unH1 j

¢ x ¢

C C ¥ I

˜ ˜

}

)xCX D h

)

}

This allows to be computed as a linear combination of the polynomials . Thus, )xCX

p c

v

we can obtain the desired least-squares polynomials from the sequence of orthogonal poly-

nomials which satisfy a three-term recurrence of the form:

¢

C

}

}

}

}

d t e D wt ) x c v i 7 ) x i $6 ) x D ) x c s¢ c s7

rC

rC

CC

CC 2¢iit

hhhh

From this, the following recurrence for the ™s can be derived: CX

}

}

}

}

}

d ’f$et ¢ x C t ) x c v C X C 7 ) x C X C 6 ‚) x D ) x c sC X c rsC 7

r

te Dw i¢it

hhhh

The weight function is chosen so that the three-term recurrence of the orthogonal

£

polynomials is known explicitly and/or is easy to generate. An interesting class of weightC

functions that satisfy this requirement is considered next.

Choice of the weight functions This section assumes that and . Consider

¢D6 `7

D e

the Jacobi weights

An4 j

e ¥ I

˜ ˜

¥}

¤

) e x c v £) D } ) x £ where and

¢q¦

¢ §

t h

d

For these weight functions, the recurrence relations are known explicitly for the polyno-

)x £ ¨ )

) x £¡) } ) x £ } }

mials that are orthogonal with respect to , , or . This allows the use of

}

any of the three methods described in the previous section for computing . More- )x c p

v

˜¦ ˜ }

over, it has been shown [129] that the preconditioned matrix is Symmetric Positive xp

˜ c

De¬nite when is Symmetric Positive De¬nite, provided that . ¨ ¨ `

§ e

}

The following explicit formula for can be derived easily from the explicit ex- )x p #

pression of the Jacobi polynomials and the fact that is orthogonal with respect to the "F

# ap

}

weight :

) x ¡)

£

p

An j

) } ) ) p } )0 e r so © ‚ D } ) p #

p ¥ I

˜ ˜

xv x) x

)

c)

§ t W v p r qo

p

w

2 )0 D ) ©

h

¦ t e t w sC

2 ) x p # e x D } ) x c p

e}

}

Using (12.13), the polynomial can be derived easily “by hand”

)

v

for small degrees; see Exercise 4.

™

—— p z7 w p w {x U |gt

˜ { { | p p #U

£¤¢

¡ 9

§ "

! "

¡

£

¢ ¥

¤

# % # ! ©¦§¥£¡¥

¨ ¤¢

As an illustration, we list the least-squares polynomials for ,, p ‚

D e i¢h

hh

c c

, obtained for the Jacobi weights with and . The polynomials listed are for ¦ ¨ fDQ§

D

¨

the interval as this leads to integer coef¬cients. For a general interval , the best

t ¢ –

l 8¢ –

l 7t

} }

polynomial of degree is . Also, each polynomial is rescaled by to

7 “ x p

) p

t x

d

simplify the expressions. However, this scaling factor is unimportant if these polynomials

are used for preconditioning.