Solve a linear system by Householder transformations
Matrix and right handside:
Compute the Householder transformation by the following steps:
- Define the direction v (unscaled and simple) and its norm N
- Compute w:=v^T*A and then w:=2*w/N^2.
- Subtract A-v*w to give the transformed A
First step:
x and beta, v and its norm:
Matrices A, v*w and the new A:=A-v*w:
Second step:
x and beta, v and its norm (Note the 0 in the first component of x):
Matrices A, v*w and the new A:=A-v*w:
Some tests:
The 2 transformation matrices and their product Q:
Indeed: Q*A=R:
Q is unitary (matrices scaled with 15):