トピック / GCSE / Mathematics / Algebra Iteration and numerical solutions 遊んで学ぼう
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レッスンノート What iteration is Iteration repeatedly applies a formula. A suitable formula and starting value may converge to a root; convergence is not guaranteed. The notation is xn+1 = f(xn ), starting from x0 or x1 . Using iteration Rearrange an equation into the form x = f(x). For x3 - 3x - 1 = 0: x = \frac{x3 - 1}{3} or x = \√[3]{3x + 1}. For xₙ₊₁ = ∛(3xₙ + 1), starting x₁ = 2 gives x₂ ≈ 1.912931 and x₃ ≈ 1.888835. Continuing converges to about 1.879385; retain calculator precision between steps. Use the calculator's ANS key to repeat quickly, and keep full calculator accuracy until you round at the end. Decimal search and change of sign If f(a) and f(b) have opposite signs, a root of f(x) = 0 lies between a and b (for a continuous function). Narrow down by testing the midpoint (the decimal search or interval bisection). To show a root is 1.88 to 2 d.p., show f(1.875) and f(1.885) have opposite signs. Stating the answer Give the number of decimal places asked, but check the rounding by looking at when successive iterations agree. Agreement between a few rounded iterates supports an estimate but is not a proof of accuracy. For a continuous function, a sign-changing interval within the required rounding limits brackets a root. スライド Sign up free to view the lesson slides
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練習問題 無料プレビュー — 33問中8問。すべて見るには登録を。 1. When using iteration to approximate a root, what process is repeated?
Easy A solves exactly by factorisingB repeats a formula to get closer to a solutionC draws a graphD expands brackets2. If xn+1 = xn + 3 and x1 = 2, what is x2 ?
Easy 3. If xn+1 = 2xn and x1 = 3, what is x3 ?
Easy 4. If xn+1 = \frac{xn + 10}{2} and x1 = 4, what is x2 ?
Easy 5. If xn+1 = \√{xn + 6} and x1 = 3, what is x2 ?
Easy 6. If xn+1 = 1 + \frac{1}{xn } and x1 = 1, what is x2 ?
Easy 7. For f(x) = x2 - 5, f(2) = -1 and f(3) = 4. What does this show?
Easy A a root between 2 and 3B no rootsC f(2.5) = 0D the root is 28. f(x) = x3 + x - 3. Which pair shows a root between 1 and 2?
Easy A f(1) = -1, f(2) = 7B f(1) = 1, f(2) = 7C f(1) = 3, f(2) = 7D f(1) = -1, f(2) = -7もっと問題 9. Which equation can be rearranged to x = \frac{x3 - 1}{3}?
Medium A x3 - 3x - 1 = 0B x3 + 3x - 1 = 0C x3 - 3x + 1 = 0D 3x3 - 1 = 010. When seeking the positive root of x² − 2x − 4 = 0, is x = √(2x + 4) a suitable rearrangement?
Medium A yesB no, it should be \√{2x - 4}C no, it needs a cube rootD no, it should be -\√{2x}11. Using xn+1 = \√[3]{3xn + 1} with x1 = 2, what is x2 to 3 d.p.?
Medium 12. Why do we keep full calculator accuracy until the end of an iteration?
Medium A to avoid rounding errors building upB to make the answer longerC to find exact surdsD it makes no difference13. If successive iterations are 1.8794, 1.8794 and 1.8794, the root to 3 d.p. is about:
Medium A 1.879B 1.880C 1.88 to 1 d.p.D 214. For a function continuous on the stated interval: f(1.87) < 0 and f(1.88) > 0. What can you conclude?
Medium A the root is 1.87 to 2 d.p.B the root is between 1.87 and 1.88C there are no rootsD the root is 1.915. For a continuous function with a single root near 1.88, which pair of values can bracket the root to show it rounds to 1.88 to 2 decimal places?
Medium A f(1.8) and f(1.9)B f(1.875) and f(1.885)C f(1.88) onlyD f(0)16. If f(x) = x3 - 4x - 1, which interval contains a root: f(2) = -1, f(3) = 14?
Medium A between 2 and 3B between 0 and 1C between 3 and 4D none17. Starting at x1 = 1, which formula gives 1, 3, 7, 15?
Medium A xn+1 = 2xn + 1B xn+1 = xn + 2C xn+1 = 3xn D xn+1 = xn 2 18. What is the ANS key used for in iteration?
Easy A to store the previous answer for repeatingB to square a numberC to clear the screenD to round19. Which is a suitable decimal search to improve an estimate of a root in [1, 2]?
Medium A test f(1.5)B test f(5)C test f(-1)D test f(100)20. For a function continuous on the stated interval: If f(1.5) > 0 and f(1) < 0, the root is between:
Easy A 1 and 1.5B 1.5 and 2C 0 and 1D 2 and 321. xn+1 = 3 - \frac{1}{xn }, x1 = 2. What is x2 ?
Medium 22. Which equation has an iteration xn+1 = \frac{5}{xn + 1}?
Medium A x2 + x - 5 = 0B x2 - x - 5 = 0C x2 + 5x = 1D x2 = 5x + 123. For an iteration that converges, the sequence of values:
Medium A grows without limitB gets closer to a fixed valueC alternates between 0 and 1D is always negative24. Iteration is used when an equation:
Easy A cannot easily be solved algebraicallyB is linearC has no unknownD is already solved25. What is the first step for iteration on x2 = 3x + 1?
Easy A rearrange into x = \ldotsB draw a circleC divide by zeroD square root only26. Iteration repeats a formula using each result as the next input.
Easy True or false?
27. If f is continuous on [a, b] and f(a) and f(b) have opposite signs, there is at least one root strictly between a and b.
Easy True or false?
28. Iteration always gives an exact answer.
Easy True or false?
29. You should round at every step of an iteration.
Easy True or false?
30. If f(1) > 0 and f(2) > 0 there must be a root between 1 and 2.
Medium True or false?
31. For f continuous on [1, 2], which endpoint pairs guarantee at least one root between 1 and 2?
Medium A f(1) = -2, f(2) = 3B f(1) = 4, f(2) = -1C f(1) = 2, f(2) = 3D f(1) = -3, f(2) = -1E f(1) = -1, f(2) = 132. Which practices are useful when approximating a nonlinear root by iteration? Select all that apply.
Medium A solving x3 + x - 3 = 0 numericallyB solving x + 2 = 5C finding a root to given accuracyD expanding bracketsE using ANS repeatedly33. Show that x3 - 3x - 1 = 0 has a root between 1 and 2.
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