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We selected and only because these choices lead to a very simple re-

—¦ p§

D D ¨

¨

currence for the polynomials , which are the Chebyshev polynomials of the ¬rst kind.

C

Theoretical considerations An interesting theoretical question is whether the least-

squares residual polynomial becomes small in some sense as its degree increases. Consider

¬rst the case . Since the residual polynomial minimizes the norm as- ! #

@16 0¢ )

7) # p

}

sociated with the weight , over all polynomials of degree such that , the

£¨ # ¢x #

U fD

e

¨

pj}} }

polynomial with satis¬es

) x #e x d 7 t 6x D

¥ 22

2p

2

2p

p 2

2

2

6 7 © D 2 45 z 2 U 2 ) 2

3 e 22 U ! p #

2

2

2

!2 ¦

4 t z 2

!2

6 t 7

©

where is the -norm of the function unity on the interval . The norm of will £

l7 t 6 – # p

tend to zero geometrically as tends to in¬nity, provided .

¢q6

Consider now the case , and the Jacobi weight (12.15). For this choice

e D 7 ¢ p6 D

¢ x p I } ) x p y }

of the weight function, the least-squares residual polynomial is known to be y

7Q

6 }

where is the degree Jacobi polynomial associated with the weight function D ) x 9£ 8

py

¤} ) e ) . It can be shown that the 2-norm of such a residual polynomial with respect to

x¢

this weight is given by

B B B

§ t x e t ¦x ¨

} } }

e t x e

@¨ } t

D A! ¢ x p p b

B} B

y

y

} }

e t § t ¦ t x x e t § t ¦ t dx t ¦ t x e

B

c c

in which is the Gamma function. For the case and , this becomes

5¦

GD‚§ D

¨ ¨

¨ l } ¤ B–

C ¨}

¨x D @ ! ¢ x p p b

y

y

D }c h

¨}e

}

t dx d ¨ t x e t dx

Therefore, the -norm of the least-squares residual polynomial converges to zero like

8£ ’ue

as the degree increases (a much slower rate than when ). However, note that the

¢q6

}

condition implies that the polynomial must be large in some interval around the

e D ¢x y

—U™

—t

7 —— p w7 z p — |w z

˜ { { | |

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¢ ¢

¡

4 ¢

$#

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origin.

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3 3 5@ I )

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Given a set of approximate eigenvalues of a nonsymmetric matrix , a simple region can 5

be constructed in the complex plane, e.g., a disk, an ellipse, or a polygon, which encloses

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