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This chapter documents the linear algebra functions of Octave. Reference material for many of these options may be found in Golub and Van Loan, Matrix Computations, 2nd Ed., Johns Hopkins, 1989, and in LAPACK Users' Guide, SIAM, 1992.
balance
aa = balance (a, opt) [dd, aa] = balance(a, opt) [dd, aa] = balance (a, opt) [cc, dd, aa, bb] = balance (a, b, opt)
[dd, aa] = balance (a) returns aa = dd \ a * dd.
aa is a matrix whose row/column norms are roughly equal in
magnitude, and dd = p * d, where p is a permutation
matrix and d is a diagonal matrix of powers of two. This allows
the equilibration to be computed without roundoff. Results of
eigenvalue calculation are typically improved by balancing first.
[cc, dd, aa, bb] = balance (a, b) returns aa (bb)
= cc*a*dd (cc*b*dd)), where aa and bb have
non-zero elements of approximately the same magnitude and cc
and dd are permuted diagonal matrices as in dd for
the algebraic eigenvalue problem.
The eigenvalue balancing option opt is selected as follows:
"N", "n"
"P", "p"
"S", "s"
"B", "b"
Algebraic eigenvalue balancing uses standard LAPACK routines.
Generalized eigenvalue problem balancing uses Ward's algorithm (SIAM Journal on Scientific and Statistical Computing, 1981).
cond (a) is
defined as norm (a) * norm (inv (a)), and is computed via a
singular value decomposition.
= eig (a)
[v, lambda] = eig (a)
The eigenvalues (and eigenvectors) of a matrix are computed in a several
step process which begins with a Hessenberg decomposition (see
hess), followed by a Schur decomposition (see schur), from
which the eigenvalues are apparent. The eigenvectors, when desired, are
computed by further manipulations of the Schur decomposition.
See also: hess, schur.
[c, s] = givens (x, y) G = givens (x, y)
G = givens(x, y) returns a
orthogonal matrix G = [c s; -s' c] such that
G [x; y] = [*; 0] (x, y scalars)
p = 2 is assumed.
If a is a matrix:
1
2
Inf
"fro"
sqrt (sum (diag (a' * a))).
If a is a vector or a scalar:
Inf
max (abs (a)).
-Inf
min (abs (a)).
(sum (abs (a) .^ p)) ^ (1/p).
The dimension of the null space is taken as the number of singular values of a not greater than tol. If the argument tol is missing, it is computed as
max (size (a)) * max (svd (a)) * eps
The dimension of the range space is taken as the number of singular values of a greater than tol. If the argument tol is missing, it is computed as
max (size (a)) * max (svd (a)) * eps
If the second argument is omitted, it is assumed that
tol = max (size (X)) * sigma_max (X) * eps,
where sigma_max (X) is the maximal singular value of X.
tol = max (size (a)) * sigma (1) * eps;
where eps is machine precision and sigma is the largest
singular value of a.
sum (diag (a)).
chol (a)
hess (a)
h = hess (a)
[p, h] = hess (a)
The Hessenberg decomposition is usually used as the first step in an
eigenvalue computation, but has other applications as well (see Golub,
Nash, and Van Loan, IEEE Transactions on Automatic Control, 1979. The
Hessenberg decomposition is p * h * p' = a where p is a
square unitary matrix (p' * p = I, using complex-conjugate
transposition) and h is upper Hessenberg
(i >= j+1 => h (i, j) = 0).
lu (a)
a = [1, 2; 3, 4],
[l, u, p] = lu (a)
returns
l = 1.00000 0.00000 0.33333 1.00000 u = 3.00000 4.00000 0.00000 0.66667 p = 0 1 1 0
qr (a)
a = [1, 2; 3, 4],
[q, r] = qr (a)
returns
q = -0.31623 -0.94868 -0.94868 0.31623 r = -3.16228 -4.42719 0.00000 -0.63246
The qr factorization has applications in the solution of least
squares problems
for overdetermined systems of equations (i.e.,
is a tall, thin matrix). The qr factorization is q * r = a
where q is an orthogonal matrix and r is upper triangular.
The permuted qr factorization [q, r, pi] = qr (a) forms
the qr factorization such that the diagonal entries of r
are decreasing in magnitude order. For example, given the matrix
a = [1, 2; 3, 4],
[q, r, pi] = qr(a)
returns
q = -0.44721 -0.89443 -0.89443 0.44721 r = -4.47214 -3.13050 0.00000 0.44721 p = 0 1 1 0
The permuted qr factorization [q, r, pi] = qr (a)
factorization allows the construction of an orthogonal basis of
span (a).
schur
[u, s] = schur (a, opt) opt = "a", "d", or "u"
s = schur (a)
The Schur decomposition is used to compute eigenvalues of a
square matrix, and has applications in the solution of algebraic
Riccati equations in control (see are and dare).
schur always returns
where
is a unitary matrix
is identity)
and
is upper triangular. The eigenvalues of
are the diagonal elements of
If the matrix
is real, then the real Schur decomposition is computed, in which the
matrix
is orthogonal and
is block upper triangular
with blocks of size at most
blocks along the diagonal. The diagonal elements of
(or the eigenvalues of the
blocks, when
appropriate) are the eigenvalues of
and
The eigenvalues are optionally ordered along the diagonal according to
the value of opt. opt = "a" indicates that all
eigenvalues with negative real parts should be moved to the leading
block of
(used in are), opt = "d" indicates that all eigenvalues
with magnitude less than one should be moved to the leading block of
(used in dare), and opt = "u", the default, indicates that
no ordering of eigenvalues should occur. The leading
columns of
always span the
subspace corresponding to the
leading eigenvalues of
svd (a)
The function svd normally returns the vector of singular values.
If asked for three return values, it computes
For example,
svd (hilb (3))
returns
ans = 1.4083189 0.1223271 0.0026873
and
[u, s, v] = svd (hilb (3))
returns
u = -0.82704 0.54745 0.12766 -0.45986 -0.52829 -0.71375 -0.32330 -0.64901 0.68867 s = 1.40832 0.00000 0.00000 0.00000 0.12233 0.00000 0.00000 0.00000 0.00269 v = -0.82704 0.54745 0.12766 -0.45986 -0.52829 -0.71375 -0.32330 -0.64901 0.68867
If given a second argument, svd returns an economy-sized
decomposition, eliminating the unnecessary rows or columns of u or
v.
expm
expm (a)
Returns the exponential of a matrix, defined as the infinite Taylor series The Taylor series is not the way to compute the matrix exponential; see Moler and Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review, 1978. This routine uses Ward's diagonal approximation method with three step preconditioning (SIAM Journal on Numerical Analysis, 1977).
Diagonal approximations are rational polynomials of matrices whose Taylor series matches the first terms of the Taylor series above; direct evaluation of the Taylor series (with the same preconditioning steps) may be desirable in lieu of the approximation when is ill-conditioned.
logm (a)
sqrtm (a)
kron (a, b)
Form the kronecker product of two matrices, defined block by block as
x = [a(i, j) b]
qzhess (a, b)
(a, b). This function returns aa = q * a * z,
bb = q * b * z, q, z orthogonal. For example,
[aa, bb, q, z] = qzhess (a, b)
The Hessenberg-triangular decomposition is the first step in Moler and Stewart's QZ decomposition algorithm. (The QZ decomposition will be included in a later release of Octave.)
Algorithm taken from Golub and Van Loan, Matrix Computations, 2nd edition.
qzval (a, b)
syl (a, b, c)
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