fortran-lang
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diff --git a/‎example/linalg/CMakeLists.txt‎
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diff --git a/‎example/linalg/example_solve_cg.f90‎
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+---
+title: linalg_iterative_solvers
+---
+
+# The `stdlib_linalg_iterative_solvers` module
+
+[TOC]
+
+## Introduction
+
+The `stdlib_linalg_iterative_solvers` module provides base implementations for known iterative solver methods. Each method is exposed with two procedure flavors: 
+
+* A `solve_<method>_kernel` which holds the method's base implementation. The linear system argument is defined through a `linop` derived type which enables extending the method for implicit or unknown (by `stdlib`) matrices or to complex scenarios involving distributed parallelism for which the user shall extend the `inner_product` and/or matrix-vector product to account for parallel syncrhonization.
+
+* A `solve_<method>` which proposes an off-the-shelf ready to use interface for `dense` and `CSR_<kind>_type` matrices for all `real` kinds.
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### The `linop` derived type
+
+The `stdlib_linop_<kind>_type` derive type is an auxiliary class enabling to abstract the definition of the linear system and the actual implementation of the solvers.
+
+#### Type-bound procedures
+
+The following type-bound procedure pointers enable customization of the solver:
+
+##### `matvec`
+
+Proxy procedure for the matrix-vector product $y = alpha * op(M) * x + beta * y$.
+
+#### Syntax
+
+`call ` [[stdlib_iterative_solvers(module):matvec(interface)]] ` (x,y,alpha,beta,op)`
+
+###### Class
+
+Subroutine
+
+###### Argument(s)
+
+`x`: 1-D array of `real(<kind>)`. This argument is `intent(in)`.
+
+`y`: 1-D array of `real(<kind>)`. This argument is `intent(inout)`.
+
+`alpha`: scalar of `real(<kind>)`. This argument is `intent(in)`.
+
+`beta`: scalar of `real(<kind>)`. This argument is `intent(in)`.
+
+`op`: `character(1)` scalar which can be have any of the following values: `N` (no transpose), `T` (transpose) or `H` (conjugate transpose). This argument is `intent(in)`.
+
+##### `inner_product`
+
+Proxy procedure for the `dot_product`.
+
+#### Syntax
+
+`res = ` [[stdlib_iterative_solvers(module):inner_product(interface)]] ` (x,y)`
+
+###### Class
+
+Pure function
+
+###### Argument(s)
+
+`x`: 1-D array of `real(<kind>)`. This argument is `intent(in)`.
+
+`y`: 1-D array of `real(<kind>)`. This argument is `intent(in)`.
+
+###### Output value or Result value
+
+The output is a scalar of `type` and `kind` same as to that of `x` and `y`.
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### The `solver_workspace` derived type
+
+The `stdlib_solver_workspace_<kind>_type` derive type is an auxiliary class enabling to hold the data associated to the working arrays needed by the solvers to operate.
+
+#### Type-bound procedures
+
+- `callback`: null pointer procedure enabling to pass a callback at each iteration to check on the solvers status.
+
+##### Class
+
+Subroutine
+
+##### Argument(s)
+
+`x`: 1-D array of `real(<kind>)` type with the current state of the solution vector. This argument is `intent(in)` as it should not be modified by the callback.
+
+`norm_sq`: scalar of `real(<kind>)` type representing the squared norm of the residual at the current iteration. This argument is `intent(in)`.
+
+`iter`: scalar of `integer` type giving the current iteration counter. This argument is `intent(in)`.
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `stdlib_solve_cg_kernel` subroutine
+
+#### Description
+
+Implements the Conjugate Gradient (CG) method for solving the linear system \( Ax = b \), where \( A \) is a symmetric positive-definite linear operator defined via the `stdlib_linop` type. This is the core implementation, allowing flexibility for custom matrix types or parallel environments.
+
+#### Syntax
+
+`call ` [[stdlib_iterative_solvers(module):stdlib_solve_cg_kernel(interface)]] ` (A, b, x, tol, maxiter, workspace)`
+
+#### Status
+
+Experimental
+
+#### Class
+
+Subroutine
+
+#### Argument(s)
+
+`A`: `class(stdlib_linop_<kind>_type)` defining the linear operator. This argument is `intent(in)`.
+
+`b`: 1-D array of `real(<kind>)` defining the loading conditions of the linear system. This argument is `intent(in)`.
+
+`x`: 1-D array of `real(<kind>)` which serves as the input initial guess and the output solution. This argument is `intent(inout)`.
+
+`rtol` and `atol`: scalars of type `real(<kind>)` specifying the convergence test. For convergence, the following criterion is used \( || b - Ax ||^2 <= max(rtol^2 * || b ||^2 , atol^2 ) \). These arguments are `intent(in)`.
+
+`maxiter`: scalar of type `integer` defining the maximum allowed number of iterations. This argument is `intent(in)`.
+
+`workspace`: scalar derived type of `type(stdlib_solver_workspace_<kind>_type)` holding the work array for the solver. This argument is `intent(inout)`.
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `solve_cg` subroutine
+
+#### Description
+
+Provides a user-friendly interface to the CG method for solving \( Ax = b \), supporting `dense` and `CSR_<kind>_type` matrices. It handles workspace allocation and optional parameters for customization.
+
+#### Syntax
+
+`call ` [[stdlib_iterative_solvers(module):solve_cg(interface)]] ` (A, b, x [, di, rtol, atol, maxiter, restart, workspace])`
+
+#### Status
+
+Experimental
+
+#### Class
+
+Subroutine
+
+#### Argument(s)
+
+`A`: `dense` or `CSR_<kind>_type` matrix defining the linear system. This argument is `intent(in)`.
+
+`b`: 1-D array of `real(<kind>)` defining the right-hand-side (or loading) of the linear system. This argument is `intent(in)`.
+
+`x`: 1-D array of `real(<kind>)` which serves as the input initial guess and as the output solution. This argument is `intent(inout)`.
+
+`di` (optional): 1-D mask array of type `logical(int8)` defining the degrees of freedom subject to dirichlet boundary conditions. The actual boundary conditions values should be stored in the `b` load array. This argument is `intent(in)`.
+
+`rtol` and `atol` (optional): scalars of type `real(<kind>)` specifying the convergence test. For convergence, the following criterion is used \( || b - Ax ||^2 <= max(rtol^2 * || b ||^2 , atol^2 ) \). Defaults values are `rtol=1.e-5` and `atol=epsilon(1._<kind>)`. These arguments are `intent(in)`.
+
+`maxiter` (optional): scalar of type `integer` defining the maximum allowed number of iterations. If no value is given, a default of `N` is set, where `N = size(b)`. This argument is `intent(in)`.
+
+`workspace` (optional): scalar derived type of `type(stdlib_solver_workspace_<kind>_type)` holding the work array for the solver. If the user passes its own `workspace`, then a pointer is set internally to it. Otherwise, memory will be internally allocated and deallocated before exiting the procedure. This argument is `intent(inout)`.
+
+#### Example
+
+```fortran
+{!example/linalg/example_solve_cg.f90!}
+```
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `stdlib_solve_pcg_kernel` subroutine
+
+#### Description
+
+Implements the Preconditioned Conjugate Gradient (PCG) method for solving the linear system \( Ax = b \), where \( A \) is a symmetric positive-definite linear operator defined via the `stdlib_linop` type. This is the core implementation, allowing flexibility for custom matrix types or parallel environments.
+
+#### Syntax
+
+`call ` [[stdlib_iterative_solvers(module):stdlib_solve_cg_kernel(interface)]] ` (A, M, b, x, tol, maxiter, workspace)`
+
+#### Status
+
+Experimental
+
+#### Class
+
+Subroutine
+
+#### Argument(s)
+
+`A`: `class(stdlib_linop_<kind>_type)` defining the linear operator. This argument is `intent(in)`.
+
+`M`: `class(stdlib_linop_<kind>_type)` defining the preconditioner linear operator. This argument is `intent(in)`.
+
+`b`: 1-D array of `real(<kind>)` defining the loading conditions of the linear system. This argument is `intent(in)`.
+
+`x`: 1-D array of `real(<kind>)` which serves as the input initial guess and the output solution. This argument is `intent(inout)`.
+
+`rtol` and `atol` (optional): scalars of type `real(<kind>)` specifying the convergence test. For convergence, the following criterion is used \( || b - Ax ||^2 <= max(rtol^2 * || b ||^2 , atol^2 ) \). These arguments are `intent(in)`.
+
+`maxiter`: scalar of type `integer` defining the maximum allowed number of iterations. This argument is `intent(in)`.
+
+`workspace`: scalar derived type  of `type(stdlib_solver_workspace_<kind>_type)` holding the work array for the solver. This argument is `intent(inout)`.
+
+#### Example
+
+```fortran
+{!example/linalg/example_solve_custom.f90!}
+```
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `stdlib_solve_pcg` subroutine
+
+#### Description
+
+Provides a user-friendly interface to the PCG method for solving \( Ax = b \), supporting `dense` and `CSR_<kind>_type` matrices. It supports optional preconditioners and handles workspace allocation.
+
+#### Syntax
+
+`call ` [[stdlib_iterative_solvers(module):stdlib_solve_pcg(interface)]] ` (A, b, x [, di, tol, maxiter, restart, precond, M, workspace])`
+
+#### Status
+
+Experimental
+
+#### Class
+
+Subroutine
+
+#### Argument(s)
+
+`A`: `dense` or `CSR_<kind>_type` matrix defining the linear system. This argument is `intent(in)`.
+
+`b`: 1-D array of `real(<kind>)` defining the loading conditions of the linear system. This argument is `intent(in)`.
+
+`x`: 1-D array of `real(<kind>)` which serves as the input initial guess and the output solution. This argument is `intent(inout)`.
+
+`di` (optional): 1-D mask array of type `logical(int8)` defining the degrees of freedom subject to dirichlet boundary conditions. The actual boundary conditions values should be stored in the `b` load array. This argument is `intent(in)`.
+
+`rtol` and `atol` (optional): scalars of type `real(<kind>)` specifying the convergence test. For convergence, the following criterion is used \( || b - Ax ||^2 <= max(rtol^2 * || b ||^2 , atol^2 ) \). Defaults values are `rtol=1.e-5` and `atol=epsilon(1._<kind>)`. These arguments are `intent(in)`.
+
+`maxiter` (optional): scalar of type `integer` defining the maximum allowed number of iterations. If no value is given, a default of `N` is set, where `N = size(b)`. This argument is `intent(in)`.
+
+`precond` (optional): scalar of type `integer` enabling to switch among the default preconditioners available with the following enum (`pc_none`, `pc_jacobi`). If no value is given, no preconditionning will be applied. This argument is `intent(in)`.
+
+`M` (optional): scalar derived type of `class(stdlib_linop_<kind>_type)` defining a custom preconditioner linear operator. If given, `precond` will have no effect, a pointer is set to this custom preconditioner.
+
+`workspace` (optional): scalar derived type of `type(stdlib_solver_workspace_<kind>_type)` holding the work temporal array for the solver. If the user passes its own `workspace`, then internally a pointer is set to it, otherwise, memory will be internally allocated and deallocated before exiting the procedure. This argument is `intent(inout)`.
+
+#### Example
+
+```fortran
+{!example/linalg/example_solve_pcg.f90!}
+```
@@ -41,6 +41,9 @@ ADD_EXAMPLE(get_norm)
 ADD_EXAMPLE(solve1)
 ADD_EXAMPLE(solve2)
 ADD_EXAMPLE(solve3)
+ADD_EXAMPLE(solve_cg)
+ADD_EXAMPLE(solve_pcg)
+ADD_EXAMPLE(solve_custom)
 ADD_EXAMPLE(sparse_from_ijv)
 ADD_EXAMPLE(sparse_data_accessors)
 ADD_EXAMPLE(sparse_spmv)
 
@@ -0,0 +1,17 @@
+program example_solve_cg
+    use stdlib_kinds, only: int8, dp
+    use stdlib_linalg_iterative_solvers, only: stdlib_solve_cg
+
+    real(dp) :: matrix(2,2)
+    real(dp) :: x(2), rhs(2)
+
+    matrix = reshape( [4, 1,&
+                       1, 3] , [2,2])
+
+    x   = dble( [2,1] ) !> initial guess
+    rhs = dble( [1,2] ) !> rhs vector
+
+    call stdlib_solve_cg(matrix, rhs, x, restart=.false.)
+    print *, x !> solution: [0.0909, 0.6364]
+
+end program