Fall 2008 Final View the exam on docuum

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1Question 1¶

1.1Solution¶

1.1.1Part (a)¶

State fixed point theorem

see htsefp

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

Relative error?

see htsefp

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

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

4Question 4¶

4.1Solution¶

4.1.1Part (a)¶

Define deg of acc

see htsefp

Find deg of acc

see htsefp

4.1.3Part (c)¶

Find constant $k$

see htsefp

4.1.4Part (d)¶

Approximate $\ln(2)$

see htsefp

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

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

7Question 7¶

Linear shooting stuff, skipping