Node:Basic Matrix Functions, Next:Matrix Factorizations, Up:Linear Algebra
aa = balance (a, opt) | Loadable Function |
[dd, aa] = balance (a, opt) | Loadable Function |
[cc, dd, aa, bb] = balance (a, b, opt) | Loadable Function |
The eigenvalue balancing option
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) | Function File |
Compute the (two-norm) condition number of a matrix. cond (a) is
defined as norm (a) * norm (inv (a)) , and is computed via a
singular value decomposition.
|
[d, rcond] = det (a) | Loadable Function |
Compute the determinant of a using LAPACK. Return an estimate of the reciprocal condition number if requested. |
dmult (a, b) | Function File |
If a is a vector of length rows (b) , return
diag (a) * b (but computed much more efficiently).
|
dot (x, y, dim) | Function File |
Computes the dot product of two vectors. If x and y are matrices, calculate the dot-product along the first non-singleton dimension. If the optional argument dim is given, calculate the dot-product along this dimension. |
lambda = eig (a) | Loadable Function |
[v, lambda] = eig (a) | Loadable Function |
The eigenvalues (and eigenvectors) of a matrix are computed in a several step process which begins with a Hessenberg decomposition, followed by a Schur decomposition, from which the eigenvalues are apparent. The eigenvectors, when desired, are computed by further manipulations of the Schur decomposition. |
g = givens (x, y) | Loadable Function |
[c, s] = givens (x, y) | Loadable Function |
Return a 2 by 2 orthogonal matrix
g = [c s; -s' c] such that
g [x; y] = [*; 0] with x and y scalars.
For example,
givens (1, 1) => 0.70711 0.70711 -0.70711 0.70711 |
[x, rcond] = inv (a) | Loadable Function |
[x, rcond] = inverse (a) | Loadable Function |
Compute the inverse of the square matrix a. Return an estimate of the reciprocal condition number if requested, otherwise warn of an ill-conditioned matrix if the reciprocal condition number is small. |
norm (a, p) | Function File |
Compute the p-norm of the matrix a. If the second argument is
missing, p = 2 is assumed.
If a is a matrix:
If a is a vector or a scalar:
|
null (a, tol) | Function File |
Return an orthonormal basis of the null space of a.
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 |
orth (a, tol) | Function File |
Return an orthonormal basis of the range space of a.
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 |
pinv (x, tol) | Loadable Function |
Return the pseudoinverse of x. Singular values less than
tol are ignored.
If the second argument is omitted, it is assumed that
tol = max (size (x)) * sigma_max (x) * eps, where |
rank (a, tol) | Function File |
Compute the rank of a, using the singular value decomposition.
The rank is taken to be the number of singular values of a that
are greater than the specified tolerance tol. If the second
argument is omitted, it is taken to be
tol = max (size (a)) * sigma(1) * eps; where |
trace (a) | Function File |
Compute the trace of a, sum (diag (a)) .
|