Fall 2008 Final

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1Question 1¶

1.1Solution¶

1.1.1Part (a)¶

State fixed point theorem

see htsefp

1.1.2Part (b) (i)¶

Show fixed point

see htsefp

1.1.3Part (b) (ii)¶

Show that this scheme satisfies conditions for FPT

see htsefp

1.1.4Part (b) (iii)¶

Order of convergence?

see htsefp

1.1.5Part (b) (iv)¶

Compute $x_n$

see htsefp

1.1.6Part (b) (v)¶

Relative error?

see htsefp

1.2Accuracy and discussion¶

2Question 2¶

2.1Solution¶

2.1.1Part (a)¶

Zeroth divided diff

see htsefp

2.1.2Part (b)¶

Define interpolating polys in Newton form

see htsefp

2.1.3Part (c)¶

Show existence of $\xi$ etc

see htsefp

2.1.4Part (d)¶

Construct table of divided diffs, find interpolating polys

see htsefp

2.2Accuracy and discussion¶

3Question 3¶

3.1Solution¶

3.1.1Part (a)¶

Use Taylor series to show something

see htsefp

3.1.2Part (b)¶

Use Richardson extrap to find some formula

see htsefp

3.1.3Part (c)¶

Find some approximation using Richardson extrap formula above

see htsefp

3.2Accuracy and discussion¶

4Question 4¶

4.1Solution¶

4.1.1Part (a)¶

Define deg of acc

see htsefp

4.1.2Part (b)¶

Find deg of acc

see htsefp

4.1.3Part (c)¶

Find constant $k$

see htsefp

4.1.4Part (d)¶

Approximate $\ln(2)$

see htsefp

4.2Accuracy and discussion¶

5Question 5¶

Simpson's rule

5.1Solution¶

5.1.1Part (a)¶

State composite rule

see htsefp

5.1.2Part (b)¶

Show composite rule satisfies something

see htsefp

5.1.3Part (c) (i)¶

Evaluate $I_h(f)$ for some $h$

see htsefp

5.1.4Part (c) (ii)¶

Value of $h$ required to get a certain deg of precision?

see htsefp

5.2Accuracy and discussion¶

6Question 6¶

Runge-Kutta

6.1Solution¶

6.1.1Part (a)¶

Define/find local truncation error, find order of method

see htsefp

6.1.2Part (b) (i)¶

Show that $w_{i+1}$ satisfies some identity

see htsefp

6.1.3Part (b) (ii)¶

Conditions on $h$ to get $\displaystyle \lim_{i \to \infty} w_i = 0$?

see htsefp

6.2Accuracy and discussion¶

7Question 7¶

Linear shooting stuff, skipping