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- Becs-114.1100 Assignment 1. Taylor series. Number representation and errors.

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Problem 1. (computer) (2 points)

Study the following central difference formula:

f 0

(x) ⇡ f(x+h) f(xh)

2h

as h ! 0. Using Taylor’s theorem, we obtain that the truncation error for this formula is

h2

6 f 000(x) for some x in the interval (xh, x+h). In a computer, the result is also subject

to a rounding error. The total error is given by the sum of these two errors.

Write and run a code which computes the approximate values of the truncation error and

the total error as a function of h for f(x) = sin(x) at x = 0.5. On the same graph, plot

the results showing the rounding error (er), the truncation error (et) and the total error

(e = er + et). Use log-scale, i.e. plot log10 h (x-axis) vs. log10 |error| (y-axis). Analyze

the results. (Hint: start with an initial value of h, e.g. h = 0.5, and divide the value of h at

each step by some factor, e.g. h h/4).

Problem 2. (pencil and paper) (0 points – do not return)

Suppose that you have a floating-point system in which numbers are represented in the

following form

x = ±0.d1d2d3d4 ⇥10n

where the exponent n is a 2-digit (decimal) integer and the normalized mantissa has four

digits d1d4. Use this system to compute the difference pq where p = 75640 and q = 4.

1

Problem 3. (pencil and paper / computer) (2 points)

(a) The harmonic series 1 + 1

2 + 1

3 + 1

4 + … is known to diverge to +•. The nth partial

sum approaches +• at the same rate as ln(n). Euler’s constant is defined to be

g = lim

n!•

” n

Â

k=1

1

k ln(n)

#

⇡ 0.57721

Write and test a program that uses a loop of 5000 steps to estimate Euler’s constant. Print

intermediate answers at every 100 steps. Compare to exact value.

(b) Prove (using pencil and paper) that Euler’s constant can also be presented by

g = lim

m!•

” m

Â

k=1

1

k ln(m+

1

2

)

#

Write and test a program that uses m = 1,2,3,…,5000 to compute g by this formula.

Compare the convergence to the formula given in part (a). For those interested, see

D.W.Temple, “A quicker convergence to Euler’s constant”, Am. Math. Monthly 100,

468 (1993).

Problem 4. (pencil and paper) (0 points – do not return)

Let us have a floating-point representation in which the numbers x are expressed in the

following form

x = ±(0.b1b2b3)2 ⇥2±m m,bi = {0,1}

List all positive numbers that are representable using this form. (You can form a table of

the mantissa and the exponent bits and convert the combination into base 10 numbers).

What if the representation is normed (the first bit b1 is always 1)?

Problem 5. (2 points) (computer)

Using a computer, examine how the computed values of the following functions behave

near the point x = 0.

(a) f1(x) = ex 1

x

(b) f2(x) = ex ex

2x

In order to avoid loss of significance as a result of subtracting almost equal numbers, expand the exponential functions into Taylor series and use the resulting approximation to

calculate the values of the functions f1 and f2 near zero. Estimate the error of the approximation by using Taylor’s theorem on the series expansion of the exponential function

(f(x) = ex):

f(x) = f(0) +

n1

Â

i=1

xi

i!

f (i)

(0) +R(n)

where the error term is

R(n) = xn

n!

f (n)

(x) 0 x x

2

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