Summary in math

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Algorithm 5.2 (CG).
Given xo;
Setro – Axo – b, po + -rok = 0;
while rk 70
Ck
(5.24a)
PÃ Apx’
Xk+1 + Xx + QkPk;
rk+1 + k + QkApk;
Bx+1
“kti”k+1
(5.24b)
(5.24c)
+
(5.24d)
Pk+1 + – +1 + Bk+1Pk;
k+ k + 1;
(5.24e)
(5.24f)
end (while)
At any given point in Algorithm 5.2 we never need to know the vectors x, r, and
p for more than the last two iterations. Accordingly, implementations of this algorithm
overwrite old values of these vectors to save on storage. The major computational tasks to be
performed at each step are computation of the matrix-vector product Apk, calculation of
the inner products pĀ (Apx) and rk+k+1, and calculation of three vector sums. The inner
product and vector sum operations can be performed in a small multiple of n floating-point
operations, while the cost of the matrix-vector product is, of course, dependent on the
problem. The CG method is recommended only for large problems; otherwise, Gaussian
elimination or other factorization algorithms such as the singular value decomposition are
to be preferred, since they are less sensitive to rounding errors. For large problems, the CG
method has the advantage that it does not alter the coefficient matrix and (in contrast to
factorization techniques) does not produce fill in the arrays holding the matrix. Another key
property is that the CG method sometimes approaches the solution quickly, as we discuss
next.

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