Table of Contents

Linear Solver by LU Factorization

Routine Name: solveLinearSystemLU

Author: Philip Nelson

Language: C++

Description

solveLinearSystemLU solves a linear system of equations \(Ax=b\) by LU Factorization. The method used is:

\[LU=PA\] \[LUx = Pb\] \[Ly = Pb\] \[Ux=y\]

Input

solveLinearSystemLU(std::array<T, M> b) is called by a Matrix<T, M, M> of type T and size MxM and requires:

• std::array<T, M> b - a column vector b of type T and size M

Output

solveLinearSystemLU returns a std::array<T,M> with the solution vector x

Code

std::array<T, M> solveLinearSystemLU(std::array<T, M> b)
{
  auto[L, U, P] = luFactorize();
  auto y = L.forwardSub(P * b);
  auto x = U.backSub(y);
  return x;
}

Example

int main()
{
  Matrix<double, 4, 4> A(1, 10); // random 4x4 with values from 0-10
  std::array<double, 4> x = {4, 7, 2, 5};
  auto b = A*x;
  std::cout << " A\n" << A << std::endl;
  std::cout << " b\n" << b << std::endl;
  std::cout << " x\n" << x << std::endl;

  std::cout << "Calculated x\n";
  std::cout << A.solveLinearSystemLU(b) << std::endl;
}

Result

 A
|         3        8        1        4 |
|         6       10        5        8 |
|         8        4        8        6 |
|         8       10        8        1 |

 b
[        90      144      106      123 ]

 Real x
[         4        7        2        5 ]

Calculated x
[         4        7        2        5 ]

Last Modification date: 07 February 2018