TUTORIAL SHEET I
ERRORS IN NUMERICAL CALCULATIONS
1.
2.
3.
4.
Find the quotient q=x/y, where x=4.536 and y=1.32, both x and y being correct to the
digits given. Find also the relative error in the result.
Find the sum of the numbers 105.5, 27.25, 6.56, 0.1568, 0.000256, 208.6, 0.0235, 0.538,
0.0571, where each number is correct to the given decimal places. Estimate the absolute
error in the sum.
Find the product of the numbers 56.54 and 12.4 which are both correct to significant
digits given.
The Maclaurin’s expression of sin(𝑥) is given by
sin(𝑥) = 𝑥 −
𝑥3
3!
+
𝑥5
5!
−
𝑥7
7!
+ …,
Where x is in radians. Use the series to compute the value of sin(25) with an accuracy
of 0.001.
5.
Find the number of terms of the exponential series such that their sum gives the value of
𝑒 𝑥 correct to five decimal places for all values of x in the range 0 ≤ 𝑥 ≤ 1.
6.
Given f(x) = sin(𝑥), construct the Taylor series approximation of order 5 at 𝑥 = 𝜋/2
to compute sin(𝜋/2).
Hint: Let 𝑥𝑖+1 =𝜋/2 and consider ℎ = 𝜋/12.
7.
If 𝑢 = 3𝑣 7 − 6𝑣, find the percentage error in u at 𝑣 = 1, if the error in ‘𝑣’ is 0.05.
8.
The strain in an axial member of a square cross-section is given by
𝐹
𝜖 = ℎ2 𝐸
Where
F=axial force in the member, N
H=length of width of the cross-section, m
E=Young’s modulus, Pa
𝐹 = 72 ± 0.9 𝑁
ℎ = 4 ± 0.1 𝑚𝑚
𝐸 = 70 ± 1.5 𝐺𝑃𝑎
Find the maximum possible error in the measured strain.
A defense system consists of an electronic detection device called range gate. It
calculates the area in the air space where it should look for a moving target object. To
find out where it should aim next, it calculates the velocity of the moving object and the
last time the radar detected the object. To store the time, the internal clock of the system
is to be measured at regular intervals. This is measured at every one-tenth of a second.
Therefore, binary the number representing one-tenth should be stored in a register whose
length is 24 (say). Compute the error in the measurement of time after 100 hours, due to
Inaccuracy in the representation of ‘one-tenth’ using 24 bits, caused due to truncation of
digits.
9.
TUTORIAL SHEET II
SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
1.
Obtain a root correct to 3 decimal places for the following polynomial.
𝑥 3 + 𝑥 2 + 𝑥 + 7 = 0.
2.
Find the root of 𝑓(𝑥) = (𝑥 − 4)2 (𝑥 + 2) = 0, in the range (-2.5, -1) using false position
method. Assume the tolerance to be 0.001
Find the real root of the equation 𝑥𝑒 𝑥 = 1, using the secant method. Assume the
tolerance to be 0.001. The search range is (0,1).
Show that in bisection method the number of steps required reduce the final search
interval to ε from the initial search range (b, a), is given by
|𝑏 − 𝑎|
𝑙𝑜𝑔𝑒 ( 𝜀 )
𝑛≥
𝑙𝑜𝑔𝑒 2
3.
4.
5.
1
𝑁
Establish the formula 𝑥𝑖+1= 2 (𝑥𝑖 + 𝑥 ) and hence compute the value of √2 correct to six
𝑖
6.
decimal places.
Thermistors are temperature measuring devices based on the principle that the
thermistor material exhibits a change in electrical resistance with a change in
temperature. By measuring the resistance of the material one can then determine the
temperature. Find the value of the resistance R for which the measured temperature
19C. The relationship between the resistance R and the temperature of the thermistor is
given by
1
= 1.129241 × 10−3 + 2.341077 × 10−4 ln(𝑅) + 8.775468 × 10−8 {ln(𝑅)}3
𝑇
Where T is in Kelvin and R is in ohms. Consider the relative error in R as 0.01.
6.. Solve the following systems of nonlinear equations using Newton-Raphson method.
Assume relative error to be 0.001. Take initial guess as (1,-1).
𝑥 2 − 𝑦 = 11
𝑥 + 𝑦2 = 7
Solve the set of nonlinear equations using Newton-Raphson method.
𝑥 3 − 2.5𝑦 2 + 4𝑥 − 3.7 = 0
0.7𝑥 2 + 3𝑦 − 1.8𝑥 + 1 = 0
Initial guess 𝑥 = 0, 𝑦 = 0. Perform two iterations.
7. The output voltage and current waveforms of a half wave rectifier circuit is drawn
below. The transcendental equation which relates the extinction angle  and the time
𝐿
constant =(𝑅) of the circuit is

𝑠𝑖𝑛 ( − ) + 𝑠𝑖𝑛 𝑒 − = 0.
R=10  and L=10 mH. Find the extinction angle  using Newton-Raphson method..
Assume relative error to be 0.001.
8. The circuit for realizing the pulse-width modulator used in the control of power
electronic converters is shown below.
Find the instant ‘t’ at which the comparator switches. The search range is
(0.0025,0.075).
9. In a single phase full-wave controlled rectifier supplying a purely resistive load, the rms
value of the output voltage is given by
𝜋 −  sin 2
)
𝑉0 = 𝑉𝑚 √(
+
2𝜋
4𝜋
where Vm is the peak value of the supply voltage and  is the firing angle for the
thyristors.
(a) Establish the formula.
(b) Find the value of  for which the output rms voltage is 175 V. The available supply
voltage is 220 V (rms). The search range is (0, 𝜋/2). Use false position method.
TUTORIAL SHEET III
SOLUTION OF SYSTEM OF LINEAR ALGEBRAIC EQUATIONS
1.
Solve the network using mesh current analysis method.
2.
Solve the following system of equations using gauss elimination method correct to three
decimal places.
9.617𝑥 + 3.502𝑦 − 2.111𝑧 = 12.345
2.972𝑥 + 4.405𝑦 − 3.954𝑧 = 8.671
3.248𝑥 + 7.199𝑦 + 1.924𝑧 = 6.227
3.
Find the inverse of the following matrix using Gauss-Jordan elimination method.
6 −4 −0
𝐴 = (−4 16 −8)
0 −8 10
4.
Find the inverse of the matrix using gauss-Jordan method.
1 1 −1
𝐴 = [ 1 2 −2]
−2 1 1
Determine whether the following matrices are Ill/well conditioned.
0.4033 0.9166
9.617 3.502 −2.111
(a)
(b)
𝐴=(
)
𝐵 = (2.972 4.402 −3.954)
0.9166 −0.4033
3.248 7.199 1.924
(c)
6 −4 −0
𝐶 = (−4 16 −8)
0 −8 10
Solve the following system of equations using Jacobi and Gauss-Seidel method correct
to three decimal places.
6𝑥1 − 4𝑥2 − 10 = 0
−4𝑥1 + 16𝑥2 − 8𝑥3 = 0
−8𝑥2 + 10𝑥3 − 15 = 0
5.
6.
7.
Solve the following system of equations using (i) Gauss-Seidel (ii) Jacobi’s method. In
each case carry your computations to two decimal places and proceed up to 10
iterations. Does the solution converge?
17 65 −13 50 𝑥1
84
𝑥2
25
12
16
17
18
[
] [𝑥 ] = [36]
3
56 23 11 −19
18
3 −5 47 10 𝑥4
Check the convergence using the following condition.
𝑎
∑𝑛𝑗=1,𝑗≠𝑖 | 𝑖𝑗 | ≤ 1, 𝑓𝑜𝑟 𝑖 = 1,2, … , 𝑛 where the ‘<’sign should be valid in case of ‘at least
𝑎𝑖𝑖
8.
9.
one equation.
Fin the ‘L’and ‘U’ components of the following matrix.
9 3 −2
𝐴 = (2 4 −3)
3 7 1
An electrical network is represented by the following system of equations.
2𝑥 + 𝑦 = 2
2𝑥 + 1.01𝑦 = 2.01
Solve for the unknowns 𝑥 and 𝑦. One of the system parameters is changed by a small
value and the resulting system is represented by the following set of equations.
2𝑥 + 𝑦 = 2
2. 01𝑥 + 𝑦 = 2.05
Solve for the unknowns 𝑥 and 𝑦. What changes you observe in the responses (x and y).
Write your comments.
TUTORIAL SHEET IV
REGRESSION
1.
For Ahmedabad the monthly averaged daily global solar radiation (Hg) is a function of
number of sun-shine hours per day (s) and is related as Hg=a+bs. At Ahmnedabad Hg
and s data for 12 months are taken as follows. Find the constants ‘a’ abd ‘b’ using
least squares method.
Month s
Hg
JAN
0.8837 0.7044
FEB
0.8978 0.7111
MAR
0.7757 0.6527
APRIL 0.7895 0.6588
MAY
0.8162 0.6723
JUNE
0.6273 0.5809
JULY
0.3391 0.4426
AUG
0.3052 0.4261
SEP
0.5606 0.5492
OCT
0.8384 0.6821
NOV
0.9007 0.7117
DEC
0.8954 0.7110
2.
A separately excited DC motor is driving a load whose load-torque characteristics is
given by TL =k1N2+ k2N where k1 is the windage coefficient and k2 is friction
coefficient. k1=2.1*10-5 k2=0.02. The motor is supplied by a 220 V DC source. The
maximum mechanical power that the motor can deliver is 3 HP. If the motor is
operating under full load, find the speed of operation. Use secant method.
3.
The speed-torque characteristic of a separately excited DC motor is given in the following
table. Obtain the speed-torque relation using linear regression.
Torque (N-m) 0
25
50
75
100
ω (rad/s)
157.0796 156.2419 154.7758 154.2522 153.2050
The motor is supplying a mechanical load whose speed-torque characteristic is given in the
following table. Obtain the speed-torque relation using Lagrangian interpolation formula.
Torque (N-m) 60
72
84
ω (rad/s)
76.5 115.6 164.79
Obtain the operating point of the system using the Newton-Raphson method with initial guess
T=80 N-m (Hint: The operating point is nothing but the point of intersection of speedtorque characteristics of motor and the load).
4.
5.
6.
The stress Vs strain data for a composite material is given in the following table.
Strain (m/m)
stress
0.0000
0.0000
1.8300 × 10−3
3.0600 × 108
3.600 × 10−3
6.1200 × 108
−3
5.3240 × 10
9.1700 × 108
7.0200 × 10−3
1.2230 × 109
8.6700 × 10−3
1.5290 × 109
−2
1.0244 × 10
1.8350 × 109
1.1744 × 10−2
2.1400 × 109
1.3290 × 10−2
2.7520 × 109
1.4790 × 10−2
2.7670 × 109
−2
1.5000 × 10
2.8960 × 109
use 𝑦 = 𝐴𝑥 model. Find the equation to solve ‘A’ using least squares criterion. Find
the longitudinal modulus E of the material using linear regression
Many patients get concerned when a test involves injection of a radioactive material.
For example for scanning a gall-bladder, a few drops of Technitium-99m isotope is
used. Half of the technetium-99m would be gone in about 6 hours. It, however, takes
about 24 hours for the radiation levels to reach what we are exposed to in day-to-day
activities. The relative intensity of radiation () as a function of time is given in the
table below.
Table1: Relative intensity of radiation as a function of time
t (hours)
0
1
3
5
7
9
1.000 0.891 0.708 0.562 0.447 0.355

If the level of the exponential is related to time via an exponential formula 𝛾 = 𝐴𝑒 𝑡 ,
find (a) the value of the regression constants A and . (b) the half-life of Technitium99m, and (c) the radiation intensity after 24 hours (c) the radiation intensity after 24
hours.
Assume the initial interval of uncertainty as (-0.120, -0.110).
Find the parameters thermal expansion model of a steel cylinder.
Coefficient
of
thermal
Temperature, T (F)
expansion,  (in/in/F)
80
6.4710-6
40
6.2410-6
-40
5.7210-6
-120
5.0910-6
-200
4.3010-6
-280
3.3310-6
-340
2.4510-6
2
Fit the above data to 𝛼 = 𝑎0 + 𝑎1 𝑇 + 𝑎2 𝑇 .
Also find the straight line model for the above data. Using the coefficient of
determination ‘r2’show that the quadratic model gives a better fit.
7.
The height of a child is measured at different ages as follows.
t (yrs)
0
5
8
12
16
18
H (in)
20
36.2
52
60
69.2
70
Estimate the height of the child as an adult of 30 years of age using the growth model
𝑎
𝐻=
1 + 𝑏𝑒 −𝑐𝑡
8.
A long hollow steel shaft called the trunnion is ‘shrink fit’ into a steel hub. The resulting steel
trunnion-hub assembly is then shrink fit into the girder of a bridge. This is done by first
immersing the trunnion in a cold medium such as dry/ice alcohol mixture or liquid nitrogen.
After the trunnion reaches the steady state temperature of the cold medium, the trunnion
outer diameter contracts. The trunnion is taken out of the medium and slid through the hole
of the hub. When the trunnion heats up to reach the room temperature of 80 F (Troom), it
expands and creates an interference fit with the hub.
A hollow trunnion of outside diameter (D) 12.363” is to be fitted in a hub of inner diameter
12.358”. The trunnion was put in the cold medium at a temperature of -108 F (Tfluid) to
contract the trunnion so that it can be slid through the hole of hub. To slide the trunnion
without sticking, a diametrical clearance of at least 0.01” is required between the trunnion
and the hub. The required contraction is given by
D = trunnion outside diameter – hub inner diameter + diametric clearance
Formula for calculating the estimated contraction is given by
∆𝐷 = 𝐷𝛼(∆𝑇)
Where ∆𝑇 = 𝑇𝑓𝑙𝑢𝑖𝑑 − 𝑇𝑟𝑜𝑜𝑚
Α = 6.47 × 10-6 in/in/F at room temperature. Using this model estimate the contraction.
The process failed when this approximate model is used.
What could be the resons?
Try to find D using the formula
𝑇𝑓𝑙𝑢𝑖𝑑
∆𝐷 = ∫
𝑇𝑟𝑜𝑜𝑚
For this α as s function of T is required.
∝ 𝑑𝑇
We can use the following experimental data.
Temperature
F
80
60
40
20
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200
-220
-240
-260
-280
-300
-320
-340
Instantaneous Thermal Expansion
µin/in/ F
6.47
6.36
6.24
6.12
6.00
5.86
5.72
5.58
5.43
5.28
5.09
4.91
4.72
4.52
4.30
4.08
3.83
3.58
3.33
3.07
2.76
2.45