[doc] Update contents in linear_solver.md (#7967)

WIP. Don't merge.

Issue: #7837 
Issue: #7921 

### Brief Summary
Update contents in `linear_solver.md` to reflect the latest changes and
development progress on the linear solver.

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

docs/lang/articles/math/linear_solver.md

@@ -6,12 +6,19 @@ sidebar_position: 3
 
 Solving linear equations is a common task in scientific computing. Taichi provides basic direct and iterative linear solvers for
 various simulation scenarios. Currently, there are two categories of linear solvers available:
-1. Solvers built for `SparseMatrix`
-2. Solvers built for `ti.field`
+1. Solvers built for `SparseMatrix`, including:
+- Direct solver `SparseSolver`
+- Iterative (conjugate-gradient method) solver `SparseCG`
+2. Solvers built for `LinearOperator`
+- Iterative (matrix-free conjugate-gradient method) solver `MatrixfreeCG`
+
+It's important to understand that those solvers are built for specific types of matrices. For example, if you built a coefficient matrix `A` as a `SparseMatrix`, then you can only use `SparseSolver` or `SparseCG` to solve the corresponding linear system. Below we will explain the usage of each type of solvers.
 
 ## Sparse linear solver
-You may want to solve some linear equations using sparse matrices.
-Then, the following steps could help:
+There are two types of linear solvers available for `SparseMatrix`, direct solver and iterative solver.
+
+### Sparse direct solver
+To solve a linear system whose coefficient matrix is a `SparseMatrix` using a direct method, follow the steps below:
 1. Create a `solver` using `ti.linalg.SparseSolver(solver_type, ordering)`. Currently, the factorization types supported on CPU backends are `LLT`, `LDLT`, and `LU`, and supported orderings include `AMD` and `COLAMD`. The sparse solver on CUDA supports the `LLT` factorization type only.
 2. Analyze and factorize the sparse matrix you want to solve using `solver.analyze_pattern(sparse_matrix)` and `solver.factorize(sparse_matrix)`
 3. Call `x = solver.solve(b)`, where `x` is the solution and `b` is the right-hand side of the linear system. On CPU backends, `x` and `b` can be NumPy arrays, Taichi Ndarrays, or Taichi fields. On the CUDA backend, `x` and `b` *must* be Taichi Ndarrays.
@@ -66,8 +73,122 @@ print(f">>>> Computation succeed: {success}")
 # [0.5 0.  0.  0.5]
 # >>>> Computation was successful?: True
 ```
-## Examples
 
 Please have a look at our two demos for more information:
 + [Stable fluid](https://github.com/taichi-dev/taichi/blob/master/python/taichi/examples/simulation/stable_fluid.py): A 2D fluid simulation using a sparse Laplacian matrix to solve Poisson's pressure equation.
 + [Implicit mass spring](https://github.com/taichi-dev/taichi/blob/master/python/taichi/examples/simulation/implicit_mass_spring.py): A 2D cloth simulation demo using sparse matrices to solve the linear systems.
+
+### Sparse iterative solver
+To solve a linear system whose coefficient matrix is a `SparseMatrix` using a iterative (conjugate-gradient) method, follow the steps below:
+1. Create a `solver` using `ti.linalg.SparseCG(A, b, x0, max_iter, atol)`, where `A` is a `SparseMatrix` that stores the coefficient matrix of the linear system, `b` is the right-hand side of the equations, `x0` is the initial guess and `atol` is the absolute tolerance threshold.
+2. Call `x, exit_code = solver.solve()` to obtain the solution `x` along with the `exit_code` that indicates the status of the solution. `exit_code` should be `True` if the solving was successful. Here is an example:
+
+```python
+import taichi as ti
+
+ti.init(arch=ti.cpu)
+
+n = 4
+
+K = ti.linalg.SparseMatrixBuilder(n, n, max_num_triplets=100)
+b = ti.ndarray(ti.f32, shape=n)
+
+@ti.kernel
+def fill(A: ti.types.sparse_matrix_builder(), b: ti.types.ndarray(), interval: ti.i32):
+    for i in range(n):
+        A[i, i] += 2.0
+        if i % interval == 0:
+            b[i] += 1.0
+
+fill(K, b, 3)
+
+A = K.build()
+print(">>>> Matrix A:")
+print(A)
+print(">>>> Vector b:")
+print(b.to_numpy())
+# outputs:
+# >>>> Matrix A:
+# [2, 0, 0, 0]
+# [0, 2, 0, 0]
+# [0, 0, 2, 0]
+# [0, 0, 0, 2]
+# >>>> Vector b:
+# [1. 0. 0. 1.]
+solver = ti.linalg.SparseCG(A, b)
+x, exit_code = solver.solve()
+print(">>>> Solve sparse linear systems Ax = b with the solution x:")
+print(x)
+print(f">>>> Computation was successful?: {exit_code}")
+# outputs:
+# >>>> Solve sparse linear systems Ax = b with the solution x:
+# [0.5 0.  0.  0.5]
+# >>>> Computation was successful?: True
+```
+Note that the building process of `SparseMatrix` `A` is exactly the same as in the case of `SparseSolver`, the only difference here is that we created a `solver` whose type is `SparseCG` instead of `SparseSolver`.
+
+## Matrix-free iterative solver
+Apart from `SparseMatrix` as an efficient representation of matrices, Taichi also support the `LinearOperator` type, which is a matrix-free representation of matrices.
+Keep in mind that matrices can be seen as a linear transformation from an input vector to a output vector, it is possible to encapsulate the information of a matrice as a `LinearOperator`.
+
+To create a `LinearOperator` in Taichi, we first need to define a kernel that represent the linear transformation:
+```python
+import taichi as ti
+from taichi.linalg import LinearOperator
+
+ti.init(arch=ti.cpu)
+
+@ti.kernel
+def compute_matrix_vector(v:ti.template(), mv:ti.template()):
+    for i in v:
+        mv[i] = 2 * v[i]
+```
+In this case, `compute_matrix_vector` kernel accepts an input vector `v` and calculates the corresponding matrix-vector product `mv`. It is mathematically equal to a matrice whose diagonal elements are all 2. In the case of `n=4`, the equivalent matrice `A` is:
+```python cont
+# >>>> Matrix A:
+# [2, 0, 0, 0]
+# [0, 2, 0, 0]
+# [0, 0, 2, 0]
+# [0, 0, 0, 2]
+```
+Then we can create the `LinearOperator` as follows:
+```python cont
+A = LinearOperator(compute_matrix_vector)
+```
+To solve a system of linear equations represented by this `LinearOperator`, we can use the built-in matrix-free solver `MatrixFreeCG`. Here is a full example:
+
+```python
+import taichi as ti
+import math
+from taichi.linalg import MatrixFreeCG, LinearOperator
+
+ti.init(arch=ti.cpu)
+
+GRID = 4
+Ax = ti.field(dtype=ti.f32, shape=(GRID, GRID))
+x = ti.field(dtype=ti.f32, shape=(GRID, GRID))
+b = ti.field(dtype=ti.f32, shape=(GRID, GRID))
+
+@ti.kernel
+def init():
+    for i, j in ti.ndrange(GRID, GRID):
+        xl = i / (GRID - 1)
+        yl = j / (GRID - 1)
+        b[i, j] = ti.sin(2 * math.pi * xl) * ti.sin(2 * math.pi * yl)
+        x[i, j] = 0.0
+
+@ti.kernel
+def compute_Ax(v: ti.template(), mv: ti.template()):
+    for i, j in v:
+        l = v[i - 1, j] if i - 1 >= 0 else 0.0
+        r = v[i + 1, j] if i + 1 <= GRID - 1 else 0.0
+        t = v[i, j + 1] if j + 1 <= GRID - 1 else 0.0
+        b = v[i, j - 1] if j - 1 >= 0 else 0.0
+        # Avoid ill-conditioned matrix A
+        mv[i, j] = 20 * v[i, j] - l - r - t - b
+
+A = LinearOperator(compute_Ax)
+init()
+MatrixFreeCG(A, b, x, maxiter=10 * GRID * GRID, tol=1e-18, quiet=True)
+print(x.to_numpy())
+```