fix(math): preserve certified sums and complete MKL algebra

src/fesa/math/dense_blas_internal.h

@@ -7,6 +7,7 @@
 
 #include "fesa/core/status.h"
 #include "fesa/math/matrix.h"
+#include "fesa/math/vector.h"
 
 namespace fesa::dense_blas_internal {
 
@@ -18,6 +19,19 @@ Result<MKL_INT> ToMklSize(std::size_t size);
 /// @throws std::invalid_argument if the row counts differ.
 Matrix MultiplyTransposeLeft(const Matrix& left, const Matrix& right);
 
+/// @brief Multiplies a compact diagonal matrix by a vector through MKL.
+/// @throws std::invalid_argument if shapes differ or inputs are nonfinite.
+/// @throws std::length_error if the size exceeds the backend integer range.
+Vector MultiplyDiagonal(const Vector& diagonal, const Vector& rhs);
+
+/// @brief Computes a lower Cholesky factor and its inverse with MKL LAPACK.
+/// @return False for a nonpositive pivot or nonfinite factor/inverse.
+/// @note Outputs are replaced only after both finite candidates succeed.
+/// @throws std::invalid_argument for a nonsquare or nonfinite input.
+/// @throws std::length_error for a backend size overflow.
+bool CholeskyLowerAndInverse(const Matrix& matrix, Matrix& lower,
+                            Matrix& inverse_lower);
+
 /// @brief Copies contiguous double values through the private BLAS backend.
 /// @param source Readable storage for size values when size is nonzero.
 /// @param size Number of contiguous values to copy.