WY £ WX

² £X

a± „ ®

i

¡

¬ ¥

. .

.. .. ¦ H4 Ry

¢

. .

. . £R

. . ¢

. W “ ¢¢¢

. ¡¡¡

‚

²

. ¢X ¢X

² ¢X

Y

’U

W

The diagonal matrix in the right-hand side scales the columns of the matrix. It is easy to see

that it has no effect on the determination of the rotations. Ignoring this scaling, the above

matrix becomes, after rotations,

¡

„

.. ..

. .

uA

Y

²

uX uX

’U

W "U

W

.. ..

. .

²

‚ ¢X ¢X

² ¢X

"U

W

The next rotation is then determined by,

² ²¬

uX uA uX

¢

u

¬ ¬

u u

¢ ’U £

W ( ( ’U u

W

£X £

£X A

"U

W "U

W "U

W

u w uA u w uA

"U

W ’U

W

T –X

u

In addition, after this rotation is applied to the above matrix, the diagonal element X ’U

W

which is in position is transformed into w w Y(

y y

a± „ ®

i

uXuA

u¢uA

u u

¬ ¥ ¬ ²¬ ²¬

uA uX u uA ¢

£ "U £

W

W"U ’U

W "U

W ’U

W "U

W "U

W

uX w uA

"U

W

¢

The above relations enable us to update the direction and the required quantities and £ ¢

. Since only the squares of these scalars are invoked in the update of the direction , £

¢¦ ¢

’U

W

a recurrence for their absolute values is suf¬cient. This gives the following recurrences

which will be used in the algorithm: T

£ X

¢

¬ £ ¢

w¢

£ £

¢ ¢ ¢¦ ¢

W"U

¢

¬ ¢X A

¢ ¢

W"U £ ¢ "U

W i

¢ ¬ w “

¢

W"U ¢y " U

W

¬

¢A ¢ ¢A ¢

£

W"U W"U W"U

¬ ¢

¢¦ ¢

W"U W"U

Before writing down the algorithm, a few relations must be exploited. Since the vectors

are no longer the actual residuals in the algorithm, we change the notation to . These

¨ ¥

¢ ¢

residual vectors can be updated by the formula

¬ §W “ ² W “ £