Basic Matrix Functions Next: Up: Linear Algebra


Basic Matrix Functions

aa = balance (a opt) Loadable Function
[dd aa] = balance (a, opt) Loadable Function
[cc dd, aa, bb] = balance (a, b, opt) Loadable Function

[dd aa] = balance (a) returns aa = dd \ a * dd. aa is a matrix whose row and 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 = cc*a*dd and bb = 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"
No balancing; arguments copied transformation(s) set to identity.
"P" "p"
Permute argument(s) to isolate eigenvalues where possible.
"S" "s"
Scale to improve accuracy of computed eigenvalues.
"B" "b"
Permute and scale in that order. Rows/columns of a (and b) that are isolated by permutation are not scaled. This is the default behavior.

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:

p = 1
1-norm the largest column sum of the absolute values of a.
p = 2
Largest singular value of a.
p = Inf
Infinity norm the largest row sum of the absolute values of a.
p = "fro"
Frobenius norm of a sqrt (sum (diag (a' * a))).

If a is a vector or a scalar:

p = Inf
max (abs (a)).
p = -Inf
min (abs (a)).
other
p-norm of a (sum (abs (a) .^ p)) ^ (1/p).

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 sigma_max (x) is the maximal singular value of x.

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 eps is machine precision and sigma(1) is the largest singular value of a.

trace (a) Function File
Compute the trace of a sum (diag (a)).