Note: remember “m” was the dimension of the linear system. using LinearMapsįinally, for calling unrestarted GMRES with a relative tolerance of 1e-8, we do: x_gmres = gmres (Ax ,F ,reltol = 1e - 8 ,restart =m ) Fortunately, we can build this LinearMaps objects from functions like iter_A, with LinearMaps.jl. The IterativeSolvers packages requires that we define a “LinearMap” satisfying some basic characteristics. In my case, I get zero or numbers like 4.6 e-21, so I’m confident I didn’t introduce any typos into my function.įinally, let’s use GMRES to solve this large matrix problem. We can do so in a simple one-liner: xt = rand (m ) Īnd check if the result is comparable to machine accuracy. In this case we would need to test whether the function iter_A effectively computes the matrix product A*x. It is considered good practice to write “tests” for our numerical routines. Given the very simple structure of the matrix under consideration, this can be done quite simply: iter_A (x ) = + x x - 2x + x - 2x + x ] We first need to write a function that computes $ Ax $ for any given vector $ x $. using IterativeSolversįor solving our example problem, we will be relying on GMRES, a general purpose iterative solver of linear systems. The IterativeSolvers.jl package contains a number of popular iterative algorithms for solving linear systems, eigensystems, and singular value problems. Iterative solvers are also very useful in cases where it is possible to speed up the application of $ Ax $ for any given vector $ x $, without explicitly constructing the matrix $ A $. Iterative methodsĪnother alternative for very large matrices is to rely upon iterative solvers. Solving By Elimination: 3 equations in 3 variables. We see that, in the present test case for m=1000, the sparse LU-solve is 13X faster than the dense-matrix version of the same back-substitution routines. So we have a system of equations (that are linear): d 0.2t d 0.5(t6) We can solve it on a graph. Julia will be internally calling the UMFPACK library.įor exmple, let’s just run using LinearAlgebraĪs for timings, this is what I got in my PC: Method We can do an LU factorization of a SparseMatrixCSC object, by resorting to the LinearAlgebra.jl package. Push ! (II ,i ) push ! (JJ ,i - 1 ) push ! (V, 1 ) # Sub-Diagonal end end return II ,JJ ,VĪs for timings, I’ve arrived at the following: MethodĬlearly, dealing with sparse matrices requires some extra care, for optimal performance. Push ! (II ,i ) push ! (JJ ,i ) push ! (V, - 2 ) if i 1 U_ (undef, 0 ) for _ in 1 : 3 ] for i = 1 :m Example ProblemĪs we all learn by example, let’s focus on a classical Laplacian problem in one dimension: In such cases, the dimension of the problem grows with the need of representing then fine-scale features present in a given physical model. ![]() We often arrive at large systems of linear equations when a continuous problem is discretized. ![]() Linear system of the form $ Ax = b $ are ubiquitous in engineering and scientific practice.Īfter the modelling is done, and a discretization strategy is selected, we are usually left with matrix computations.
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