A sufficient ( but not necessary ) condition for the method to converge is that the matrix " A " is strictly or irreducibly diagonally dominant.
12.
As the coefficients of the discretised equation are always positive hence satisfying the requirements for boundedness and also the coefficient matrix is diagonally dominant therefore no irregularities occur in the solution.
13.
We say A is SDD if all of its rows are SDD . "'Weakly diagonally dominant "'( WDD ) is defined with \ geq instead.
14.
P-SHAKE computes and updates a pre-conditioner which is applied to the constraint gradients before the SHAKE iteration, causing the Jacobian \ mathbf J _ \ sigma to become diagonal or strongly diagonally dominant.
15.
*PM : proof of determinant lower bound of a strict diagonally dominant matrix, id = 9304 new !-- WP guess : proof of determinant lower bound of a strict diagonally dominant matrix-- Status:
16.
*PM : proof of determinant lower bound of a strict diagonally dominant matrix, id = 9304 new !-- WP guess : proof of determinant lower bound of a strict diagonally dominant matrix-- Status:
17.
For a matrix with polynomial entries, one sensible definition of diagonal dominance is if the highest power of q appearing in each row appears only on the diagonal . ( The evaluations of such a matrix at large values of q are diagonally dominant in the above sense .)
18.
Recalling that an irreducible matrix is one whose associated directed graph is strongly connected, a trivial corollary of the above is that an " irreducibly diagonally dominant matrix " ( i . e ., an irreducible WDD matrix with at least one SDD row ) is nonsingular.
19.
Thomas'algorithm is not stable in general, but is so in several special cases, such as when the matrix is diagonally dominant ( either by rows or columns ) or symmetric positive definite; If stability is required in the general case, Gaussian elimination with partial pivoting ( GEPP ) is recommended instead.
20.
Similarly, an Hermitian strictly diagonally dominant matrix with real positive diagonal entries is positive definite, as it equals to the sum of some Hermitian diagonally dominant matrix A with real non-negative diagonal entries ( which is positive semidefinite ) and xI for some positive real number x ( which is positive definite ).
