Page "Conjugate gradient method" Paragraph 115

As an iterative method, it is not necessary to form A < sup > T </ sup > A explicitly in memory but only to perform the matrix-vector and transpose matrix-vector multiplications.

Therefore CGNR is particularly useful when A is a sparse matrix since these operations are usually extremely efficient.

However the downside of forming the normal equations is that the condition number κ ( A < sup > T </ sup > A ) is equal to κ < sup > 2 </ sup >( A ) and so the rate of convergence of CGNR may be slow and the quality of the approximate solution may be sensitive to roundoff errors.

Finding a good preconditioner is often an important part of using the CGNR method.