Numerical Methods in Context

01Numerical Methods in Context

Solving Model Equations

This section covers forming model equations, applying numerical methods and checking a solution in context.

Turn the context into an equation

Numerical methods provide approximate solutions to model equations when the usual algebraic methods are not practical. The context may involve motion, temperature, populations or a changing value.

Identify the unknown, the given parameter values and any allowed range.
Write the required condition as an equation. For root-finding, put it in the form $f(x)=0$ and use the specified method and starting value.
Check the requested accuracy, then state what the solution represents in the original situation.

A question may combine methods: an iteration gives an approximation, while a change of sign can verify its accuracy.

Worked example: an angle in an orbital model

A model relates two angles, $M$ and $E$ , measured in radians:

$M=E-0.1\sin E.$

For $M=\frac{\pi}{6}$ , determine $k$ in $f(x)=x-0.1\sin x-k$ , use one Newton-Raphson step from $x_0=0.6$ , and justify the value of $E$ to three decimal places.

Set up the equation. Substituting the given value of $M$ gives

$E-0.1\sin E-\frac{\pi}{6}=0,\qquad k=\frac{\pi}{6}.$

Calculate the next approximation. With $f(x)=x-0.1\sin x-\frac{\pi}{6}$ ,

$f'(x)=1-0.1\cos x,$

$\begin{aligned} x_1&=0.6-\frac{0.6-0.1\sin(0.6)-\pi/6}{1-0.1\cos(0.6)}\\ &=0.5782695\ldots. \end{aligned}$

Verify the rounding. Test the two boundaries of the interval that rounds to $0.578$ :

Boundary $x$	$f(x)$ , approximately	Sign
$0.5775$	$-0.00069188$	Negative
$0.5785$	$0.00022436$	Positive

The function is continuous between these boundaries, so the sign change establishes a root with $0.5775\lt E\lt0.5785$ . Therefore, the model gives $E\approx0.578$ radians, correct to three decimal places.

Exam Tip: When $x_0$ is the first approximation, the second approximation is $x_1$ : apply the formula once. Keep the calculator in radians for this model, and show the two signed function values when justifying the stated accuracy.

02Numerical Methods in Context

Numerical Integration in Context

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Give the area a meaning

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A boat journey from speed readings

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Time $t$ (s)	Speed $v$ ( $\mathrm{m\,s^{-1}}$ )
$0$	$2$
$5$	$5$
$10$	$10$
$15$	$18$
$20$	$28$
$25$	$42$

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$h=\frac{25-0}{5}=5\text{ s}.$

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$\begin{aligned} \text{distance}&\approx\frac{5}{2}\bigl[(2+42)\\ &\qquad{}+2(5+10+18+28)\bigr]\\ &=\frac{5}{2}(166)=415\text{ m}. \end{aligned}$

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Judge the estimate from the curve

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03Numerical Methods in Context

Interpreting Results

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Distinguish the value from the step number

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A population falling below half its initial value

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$x_{n+1}=\frac{27\sqrt{x_n}+x_n}{0.75e^2},\qquad x_0=90.$

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$x_1=\frac{27\sqrt{90}+90}{0.75e^2}=62.4607502\ldots.$

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Year $n$	$x_n$ (millions)	Below half?
$0$	$90.00$	Pj
$1$	$62.46$	Oq
$2$	$49.78$	Ei
$3$	$43.36$	Ouy

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04Numerical Methods in Context

Model Limitations

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Check whether the prediction is sensible

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Separate the model from the calculation

Numerical-method limitation

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Model limitation

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