Multiplying the system by turns it into . For nonsingular , compute the transformed right side and solve the upper triangular system by back substitution. Orthogonal transformations preserve Euclidean lengths and do not amplify errors in that transformation.
If is singular,
(1)
Since , at least one diagonal entry of is zero. The resulting triangular system must then be checked for consistency and any free variables, rather than applying divisions by zero.

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