Iteration

    OCR
    GCSE

    Iteration is a numerical method used to approximate solutions to equations where analytical derivation is impossible or inefficient. The process requires rearranging an equation f(x)=0 into the form x=g(x) to generate a sequence of approximations via the recursive formula x_{n+1} = g(x_n). Mastery involves executing recursive algorithms, analyzing convergence through cobweb and staircase diagrams, and verifying roots using sign-change intervals. Candidates must rigorously apply accuracy bounds to justify the precision of the final solution.

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    Objectives
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    Exam Tips
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    Pitfalls
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    Key Terms
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    Mark Points

    What You Need to Demonstrate

    Key skills and knowledge for this topic

    • Award M1 for a correct first step in rearranging the equation f(x)=0, such as isolating the highest power or a specific x term
    • Award A1 for the fully correct derivation of the iterative formula x_{n+1} = g(x_n) with all signs and powers correct
    • Award M1 for substituting the starting value x_0 correctly to find the first iteration x_1 (allow follow through if formula is incorrect but method is valid)
    • Award A1 for the final root stated to the required accuracy (e.g., 3 significant figures), supported by a sequence of converging values

    Example Examiner Feedback

    Real feedback patterns examiners use when marking

    • "You have correctly found x_1, but you rounded it too early. Keep the full value in your calculator for the next step."
    • "Your rearrangement has a sign error in the second step. Check your inverse operations when moving terms across the equals sign."
    • "Good use of the iterative method. To ensure full marks, explicitly state the final answer to the requested significant figures."
    • "You didn't show the intermediate steps for the rearrangement. 'Show that' questions require a clear logical chain of algebra."

    Marking Points

    Key points examiners look for in your answers

    • Award M1 for a correct first step in rearranging the equation f(x)=0, such as isolating the highest power or a specific x term
    • Award A1 for the fully correct derivation of the iterative formula x_{n+1} = g(x_n) with all signs and powers correct
    • Award M1 for substituting the starting value x_0 correctly to find the first iteration x_1 (allow follow through if formula is incorrect but method is valid)
    • Award A1 for the final root stated to the required accuracy (e.g., 3 significant figures), supported by a sequence of converging values

    Examiner Tips

    Expert advice for maximising your marks

    • 💡Use the 'Ans' key on your calculator to automate the iteration process; this reduces transcription errors and ensures full precision is maintained
    • 💡When asked to 'Show that' an equation rearranges to a specific form, you must show every algebraic step clearly—do not skip lines or work backwards
    • 💡Always write down the values of x_1, x_2, and x_3 to at least 5 decimal places to demonstrate to the examiner that you are performing the iteration correctly
    • 💡Check your final answer by substituting it back into the original equation f(x)=0; the result should be very close to zero

    Common Mistakes

    Pitfalls to avoid in your exam answers

    • Rounding intermediate values (e.g., x_1, x_2) which introduces cumulative error and prevents the sequence from converging to the correct accuracy
    • Incorrect algebraic manipulation when rearranging the original equation, particularly when dealing with negative signs or dividing by variables
    • Stopping the iteration process too early before the values have stabilised to the required number of decimal places
    • Failing to equate the original expression to zero before attempting to rearrange it into the iterative form

    Study Guide Available

    Comprehensive revision notes & examples

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    Key Terminology

    Essential terms to know

    Rearrangement of algebraic equations to iterative formulae
    Recursive sequence generation and convergence
    Graphical representation (Cobweb and Staircase diagrams)
    Interval bisection and sign change verification
    Error bounds and degree of accuracy

    Likely Command Words

    How questions on this topic are typically asked

    Show that
    Use
    Find
    Calculate
    Solve
    Determine

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