1.1Estimate a Function with Newton's Method
For $f(x)=x^2-2$, write one Newton step starting from $x_0=1$.
Solution
$x_1=x_0-f(x_0)/f'(x_0)=1-(-1)/2=1.5$.
Numerical Methods Practice
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Difficulty
Direct Practice
1.2Identify Absolute Error
An approximation is $3.14$ and the exact value is $\pi$. Write its absolute error.
Solution
The absolute error is $|3.14-\pi|\approx0.00159265$.
Difficulty
Integrated Practice
2.1Apply Forward Euler
Use one forward-Euler step with $\Delta t=0.1$ for $y'=-2y$, $y(0)=3$.
Solution
$y_1=y_0+\Delta t(-2y_0)=3+0.1(-6)=2.4$.
2.2Check a Finite-Difference Units
What units must $\Delta x^2$ have if it appears in the denominator of a second spatial derivative approximation?
Solution
If $x$ is measured in meters, $\Delta x^2$ has units $\mathrm{m^2}$, so the approximation has the units of the field divided by $\mathrm{m^2}$.
Difficulty
Applied Problems
3.1Interpret Convergence
Why does reducing a discretization step not by itself prove that a numerical result is physically correct?
Solution
Refinement addresses discretization error, but model error, incorrect boundary data, instability, and implementation errors can remain.