Fall 2005 Final

View the exam on docuum

Coming soon.

1Question 1¶

1.1Solution¶

1.1.1Part (a)¶

State the fixed point theorem

ez

1.1.2Part (b)¶

Find an interval and starting point on which some iterative scheme for finding something satisfies the conditions of the theorem, and find the rate of convergence.

Choose an interval that satisfies the FPT. Rate of convergence is probably linear cus the first deriv is non-zero

1.1.3Part (c)¶

Aitken's method.

Hopefully it's not on our syllabus because I'm not going to cover it.

2Question 2¶

2.1Solution¶

2.1.1Part (a)¶

Define fundamental Lagrange polys

See the HTSEFP

Show uniqueness

See the HTSEFP

2.1.3Part (c)¶

Find the interpolating poly

Construct the table of divided differences (remember to DIVIDE ... DIVIDED differences)

$f[x_i]$ $f[x_{i-1}, x_i]$ $f[x_{i-2}, x_{i-1}, x_i]$ $f[x_{i-3}, x_{i-2}, x_{i-1}, x]$
$x_0 = 0$ 0
$x_1 = 1$ 0 0
$x_2 = 2$ 4 4 2
$x_3 = 3$ 6 2 -1 -1

$$p_3(x) = 0 + 0(x -0) + 2(x-0)(x-1) -1(x-0)(x-1)(x-2) = 2x(x-1) - x(x-1)(x-2)$$

Find the error bounds

Use the given formula. Should get $10/24 * (2.5-0)(2.5-1)(2.5-2)(2.5-3)$ so the error $\leq 0.390625$

2.2Accuracy and discussion¶

No solutions for this that I can find. Not sure if right

3Question 3¶

3.1Solution¶

3.1.1Part (a)¶

Diff between Lagrange/Hermite? Clamped/natural cubic spline?

see htsefp

3.1.2Part (b)¶

Find constants for some natural cubic spline

see htsefp

3.1.3Part (c)¶

Bezier curve

same as 2009 q3 (c)

4Question 4¶

4.1Solution¶

4.1.1Part (a)¶

Use quadratic formula to approximate roots

idk, just use it, not really worth thinking about

4.1.2Part (b)¶

Use centred-difference expression for approximating second derivative

basically the same as 2009

5Question 5¶

5.1Solution¶

5.1.1Part (a)¶

Define deg of acc for quadrature

see htsefp

5.1.2Part (b)¶

Find constants to maximise deg of acc

see htsefp

5.1.3Part (c)¶

Find deg of acc for previous method, and find the constant $k$

see htsefp

6Question 6¶

6.1Solution¶

6.1.1Part (a)¶

Use composite trapezoidal rule to approximate something

see htsefp

6.1.2Part (b)¶

Derive error bound

see htsefp

6.1.3Part (c)¶

Obtain upper bounds for error for previous approximation

First, compute the second derivative:

$$f'(x) = 2xe^{x^2} \quad f''(x) = 2e^{x^2} + 4x^2e^{x^2} = 2e^{x^2}(1+2x^2)$$

Clearly $f''$ is a non-decreasing function for the interval $[0, 1]$. So the maximum $f''$-value is when $x = 1$, at which point $f''(1) = 2e(1+2) = 6e$.

For $h=0.25$:

$$I(f) - I_h(f) \leq -\frac{1-0}{12}(0.25)^2 \max_{\xi \in [0, 1]} f''(\xi) = -\frac{6e}{192} \approx -0.084946$$

For $h=0.5$

$$I(f) - I_h(f) \leq -\frac{1-0}{12}(0.5)^2 \max_{\xi \in [0, 1]} f''(\xi) = -\frac{6e}{48} \approx -0.0212365$$

6.1.4Part (d)¶

Apply one step of Richardson extrap

if this shows up on the final im just gonna wing it

7Question 7¶

Runge-Kutta

7.1Solution¶

7.1.1Part (a)¶

Define local truncation error, find order of method

see htsefp

7.1.2Part (b) (i)¶

Show some identity for $w_{i+1}$

see htsefp

7.1.3Part (b) (ii)¶

What conditions on $h$ are required to get the limit of $w_i = 0$ as $i \to \infty$

see htsefp

8Question 8¶

Linear shooting stuff, not gonna bother