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Input: initial guess x (0) to the solution, (diagonal dominant) matrix A, right-hand side vector b, convergence criterion Output: solution when convergence is reached Comments: pseudocode based on the element-based formula above k = 0 while convergence not reached do for i := 1 step until n do σ = 0 for j := 1 step until n do if j ≠ i then ...
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
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]
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 =.
The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB. [1] Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, [2] to represent the composition of linear maps ...
Lemma 1. ′ =, where ′ is the differential of . This equation means that the differential of , evaluated at the identity matrix, is equal to the trace.The differential ′ is a linear operator that maps an n × n matrix to a real number.
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 [ 10 ] considers the matrix equation A X = B {\displaystyle AX=B}
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. (There, "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 ...