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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