feat: Sherman–Morrison formula

config.yml

@@ -959,6 +959,9 @@ notebook:
     - mat_det_rd: 求矩阵的行列式(随机化算法)
       code_ext: hpp
       test_ext: cpp
+    - sherman_morrison: Sherman–Morrison 公式
+      code_ext: hpp
+      test_ext: cpp
     - bitmat: 静态 Bool 矩阵
       code_ext: hpp
       test_ext: cpp

src/code/lalg/sherman_morrison.hpp

@@ -0,0 +1,26 @@
+#ifndef TIFALIBS_LALG_SHERMAN_MORRISON
+#define TIFALIBS_LALG_SHERMAN_MORRISON
+
+#include "../util/util.hpp"
+
+namespace tifa_libs::math {
+
+template <class Mat, class T = typename Mat::value_type>
+requires std::same_as<T, typename Mat::value_type>
+constexpr void sherman_morrison(Mat &inv_A, vec<T> const &u, vec<T> const &v) {
+  u32 n = inv_A.row();
+  assert(n == inv_A.col());
+  vec<T> x(n), y(n);
+  for (u32 i = 0; i < n; ++i)
+    for (u32 j = 0; j < n; ++j)
+      x[i] += inv_A[i][j] * u[j], y[j] += v[i] * inv_A[i][j];
+  T sum = 1;
+  for (u32 i = 0; i < n; ++i)
+    for (u32 j = 0; j < n; ++j) sum += v[i] * inv_A[i][j] * u[j];
+  for (u32 i = 0; i < n; ++i)
+    for (u32 j = 0; j < n; ++j) inv_A[i][j] -= x[i] * y[j] / sum;
+}
+
+}  // namespace tifa_libs::math
+
+#endif

src/doc_tex/lalg/sherman_morrison.tex

@@ -0,0 +1,5 @@
+根据方阵 \(A^{-1}\) 和列向量 \(u,v\) 计算 \((A+uv^T)^{-1}\)
+
+\[
+    \left(A+uv^T\right)^{-1}=A^{-1}-\frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}
+\]