# Iterative Sparse Solvers in Visual Basic QuickStart Sample

Illustrates the use of iterative sparse solvers and preconditioners for efficiently solving large, sparse systems of linear equations in Visual Basic.

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``````Option Infer On

Imports Extreme.Data.Text
Imports Extreme.Mathematics
Imports Extreme.Mathematics.LinearAlgebra
Imports Extreme.Mathematics.LinearAlgebra.IterativeSolvers
Imports Extreme.Mathematics.LinearAlgebra.IterativeSolvers.Preconditioners

' Illustrates the use of iterative sparse solvers for efficiently
' solving large, sparse systems of linear equations using the
' iterative sparse solver and preconditioner classes from the
' Extreme.Mathematics.LinearAlgebra.IterativeSolvers namespace of Extreme Numerics.NET.
Module IterativeSparseSolvers

Sub Main()
' The license is verified at runtime. We're using
' https://numerics.net/trial-key

' This QuickStart Sample illustrates how to solve sparse linear systems
' using iterative solvers.

' IterativeSparseSolver is the base class for all
' iterative solver classes:
Dim solver As IterativeSparseSolver(Of Double)

'
' Non-symmetric systems
'

Console.WriteLine("Non-symmetric systems")

' We load a sparse matrix and right-hand side from a data file:
"..\..\..\..\data\sherman3.mtx"), SparseMatrix(Of Double))
"..\..\..\..\data\sherman3_rhs1.mtx")

Console.WriteLine("Solve Ax = b")
Console.WriteLine("A is {0}x{1} with {2} nonzeros.", A.RowCount, A.ColumnCount, A.NonzeroCount)

' Some solvers are suitable for symmetric matrices only.
' Our matrix is not symmetric, so we need a solver that
' can handle this:

' #Region DOCIterativeSparseSolvers1

solver.Solve(b)
Console.WriteLine("Solved in {0} iterations.", solver.IterationsNeeded)
Console.WriteLine("Estimated error: {0}", solver.SolutionReport.Error)
' #End Region

' Using a preconditioner can improve convergence. You can use
' one of the predefined preconditioners, or supply your own.

' With incomplete LU preconditioner
solver.Preconditioner = New IncompleteLUPreconditioner(Of Double)(A)
solver.Solve(b)
Console.WriteLine("Solved in {0} iterations.", solver.IterationsNeeded)
Console.WriteLine("Estimated error: {0}", solver.EstimatedError)

'
' Symmetrical systems
'

Console.WriteLine("Symmetric systems")

' In this example we solve the Laplace equation on a rectangular grid
' with Dirichlet boundary conditions.

' We create 100 divisions in each direction, giving us 99 interior points
' in each direction:
Const nx = 99
Const ny = 99

' The boundary conditions are just some arbitrary functions.
Dim left = Vector.Create(ny, Function(i) (i / (nx + 1.0)) ^ 2)
Dim right = Vector.Create(ny, Function(i) 1 - (i / (nx + 1.0)))
Dim top = Vector.Create(nx, Function(i) Elementary.SinPi(5 * (i / (nx + 1.0))))
Dim bottom = Vector.Create(nx, Function(i) Elementary.CosPi(5 * (i / (nx + 1.0))))

' We discretize the Laplace operator using the 5 point stencil.
Dim laplacian = Matrix.CreateSparse(Of Double)(nx * ny, nx * ny, 5 * nx * ny)
Dim rhs = Vector.Create(Of Double)(nx * ny)
For j As Integer = 0 To ny - 1
For i As Integer = 0 To nx - 1
Dim ix As Integer = j * nx + i
If (j > 0) Then laplacian(ix, ix - nx) = 0.25
If (i > 0) Then laplacian(ix, ix - 1) = 0.25
laplacian(ix, ix) = -1.0
If (i + 1 < nx) Then laplacian(ix, ix + 1) = 0.25
If (j + 1 < ny) Then laplacian(ix, ix + nx) = 0.25
Next
Next
' We build up the right-hand sides using the boundary conditions:
For i As Integer = 0 To nx - 1
rhs(i) = -0.25 * top(i)
rhs(nx * (ny - 1) + i) = -0.25 * bottom(i)
Next
For j As Integer = 0 To ny - 1
rhs(j * nx) -= 0.25 * left(j)
rhs(j * nx + nx - 1) -= 0.25 * right(j)
Next

Console.WriteLine("A is {0}x{1} with {2} nonzeros.", laplacian.RowCount, laplacian.ColumnCount, laplacian.NonzeroCount)

' Finally, we create an iterative solver suitable for
' symmetric systems...
solver = New QuasiMinimalResidualSolver(Of Double)(laplacian)
' and solve using the right-hand side we just calculated:
solver.Solve(rhs)

Console.WriteLine("Solved in {0} iterations.", solver.IterationsNeeded)
Console.WriteLine("Estimated error: {0}", solver.EstimatedError)

Console.Write("Press Enter key to exit...")