# Suppose a computer program is available which yields values for

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```					                                Take Home Exercises
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Book: Burlish & Stoer
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1- Suppose a computer program is available which yields values for arcsin y in floating-
point representation whit t decimal mantissa places and for           y ≤ 1 subject to a relative
error ε whit .In view of the relation
x
arctan x = arcsin
1+ x2
This program could also be used to evaluate arctan x . Determine for which values x this
procedure is numerically stable by estimating the relative error.

******************

⎛z⎞
2- For given z ,the function tan ⎜ ⎟ can be computed according to the formula
⎝2⎠
1
⎛z⎞    ⎛ 1 − cos z ⎞ 2
tan⎜ ⎟ = ±⎜           ⎟ .
⎝2⎠    ⎝ 1 + cos z ⎠
π
Is this method of evaluation numerically stable for z ≈ 0, z ≈                 ? If necessary, give
2
numerically stable alternatives.

******************

3- The variance of a set of observation is to be determined. Which of the formulas
1 ⎛ n 2           ⎞
S2 =         ⎜ ∑ xi − n~ 2 ⎟,
x
n − 1 ⎝ i =1        ⎠
2
1 n                       n
S =2               ~ ) ; ~ := 1 x
∑ ( xi − x x n ∑ i
n − 1 i =1                i =1
is numerically more trustworthy?

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4- Let Li ( x ) be the Lagrange polynomials (2.1.1.3) for pair wise different support abscissas
x0 ,..., x n and let . Show that
⎧        1            ;j=0
n
⎪
(a) ∑ ci xi = ⎨ j
0        ; j = 1,2,..., n
i =0      ⎪(−1) n x x ...x ; j = n + 1
⎩        0 1    n

1
n
(b)    ∑ L ( x) ≡ 1
i =0
i

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5- (a) The Bessel function of order zero,
1    π

π∫
J 0 ( x) =                 cos( x sin t )dt ,
0

is to be tabulated at equidistant arguments xi = x0 + ih, i = 0,1,2,... .How small must the increment
h be chosen so that the interpolation error remains below 10 −6 if linear interpolation is used?
(b) What is the behavior of the maxsimal interpolation error
max Pn ( x) − J 0 ( x)
0≤ x ≤1

i
as n → ∞ ,if Pn (x) interpolates at x = xi( n ) :=       , i = 0,1,..., n ?
n
Hint: It suffices to show that J 0 k ) ( x ) ≤ 1 for i = 0,1,...
(

(c) Compare the above result whit the behavior of the error
max S Δ n ( x) − J 0 ( x)
0≤ x ≤1

as n → ∞ ,where S Δ n (x ) is the interpolating spline function whit knot set Δ n = {xi(n ) } and
S ' Δ n ( x ) = J 0 ( x ) for x = 0,1 .
'

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6- Calculation the inverse and reciprocal differences for the support points

xi 0             1     -1           2   -2
fi                     3                3
1           3                  3
5                5

and use them to determine the rational expression Φ 2, 2 ( x) whose numerator and denominator
are quadratic polynomials and for which Φ 2, 2 ( x) = f i , first in continued-fraction from and then
as the ration of polynomials.

******************

7- Suppose S Δ (x) is a spline function whit the set of knots
Δ = {a = x0 < x1 < ... < x n = b}
Interpolating f ∈ κ 4 (a, b) . Show that

2
b
f − SΔ       = ∫ ( f ( x) − S Δ ( x)) f ( 4) ( x)dx
2
a
if any one of the following additional conditions is met:
(a) f ' ( x) = S Δ ( x) for x = a, b .
'

(b) f " ( x) = S Δ ( x) for x = a, b .
"

(c) S Δ is periodic and f ∈ κ p (a, b).
4

3

```
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