Question

Difference between classical and modified gram schmidt

Answer 1

The classical Gram-Schmidt (CGS) and modified Gram-Schmidt (MGS) processes lead to the same result in exact precision arithmetic. In finite-precision arithmetic, MGS is more numerically stable than the classic process.

For a given square matrix AA of size nn, with linearly independent columns a1,...,ana1,...,an, the algorithms can be written is pseudocode as:

CGS vs. MGS

Note that for n=2n=2, the methods coincide even in finite-precision arithmetic. The error in enforcing orthogonality of vectors qiqi found by MGS depends not only on machine precision but also on the condition number κ(A)κ(A) of the matrix.

For ill-conditioned matrices other orthogonalization methods are recommended, such as the Householder of the Givens algorithm. These methods produce vectors that are orthogonal up to machine precision. For further information on error bounds of orthogonalization schemes, see The influence of orthogonality on the Arnoldi method by Braconnier, Langlois and Rioual (Linear Algebra and its Applications, 2000).