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In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
A singular solution y s (x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the ...
are linearly independent for any choice of n distinct points x 1, x 2... x n in Ω. Equivalently, the collection is unisolvent if the matrix F with entries f i (x j) has nonzero determinant: det(F) ≠ 0 for any choice of distinct x j 's in Ω. Unisolvency is a property of vector spaces, not just particular sets of functions.
Moreover, they both take the value when is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim. [8] A matrix with entries in a field is invertible precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the ...
This matrix has variables in its first row and numbers in all other rows, so the determinant is visibly a linear combination of the six monomials of degree at most 2. Also, the resulting polynomial clearly vanishes at the five input points (when (,) = (,)), as the matrix has then a repeated row.
Therefore, to find the unique LU decomposition, it is necessary to put some restriction on L and U matrices. For example, we can conveniently require the lower triangular matrix L to be a unit triangular matrix, so that all the entries of its main diagonal are set to one. Then the system of equations has the following solution:
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edges attached to each vertex. [1]