Lab 3: Taylor Series Lab · Numerical Methods

Pick a function and a degree. The orange dashed curve is the Taylor polynomial about 0. Move the evaluation point and compare the actual error with the first neglected term.

Order of a difference formula

Try this

Run it. Do the central-difference ratios approach 4?
Change f to math.exp and exact to math.exp(1.0). Same ratios?
Add h = 1e-6 and 1e-9 to the list. What happens to the central error, and why?

Sine series with error bound

Try this

Run it. Is the error always below the bound?
Change x to 3. How many terms until the error is below 1e-6?

How many terms for e?

Try this

Run it. Compare with the 10 terms predicted on the slide "How many terms do we need?".
Change the tolerance to 1e-12. How many terms?

Q1 Write the Taylor series of

\ln x

about

x = 1

up to the

(x-1)^3

term.

\ln x = (x-1) - \tfrac{(x-1)^2}{2} + \tfrac{(x-1)^3}{3} - \cdots

Q2 Approximate

\cos(0.2)

with the first three non-zero terms and bound the error.

1 - 0.02 + 0.0000667 = 0.9800667

. Next term

x^6/720 = 8.9\times10^{-8}

bounds the error. True value

0.9800666

.

Q3 Derive the central difference formula for

f'(x)

and state its order.

Expand

f(x\pm h)

, subtract:

f(x+h)-f(x-h) = 2hf'(x) + \tfrac{h^3}{3}f'''(x)+\cdots

. So

f'(x) = \tfrac{f(x+h)-f(x-h)}{2h} - \tfrac{h^2}{6}f'''(x)

: order 2.

Q4 Errors of a method are

2.0\times10^{-3}

for

h = 0.2

and

5.1\times10^{-4}

for

h = 0.1

. Estimate its order.

Ratio

\approx 3.92 \approx 4 = 2^2

, so order

p \approx 2

(

\log_2 3.92 = 1.97

).

Q5 Why does the error of

(f(x+h) - f(x))/h

increase again when

h

is very small?

Round-off:

f(x+h)

and

f(x)

each have an error of about

\varepsilon|f|

; their difference is divided by

h

, giving an error of about

\varepsilon|f|/h

that grows as

h \to 0

. The total error is minimised near

h \approx \sqrt{\varepsilon}

.