The BiConjugate Gradient STABilization (BiCGSTAB) and Generalized Minimum RESidual (GMRES) methods for calculating linear sparse systems
of equations are presented, and the application of these methods to the calculation of three-dimensional fields is shown.

 Fortran codes are based on published algorithms:

• the algorithm BCGSTAB is presented in Xianyi Zeng's lectures : Algorithm 2.3 Lecture Note 7 https://utminers.utep.edu/xzeng/2017spring_math5330/MATH_5330_Computational_Methods_of_Linear_Algebra_files/ln07.pdf
• the algorithm GMRES is presented in Wikipedia (Matlab/Octave version): https://en.wikipedia.org/wiki/Generalized_minimal_residual_method

Repository Structure

  The /SOLVERS directory contains :

• /src
  • file SOLVERS_base.f90 contains an implementation of the BCGSTAB or GMRES methods for solving a linear sparse system: Ax = B.
  • file SOLVERS.f90 oriented on implementation of BCGSTAB or GMRES for calculating 3D fields. Explanations are given below.
  • file utilites.f90 contains subroutines for calculating the product of a sparse matrix by a vector,
    converting a matrix from full format to sparse CSR format, inverting a matrix and a subroutine for outputting
    calculation results to csv and vtk files. Paraview may be required to visualization vtk (free download: https://www.paraview.org/).
• /example - contains example_SOLVERS_base.f90 A simple example of solve a sparse system Ax=B using GMRES or BCGSTAB methods.
  A small matrix is entered in full format, which is automatically converted to CSR format.
• /test - contains test_SOLVERS.f90 this is a test for solving systems of equations for calculating
  three-dimensional vector fields using the BCGSTAB or GMRES methods. Here the field equations are presented
  in the format: $L(X) = B,$ where $L()$ is a linear operator corresponding to the finite-difference approximation of the PDE.
  This version uses a finite-difference approximation of the Laplace equation $L(X) = Laplacian(X)$ and
  a finite-difference approximation of the equation $L(X) = grad(div(X)) - Laplacian(X)$ (this is equivalent to $L(X)=Curl(Curl(X))$
  without using a matrix. Alternatively, for the Laplacian, on can use a sparse matrix,
  which is automatically generated to solve the equation $Ax=B$.
    All files contain comments.