = gallery ("cauchy", X, Y)
Create a Cauchy matrix.
-- : C = gallery ("chebspec", N)
-- : C = gallery ("chebspec", N, K)
Create a Chebyshev spectral differentiation matrix.
-- : C = gallery ("chebvand", P)
-- : C = gallery ("chebvand", M, P)
Create a Vandermonde-like matrix for the Chebyshev polynomials.
-- : A = gallery ("chow", N)
-- : A = gallery ("chow", N, ALPHA)
-- : A = gallery ("chow", N, ALPHA, DELTA)
Create a Chow matrix – a singular Toeplitz lower Hessenberg matrix.
-- : C = gallery ("circul", V)
Create a circulant matrix.
-- : A = gallery ("clement", N)
-- : A = gallery ("clement", N, K)
Create a tridiagonal matrix with zero diagonal entries.
-- : C = gallery ("compar", A)
-- : C = gallery ("compar", A, K)
Create a comparison matrix.
-- : A = gallery ("condex", N)
-- : A = gallery ("condex", N, K)
-- : A = gallery ("condex", N, K, THETA)
Create a ‘counterexample’ matrix to a condition estimator.
-- : A = gallery ("cycol", [M N])
-- : A = gallery ("cycol", N)
-- : A = gallery (..., K)
Create a matrix whose columns repeat cyclically.
-- : [C, D, E] = gallery ("dorr", N)
-- : [C, D, E] = gallery ("dorr", N, THETA)
-- : A = gallery ("dorr", ...)
Create a diagonally dominant, ill-conditioned, tridiagonal matrix.
-- : A = gallery ("dramadah", N)
-- : A = gallery ("dramadah", N, K)
Create a (0, 1) matrix whose inverse has large integer entries.
-- : A = gallery ("fiedler", C)
Create a symmetric Fiedler matrix.
-- : A = gallery ("forsythe", N)
-- : A = gallery ("forsythe", N, ALPHA)
-- : A = gallery ("forsythe", N, ALPHA, LAMBDA)
Create a Forsythe matrix (a perturbed Jordan block).
-- : F = gallery ("frank", N)
-- : F = gallery ("frank", N, K)
Create a Frank matrix (ill-conditioned eigenvalues).
-- : C = gallery ("gcdmat", N)
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)’.
-- : A = gallery ("gearmat", N)
-- : A = gallery ("gearmat", N, I)
-- : A = gallery ("gearmat", N, I, J)
Create a Gear matrix.
-- : G = gallery ("grcar", N)
-- : G = gallery ("grcar", N, K)
Create a Toeplitz matrix with sensitive eigenvalues.
-- : A = gallery ("hanowa", N)
-- : A = gallery ("hanowa", N, D)
Create a matrix whose eigenvalues lie on a vertical line in the
complex plane.
-- : V = gallery ("house", X)
-- : [V, BETA] = gallery ("house", X)
Create a householder matrix.
-- : A = gallery ("integerdata", IMAX, [M N ...], J)
-- : A = gallery ("integerdata", IMAX, M, N, ..., J)
-- : A = gallery ("integerdata", [IMIN, IMAX], [M N ...], J)
-- : A = gallery ("integerdata", [IMIN, IMAX], M, N, ..., J)
-- : A = gallery ("integerdata", ..., "CLASS")
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".
-- : A = gallery ("invhess", X)
-- : A = gallery ("invhess", X, Y)
Create the inverse of an upper Hessenberg matrix.
-- : A = gallery ("invol", N)
Create an involutory matrix.
-- : A = gallery ("ipjfact", N)
-- : A = gallery ("ipjfact", N, K)
Create a Hankel matrix with factorial elements.
-- : A = gallery ("jordbloc", N)
-- : A = gallery ("jordbloc", N, LAMBDA)
Create a Jordan block.
-- : U = gallery ("kahan", N)
-- : U = gallery ("kahan", N, THETA)
-- : U = gallery ("kahan", N, THETA, PERT)
Create a Kahan matrix (upper trapezoidal).
-- : A = gallery ("kms", N)
-- : A = gallery ("kms", N, RHO)
Create a Kac-Murdock-Szego Toeplitz matrix.
-- : B = gallery ("krylov", A)
-- : B = gallery ("krylov", A, X)
-- : B = gallery ("krylov", A, X, J)
Create a Krylov matrix.
-- : A = gallery ("lauchli", N)
-- : A = gallery ("lauchli", N, MU)
Create a Lauchli matrix (rectangular).
-- : A = gallery ("lehmer", N)
Create a Lehmer matrix (symmetric positive definite).
-- : T = gallery ("lesp", N)
Create a tridiagonal matrix with real, sensitive eigenvalues.
-- : A = gallery ("lotkin", N)
Create a Lotkin matrix.
-- : A = gallery ("minij", N)
Create a symmetric positive definite matrix MIN(i,j).
-- : A = gallery ("moler", N)
-- : A = gallery ("moler", N, ALPHA)
Create a Moler matrix (symmetric positive definite).
-- : [A, T] = gallery ("neumann", N)
Create a singular matrix from the discrete Neumann problem
(sparse).
-- : A = gallery ("normaldata", [M N ...], J)
-- : A = gallery ("normaldata", M, N, ..., J)
-- : A = gallery ("normaldata", ..., "CLASS")
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".
-- : Q = gallery ("orthog", N)
-- : Q = gallery ("orthog", N, K)
Create orthogonal and nearly orthogonal matrices.
-- : A = gallery ("parter", N)
Create a Parter matrix (a Toeplitz matrix with singular values near
pi).
-- : P = gallery ("pei", N)
-- : P = gallery ("pei", N, ALPHA)
Create a Pei matrix.
-- : A = gallery ("Poisson", N)
Create a block tridiagonal matrix from Poisson’s equation (sparse).
-- : A = gallery ("prolate", N)
-- : A = gallery ("prolate", N, W)
Create a prolate matrix (symmetric, ill-conditioned Toeplitz
matrix).
-- : H = gallery ("randhess", X)
Create a random, orthogonal upper Hessenberg matrix.
-- : A = gallery ("rando", N)
-- : A = gallery ("rando", N, K)
Create a random matrix with elements -1, 0 or 1.
-- : A = gallery ("randsvd", N)
-- : A = gallery ("randsvd", N, KAPPA)
-- : A = gallery ("randsvd", N, KAPPA, MODE)
-- : A = gallery ("randsvd", N, KAPPA, MODE, KL)
-- : A = gallery ("randsvd", N, KAPPA, MODE, KL, KU)
Create a random matrix with pre-assigned singular values.
-- : A = gallery ("redheff", N)
Create a zero and ones matrix of Redheffer associated with the
Riemann hypothesis.
-- : A = gallery ("riemann", N)
Create a matrix associated with the Riemann hypothesis.
-- : A = gallery ("ris", N)
Create a symmetric Hankel matrix.
-- : A = gallery ("smoke", N)
-- : A = gallery ("smoke", N, K)
Create a complex matrix, with a ‘smoke ring’ pseudospectrum.
-- : T = gallery ("toeppd", N)
-- : T = gallery ("toeppd", N, M)
-- : T = gallery ("toeppd", N, M, W)
-- : T = gallery ("toeppd", N, M, W, THETA)
Create a symmetric positive definite Toeplitz matrix.
-- : P = gallery ("toeppen", N)
-- : P = gallery ("toeppen", N, A)
-- : P = gallery ("toeppen", N, A, B)
-- : P = gallery ("toeppen", N, A, B, C)
-- : P = gallery ("toeppen", N, A, B, C, D)
-- : P = gallery ("toeppen", N, A, B, C, D, E)
Create a pentadiagonal Toeplitz matrix (sparse).
-- : A = gallery ("tridiag", X, Y, Z)
-- : A = gallery ("tridiag", N)
-- : A = gallery ("tridiag", N, C, D, E)
Create a tridiagonal matrix (sparse).
-- : T = gallery ("triw", N)
-- : T = gallery ("triw", N, ALPHA)
-- : T = gallery ("triw", N, ALPHA, K)
Create an upper triangular matrix discussed by Kahan, Golub, and
Wilkinson.
-- : A = gallery ("uniformdata", [M N ...], J)
-- : A = gallery ("uniformdata", M, N, ..., J)
-- : A = gallery ("uniformdata", ..., "CLASS")
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".
-- : A = gallery ("wathen", NX, NY)
-- : A = gallery ("wathen", NX, NY, K)
Create the Wathen matrix.
-- : [A, B] = gallery ("wilk", N)
Create various specific matrices devised/discussed by Wilkinson.
-- : hadamard (N)
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’ and ‘Hn(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 with
‘abs (A(i, j)) <= 1’.
• Multiplying any row or column by -1 and the matrix will remain
a Hadamard matrix.
DONTPRINTYET See also: 
compan XREFcompan, hankel XREFhankel, *noteDONTPRINTYET See also: compan XREFcompan, hankel XREFhankel,
toeplitz XREFtoeplitz.
-- : hankel (C)
-- : hankel (C, R)
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
DONTPRINTYET See also: hadamard XREFhadamard, *notetoeplitz:
DONTPRINTYET See also: hadamard XREFhadamard, toeplitz
XREFtoeplitz.
-- : hilb (N)
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+05
See also: invhilb XREFinvhilb.
-- : invhilb (N)
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 XREFhilb.
-- : magic (N)
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 a scalar greater than or equal to 3. If you supply
N less than 3, magic returns either a nonmagic square, or else the
degenerate magic squares 1 and [].
-- : pascal (N)
-- : pascal (N, T)
Return the Pascal matrix of order N if ‘T = 0’.
The default value of T is 0.
When ‘T = 1’, return the pseudo-lower triangular Cholesky factor of
the Pascal matrix (The sign of some columns may be negative). This
matrix is its own inverse, that is ‘pascal (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 of ‘pascal (N,
1)’, which is the cube root of the identity matrix. That is,
‘pascal (N, 2) ^ 3 == eye (N)’.
See also: chol XREFchol.
-- : rosser ()
Return the Rosser matrix.
This is a difficult test case used to evaluate eigenvalue
algorithms.
See also: wilkinson XREFwilkinson, eig XREFeig.
-- : toeplitz (C)
-- : toeplitz (C, R)
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 XREFhankel.
-- : vander (C)
-- : vander (C, N)
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) 1
See also: polyfit XREFpolyfit.
-- : wilkinson (N)
Return the Wilkinson matrix of order N.
Wilkinson matrices are symmetric and tridiagonal with pairs of
nearly, but not exactly, equal eigenvalues. They are useful in
testing the behavior and performance of eigenvalue solvers.
See also: rosser XREFrosser, eig XREFeig.
Info Catalog
octave: Special Utility Matrices
octave: Matrix Manipulation