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The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations A T A and right-hand side vector A T b, since A T A is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGN or CGNR). A T Ax = A T b
The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1: ((+)) < A sufficient (but not necessary) condition for the method to converge is that the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute ...
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
To solve a linear system Ax = b with a preconditioner K = K 1 K 2 ≈ A, preconditioned BiCGSTAB starts with an initial guess x 0 and proceeds as follows: r 0 = b − Ax 0 Choose an arbitrary vector r̂ 0 such that ( r̂ 0 , r 0 ) ≠ 0 , e.g., r̂ 0 = r 0
Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in his work dated 1910. It is similar to the Jacobi and Gauss–Seidel method. We seek the solution to a set of linear equations, expressed in matrix terms as =.
where is the matrix formed by replacing the i-th column of A by the column vector b. A more general version of Cramer's rule [ 13 ] considers the matrix equation A X = B {\displaystyle AX=B}
Given two square complex matrices A and B, of size n and m, and a matrix C of size n by m, then one can ask when the following two square matrices of size n + m are similar to each other: [] and []. The answer is that these two matrices are similar exactly when there exists a matrix X such that AX − XB = C .
The linear transformation mapping x to Ax is bijective; that is, the equation Ax = b has exactly one solution for each b in K n. (Here, "bijective" can equivalently be replaced with "injective" or "surjective") The columns of A form a basis of K n. (In this statement, "basis" can equivalently be replaced with either "linearly independent set ...