Rukasha Maths

GCSE maths Beta

Iteration and numerical solutions · Revision guide

Higher · Algebra · 1 page in the PDF

Download PDF1 pageRead the study notes

Learning objectives

  • Use a starting value to generate successive estimates.
  • Recognise a fixed point and keep unrounded intermediate values.

Key ideas

The next estimate is calculated from the latest estimate.

xn+1=f(xn),x0=starting valuex_{n+1}=f(x_n),\qquad x_0=\text{starting value}

Keep full calculator values between iterations; round only the reported answers.

A fixed point satisfies this equation; iteration does not always converge.

x=f(x)x=f(x)

A sign change for a continuous function guarantees at least one root between the two inputs.

Worked examples

Example 1

Start at two and use the recurrence. Report the first three estimates to three decimal places.

xn+1=xn+4,x0=2x_{n+1}=\sqrt{x_n+4},\qquad x_0=2
x1=6=2.449489742…x_1=\sqrt6=2.449489742\ldots
x2=x1+4=2.539584561…x_2=\sqrt{x_1+4}=2.539584561\ldots
x3=x2+4=2.557261144…x_3=\sqrt{x_2+4}=2.557261144\ldots
x1≈2.449,x2≈2.540,x3≈2.557x_1\approx2.449,\quad x_2\approx2.540,\quad x_3\approx2.557

Example 2

Start at four and calculate the second estimate.

xn+1=0.5xn+3,x0=4⇒x1=5x_{n+1}=0.5x_n+3,\quad x_0=4\quad\Rightarrow\quad x_1=5
x2=0.5(5)+3=5.5x_2=0.5(5)+3=5.5

Watch out

Use the full latest estimate as the next input, not its rounded display value. Check whether the values actually settle.