Numerical methods flashcards

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Root of an equation f(x) = 0
A value of x for which f(x) = 0: an x-intercept of the graph of y = f(x).
Continuous function (on an interval)
A function whose graph has no breaks, jumps or asymptotes on that interval: it can be drawn without lifting the pen.
Iteration formula
A rule x(n+1) = g(x(n)) that produces each term of a sequence from the previous one, starting from a chosen x₀.
Convergent iteration
One whose iterates get closer and closer to a fixed value α; then α = g(α), so α is a root of the original equation.
Divergent iteration
One whose iterates do not settle on the root: they move further away from it (or head for another root, or become undefined).
Staircase diagram
Path between y = g(x) and y = x in which the iterates approach (or leave) the root from one side only. Occurs when g is increasing near the root.
Cobweb diagram
Path between y = g(x) and y = x that spirals round the root, the iterates landing alternately above and below it. Occurs when g is decreasing near the root.
Change-of-sign rule
If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then f(x) = 0 has a root between a and b. (No — learn it)
Fixed-point iteration
x(n+1) = g(x(n)), from a rearrangement x = g(x) of f(x) = 0. (No — learn it)
Bounds for showing α = c to n decimal places
Test f at c − 0.5 × 10⁻ⁿ and c + 0.5 × 10⁻ⁿ (e.g. 2.2785 and 2.2795 for 2.279). (No — learn it)
Convergence guide (qualitative)
Near the root, if y = g(x) is less steep than y = x (gradient between −1 and 1) the iteration converges; if steeper, it diverges. (No — learn it)
Undoing an exponential or a logarithm
e^(ln x) = x and ln(eˣ) = x: e.g. e^(x/2) = 3x − 1 ⟺ x = 2 ln(3x − 1). (No — learn it)
Where is the root on a graph of y = g(x) and y = x?
Where the curve y = g(x) crosses the line y = x.
Drawing a staircase or cobweb from x₀
Go vertically from x₀ to the curve y = g(x), then horizontally to y = x; repeat. The x-coordinates of the corners on y = x are x₁, x₂, …
Graph with a vertical asymptote inside [a, b]
f(a) and f(b) can have opposite signs although the graph never meets the x-axis there.
Graph touching the x-axis at a repeated root
The curve does not cross the axis, so f has the same sign on both sides: no sign change.
Two roots close together inside [a, b]
The curve goes through the axis and back, so f(a) and f(b) have the same sign and the test misses both roots.
A cobweb that unwinds outwards
The iteration diverges: y = g(x) is decreasing and steeper than y = x near the root.
Why must f be continuous for the change-of-sign test?
Only an unbroken graph has to cross the axis to get from negative to positive; a graph with a break can jump across.
Why do iterates agreeing to 3 d.p. not prove a root to 3 d.p.?
Agreement is only evidence; the proof is a change of sign between the bounds, e.g. 2.2785 and 2.2795.
Why can two rearrangements of the same equation behave differently?
They have the same root, but different gradients near it: a shallow g converges, a steep g diverges.
Why use the Ans key when iterating?
It keeps full calculator accuracy. Re-typing rounded values can change later decimal places.
What can a divergent iteration do instead of finding the root?
Move steadily away, oscillate with growing size, converge to a different root, or become undefined (e.g. log of a negative).
Show x³ − 3x − 5 = 0 has a root in [2.2, 2.3].
f(2.2) = −0.952 < 0, f(2.3) = 0.267 > 0; change of sign and f continuous, so a root lies in [2.2, 2.3].
x(n+1) = ∛(3x(n) + 5), x₀ = 2. Find x₁ and x₂ to 4 d.p.
x₁ = ∛11 = 2.2240, x₂ = 2.2684.
Show the root β of e^(x/2) − 3x + 1 = 0 is 5.470 to 3 d.p.
f(5.4695) = −0.00261 < 0, f(5.4705) = 0.00210 > 0; change of sign, f continuous, so 5.4695 < β < 5.4705.
Rearrange ln x + 2x − 4 = 0 into x = 2 − ½ ln x.
2x = 4 − ln x, so x = 2 − ½ ln x.
x(n+1) = 2 sin x(n) (radians), x₀ = 2. Find x₁ and x₂ to 4 d.p.
x₁ = 1.8186, x₂ = 1.9389 (alternating either side of the root ≈ 1.8955: a cobweb).
Will x = (x³ − 5)/3 find the root 2.279 of x³ − 3x − 5 = 0?
No: near the root its gradient is x² ≈ 5.2, much steeper than y = x, so the iterates move away.
f(x) = (2x + 1)/(x − 3): f(2) = −5, f(4) = 9. Is there a root in (2, 4)?
No: f is not continuous at x = 3 (asymptote). The only root is x = −½.
Full sign-change conclusion
Give both values with signs, say “change of sign and f is continuous”, then state the interval containing the root.
Trig functions in an iteration
Use radians. Degree mode gives completely different values and loses every mark.
Writing iterates to 4 d.p.
Keep trailing zeros: 2.2240, not 2.224.
Wrong bounds for 3 d.p.
Testing 2.278 and 2.280 does not prove 2.279; use 2.2785 and 2.2795.
“Comment on” a divergent iteration
Say what the values do relative to the root, then conclude it diverges and cannot find that root.

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