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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 ...
We seek the solution to a set of linear equations, expressed in matrix terms as =. The Richardson iteration is (+) = + (()), where is a scalar ...
In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations A x = b . {\displaystyle Ax=b.\,} Unlike the conjugate gradient method , this algorithm does not require the matrix A {\displaystyle A} to be self-adjoint , but instead one needs to perform ...
When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]
The minimum can be computed using a QR decomposition: find an (n + 1)-by-(n + 1) orthogonal matrix Ω n and an (n + 1)-by-n upper triangular matrix ~ such that ~ = ~. The triangular matrix has one more row than it has columns, so its bottom row consists of zero.
Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b', where b' is the projection of b onto the column space of A. The best ...
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}