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This algorithm takes a finite number of steps to reach a solution and smoothly improves its candidate solution as it goes (so it can find good approximate solutions when cut off at a reasonable number of iterations), but is very slow in practice, owing largely to the computation of the pseudoinverse ((A P) T A P) −1. [1]
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges.
The solution is obtained iteratively via (+) = (), where the matrix is decomposed into a lower triangular component , and a strictly upper triangular component such that = +. [4] More specifically, the decomposition of A {\displaystyle A} into L ∗ {\displaystyle L_{*}} and U {\displaystyle U} is given by:
a k is the "contrapoint," i.e., a point such that f(a k) and f(b k) have opposite signs, so the interval [a k, b k] contains the solution. Furthermore, |f(b k)| should be less than or equal to |f(a k)|, so that b k is a better guess for the unknown solution than a k. b k−1 is the previous iterate (for the first iteration, we set b k−1 = a 0).
For example, in the MATLAB or GNU Octave function pinv, the tolerance is taken to be t = ε⋅max(m, n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implementation ...
NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
If these fall below some tolerance, then the cycle is stopped and the lowest point in the simplex returned as a proposed optimum. Note that a very "flat" function may have almost equal function values over a large domain, so that the solution will be sensitive to the tolerance. Nash adds the test for shrinkage as another termination criterion. [6]
If is an Toeplitz matrix, then the system has at most only unique values, rather than . We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case. We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.