5 Linear Algebra

5.1 Transpose

• Caution: Transpose does not consume the original value.

  fn main() {
      let a = matrix!(1;4;1, 2, 2, Row);
      a.transpose().print();
      // Or you can use shorter one
      a.t().print();
      //      c[0] c[1]
      // r[0]    1    3
      // r[1]    2    4
  }

5.2 LU Decomposition

• Peroxide uses complete pivoting for LU decomposition - Very stable

• Since there are lots of causes to generate error, you should use Option

• lu returns Option<PQLU>

  • PQLU has four field - p, q, l , u
  • p means row permutations
  • q means column permutations
  • l means lower triangular matrix
  • u menas upper triangular matrix

• The structure of PQLU is as follows:

  #[derive(Debug, Clone)]
  pub struct PQLU {
      pub p: Perms,
      pub q: Perms,
      pub l: Matrix,
      pub u: Matrix,
  }
  
  pub type Perms = Vec<(usize, usize)>;

Example of LU decomposition:

fn main() {
    let a = matrix(c!(1,2,3,4), 2, 2, Row);
    let pqlu = a.lu().unwrap(); // unwrap because of Option
    let (p,q,l,u) = (pqlu.p, pqlu.q, pqlu.l, pqlu.u);
    assert_eq!(p, vec![(0,1)]); // swap 0 & 1 (Row)
    assert_eq!(q, vec![(0,1)]); // swap 0 & 1 (Col)
    assert_eq!(l, matrix(c!(1,0,0.5,1),2,2,Row));
    //      c[0] c[1]
    // r[0]    1    0
    // r[1]  0.5    1
    assert_eq!(u, matrix(c!(4,3,0,-0.5),2,2,Row));
    //      c[0] c[1]
    // r[0]    4    3
    // r[1]    0 -0.5
}