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The Conjugate Gradient Squared algorithm was developed by Sonneveld in 1984 [201],

§

mainly to avoid using the transpose of in the BCG and to gain faster convergence for

roughly the same computational cost. The main idea is based on the following simple

observation. In the BCG algorithm, the residual vector at step can be expressed as T

r± 0 „ ®

i

uD X

¬ §

u¨ ¨ T

uD uD ¬ XY

T

where is a certain polynomial of degree satisfying the constraint . Similarly,

X y

¡ u

the conjugate-direction polynomial is given by T

p±q0 „ ®

°i

X

§ u ¢¬

u ¡ (t ¨

¡

u

¡ ¨u

in which is a polynomial of degree . From the algorithm, observe that the directions

¨u ¨ §

u¨ u

and are de¬ned through the same recurrences as and in which is replaced by

¡ ¡

§ and, as a result, T T

uD X X

¨u ¨ ( ¨ ¨

¨u ¨ ¨

¬ § § u¡ ¬

¡

Tu

Also, note that the scalar in BCG is given by T T

T T

¨ ¨ ( ¨ X T § £u D T X

D

X¨ X T uD X

§ §

(¨ T u T

¨X

§ ¡§ ¬ ¬

u ¨ ¨ p ¨ X § £ u ¡ §

u ¨ X u

X¨ X

§¡

¨ ( ( T T

£D

§ £u ¡

X X

§

which indicates that if it is possible to get a recursion for the vectors and , u

¨ ¨

u uS

then computing and, similarly, causes no problem. Hence, the idea of seeking an

algorithm which would give a sequence of iterates whose residual norms satisfy ¤u

¨

T

r±0 0 „ ®

i

£D X

¬ §

¤u u t ¨

¨

The derivation of the method relies on simple algebra only. To establish the desired

uD u

¡

recurrences for the squared polynomials, start with the recurrences that de¬ne and ,

which are, T T

T

r±a 0 „ ®

i

XT XT u u T X u D

uD ¬ X ² ¡

(

± ¯ 0 „ ®

i

X u u w X

’U u

W uD

¬

¡ ¡

S

W’U "U

W

£

(u u u£

£w

¢ ¬ W“ “

¨ £

. This de¬nition leads to the relations u uS w u u

¢W ¬

A slight simpli¬cation to the algorithm can be made by using the auxiliary vector

W"U W’U

¡ ¡

S ˜S ¨ .

¬ u

Xu u w u¢ u w Wu

¨ ( ¨ ¨ ( ’U u ¨ ¬ u S

X ¨ uT X ¨ ’U

W

¨

u £ T § u ² T ¬ W ¨

£ w u ¦¬ u ©

©u ’U u

Wu u

“ W“

¡

u §u ² ¢ S w u¨ ¬ u¢

W “ Wu “ u

¡

u §u ² u¢ u S˜ w u¨˜ ¬ u£

¡

¨ ¨ ( u § ¨ ( ¨ ¬ u

X¨ u

¡

T

¨

.

,

,

,

solution, starting with Y`¬ S Y`¬ ¢ ¬ ty¨ ¬ ¨ T

© § ²X

3 3 3 3

With this we obtain the following sequence of operations to compute the approximate

¡