calo::multifit Namespace Reference

Classes
struct	MapMForPM
struct	MapSymM

Typedefs
template<int NROWS, int NCOLS>
using	ColMajorMatrix = Eigen::Matrix< float, NROWS, NCOLS, Eigen::ColMajor >
template<int SIZE, typename T = float>
using	ColumnVector = Eigen::Matrix< T, SIZE, 1 >
template<int NROWS, int NCOLS>
using	RowMajorMatrix = Eigen::Matrix< float, NROWS, NCOLS, Eigen::RowMajor >
template<int SIZE, typename T = float>
using	RowVector = Eigen::Matrix< T, 1, SIZE >

Functions
template<typename MatrixType1 , typename MatrixType2 , typename MatrixType3 , typename MatrixType4 >
EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void	calculateChiSq (MatrixType1 const &matrixL, MatrixType2 const &pulseMatrixView, MatrixType3 const &resultAmplitudesVector, MatrixType4 const &inputAmplitudesView, float &chi2)
template<typename MatrixType1 , typename MatrixType2 >
EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void	compute_decomposition (MatrixType1 &L, MatrixType2 const &M, int const N)
template<typename MatrixType1 , typename MatrixType2 , typename VectorType >
EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void	compute_decomposition_forwardsubst_with_offsets (MatrixType1 &L, MatrixType2 const &M, float b[MatrixType1::stride], VectorType const &Atb, int const N, ColumnVector< MatrixType1::stride, int > const &pulseOffsets)
template<typename MatrixType1 , typename MatrixType2 >
EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void	compute_decomposition_unrolled (MatrixType1 &L, MatrixType2 const &M)
template<typename MatrixType , typename VectorType >
EIGEN_DEVICE_FUNC void	fnnls (MatrixType const &AtA, VectorType const &Atb, VectorType &solution, int &npassive, ColumnVector< VectorType::RowsAtCompileTime, int > &pulseOffsets, MapSymM< float, VectorType::RowsAtCompileTime > &matrixL, double eps, const int maxIterations, const int relaxationPeriod, const int relaxationFactor)
template<typename MatrixType1 , typename MatrixType2 , typename MatrixType3 >
EIGEN_DEVICE_FUNC void	solve_forward_subst_matrix (MatrixType1 &A, MatrixType2 const &pulseMatrixView, MatrixType3 const &matrixL)
template<typename MatrixType1 , typename MatrixType2 >
EIGEN_DEVICE_FUNC void	solve_forward_subst_vector (float reg_b[MatrixType1::RowsAtCompileTime], MatrixType1 inputAmplitudesView, MatrixType2 matrixL)
template<typename MatrixType1 , typename MatrixType2 , typename VectorType >
EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void	update_decomposition_forwardsubst_with_offsets (MatrixType1 &L, MatrixType2 const &M, float b[MatrixType1::stride], VectorType const &Atb, int const N, ColumnVector< MatrixType1::stride, int > const &pulseOffsets)

Typedef Documentation

ColMajorMatrix

template<int NROWS, int NCOLS>

using calo::multifit::ColMajorMatrix = typedef Eigen::Matrix<float, NROWS, NCOLS, Eigen::ColMajor>

Definition at line 16 of file MultifitComputations.h.

ColumnVector

template<int SIZE, typename T = float>

using calo::multifit::ColumnVector = typedef Eigen::Matrix<T, SIZE, 1>

Definition at line 22 of file MultifitComputations.h.

RowMajorMatrix

template<int NROWS, int NCOLS>

using calo::multifit::RowMajorMatrix = typedef Eigen::Matrix<float, NROWS, NCOLS, Eigen::RowMajor>

Definition at line 19 of file MultifitComputations.h.

RowVector

template<int SIZE, typename T = float>

using calo::multifit::RowVector = typedef Eigen::Matrix<T, 1, SIZE>

Definition at line 25 of file MultifitComputations.h.

Function Documentation

calculateChiSq()

template<typename MatrixType1 , typename MatrixType2 , typename MatrixType3 , typename MatrixType4 >

EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void calo::multifit::calculateChiSq	(	MatrixType1 const &	matrixL,
		MatrixType2 const &	pulseMatrixView,
		MatrixType3 const &	resultAmplitudesVector,
		MatrixType4 const &	inputAmplitudesView,
		float &	chi2
	)

Definition at line 289 of file MultifitComputations.h.

References cms::cudacompat::__ldg(), nano_mu_local_reco_cff::chi2, CMS_UNROLL_LOOP, ALPAKA_ACCELERATOR_NAMESPACE::brokenline::constexpr(), counter, mps_fire::i, and mysort::results.

Referenced by ALPAKA_ACCELERATOR_NAMESPACE::ecal::multifit::Kernel_minimize::operator()(), and ALPAKA_ACCELERATOR_NAMESPACE::hcal::reconstruction::mahi::Kernel_minimize< NSAMPLES, NPULSES >::operator()().

                                                                           {
      // FIXME: this assumes pulses are on columns and samples on rows
      constexpr auto NPULSES = MatrixType2::ColsAtCompileTime;
      constexpr auto NSAMPLES = MatrixType2::RowsAtCompileTime;

      // replace pulseMatrixView * resultAmplitudesVector - inputAmplitudesView
      // NOTE:
      float accum[NSAMPLES];
      {
        float results[NPULSES];

        // preload results and permute according to the pulse offsets /////////////// ??? this is not done in ECAL
        CMS_UNROLL_LOOP
        for (int counter = 0; counter < NPULSES; counter++) {
          results[counter] = resultAmplitudesVector[counter];
        }

        // load accum
        CMS_UNROLL_LOOP
        for (int counter = 0; counter < NSAMPLES; counter++)
          accum[counter] = -inputAmplitudesView(counter);

        // iterate
        for (int icol = 0; icol < NPULSES; icol++) {
          float pm_col[NSAMPLES];

          // preload a column of pulse matrix
          CMS_UNROLL_LOOP
          for (int counter = 0; counter < NSAMPLES; counter++)
#ifdef __CUDA_ARCH__
            pm_col[counter] = __ldg(&pulseMatrixView.coeffRef(counter, icol));
#else
            pm_col[counter] = pulseMatrixView.coeffRef(counter, icol);
#endif

          // accum
          CMS_UNROLL_LOOP
          for (int counter = 0; counter < NSAMPLES; counter++)
            accum[counter] += results[icol] * pm_col[counter];
        }
      }

      // compute chi2 and check that there is no rotation
      // chi2 = matrixDecomposition
      //    .matrixL()
      //    . solve(mapAccum)
      //            .solve(pulseMatrixView * resultAmplitudesVector - inputAmplitudesView)
      //    .squaredNorm();

      {
        float reg_L[NSAMPLES];
        float accumSum = 0;

        // preload a column and load column 0 of cholesky
        CMS_UNROLL_LOOP
        for (int i = 0; i < NSAMPLES; i++) {
          reg_L[i] = matrixL(i, 0);
        }

        // compute x0 and store it
        auto x_prev = accum[0] / reg_L[0];
        accumSum += x_prev * x_prev;

        // iterate
        CMS_UNROLL_LOOP
        for (int iL = 1; iL < NSAMPLES; iL++) {
          // update accum
          CMS_UNROLL_LOOP
          for (int counter = iL; counter < NSAMPLES; counter++)
            accum[counter] -= x_prev * reg_L[counter];

          // load the next column of cholesky
          CMS_UNROLL_LOOP
          for (int counter = iL; counter < NSAMPLES; counter++)
            reg_L[counter] = matrixL(counter, iL);

          // compute the next x for M(iL, icol)
          x_prev = accum[iL] / reg_L[iL];

          // store the result value
          accumSum += x_prev * x_prev;
        }

        chi2 = accumSum;
      }
    }

compute_decomposition()

template<typename MatrixType1 , typename MatrixType2 >

EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void calo::multifit::compute_decomposition	(	MatrixType1 &	L,
		MatrixType2 const &	M,
		int const	N
	)

Definition at line 99 of file MultifitComputations.h.

References mps_fire::i, dqmiolumiharvest::j, dqmdumpme::k, dttmaxenums::L, N, and mathSSE::sqrt().

                                                                                  {
      auto const sqrtm_0_0 = std::sqrt(M(0, 0));
      L(0, 0) = sqrtm_0_0;
      using T = typename MatrixType1::base_type;

      for (int i = 1; i < N; i++) {
        T sumsq{0};
        for (int j = 0; j < i; j++) {
          T sumsq2{0};
          auto const m_i_j = M(i, j);
          for (int k = 0; k < j; ++k)
            sumsq2 += L(i, k) * L(j, k);

          auto const value_i_j = (m_i_j - sumsq2) / L(j, j);
          L(i, j) = value_i_j;

          sumsq += value_i_j * value_i_j;
        }

        auto const l_i_i = std::sqrt(M(i, i) - sumsq);
        L(i, i) = l_i_i;
      }
    }

compute_decomposition_forwardsubst_with_offsets()

template<typename MatrixType1 , typename MatrixType2 , typename VectorType >

EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void calo::multifit::compute_decomposition_forwardsubst_with_offsets	(	MatrixType1 &	L,
		MatrixType2 const &	M,
		float	b[MatrixType1::stride],
		VectorType const &	Atb,
		int const	N,
		ColumnVector< MatrixType1::stride, int > const &	pulseOffsets
	)

Definition at line 126 of file MultifitComputations.h.

References b, mps_fire::i, dqmiolumiharvest::j, dqmdumpme::k, dttmaxenums::L, SiStripPI::max, SiStripPI::min, N, HLT_2024v13_cff::pulseOffsets, mathSSE::sqrt(), and dqmMemoryStats::total.

Referenced by fnnls().

                                                                  {
      auto const real_0 = pulseOffsets(0);
      auto const sqrtm_0_0 = std::sqrt(M(real_0, real_0));
      L(0, 0) = sqrtm_0_0;
      using T = typename MatrixType1::base_type;
      b[0] = Atb(real_0) / sqrtm_0_0;

      for (int i = 1; i < N; i++) {
        auto const i_real = pulseOffsets(i);
        T sumsq{0};
        T total = 0;
        auto const atb = Atb(i_real);
        for (int j = 0; j < i; j++) {
          auto const j_real = pulseOffsets(j);
          T sumsq2{0};
          auto const m_i_j = M(std::max(i_real, j_real), std::min(i_real, j_real));
          for (int k = 0; k < j; ++k)
            sumsq2 += L(i, k) * L(j, k);

          auto const value_i_j = (m_i_j - sumsq2) / L(j, j);
          L(i, j) = value_i_j;

          sumsq += value_i_j * value_i_j;
          total += value_i_j * b[j];
        }

        auto const l_i_i = std::sqrt(M(i_real, i_real) - sumsq);
        L(i, i) = l_i_i;
        b[i] = (atb - total) / l_i_i;
      }
    }

compute_decomposition_unrolled()

template<typename MatrixType1 , typename MatrixType2 >

EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void calo::multifit::compute_decomposition_unrolled	(	MatrixType1 &	L,
		MatrixType2 const &	M
	)

Definition at line 73 of file MultifitComputations.h.

References CMS_UNROLL_LOOP, mps_fire::i, dqmiolumiharvest::j, dqmdumpme::k, dttmaxenums::L, mathSSE::sqrt(), and riemannFit::stride.

Referenced by ALPAKA_ACCELERATOR_NAMESPACE::ecal::multifit::Kernel_minimize::operator()(), and ALPAKA_ACCELERATOR_NAMESPACE::hcal::reconstruction::mahi::Kernel_minimize< NSAMPLES, NPULSES >::operator()().

                                                                                                                    {
      auto const sqrtm_0_0 = std::sqrt(M(0, 0));
      L(0, 0) = sqrtm_0_0;
      using T = typename MatrixType1::base_type;

      CMS_UNROLL_LOOP
      for (int i = 1; i < MatrixType1::stride; i++) {
        T sumsq{0};
        for (int j = 0; j < i; j++) {
          T sumsq2{0};
          auto const m_i_j = M(i, j);
          for (int k = 0; k < j; ++k)
            sumsq2 += L(i, k) * L(j, k);

          auto const value_i_j = (m_i_j - sumsq2) / L(j, j);
          L(i, j) = value_i_j;

          sumsq += value_i_j * value_i_j;
        }

        auto const l_i_i = std::sqrt(M(i, i) - sumsq);
        L(i, i) = l_i_i;
      }
    }

fnnls()

template<typename MatrixType , typename VectorType >

EIGEN_DEVICE_FUNC void calo::multifit::fnnls	(	MatrixType const &	AtA,
		VectorType const &	Atb,
		VectorType &	solution,
		int &	npassive,
		ColumnVector< VectorType::RowsAtCompileTime, int > &	pulseOffsets,
		MapSymM< float, VectorType::RowsAtCompileTime > &	matrixL,
		double	eps,
		const int	maxIterations,
		const int	relaxationPeriod,
		const int	relaxationFactor
	)

Definition at line 382 of file MultifitComputations.h.

References simBeamSpotPI::alpha, CMS_UNROLL_LOOP, compute_decomposition_forwardsubst_with_offsets(), ALPAKA_ACCELERATOR_NAMESPACE::brokenline::constexpr(), HLT_2024v13_cff::eps, mps_fire::i, CommonMethods::isnan(), dqmiolumiharvest::j, SiStripPI::max, HLT_2024v13_cff::maxIterations, HLT_2024v13_cff::pulseOffsets, particleFlowDisplacedVertex_cfi::ratio, alignCSCRings::s, edm::swap(), dqmMemoryStats::total, update_decomposition_forwardsubst_with_offsets(), and w().

Referenced by ALPAKA_ACCELERATOR_NAMESPACE::ecal::multifit::Kernel_minimize::operator()(), and ALPAKA_ACCELERATOR_NAMESPACE::hcal::reconstruction::mahi::Kernel_minimize< NSAMPLES, NPULSES >::operator()().

                                                             {  // multiply "eps" by "relaxationFactor"
      // constants
      constexpr auto NPULSES = VectorType::RowsAtCompileTime;

      // to keep track of where to terminate if converged
      Eigen::Index w_max_idx_prev = 0;
      float w_max_prev = 0;
      bool recompute = false;

      // used throughout
      VectorType s;
      float reg_b[NPULSES];
      //float matrixLStorage[MapSymM<float, NPULSES>::total];
      //MapSymM<float, NPULSES> matrixL{matrixLStorage};

      int iter = 0;
      while (true) {
        if (iter > 0 || npassive == 0) {
          auto const nactive = NPULSES - npassive;
          // exit if there are no more pulses to constrain
          if (nactive == 0)
            break;

          // compute the gradient
          //w.tail(nactive) = Atb.tail(nactive) - (AtA * solution).tail(nactive);
          Eigen::Index w_max_idx = 0;
          float w_max = -std::numeric_limits<float>::max();
          for (int icol = npassive; icol < NPULSES; icol++) {
            auto const icol_real = pulseOffsets(icol);
            auto const atb = Atb(icol_real);
            float sum = 0;
            CMS_UNROLL_LOOP
            for (int counter = 0; counter < NPULSES; counter++)
              sum += counter > icol_real ? AtA(counter, icol_real) * solution(counter)
                                         : AtA(icol_real, counter) * solution(counter);

            auto const w = atb - sum;
            if (w > w_max) {
              w_max = w;
              w_max_idx = icol - npassive;
            }
          }

          // check for convergence
          if (w_max < eps || (w_max_idx == w_max_idx_prev && w_max == w_max_prev))
            break;

          if (iter >= maxIterations)
            break;

          w_max_prev = w_max;
          w_max_idx_prev = w_max_idx;

          // move index to the right part of the vector
          w_max_idx += npassive;

          Eigen::numext::swap(pulseOffsets.coeffRef(npassive), pulseOffsets.coeffRef(w_max_idx));
          ++npassive;
        }

        // inner loop
        while (true) {
          if (npassive == 0)
            break;

          //s.head(npassive)
          //auto const& matrixL =
          //    AtA.topLeftCorner(npassive, npassive)
          //        .llt().matrixL();
          //.solve(Atb.head(npassive));
          if (recompute || iter == 0)
            compute_decomposition_forwardsubst_with_offsets(matrixL, AtA, reg_b, Atb, npassive, pulseOffsets);
          else
            update_decomposition_forwardsubst_with_offsets(matrixL, AtA, reg_b, Atb, npassive, pulseOffsets);

          // run backward substituion
          s(npassive - 1) = reg_b[npassive - 1] / matrixL(npassive - 1, npassive - 1);
          for (int i = npassive - 2; i >= 0; --i) {
            float total = 0;
            for (int j = i + 1; j < npassive; j++)
              total += matrixL(j, i) * s(j);

            s(i) = (reg_b[i] - total) / matrixL(i, i);
          }

          // done if solution values are all positive
          bool hasNegative = false;
          bool hasNans = false;
          for (int counter = 0; counter < npassive; counter++) {
            auto const s_ii = s(counter);
            hasNegative |= s_ii <= 0;
            hasNans |= std::isnan(s_ii);
          }

          // FIXME: temporary solution. my cholesky impl is unstable yielding nans
          // this check removes nans - do not accept solution unless all values
          // are stable
          if (hasNans)
            break;
          if (!hasNegative) {
            for (int i = 0; i < npassive; i++) {
              auto const i_real = pulseOffsets(i);
              solution(i_real) = s(i);
            }
            //solution.head(npassive) = s.head(npassive);
            recompute = false;
            break;
          }

          // there were negative values -> have to recompute the whole decomp
          recompute = true;

          auto alpha = std::numeric_limits<float>::max();
          Eigen::Index alpha_idx = 0, alpha_idx_real = 0;
          for (int i = 0; i < npassive; i++) {
            if (s[i] <= 0.) {
              auto const i_real = pulseOffsets(i);
              auto const ratio = solution[i_real] / (solution[i_real] - s[i]);
              if (ratio < alpha) {
                alpha = ratio;
                alpha_idx = i;
                alpha_idx_real = i_real;
              }
            }
          }

          // upadte solution
          for (int i = 0; i < npassive; i++) {
            auto const i_real = pulseOffsets(i);
            solution(i_real) += alpha * (s(i) - solution(i_real));
          }
          //solution.head(npassive) += alpha *
          //    (s.head(npassive) - solution.head(npassive));
          solution[alpha_idx_real] = 0;
          --npassive;

          Eigen::numext::swap(pulseOffsets.coeffRef(npassive), pulseOffsets.coeffRef(alpha_idx));
        }

        // as in cpu
        ++iter;
        if (iter % relaxationPeriod == 0)
          eps *= relaxationFactor;
      }
    }

solve_forward_subst_matrix()

template<typename MatrixType1 , typename MatrixType2 , typename MatrixType3 >

EIGEN_DEVICE_FUNC void calo::multifit::solve_forward_subst_matrix	(	MatrixType1 &	A,
		MatrixType2 const &	pulseMatrixView,
		MatrixType3 const &	matrixL
	)

Definition at line 197 of file MultifitComputations.h.

References cms::cudacompat::__ldg(), A, CMS_UNROLL_LOOP, ALPAKA_ACCELERATOR_NAMESPACE::brokenline::constexpr(), and mps_fire::i.

Referenced by ALPAKA_ACCELERATOR_NAMESPACE::ecal::multifit::Kernel_minimize::operator()(), and ALPAKA_ACCELERATOR_NAMESPACE::hcal::reconstruction::mahi::Kernel_minimize< NSAMPLES, NPULSES >::operator()().

                                                                                  {
      // FIXME: this assumes pulses are on columns and samples on rows
      constexpr auto NPULSES = MatrixType2::ColsAtCompileTime;
      constexpr auto NSAMPLES = MatrixType2::RowsAtCompileTime;

      CMS_UNROLL_LOOP
      for (int icol = 0; icol < NPULSES; icol++) {
        float reg_b[NSAMPLES];
        float reg_L[NSAMPLES];

        // preload a column and load column 0 of cholesky
        CMS_UNROLL_LOOP
        for (int i = 0; i < NSAMPLES; i++) {
#ifdef __CUDA_ARCH__
          // load through the read-only cache
          reg_b[i] = __ldg(&pulseMatrixView.coeffRef(i, icol));
#else
          reg_b[i] = pulseMatrixView.coeffRef(i, icol);
#endif  // __CUDA_ARCH__
          reg_L[i] = matrixL(i, 0);
        }

        // compute x0 and store it
        auto x_prev = reg_b[0] / reg_L[0];
        A(0, icol) = x_prev;

        // iterate
        CMS_UNROLL_LOOP
        for (int iL = 1; iL < NSAMPLES; iL++) {
          // update accum
          CMS_UNROLL_LOOP
          for (int counter = iL; counter < NSAMPLES; counter++)
            reg_b[counter] -= x_prev * reg_L[counter];

          // load the next column of cholesky
          CMS_UNROLL_LOOP
          for (int counter = iL; counter < NSAMPLES; counter++)
            reg_L[counter] = matrixL(counter, iL);

          // compute the next x for M(iL, icol)
          x_prev = reg_b[iL] / reg_L[iL];

          // store the result value
          A(iL, icol) = x_prev;
        }
      }
    }

solve_forward_subst_vector()

template<typename MatrixType1 , typename MatrixType2 >

EIGEN_DEVICE_FUNC void calo::multifit::solve_forward_subst_vector	(	float	reg_b[MatrixType1::RowsAtCompileTime],
		MatrixType1	inputAmplitudesView,
		MatrixType2	matrixL
	)

Definition at line 248 of file MultifitComputations.h.

References CMS_UNROLL_LOOP, ALPAKA_ACCELERATOR_NAMESPACE::brokenline::constexpr(), and mps_fire::i.

Referenced by ALPAKA_ACCELERATOR_NAMESPACE::ecal::multifit::Kernel_minimize::operator()(), and ALPAKA_ACCELERATOR_NAMESPACE::hcal::reconstruction::mahi::Kernel_minimize< NSAMPLES, NPULSES >::operator()().

                                                                           {
      constexpr auto NSAMPLES = MatrixType1::RowsAtCompileTime;

      float reg_b_tmp[NSAMPLES];
      float reg_L[NSAMPLES];

      // preload a column and load column 0 of cholesky
      CMS_UNROLL_LOOP
      for (int i = 0; i < NSAMPLES; i++) {
        reg_b_tmp[i] = inputAmplitudesView(i);
        reg_L[i] = matrixL(i, 0);
      }

      // compute x0 and store it
      auto x_prev = reg_b_tmp[0] / reg_L[0];
      reg_b[0] = x_prev;

      // iterate
      CMS_UNROLL_LOOP
      for (int iL = 1; iL < NSAMPLES; iL++) {
        // update accum
        CMS_UNROLL_LOOP
        for (int counter = iL; counter < NSAMPLES; counter++)
          reg_b_tmp[counter] -= x_prev * reg_L[counter];

        // load the next column of cholesky
        CMS_UNROLL_LOOP
        for (int counter = iL; counter < NSAMPLES; counter++)
          reg_L[counter] = matrixL(counter, iL);

        // compute the next x for M(iL, icol)
        x_prev = reg_b_tmp[iL] / reg_L[iL];

        // store the result value
        reg_b[iL] = x_prev;
      }
    }

update_decomposition_forwardsubst_with_offsets()

template<typename MatrixType1 , typename MatrixType2 , typename VectorType >

EIGEN_ALWAYS_INLINE EIGEN_DEVICE_FUNC void calo::multifit::update_decomposition_forwardsubst_with_offsets	(	MatrixType1 &	L,
		MatrixType2 const &	M,
		float	b[MatrixType1::stride],
		VectorType const &	Atb,
		int const	N,
		ColumnVector< MatrixType1::stride, int > const &	pulseOffsets
	)

Definition at line 165 of file MultifitComputations.h.

References b, mps_fire::i, dqmiolumiharvest::j, dqmdumpme::k, dttmaxenums::L, SiStripPI::max, SiStripPI::min, N, HLT_2024v13_cff::pulseOffsets, mathSSE::sqrt(), and dqmMemoryStats::total.

Referenced by fnnls().

                                                                  {
      using T = typename MatrixType1::base_type;
      auto const i = N - 1;
      auto const i_real = pulseOffsets(i);
      T sumsq{0};
      T total = 0;
      for (int j = 0; j < i; j++) {
        auto const j_real = pulseOffsets(j);
        T sumsq2{0};
        auto const m_i_j = M(std::max(i_real, j_real), std::min(i_real, j_real));
        for (int k = 0; k < j; ++k)
          sumsq2 += L(i, k) * L(j, k);

        auto const value_i_j = (m_i_j - sumsq2) / L(j, j);
        L(i, j) = value_i_j;
        sumsq += value_i_j * value_i_j;

        total += value_i_j * b[j];
      }

      auto const l_i_i = std::sqrt(M(i_real, i_real) - sumsq);
      L(i, i) = l_i_i;
      b[i] = (Atb(i_real) - total) / l_i_i;
    }