MATLAB Function Reference Volume 1: A - E - Bad Request

Arithmetic Operators + - * / \ ^ '
Algorithm The specific algorithm used for solving the simultaneous linear equations
denoted by X = A\B and X = B/A depends upon the structure of the coefficient
matrix A.
2-14
Matrix Operations Array Operations
2^x Error 2.^x 2
4
8
(x+i∗y)' 1 – 4i 2 – 5i 3 – 6i
(x+i∗y).' 1 + 4i 2 + 5i 3 + 6i
• If A is a triangular matrix, or a permutation of a triangular matrix, then X
can be computed quickly by a permuted backsubstitution algorithm. The
check for triangularity is done for full matrices by testing for zero elements
and for sparse matrices by accessing the sparse data structure. Most
nontriangular matrices are detected almost immediately, so this check
requires a negligible amount of time.
• If A is symmetric, or Hermitian, and has positive diagonal elements, then a
Cholesky factorization is attempted (see chol). If A is found to be positive
definite, the Cholesky factorization attempt is successful and requires less
than half the time of a general factorization. Nonpositive definite matrices
are usually detected almost immediately, so this check also requires little
time. If successful, the Cholesky factorization is
A = R'∗R
where R is upper triangular. The solution X is computed by solving two
triangular systems,
X = R\(R'\B)
If A is sparse, a symmetric minimum degree preordering is applied (see
symmmd and spparms). The algorithm is:
perm = symmmd(A); % Symmetric minimum degree reordering
R = chol(A(perm,perm)); % Cholesky factorization
y = R'\B(perm); % Lower triangular solve
X(perm,:) = R\y; % Upper triangular solve