Numerical Solutions of Equations

Numerical Solutions of Equations

Finding Starting Point

Learning Objectives

• locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change

• understand the idea of, and use the notation for, a sequence of approximations that converges to a root of an equation

• understand how a given simple iterative formula of the form x_n+1 = F(x_n) relates to the equation being solved

• use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy.

Prerequisite Knowledge (1)

• Substitute numbers for letters in complicated formulae and algebraic expressions

• Evaluate the following expression when x = 1.5

•  $\frac{2}{3}\left(x+\frac{1}{x^2}\right)\$  

•  $\sqrt{\ln x+3}$  

Prerequisite Knowledge (2)

• Rearrange complicated formulae and equations.

• Rearrange each formula so that x

  x is the subject.

•  $2=\frac{y-3x+7}{w}$              ---->  $x=\frac{2w-7-y}{3}$  

•  $y=2x^2+13,\ x>0$     ---->  $x=\sqrt{\frac{y-13}{2}}$  

Prerequisite Knowledge (3)

• Understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations.

• Show that the x -coordinates of the solutions to the simultaneous equations:     $y=x^2-6x+10$   and  $y=x$       also satisfies the equation   $x^2-7x+10=0$  

Prerequisite Knowledge (4)

• Know that the values of x that make both sides of an equation equal are called the roots of the equation.

• Also, the roots of f(x)=0 are the x -intercepts on the graph of y=f(x)
  

• Find the roots of the equation   $x^2-7x+10=0$  

Prerequisite Knowledge (5)

Sketch graphs of quadratic or cubic functions or the trigonometric functions sine, cosine, tangent.

Prerequsite Knowledege (6)

Sketch graphs of exponential or logarithmic functions or the trigonometric functions cosecant, secant, cotangent.

Multiple Choice

What is the value of f(x) when x = 2, for the expression

 $4\left(x-\frac{1}{x^2}\right)\$  ?

• 28

•  $\frac{1}{4}$  

•  $\frac{7}{4}$  

• 7

Multiple Choice

Which of the followings is the rearrangement of the expression so that x is the subject

 $y=\frac{3-2x}{5}$  

•  $x=\frac{3-2y}{5}$  

•  $x=\frac{5y-3}{2}$  

•  $x=\frac{3-5y}{2}$  

Multiple Choice

Which equations that has the same solution with x-coordinate of the solution of simultaneous equations:

 $y=x^2-2x+10$   and   $y=x$  

•  $x^2-x+10=0$  

•  $x^2+x+10=0$  

•  $x^2-3x+10=0$  

Multiple Choice

Which of the followings are the roots of the equation :

  $x^2-2x-8=0$  

• x = 2 or x = -4

• x= -2 or x = 4

• x = 2 or x = 4

• x = -2 or x = -4

Why solve equations numerically?

• You are used to solving equations using direct, algebraic methods such as factorising or using the quadratic formula.

• ou might be surprised to learn that not all equations can be solved using a direct, algebraic process.

• Numerical methods are ways of calculating approximate solutions to equations. They are extremely powerful problem-solving tools and many are available.

• They are widely used in engineering, computing, finance and many other application

Why do we need to understand how the method works?

• understand when the results are likely to be reasonable

• understand how to use available software correctly

• select an appropriate method when choices are available

• write our own programs when we need to do so, if we have the programming skills!

Example 1

Show that x³ + x − 4 = 0 has a root α between 1 and 2

Graphical approach

• By sketching graphs of y=x³ and y=4−x, show that the equation x³ +x−4=0 has a root α between 1 and 2

• There is one point of intersection and its x-coordinate is between 1 and 2. At this point x³ = 4 − x.

• This means that the equation x³ = 4 − x has one root between 1 and 2.

• Rearranging this equation gives x³ + x − 4 = 0.

Example 2

Show, by calculation, that the equation f(x) = x⁵ + x − 1 = 0 has a root α between 0 and 1

The change of sign

• f(x) = x⁵ + x − 1 = 0 has a root α between 0 and 1

• Find the value of α such that f(α) = 0.

• f(0) = 0⁵+0-1 = -1 f(1) = 1⁵+1-1 = 1

• The change of sign indicates the presence of a root, so 0<α<1

Example 3

By sketching a suitable pair of graphs, show that the equation cosx = 2x−1 (where x is in radians) has only one root for 0 ⩽ x ⩽ 1/2 π

Graphical approach

• Draw the graphs of y = cosx and y = 2x−1.

• The graphs intersect once only and so the equation cos x = 2x−1 has only one root for 0 ⩽ x ⩽ 1/2 π .

The sign of change

• cos x = 2x−1 so cos x − 2x + 1 = 0

• Let f(x)=cosx−2x+1 and so f(x)=0 then

• f(0.8) = cos 0.8−2(0.8)+1=0.0967…

  f(0.9) = cos0.9−2(0.9)+1=−0.1783…

• Change of sign indicates the presence of a root.