TT Q

)

e ic r

}

D h

x

' XQ

W

The matrix operation with the preconditioned matrix can be dif¬cult numerically for large

˜

c

. If the original matrix is Symmetric Positive De¬nite, then is not symmetric, but v

it is self-adjoint with respect to the -inner product; see Exercise 1. „

)0©§

¨ )0I

¨

! #$&#$"

!

% 2

B CA@

97

B

( 9 PH5 G3

F

IA

The polynomial can be selected to be optimal in some sense, and this leads to the use of

˜˜

}

Chebyshev polynomials. The criterion that is used makes the preconditioned matrix x

as close as possible to the identity matrix in some sense. For example, the spectrum of the

preconditioned matrix can be made as close as possible to that of the identity. Denoting by

˜ ˜

}

the spectrum of , and by the space of polynomials of degree not exceeding , p

x

the following may be solved.

Find which minimizes:

p

un1 j

˜

}

h P ) x )0 e P " %%#

'!

r (o & $

˜

Unfortunately, this problem involves all the eigenvalues of and is harder to solve than

the original problem. Usually, problem (12.4) is replaced by the problem

Find which minimizes:

p

An4 j

˜

}

tfP ) x )0 efP " 2#

! %$

3

˜

}

which is obtained from replacing the set by some continuous set that encloses it. 5

x

˜

Thus, a rough idea of the spectrum of the matrix is needed. Consider ¬rst the particular

˜

case where is Symmetric Positive De¬nite, in which case can be taken to be an interval 5

˜

containing the eigenvalues of .

8A6 –

l 7t

A variation of Theorem 6.4 is that for any real scalar such with , the minimum 9 @59

6U

}

B

A Yx Py 2V " T S$ Q

X W !U R

‘ P

G

c r IPo g HEFD$ g

GC

is reached for the shifted and scaled Chebyshev polynomial of the ¬rst kind,

T Q

V t e cp a efT

b

`a v

d

¤} v

Yx p h

X T

a

e IT

V t e cp

b v

d

v

X

Yx p p c v p D Xx p Xx c r p

¨c d x XYx c r p

} } }

¤} ¤ ¤} ¤ v

c

¨c

p p

and the relations (12.8) yield The identity

dx D e }

h Y

}

˜ ˜

Y e} x p ¤ } x c r p ¤ x D p W c r p Y

between two successive residual vectors is given by

p

is the polynomial de¬ned by the above recurrence. The difference

where ˜

¤ Y} x c rp ¤

D c rpY

The goal is to obtain an iteration that produces a residual vector of the form

v

.

, so that c d x uefD

}

v

¢ ¤c

¢D

$5 and

provided we set

Observe that formulas (12.7“12.8) can be started at

¢ ¤c

¤

©

v

Yx v