Chebyshev interpolation and quadrature formulas of very high degree

✍ Scribed by Salzer, Herbert E.

• Book ID: 121869202
• Publisher: Association for Computing Machinery
• Year: 1969
• Tongue: English
• Weight: 73 KB
• Volume: 12
• Category: Article
• ISSN: 0001-0782
• DOI: 10.1145/362946.362980

✦ Synopsis

All the zeros
x
2

m
,
i
,
i
= 1(1)2

m
, of the Chebyshev polynomials
T
2

m
(
x
),
m
= 0(1)
n
, are found recursively just by taking
n
2

n
-1

real square roots. For interpolation or integration of ƒ(
x
), given ƒ(
x
2

m
,
i
), only
x
2

m
,
i
is needed to calculate (a) the (2

m

• 1)-th degree Lagrange interpolation polynomial, and (b) the definite integral over [-1, 1], either with or without the weight function (1 -
  x
  2
  )
  -1/2
  , the former being exact for ƒ(
  x
  ) of degree 2

m
+1