Sign-Change Methods (OCR GCSE Maths)

Sign-Change Methods

What is iteration?

• Iteration means finding an approximation to the solution using a method that gets closer and closer (converges) to the real solution

• It is often used for equations which don't have nice solutions

  • solutions themselves might be irrational numbers

How do I show that there is a solution between two values?

• To find x₀ (the initial / starting value), you are often asked to show that there is a solution between two values

• For example, show that there is a solution to x³ + x = 7 between 1 and 2

• Method 1:

  • Leave a constant term (e.g. the 7) on the right

  • substitute x = 1 and x = 2 into the left and show that this gives values below and above 7

    • 1³ + 1 = 2 and 2³ + 2 = 10 which are below and above 7

    • A solution therefore lies between 1 and 2

• Method 2:

  • Use "0" as your constant term on the right (by rearranging the equation into "... = 0")

  • then substitute in x = 1 and x = 2, showing this gives values below and above 0, i.e. negative and positive

  • this is called a change of sign between 1 and 2

    • x³ + x - 7 = 0

    • Substitute x = 1 into the left-hand side: 1³ + 1 - 7 = -5 (negative)

    • Substitute x = 2 into the left-hand side:  2³ + 2 - 7 = 3 (positive)

    • A solution lies between 1 and 2 as there is a change of sign

How do I find an approximate solution to an equation?

• You must know either the Decimal Search method or the Interval Bisection method

  • exam questions will not force you to use a particular method

Decimal Search

• Let’s say you know that a solution to an equation lies between 1 and 2 and you want to find it to 1 decimal place

  • The equation must be rearranged to have “= 0” on one side

• Search every decimal, by substituting 1.0, 1.1, 1.2, 1.3, 1.4, … into the equation until you see a change of sign

  • Let’s say there was a change of sign between 1.6 and 1.7

  • This is not yet enough information to be able to round the solution to 1 decimal place

• Substitute in 1.65, the value exactly in the middle of 1.6 and 1.7

  • If the sign change is between 1.6 and 1.65, the solution rounds down to 1.6 to 1 decimal place

  • If the sign change is between 1.65 and 1.7, the solution rounds up to 1.7 to 1 decimal place

Interval Bisection

• Let’s say you know that a solution to an equation lies between 1 and 2 and you want to find it to 1 decimal place

  • The equation must be rearranged to have “= 0” on one side

• Bisect (find the mid-point of) the interval from 1 to 2 to get 1.5

  • Substitute 1, 1.5 and 2 into the equation and see where the sign change is

• Let’s say the sign change is between 1.5 and 2

  • Bisect this interval to get 1.75 and substitute this value in

• Let’s say the sign change is now between 1.5 and 1.75

  • Bisect this interval to get 1.625 and substitute this value in

• Keep going until your interval is small enough to give you only one option to round your solution to

  • For example, 1.65625 < x < 1.6875 means that the solution is 1.7 to 1 decimal place