20.11 Exercises 593
16.Use linear interpolation to computef(0.04), f(0.26), f(0.5), f(0.81).
17.Use quadratic interpolation to computef(0.04), f(0.26), f(0.81).
18.Construct the finite difference interpolation table, and use it to computef(0.26).
Section 20.5
Given in Exercises 19–21:
19.Find estimates of the integralIby means of the trapezoidal rule (20.32) and the error
formula (20.34), starting with one strip (n 1 = 11 ) and doubling the number of strips
(n 1 = 1 2, 4, 8, =) until 4-figure accuracy is obtained.
20.Use Simpson’s rule (20.35) for 2 n 1 = 1 2, 4, 8, =to find the value of the integral Ito 4
decimal places
- (i)The error in Simpson’s rule isAh
4
when his small enough, and A is a constant. If
S(2n)is the Simpson formula for 2 nintervals, show that
(Richardson’s extrapolation).
(ii)Use this and the results of Exercise 20 forn 1 = 18 to estimate a more accurate
value of the integral I. Check the accuracy by comparing with the exact value
(ln 1 2.6).
Use Simpson’s rule for 2 n 1 = 1 2, 4, 8, =to find the values of the following integrals to 5
decimal places:
- (i)Use the Euler–MacLaurin formula to calculate the sum to
six decimal places. (ii)Use the value ofS(10) in Example 20.12 to verify that
S(11) 1 = 1 S(10) 1 − 1 0.01.
25.Use the results of Exercise 24 to calculateS(1) to six significant figures.
26.Calculate ln 1 1!, ln 1 2! and ln 1 3!(i) using the ‘large-number’ formulan 1 ln 1 n 1 − 1 n, (ii)from
the more accurate formula given in Example 20.14. (iii)Exactly.
Section 20.7
Solve the following systems of equations by the Gauss elimination method:
- x
1
− 14 x
2
= 1 − 2 28. 2 x
1
- x
2
−x
3
1 = 6
3 x
1
1 + x
2
= 74 x
1
− x
3
1 = 6
− 8 x
1
1 + 12 x
2
1 + 12 x
3
1 = 1 − 8
- x 1 + y 1 + z 1 = 12 30. w 1 + 12 x 1 + 13 y 1 + z= 15
−x 1 + 12 y 1 − 13 z 1 = 132 2 w 1 + x 1 + y 1 + z= 13
3 x − 14 z 1 = 117 w 1 + 12 x 1 + y = 14
x 1 + y 1 + 12 z 1 = 10
Sm
m() ( )11 1 11
2
0
=+
=
∑
∞
erf()zedxis the error function
zx=
−
2
0
2π
Z
Z
0
2
2edx
−xSi() is the Sine integral
sin
z
x
x
dx
z=
Z
0
Z
0
1
sinx
x
dx
Z
abfxdx() ≈−[ ( ) ()]S n Sn
1
15
16 2
I
dx
x
=Z