Page 89 - Standard Handbook Of Petroleum & Natural Gas Engineering

78 Mathematics

Table 1-14
Chebyshev Polynomials
T,(x) = 1
T,(x) = x
T,(x) = 2~’- 1
T3(x) = 4x3 - 3~
T,(x) = 8x4 - 8x2 + 1
T,(x) = 16x5 - 20x3 + 5x
T6(x) = 32x6 - 48x4 + 18X2 - 1
T,(x) = 64x7 - 112x5 + 56x3 - 7x
T,(x) = 128~’ - 256x6 + 160~~ - 32~’ + 1

Table 1-15
Inverted Chebyshev Polynomials
1 = To

x = T,

1
-w, + T3)
4
1
i(3To + 4T, + T4)
1
-(10T, + 5T, + T,)
16
1
-(lOTo + 15T, + 6T4 + TJ
32
1
-(35T, + 21T3 + 7T, +T7)
64
1
-(35T, + 56T, + 28T, + 8T, + TJ
128

and the inverted Chebyshev polynomials can be substituted for powers of x in
a power series representing any function f(x). Since the maximum magnitude
for Tn = 1 because of the interval, the sum of the magnitudes of lower-order
terms is relatively small. Therefore, even with truncation of the series after
comparatively few terms, the altered series can provide sufficient accuracy.
See also the discussion on cubic splines in “Interpolation.”
Numerical integration

By assuming that a function can be replaced over a limited range by a simpler
function and by first considering the simplest function, a straight line, the areas