Version: v1.5.0

# 稀疏矩阵

Sparse matrices are frequently involved in solving linear systems in science and engineering. Taichi provides useful APIs for sparse matrices on the CPU and CUDA backends.

To use sparse matrices in Taichi programs, follow these three steps:

1. Create a `builder` using `ti.linalg.SparseMatrixBuilder()`.
2. Call `ti.kernel` to fill the `builder` with your matrices' data.
3. Build sparse matrices from the `builder`.
##### WARNING

The sparse matrix feature is still under development. 以下是一些限制：

• The sparse matrix data type on the CPU backend only supports `f32` and `f64`.
• The sparse matrix data type on the CUDA backend only supports `f32`.

Here's an example:

``import taichi as tiarch = ti.cpu # or ti.cudati.init(arch=arch)n = 4# step 1: create sparse matrix builderK = ti.linalg.SparseMatrixBuilder(n, n, max_num_triplets=100)@ti.kerneldef fill(A: ti.types.sparse_matrix_builder()):    for i in range(n):        A[i, i] += 1  # Only +=  and -= operators are supported for now.# step 2: 填充 builderfill(K)print(">>>> K.print_triplets()")K.print_triplets()# outputs:# >>>> K.print_triplets()# n=4, m=4, num_triplets=4 (max=100)(0, 0) val=1.0(1, 1) val=1.0(2, 2) val=1.0(3, 3) val=1.0# step 3: 由builder创建稀疏矩阵.A = K.build()print(">>>> A = K.build()")print(A)# outputs:# >>>> A = K.build()# [1, 0, 0, 0]# [0, 1, 0, 0]# [0, 0, 1, 0]# [0, 0, 0, 1]``

``print(">>>> Summation: C = A + A")C = A + Aprint(C)# outputs:# >>>> Summation: C = A + A# [2, 0, 0, 0]# [0, 2, 0, 0]# [0, 0, 2, 0]# [0, 0, 0, 2]print(">>>> Subtraction: D = A - A")D = A - Aprint(D)# outputs:# >>>> Subtraction: D = A - A# [0, 0, 0, 0]# [0, 0, 0, 0]# [0, 0, 0, 0]# [0, 0, 0, 0]print(">>>> Multiplication with a scalar on the right: E = A * 3.0")E = A * 3.0print(E)# outputs:# >>>> Multiplication with a scalar on the right: E = A * 3.0# [3, 0, 0, 0]# [0, 3, 0, 0]# [0, 0, 3, 0]# [0, 0, 0, 3]print(">>>> Multiplication with a scalar on the left: E = 3.0 * A")E = 3.0 * Aprint(E)# outputs:# >>>> Multiplication with a scalar on the left: E = 3.0 * A# [3, 0, 0, 0]# [0, 3, 0, 0]# [0, 0, 3, 0]# [0, 0, 0, 3]print(">>>> Transpose: F = A.transpose()")F = A.transpose()print(F)# outputs:# >>>> Transpose: F = A.transpose()# [1, 0, 0, 0]# [0, 1, 0, 0]# [0, 0, 1, 0]# [0, 0, 0, 1]print(">>>> Matrix multiplication: G = E @ A")G = E @ Aprint(G)# outputs:# >>>> Matrix multiplication: G = E @ A# [3, 0, 0, 0]# [0, 3, 0, 0]# [0, 0, 3, 0]# [0, 0, 0, 3]print(">>>> Element-wise multiplication: H = E * A")H = E * Aprint(H)# outputs:# >>>> Element-wise multiplication: H = E * A# [3, 0, 0, 0]# [0, 3, 0, 0]# [0, 0, 3, 0]# [0, 0, 0, 3]print(f">>>> Element Access: A[0,0] = {A[0,0]}")# outputs:# >>>> Element Access: A[0,0] = 1.0``

## 稀疏线性求解器

1. 使用 `ti.linalg.SparseSolver(solver_type, ordering)` 创建一个`solver`。 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. 使用 `solver.analyze_pattern(Sparse_matrix)``solver.factorize(Sparse_matrix)` 分析和因式分解您想要求解的稀疏矩阵。
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.
4. 调用 `solver.info()` 来检查求解过程是否成功。

``import taichi as tiarch = ti.cpu # or ti.cudati.init(arch=arch)n = 4K = ti.linalg.SparseMatrixBuilder(n, n, max_num_triplets=100)b = ti.ndarray(ti.f32, shape=n)@ti.kerneldef fill(A: ti.types.sparse_matrix_builder(), b: ti.template(), interval: ti.i32):    for i in range(n):        A[i, i] += 2.0        if i % interval == 0:            b[i] += 1.0fill(K, b, 3)A = K.build()print(">>>> Matrix A:")print(A)print(">>>> Vector b:")print(b)# 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.SparseSolver(solver_type="LLT")solver.analyze_pattern(A)solver.factorize(A)x = solver.solve(b)isSuccess = solver.info()print(">>>> Solve sparse linear systems Ax = b with the solution x:")print(x)print(f">>>> Computation was successful?: {isSuccess}")# outputs:# >>>> Solve sparse linear systems Ax = b with the solution x:# [0.5 0.  0.  0.5]# >>>> Computation was successful?: True``

## 示例

• Stable fluid: 使用 稀疏 Laplacian 矩阵来求解 Poisson 压力方程的一个二维流体模拟项目。
• Implicit mass spring: 一个用稀疏矩阵求解线性系统的二维布料仿真项目。