EB09 Equation Solving

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Equation Solving

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Iterative Techniques for solving

equations

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There are situations when finding an exact

solution to a given equation is not easy or

sometimes not possible. For example,

0xx sin

x-0.75x-4.5x+4.75=0

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For such equations we use iterative methods

for finding approximate solutions.

The methods discussed in this lecture are:

1. Bisection Method

2. Newton-Raphson Method

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Theorem

Let be a continuous function on such

that then there exist a number

such that .

f ba, 0bfaf bac , 0cf

a b

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Newton Raphson Method

1x2x3x

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Step 1: Start with an initial guess, .

Step 2: Calculate

Step 3: Calculate

and so on.

The nth iteration is given by

1x

1

1

12

xfxfxx

2

2

23

xf

xfxx

n

n

nnxf

xfxx 1

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The iterations should be done until we get the

solution of desired accuracy.

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While solving problems in business and

finance we need to solve several type of

mathematical equations or system of

equations.

Example:

A person selects 5 stocks and invest equal

money in them for equal time period. The

returns from first 4 stocks are 20%, 30%, -15%

and 35%. What should be the return from the

5th stock so that the average return from the 5

stocks is 25%?

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OpenOffice.org Calc provides a tool to solve such

equations. This tool is known as Goal Seek.

Tools

Goal Seek

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To solve the above equation using Goal Seek, open

the sheet OneVariable_1.

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Goal Seek for solving a polynomial

equation

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A person deposits Rs. 20,000 in an account everyyear at a certain interest rate. If the money is

deposited at the beginning of each year then

what should be the interest rate so that hereceive 1,50,000 at the end of 5th year.

Example:

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To solve the problem we need to set up the

following equation:

150000120000120000

120000120000120000

2

345

rr

rrr

To find the value ofr we need to solve a 5th

degree polynomial equation which could be

very difficult.

Let us see how to use Goal Seek to solve such

an equation. Go to the sheet OneVariable_3.

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Solving System of Linear Equations

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Consider a system of linear equations:

To solve this system of linear equations means

to find the values ofx,y andz which satisfy all

the above equations.

3333

2222

1111

dzcybxa

dzcybxa

dzcybxa

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Matrix Method for Solving System of

Linear Equations

Step 1: Consider the matrices:

Step 2: The system of equations can be written

as

3

2

1

333

222

111

d

d

d

B

z

y

x

X

cba

cba

cba

A ,,

BAX 20 MSF


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This function computes the inverse of a

matrix.

Suppose a matrix is stored in cells B16:C17

then its inverse can be computed by

=MINVERSE(B16:C17)

The MINVERSE Function

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The MINVERSE Function

In order to execute this function you first

select a 2x2 range and then pressCTRL+SHIFT+ENTER.

Pressing ENTER alone will not work!

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This function computes the product of two

matrices.

Suppose we have two matrices B16:C17 and

H16:H17 then their product is computed as

=MMULT(B16:C17,H16:H17)

The MMULT Function

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The MMULT Function

In order to execute this function you first

select a 2x1 range and then pressCTRL+SHIFT+ENTER.

Pressing ENTER alone will not work!

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A company wants to pay an obligation of

Rs 2 million after 10 years. In order to do so it

decides to invest in the following bonds:

An Immunization Problem

Bond Coupon YTM Maturity Face Value

A 8% 8% 13 1000

B 6% 8% 20 1000

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How much should be invested in each of the

bonds so that the obligation is metirrespective of the change in the yield?

An Immunization Problem

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The Strategy

Let VA and VB be the amounts invested in

bonds A and B respectively. Then

DDVVD

VV

V

V

V

V

BB

AA

BA

1

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The Strategy (Cont)

Here DA and DB are the durations of bonds A

and B respectively, and V is the present value

of the obligation

DDV

VD

V

V

V

V

V

V

B

B

A

A

BA

1

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EB09 Equation Solving