d point iteration (A-level only) Flashcards","topicId":"41eecc2e-feda-4555-a91f-e0680ead88c8","topicSlug":"fixed-point-iteration","cards":[{"id":"632e9953-c397-461b-a182-e0fc454e08c3","cardType":"cloze","front":"Before fixed point iteration, rearrange the equation into the form $\\text{[...]}$.","back":"Before fixed point iteration, rearrange the equation into the form $\\boxed{x=g(x)}$.","points":null,"mcqOptions":null},{"id":"eddbff91-a804-47f6-8556-17a8a05a1633","cardType":"cloze","front":"Starting from $x_0$, generate approximations using $\\text{[...]}$.","back":"Starting from $x_0$, generate approximations using $\\boxed{x_{n+1}=g(x_n)}$.","points":null,"mcqOptions":null},{"id":"7420f6cf-2373-4a2c-9673-01e62de4eba4","cardType":"basic","front":"How is $x_2$ obtained directly from $x_0$?","back":"Calculate $x_2=g(g(x_0))$.","points":1,"mcqOptions":null},{"id":"bbf694e0-e258-4eae-b87a-503574bc73fd","cardType":"cloze","front":"A value $\\alpha$ is a fixed point of $g$ when $\\text{[...]}$.","back":"A value $\\alpha$ is a fixed point of $g$ when $\\boxed{\\alpha=g(\\alpha)}$.","points":null,"mcqOptions":null},{"id":"ed5235de-e976-46f3-9fef-b1995428f831","cardType":"basic","front":"Why should unrounded values be used during iteration?","back":"Premature rounding propagates error and reduces final accuracy.","points":2,"mcqOptions":null},{"id":"48d3efca-7d4c-4925-af68-d36ca10d391b","cardType":"basic","front":"Why do intersections of $y=g(x)$ and $y=x$ represent fixed points?","back":"At an intersection, $g(x)=x$.","points":2,"mcqOptions":null},{"id":"87c6781b-459e-4841-9d44-3745a07d35fb","cardType":"basic","front":"Where is $x_0$ placed when beginning a staircase or cobweb diagram?","back":"On the horizontal axis.","points":1,"mcqOptions":null},{"id":"d7e9f446-e057-4ffd-b4be-7bddcdadd5d5","cardType":"cloze","front":"From $x_0$ on the horizontal axis, first move [ ].","back":"From $x_0$ on the horizontal axis, first move vertically to $y=g(x)$.","points":null,"mcqOptions":null},{"id":"13ec4d50-3195-47a2-8c30-6b5d6bc0ecac","cardType":"basic","front":"What does the horizontal move from $y=g(x)$ to $y=x$ achieve?","back":"It makes the output $x_{n+1}$ the next input.","points":2,"mcqOptions":null},{"id":"55c5742a-ce6f-4bc2-96ff-3e36dcd705db","cardType":"basic","front":"What diagram pattern commonly occurs when $g$ is increasing and the iteration converges?","back":"A staircase, usually approaching from one side.","points":2,"mcqOptions":null},{"id":"7cf73f1e-3ba3-4efc-9a0d-2ec5b1d16d4e","cardType":"basic","front":"What diagram pattern commonly occurs when $g$ is decreasing near a fixed point?","back":"A cobweb, with approximations alternating across the fixed point.","points":2,"mcqOptions":null},{"id":"c586bade-8db0-44d6-8f7d-d15e65fa46cf","cardType":"basic","front":"What does a diagram whose steps move towards an intersection indicate?","back":"The iteration converges.","points":1,"mcqOptions":null},{"id":"9756be2a-9637-4dcf-8530-17d2f0b09e82","cardType":"cloze","front":"Near a fixed point $\\alpha$, the iteration normally converges locally if $\\text{[...]}$.","back":"Near a fixed point $\\alpha$, the iteration normally converges locally if $\\boxed{\\lvert g'(\\alpha)\\rvert<1}$.","points":null,"mcqOptions":null},{"id":"a789bfc6-d8db-4ebc-b8a3-ec353140f896","cardType":"cloze","front":"Near a fixed point $\\alpha$, approximations normally move away if $\\text{[...]}$.","back":"Near a fixed point $\\alpha$, approximations normally move away if $\\boxed{\\lvert g'(\\alpha)\\rvert>1}$.","points":null,"mcqOptions":null},{"id":"3f7d5105-a9b2-42b3-a97e-494811377f5f","cardType":"bas

1.10.3 Fixed point iteration (A-level only) Flashcards

1.10.3 Fixed point iteration (A-level only) Flashcards