9709 · 2.6
Numerical solution of equations flashcards
Revision flashcards for Cambridge 9709 Numerical solution of equations (syllabus 2.6). Flip, recall, then mark a real past-paper question.
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What is a 'root' of an equation f(x) = 0?
A value 'α' such that f(α) = 0. Graphically, it is the x-coordinate where the curve y = f(x) intersects the x-axis.
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What condition guarantees a root of f(x) = 0 lies in the interval [a, b]?
If the function f(x) is continuous on [a, b] and f(a) and f(b) have opposite signs (i.e., f(a) * f(b) < 0), then there is at least one root in the interval (a, b).
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What is an iterative formula?
A formula of the form x_{n+1} = g(x_n) which generates a sequence of approximations to a root, starting from an initial value x_0.
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How do you show that a root is, for example, 2.43 correct to 2 decimal places?
You must show that the root lies in the interval [2.425, 2.435]. Do this by evaluating f(2.425) and f(2.435) and showing they have opposite signs.
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What is convergence in an iterative process?
When the sequence of values x_1, x_2, x_3, ... gets progressively closer to a fixed value, which is the root of the equation.
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What is divergence in an iterative process?
When the sequence of values x_1, x_2, x_3, ... moves further and further away from the root. The iteration fails to find a solution.
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What is a common cause of error when performing iterations?
Using a calculator in Degrees mode when the question involves trigonometric functions that require Radians mode (e.g., solving x = cos(x)). Always check your calculator's mode.
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How much working should I show for an iteration question?
Write down the iterative formula, the starting value x_0, and each subsequent value (x_1, x_2, ...) to at least two more decimal places than the final required accuracy. This demonstrates your process.
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How do I choose a starting value, x_0, for an iteration?
The question usually provides one. If not, use the interval [a, b] where you've shown a root exists. A good choice is the midpoint of the interval, (a+b)/2.
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What does the condition |g'(α)| < 1 mean, where α is the root?
This is the mathematical condition for an iterative formula x_{n+1} = g(x_n) to converge to the root α. If the magnitude of the gradient of g(x) at the root is less than 1, the iteration will converge.