diff --git a/python/taichi/examples/simulation/implicit_mass_spring.py b/python/taichi/examples/simulation/implicit_mass_spring.py
index a9cfc743d1907..293fd283b59b3 100644
--- a/python/taichi/examples/simulation/implicit_mass_spring.py
+++ b/python/taichi/examples/simulation/implicit_mass_spring.py
@@ -179,9 +179,29 @@ def update(self, h):
 
         A = self.M - h * D - h**2 * K
 
-        vel = self.vel.to_numpy().reshape(2 * self.NV)
-        force = self.force.to_numpy().reshape(2 * self.NV)
-        b = (force + h * K @ vel) * h
+        vel = ti.ndarray(ti.f32, 2 * self.NV)
+        force = ti.ndarray(ti.f32, 2 * self.NV)
+
+        @ti.kernel
+        def copy_to(des: ti.types.ndarray(), source: ti.template()):
+            for i in range(self.NV):
+                des[2 * i] = source[i][0]
+                des[2 * i + 1] = source[i][1]
+
+        copy_to(vel, self.vel)
+        copy_to(force, self.force)
+
+        @ti.kernel
+        def compute_b(b: ti.types.ndarray(), f: ti.types.ndarray(),
+                      Kv: ti.types.ndarray(), h: ti.f32):
+            for i in range(2 * self.NV):
+                b[i] = (f[i] + Kv[i] * h) * h
+
+        # b = (force + h * K @ vel) * h
+        b = ti.ndarray(ti.f32, 2 * self.NV)
+        Kv = K @ vel
+        compute_b(b, force, Kv, h)
+
         # Sparse solver
         solver = ti.linalg.SparseSolver(solver_type="LDLT")
         solver.analyze_pattern(A)
diff --git a/taichi/program/sparse_matrix.cpp b/taichi/program/sparse_matrix.cpp
index 5ed15f6b79618..718ecbf3b887e 100644
--- a/taichi/program/sparse_matrix.cpp
+++ b/taichi/program/sparse_matrix.cpp
@@ -25,6 +25,11 @@
         }                                                                      \
   }
 
+#define INSTANTIATE_SPMV(type, storage)                               \
+  template void                                                       \
+  EigenSparseMatrix<Eigen::SparseMatrix<type, Eigen::storage>>::spmv( \
+      Program *prog, const Ndarray &x, const Ndarray &y);
+
 namespace {
 using Pair = std::pair<std::string, std::string>;
 struct key_hash {
@@ -226,6 +231,31 @@ void EigenSparseMatrix<EigenMatrix>::build_triplets(void *triplets_adr) {
   }
 }
 
+template <class EigenMatrix>
+void EigenSparseMatrix<EigenMatrix>::spmv(Program *prog,
+                                          const Ndarray &x,
+                                          const Ndarray &y) {
+  size_t dX = prog->get_ndarray_data_ptr_as_int(&x);
+  size_t dY = prog->get_ndarray_data_ptr_as_int(&y);
+  std::string sdtype = taichi::lang::data_type_name(dtype_);
+  if (sdtype == "f32") {
+    Eigen::Map<Eigen::VectorXf>((float *)dY, cols_) =
+        matrix_.template cast<float>() *
+        Eigen::Map<Eigen::VectorXf>((float *)dX, cols_);
+  } else if (sdtype == "f64") {
+    Eigen::Map<Eigen::VectorXd>((double *)dY, cols_) =
+        matrix_.template cast<double>() *
+        Eigen::Map<Eigen::VectorXd>((double *)dX, cols_);
+  } else {
+    TI_ERROR("Unsupported sparse matrix data type {}!", sdtype);
+  }
+}
+
+INSTANTIATE_SPMV(float32, ColMajor)
+INSTANTIATE_SPMV(float32, RowMajor)
+INSTANTIATE_SPMV(float64, ColMajor)
+INSTANTIATE_SPMV(float64, RowMajor)
+
 std::unique_ptr<SparseMatrix> make_sparse_matrix(
     int rows,
     int cols,
@@ -652,7 +682,7 @@ std::unique_ptr<SparseMatrix> CuSparseMatrix::transpose() const {
 #endif
 }
 
-void CuSparseMatrix::spmv(Program *prog, const Ndarray &x, Ndarray &y) {
+void CuSparseMatrix::spmv(Program *prog, const Ndarray &x, const Ndarray &y) {
 #if defined(TI_WITH_CUDA)
   size_t dX = prog->get_ndarray_data_ptr_as_int(&x);
   size_t dY = prog->get_ndarray_data_ptr_as_int(&y);
diff --git a/taichi/program/sparse_matrix.h b/taichi/program/sparse_matrix.h
index 6031e402441cb..3708840e2ee9e 100644
--- a/taichi/program/sparse_matrix.h
+++ b/taichi/program/sparse_matrix.h
@@ -87,7 +87,7 @@ class SparseMatrix {
     return nullptr;
   }
 
-  inline DataType get_data_type() {
+  inline const DataType get_data_type() const {
     return dtype_;
   }
 
@@ -197,6 +197,8 @@ class EigenSparseMatrix : public SparseMatrix {
     return matrix_ * b;
   }
 
+  void spmv(Program *prog, const Ndarray &x, const Ndarray &y);
+
  private:
   EigenMatrix matrix_;
 };
@@ -265,7 +267,7 @@ class CuSparseMatrix : public SparseMatrix {
                           void *coo_values_ptr,
                           int nnz) override;
 
-  void spmv(Program *prog, const Ndarray &x, Ndarray &y);
+  void spmv(Program *prog, const Ndarray &x, const Ndarray &y);
 
   const void *get_matrix() const override {
     return &matrix_;
diff --git a/taichi/program/sparse_solver.cpp b/taichi/program/sparse_solver.cpp
index 1d3f5ba4ff211..5a55add667983 100644
--- a/taichi/program/sparse_solver.cpp
+++ b/taichi/program/sparse_solver.cpp
@@ -14,6 +14,14 @@
     }                                                                          \
   }
 
+#define INSTANTIATE_SOLVER(dt, type, order)                              \
+  using dt##type##order =                                                \
+      Eigen::Simplicial##type<Eigen::SparseMatrix<dt>, Eigen::Lower,     \
+                              Eigen::order##Ordering<int>>;              \
+  template void                                                          \
+  EigenSparseSolver<dt##type##order, Eigen::SparseMatrix<dt>>::solve_rf( \
+      Program *prog, const SparseMatrix &sm, const Ndarray &b, Ndarray &x);
+
 using Triplets = std::tuple<std::string, std::string, std::string>;
 namespace {
 struct key_hash {
@@ -34,6 +42,9 @@ namespace taichi::lang {
 template <class EigenSolver, class EigenMatrix>
 bool EigenSparseSolver<EigenSolver, EigenMatrix>::compute(
     const SparseMatrix &sm) {
+  if (!is_initialized_) {
+    SparseSolver::init_solver(sm.num_rows(), sm.num_cols(), sm.get_data_type());
+  }
   GET_EM(sm);
   solver_.compute(*mat);
   if (solver_.info() != Eigen::Success) {
@@ -44,6 +55,9 @@ bool EigenSparseSolver<EigenSolver, EigenMatrix>::compute(
 template <class EigenSolver, class EigenMatrix>
 void EigenSparseSolver<EigenSolver, EigenMatrix>::analyze_pattern(
     const SparseMatrix &sm) {
+  if (!is_initialized_) {
+    SparseSolver::init_solver(sm.num_rows(), sm.num_cols(), sm.get_data_type());
+  }
   GET_EM(sm);
   solver_.analyzePattern(*mat);
 }
@@ -66,6 +80,23 @@ bool EigenSparseSolver<EigenSolver, EigenMatrix>::info() {
   return solver_.info() == Eigen::Success;
 }
 
+template <class EigenSolver, class EigenMatrix>
+void EigenSparseSolver<EigenSolver, EigenMatrix>::solve_rf(
+    Program *prog,
+    const SparseMatrix &sm,
+    const Ndarray &b,
+    Ndarray &x) {
+  size_t db = prog->get_ndarray_data_ptr_as_int(&b);
+  size_t dX = prog->get_ndarray_data_ptr_as_int(&x);
+  Eigen::Map<Eigen::VectorXf>((float *)dX, rows_) =
+      solver_.solve(Eigen::Map<Eigen::VectorXf>((float *)db, cols_));
+}
+
+INSTANTIATE_SOLVER(float32, LLT, COLAMD)
+INSTANTIATE_SOLVER(float32, LDLT, COLAMD)
+INSTANTIATE_SOLVER(float32, LLT, AMD)
+INSTANTIATE_SOLVER(float32, LDLT, AMD)
+
 CuSparseSolver::CuSparseSolver() {
 #if defined(TI_WITH_CUDA)
   if (!CUSPARSEDriver::get_instance().is_loaded()) {
diff --git a/taichi/program/sparse_solver.h b/taichi/program/sparse_solver.h
index 917fccd0f2917..876452cf93161 100644
--- a/taichi/program/sparse_solver.h
+++ b/taichi/program/sparse_solver.h
@@ -9,8 +9,19 @@
 namespace taichi::lang {
 
 class SparseSolver {
+ protected:
+  int rows_{0};
+  int cols_{0};
+  DataType dtype_{PrimitiveType::f32};
+  bool is_initialized_{false};
+
  public:
   virtual ~SparseSolver() = default;
+  void init_solver(const int rows, const int cols, const DataType dtype) {
+    rows_ = rows;
+    cols_ = cols;
+    dtype_ = dtype;
+  }
   virtual bool compute(const SparseMatrix &sm) = 0;
   virtual void analyze_pattern(const SparseMatrix &sm) = 0;
   virtual void factorize(const SparseMatrix &sm) = 0;
@@ -46,9 +57,7 @@ class EigenSparseSolver : public SparseSolver {
   void solve_rf(Program *prog,
                 const SparseMatrix &sm,
                 const Ndarray &b,
-                Ndarray &x) override {
-    TI_NOT_IMPLEMENTED;
-  };
+                Ndarray &x) override;
 
   bool info() override;
 };
diff --git a/taichi/python/export_lang.cpp b/taichi/python/export_lang.cpp
index 4e131e2b30837..4c49ee6c0cb79 100644
--- a/taichi/python/export_lang.cpp
+++ b/taichi/python/export_lang.cpp
@@ -1219,6 +1219,7 @@ void export_lang(py::module &m) {
       .def(float##TYPE() * py::self)                                         \
       .def(py::self *py::self)                                               \
       .def("matmul", &EigenSparseMatrix<STORAGE##TYPE##EigenMatrix>::matmul) \
+      .def("spmv", &EigenSparseMatrix<STORAGE##TYPE##EigenMatrix>::spmv)     \
       .def("transpose",                                                      \
            &EigenSparseMatrix<STORAGE##TYPE##EigenMatrix>::transpose)        \
       .def("get_element",                                                    \
diff --git a/tests/python/test_sparse_linear_solver.py b/tests/python/test_sparse_linear_solver.py
index 772cfc2382d75..b0e79b66d3bd1 100644
--- a/tests/python/test_sparse_linear_solver.py
+++ b/tests/python/test_sparse_linear_solver.py
@@ -38,6 +38,39 @@ def fill(Abuilder: ti.types.sparse_matrix_builder(),
         assert x[i] == test_utils.approx(res[i], rel=1.0)
 
 
+@pytest.mark.parametrize("dtype", [ti.f32])
+@pytest.mark.parametrize("solver_type", ["LLT", "LDLT", "LU"])
+@pytest.mark.parametrize("ordering", ["AMD", "COLAMD"])
+@test_utils.test(arch=ti.cpu)
+def test_sparse_solver_ndarray_vector(dtype, solver_type, ordering):
+    n = 10
+    A = np.random.rand(n, n)
+    A_psd = np.dot(A, A.transpose())
+    Abuilder = ti.linalg.SparseMatrixBuilder(n, n, max_num_triplets=300)
+    b = ti.ndarray(ti.f32, shape=n)
+
+    @ti.kernel
+    def fill(Abuilder: ti.types.sparse_matrix_builder(),
+             InputArray: ti.types.ndarray(), b: ti.types.ndarray()):
+        for i, j in ti.ndrange(n, n):
+            Abuilder[i, j] += InputArray[i, j]
+        for i in range(n):
+            b[i] = i + 1
+
+    fill(Abuilder, A_psd, b)
+    A = Abuilder.build()
+    solver = ti.linalg.SparseSolver(dtype=dtype,
+                                    solver_type=solver_type,
+                                    ordering=ordering)
+    solver.analyze_pattern(A)
+    solver.factorize(A)
+    x = solver.solve(b)
+
+    res = np.linalg.solve(A_psd, b.to_numpy())
+    for i in range(n):
+        assert x[i] == test_utils.approx(res[i], rel=1.0)
+
+
 @test_utils.test(arch=ti.cuda)
 def test_gpu_sparse_solver():
     from scipy.sparse import coo_matrix
diff --git a/tests/python/test_sparse_matrix.py b/tests/python/test_sparse_matrix.py
index ce90d4616e15a..bcf0ed72311fc 100644
--- a/tests/python/test_sparse_matrix.py
+++ b/tests/python/test_sparse_matrix.py
@@ -376,6 +376,34 @@ def fill(Abuilder: ti.types.sparse_matrix_builder(),
             assert C[i, j] == GT[i][j]
 
 
+@pytest.mark.parametrize('dtype, storage_format', [(ti.f32, 'col_major'),
+                                                   (ti.f32, 'row_major'),
+                                                   (ti.f64, 'col_major'),
+                                                   (ti.f64, 'row_major')])
+@test_utils.test(arch=ti.cpu)
+def test_sparse_matrix_ndarray_vector_multiplication(dtype, storage_format):
+    n = 2
+    Abuilder = ti.linalg.SparseMatrixBuilder(n,
+                                             n,
+                                             max_num_triplets=100,
+                                             dtype=dtype,
+                                             storage_format=storage_format)
+    x = ti.ndarray(dtype, n)
+
+    @ti.kernel
+    def fill(Abuilder: ti.types.sparse_matrix_builder()):
+        for i, j in ti.ndrange(n, n):
+            Abuilder[i, j] += i + j
+
+    fill(Abuilder)
+    x.fill(1.0)
+    A = Abuilder.build()
+    res = A @ x
+    res_n = res.to_numpy()
+    assert res_n[0] == 1.0
+    assert res_n[1] == 3.0
+
+
 @test_utils.test(arch=ti.cuda)
 def test_gpu_sparse_matrix():
     import numpy as np