Iteration Flashcards

All 9 cards in this deck

What are the steps to show that an equation has a solution between two given values?

e.g. show $x^3+7x-5=0$ has a solution between $x=0$ and $x=1$

1. Work out $f$ at the lower value: $f(0)=-5$
2. Work out $f$ at the upper value: $f(1)=3$
3. State the sign change: as $f$ changes sign there is at least one root between $x=0$ and $x=1$

What are the steps to find $x_1$, $x_2$ and $x_3$ from an iteration formula?

e.g. $x_{n+1}=\sqrt[3]{10-2x_n}$ starting with $x_0=2$

1. Substitute $x_0$: $x_1=\sqrt[3]{10-4}=1.8171\ldots$
2. Substitute $x_1$: $x_2=1.8533\ldots$
3. Substitute $x_2$: $x_3=1.8463\ldots$

When using an iteration formula, which value do you substitute at each step, and when do you round?

Substitute the full unrounded calculator value each time; round only the final answer to the accuracy asked for.
e.g. final answer $1.4496$ to 4 decimal places.

True or false? The values $x_1$, $x_2$, $x_3$ given by an iteration formula are exact solutions of the original equation.

False. They are successive approximations to a root of the equation, because the iterative formula is a rearrangement of that equation.

What are the steps to carry out a general iterative process described in words?

e.g. "start with 10, halve the number then add 1, repeat"

1. Apply the rule to the starting value: $10\div2+1=6$
2. Put that answer back into the rule as the new input: $6\div2+1=4$
3. Repeat as many times as asked: $4\div2+1=3$

What is the relationship between the values $x_1$, $x_2$, $x_3$ produced by an iteration formula and the equation it came from?

They are successive approximations to a root of the equation. The iteration formula is a rearrangement of that equation.

True or false? If the values of an iteration get closer and closer to one number, that number is a solution of the original equation.

True. The formula $x=g(x)$ is a rearrangement of the equation, so the value it settles on is a root.

In an iterative process, what does $x_{n+1}$ stand for?

e.g. the rule "double the number, then subtract 1"

The next value, found by applying the rule to the current value $x_n$.

e.g. $x_{n+1}=2x_n-1$

True or false? In a general iterative process, the starting value $x_0$ can affect which solution the sequence ends up at.

True. Different starting values can converge to different roots (or not converge at all).