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Birla Institute of Technology and Science,Pilani-K. K. Birla Goa
Campus
SECOND SEMESTER(2011-2012)
Numerical Analysis(AAOC C341)
TUTORIAL-BOOKLET
Instructor-In-Charge
Dr. P. Dhanumjaya
Department of Mathematics
BITS, Pilani - K. K. Birla GOA Campus
January, 2012
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Table of Contents
Tutorial-1 .......1
Tutorial-2 .......3
Tutorial-3 .......5
Tutorial-4 .......7
Tutorial-5 .......9
Tutorial-6 .11
Tutorial-7 .13
Tutorial-8 .15
Tutorial-9 17
Tutorial-10 ...19
Tutorial-11 ...21
Tutorial-12 ...23
Tutorial-13 ...25
Tutorial-14 ...27
Model Test Papers ...29
Formula Sheet .42
Bibliography 46
Handout ...48
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-1
1. Compute the absolute error and relative error in
approximations of x by x
(a) x = pi, x = 3.1416
(b) x = e, x = 2.718
(c) x =2, x = 1.414
2. Use four-digit floating point arithmetic with rounding to
perform the following calcu-
lations:
(i)1314 6
7
2e 5.4 , (ii) 10pi + 6e3
62.
3. Let
f(x) =x cosx sin xx sin x .
(a) Use four-digit floating point arithmetic with rounding to
evaluate f(0.1).
(b) The actual value is f(0.1) = 1.99899998. Find the relative
error for the valueobtained in (a).
4. The Maclaurins series expansion for ex is given by
ex = 1 + x+x2
2!+x3
3!+ + x
n1
(n 1)! +xn
n!e, (0, x).
Find the number of terms n such that their sum yields the value
of ex correct up to 8
significant digits at x = 1.
5. Evaluate
f(x) = x3 6.1x2 + 3.2x+ 1.5,
at x = 4.71 using three-digit floating point arithmetic with
rounding.
1
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6. Evaluate the polynomial
f(x) = 1.1071x3 + 0.3129x2 0.0172x+ 1.1075,
at x = 0.1234 in nested form using five-digit floating point
arithmetic with chopping.
7. Find value of the polynomial at x = 0.125 in nested form
9.26 x3 3.48 x2 + 0.436 x 0.0182,
using four-digit floating point arithmetic with chopping.
8. One root of the quadratic equation
0.2x2 47.91x+ 6 = 0,
is x = 239.4. Use four-digit floating point arithmetic with
rounding to find other root.
9. Find the roots of the quadratic equation
x2 + 111.11x+ 1.2121 = 0,
using five-digit floating point arithmetic with chopping.
10. Use four-digit floating point arithmetic with rounding to
find the most accurate ap-
proximations to the roots of the following equations:
(a) 13x2 123
4x+ 1
6= 0.
(b) 1.002x2 11.01x+ 0.01265 = 0.
2
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-2
Note:All the problems use 5-digit floating point arithmetic with
rounding.
1. Use bisection method to find p3 (3rd-iteration) for
f(x) =x cosx,
on [0, 1].
2. Use bisection method to determine the number of iterations
necessary to solve
f(x) = x4 + 2x3 10 = 0
with accuracy 103 on the interval [a, b] = [1, 2].
3. Use bisection method to find the solution accurate to within
102 for
f(x) = 2 sinx ex
4 1,
on the interval [5,3].
4. Find an approximation to3 correct to within 104 using the
bisection method.
5. Find root of the equation
f(x) = x sin x+ cosx = 0,
in [2, 3] using bisection method. Perform four iterations.
6. The number of fixed points of the iterative function g(x) = x
sin pix in [0, 1]?
7. Is there a root of the equation
f(x) = ex 4 x2 = 0,
between x = 4 and x = 5? Show that we cannot find this root
using fixed-point
iteration with the natural iteration function x = 0.5 ex/2.
Can you find an iteration function which will correctly locate
this root?
3
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8. Find a suitable interval and an iterative function g(x) such
that the fixed point iteration
converges to the solution of the equation
f(x) = e2x ex 2 = 0.
Perform four iterations.
9. Verify that x =a is an unique fixed point of the function
g(x) =1
2
(x+
a
x
),
be defined on the interval [a ,a+ ] for > 0.
Determine the order of convergence and the asymptotic error
constant of the sequence
xn+1 = g(xn) toward x =a.
10. Find a function g defined on [0, 1] that satisfies none of
the hypotheses of existence
and uniqueness of fixed-point method but still has a unique
fixed point on [0, 1].
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-3
Note: All the problems use 5-digit floating point arithmetic
with rounding.
1. Find the root of the equation
f(x) = x5 3x+ 8 = 0,
lying between x = 2 and x = 1, using fixed point method. Perform
three iterations.
2. Find the zero of
f(x) = x2 + 10 cosx,
by using the fixed-point iteration method for an appropriate
iteration function g.
3. Most functions can be rearranged in several ways to give x =
g(x) with which to begin
the fixed-point method. For
f(x) = ex 2x2,one g(x) is
x = (
ex
2
).
(a) Show that this converges to the root near 1.5 if the
positive value is used and to
the root near 0.5 if the negative is used.(b) There is a third
root near 2.6. Show that we do not converge to this root even
though values near to the root are used to begin the iterations.
Where does it
converge if x0 = 2.5? If x0 = 2.7?
(c) Find another rearrangement that does converge correctly to
the third root.
4. Use fixed-point iteration method to determine a solution
accurate to within 102 for
f(x) = 2 sin pix+ x = 0.
Use p0 = 1.
5
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5. Use fixed point theorem to show that the sequence defined
by
xn =1
2xn1 +
1
xn1, for n 1,
converges to2 whenever x0 >
2.
6. For each of the following equations, determine suitable
iterative function g and an
interval [a, b] on which fixed-point iteration will converge to
a positive solution of the
equation
(a) f(x) = 3x2 ex = 0.(b) f(x) = x cosx = 0.
7. Verify that x = 1ais a fixed point of the function
g(x) = x(2 ax).
Determine the order of convergence and the asymptotic error
constant to the sequence
pn+1 = g(pn) toward x =1a.
8. Find the parameters a, b and c in the iterative function g(x)
= ax3 + bx2 + cx such
that the order of convergence for finding the root x = using
fixed point method is 3.
9. Derive a Newtons iteration formula for finding the cube root
of a positive number .
10. The function
f(x) = e2x ex 2,
has a zero on the interval [0, 1]. Find this zero correct to
three significant digits using
Newtons method. Use p0 = 0.5.
6
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-4
Note: All the problems use 5-digit floating point arithmetic
with chopping.
1. Let
f(x) = x3 cosx,
and p0 = 1. Use Newtons method to find p2. Could p0 = 0 be
used?
2. Use Newtons method to find the root of the equation
f(x) = 4x3 1 ex2
2 = 0,
near p0 = 1.0. Perform two iterations.
3. The equation
f(x) = x3 + x2 3x 3 = 0,
has a root on the interval (1, 2) namely x =3.
For n 1, compute the ratio |pnp||pn1p|2
and show that this value approaches(|f (p)|2|f (p)|
).
4. Use Newtons method to find the root of the equation
f(x) =x
1 + x2 500
841
(1 21
125x
)= 0,
near p0 = 2.0. Perform one iteration.
5. Perform three iterations of the secant method to find the
root of the equation
f(x) = ln(x) 1.04x+ 1.05 = 0,
lying on the interval (1, 2).
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6. If the secant method is applied to the equation
f(x) = x2 2 = 0
with p0 = 0 and p1 = 1, what is p2?
7. Let
f(x) = x2 6
with p0 = 2 and p1 = 3 then find p3 by using
(a) secant method
(b) method of false position
(c) which of (a) or (b) is closer to6?
8. Let
f(x) = x3 cosx
with p0 = 1 and p1 = 0 then find p3 by using
(a) secant method
(b) method of false position
9. The function
f(x) = 4 sin x ex,
has a zero on the interval [0, 0.5]. Find the root correct to
four significant digits using
secant method.
10. Find the root of the equation
f(x) = x sin x+ cosx = 0,
in [2, 3] using method of false position. Perform three
iterations.
8
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-5
Note: All the problems use 5-digit floating point arithmetic
with rounding.
1. Use Mullers method to find a root of the equation
f(x) = 4x3 3x2 + 2x 1 = 0,
starting with three initial values x = 0, 0.6, 1. Perform two
iterations.
2. Find a root of the equation
f(x) = 3x+ sin x ex = 0,
on the interval [0, 1] using Mullers method. Perform two
iterations.
3. Use Mullers method to find a root of the equation
f(x) = tan x+ 3x2 1 = 0.
Perform two iterations using three initial values x = 0, 0.8,
1.
4. Do three iterations of Newtons method to obtain the double
root of
f(x) = x3 2x2 0.75x+ 2.25 = 0,
which is close to 1 such that iterations converges
quadratically.
5. Suppose we want to solve the equation f(x) = 0, that has a
root of multiplicity m at
x = p. (Assume f(x) is sufficiently differentiable function).
Show that the Newtons
method
xn+1 = xn f(xn)f (xn)
,
will converge, but only linearly. Determine the asymptotic error
constant.
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6. Do one iteration of Newtons method to solve the system of
nonlinear equations
x2 + y2 = 4,
ex + y = 1.
Use X(0) = [1, 1]T .
7. Perform two iterations of Newtons method for the system of
nonlinear equations
4 x21 x22 = 0,4 x1 x
22 x1 = 1.
Use X(0) = [0, 1]T .
8. Do one iteration of Newtons method to solve the system of
nonlinear equations
x y2 + x2 y + x4 = 3,
x3 y5 2 x5 y x2 = 2.
Use X(0) = [1, 1]T .
9. The nonlinear system (10 marks)
5 x21 x22 = 0,x2 0.25 (sin x1 + cosx2) = 0,
has a solution near(14, 14
). Find a function G and a set D in R2 such that G : D R2
and G has a unique fixed point in D.
10. The nonlinear system
x21 10x1 + x22 + 8 = 0,x1 x
22 + x1 10x2 + 8 = 0,
can be transformed into the fixed-point problem
x1 = g1(x1, x2) =x21 + x
22 + 8
10,
x2 = g2(x1, x2) =x1 x
22 + x1 + 8
10.
Show that G = (g1, g2)T mapping D
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-6
Note: All the problems use 5-digit floating point arithmetic
with rounding.
1. Solve the system of linear equations
0.7 x1 + 1725 x2 = 1739,
0.4352 x1 5.433 x2 = 3.271,
using
(a) Gaussian elimination with no pivoting
(b) Gaussian elimination with partial pivoting
(c) Gaussian elimination with scaled partial pivoting.
Compare the results obtained from each technique with the exact
solution x1 =
20, x2 = 1 of the system. Show all intermediate matrices,
scaling factors and mul-
tipliers.
2. Solve the system of linear equations
3.41 x1 + 1.23 x2 1.09 x3 = 4.72,2.71 x1 + 2.14 x2 + 1.29 x3 =
3.10,
1.89 x1 1.91 x2 1.89 x3 = 2.91,
using
(a) Gaussian elimination with no pivoting
(b) Gaussian elimination with partial pivoting
(c) Gaussian elimination with scaled partial pivoting.
11
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Compare the results obtained from each technique with the exact
solution of the sys-
tem.
3. Solve the system of linear equations
9.3746 x1 + 3.0416 x2 2.4371 x3 = 9.67685,3.0416 x1 + 6.1832 x2
+ 1.2163 x3 = 6.74135,
2.4371 x1 + 1.2163 x2 + 8.4429 x3 = 2.3925,
using Gaussian elimination method with scaled partial
pivoting.
Show all intermediate matrices, scaling factors and
multipliers.
4. Solve the system of linear equations
x1 + x2 + 2x3 = 2,x1 + 2x3 = 1,
3x1 + 2x2 x3 = 0,
by using Crout decomposition and Doolittle decomposition
method.
5. Consider
A =
2 6 46 17 174 17 20
.
Determine directly the factorization A = LDLT , where D is
diagonal and L is unit
lower triangular matrix.
6. Find the LU-factorization of the matrix
A =
3 0 1
0 1 31 3 0
,
in which L is lower triangular and U is an unit upper triangular
matrix.
7. Prove that the matrix
A =
0 1
1 1
does not have an LU-factorization.
12
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-7
1. Are these matrices positive definite?
(i)
1 11 1
, (ii)
4 2 1
2 5 2
1 2 4
.
2. For what value(s) of is this matrix positive definite?
A =
1
1
1
.
3. Find X and X2 for the following vectors
(i)
[3, 4, 0, 3
2
]T, (ii) [2, 1, 3, 4]T .
4. Find for the following matrices
(i)
10 15
0 1
, (ii)
10 0
15 1
.
5. Compute condition numbers using norms A1, A2 and A
(i)
+ 1
1
, (ii)
0 12 0
, (iii)
s 1
1 1
.
6. Let A =
25 19
21 16
. Then find k(A) (condition number of A in maximum-norm).
13
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7. Let AX = b be any linear system. If A = A + E represents the
perturbed coefficient
matrix, and X is the solution of A X = b, then prove that
X XX k(A)
(EA
),
where k(A) is the condition number.
8. Prove that if A < 1 then
(I A)1 11 + A .
9. The linear system Ax = b given by 1 2
1.0001 2
x1x2
=
3
3.0001
has the solution [1, 1]T . Change A slightly to
1 2
0.9999 2
and consider the linear
system 1 2
0.9999 2
x1x2
=
3
3.0001
.
Compute the new solution using five-digit floating point
arithmetic with rounding. Is
A ill-conditioned?
10. Do two iterations using Gauss-Seidel method with 5-digit
floating-point arithmetic
with rounding to the following system of equations
4x1 10x2 + 5x3 = 32,5x1 4x2 + 10x3 = 39,10x1 + 5x2 4x3 = 17.
Starting vector X(0) = [ 1, 1, 1 ]T .
11. Solve the system of linear equations
3x1 + 6x2 + 2x3 = 0,
3x1 + 3x2 + 7x3 = 4,
3x1 x2 + x3 = 1,
using Jacobi iteration method with starting X(0) = [1, 1, 1]T .
Perform two iterations
using 5-digit floating-point arithmetic with chopping.
14
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-8
Note: All the problems use 5-digit floating point arithmetic
with rounding.
1. (a) The values listed in the table provide the surface
tension of mercury as a function
of temperature
Temperature (0c) 10 20 30 40 50
Surface Tension 488.55 485.48 480.36 475.23 470.11
Use Lagrange interpolating polynomial to find the surface
tension of mercury at 150c.
2. Consider the function f(x) = ex. Construct the Lagrange form
of the interpolating
polynomial for f passing through the points (1, e1), (0, e0) and
(1, e1).
3. Find the interpolating polynomial for the data
x 4 1 0 2 5f(x) 1245 33 5 9 1335
and hence find the value of the polynomial at x = 1.
4. Find the interpolating polynomial P3(x) for the data
x 3 2 0 1f(x) 23 10 4 1
Now one more data f(2) = 18 is added to get the interpolating
polynomial P4(x) =
P3(x) + g(x). Find g(x) and hence interpolate at x = 1.
5. Find the unknown in the set of data points (1, 2), (5, 8),
(7, 10), (8, ) and (10, 15).
6. For a function f , the divided differences are given by
15
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x f(x) First order Second order
x0 = 0.0 f [x0] =?
f [x0, x1] =?
x1 = 0.4 f [x1] =? f [x0, x1, x2] =507
f [x1, x2] = 10
x2 = 0.7 f [x2] = 6
Determine the missing entries in the table. Add f(0.9) = 1.8 to
the table and construct
the interpolating polynomial of degree 3.
7. If f(x) = 12x, then prove that
f [1, 2, 3, 4] = 12 (1234)
.
8. Let
g(x) = f [x0, x1, x2, , xk, x] .
Then prove that
g(x) = 2f [x0, x1, x2, , xk, x, x, x] .
9. If
P4(x) = (x 1)(x+ 2)(x 2)(x+ 1),
then find the value of f [2, 1, 1, 2, 3].
10. If function f(x) is a polynomial of degree n and is to be
interpolated by a ploynomialPn(x) of degree n. Then find the error
in interpolation.
16
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-9
1. The values listed in the table provide the surface tension of
mercury as a function of
temperature
Temperature (0c) 10 20 30 40 50
Surface Tension 488.55 485.48 480.36 475.23 470.11
Use Newtons backward interpolating polynomial to find the
surface tension of mercury
at 450c. Use 5-digit floating-point arithmetic with
rounding.
2. (a) Construct the piecewise linear interpolating polynomial
for the given data:
x 2 0 3 4 5 6f(x) 3 1 2 4 1 5
3. What should be the minimum number of tabular points required
so that the piecewise
linear interpolation for f(x) = cosx on [0, pi] yields values
correct upto 5 significant
digits.
4. Find the minimum number of equispaced tabular points required
for piecewise quadratic
interpolation of the function
f(x) = e3x +2
3(1 + x)2 ,
on the interval [0, 3] so that the values of f are correct upto
4 decimal places.
5. We define the backward difference operator
yi = yi yi1.
Then find the value of 3yi.
17
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6. A car travelling along a straight road is clocked at a number
of points. The data from
the observations are given in the following table, where the
time is in seconds, the
distance is in feet, and the speed is in feet per second
Time 0 3 5 8 13
Distance 0 225 383 623 993
Speed 75 77 80 74 72
Use cubic spline to predict the position of the car and its
speed when t = 10 seconds.
Use 5-digit floating point arithmetic with chopping.
7. Find f (x) corresponding to the data points (0, 1), (1, 1),
(2, 10), (3, 40) and (4, 85) at
x = 1.5.
8. Compute the first and second order derivatives of the
function y = f(x) at x = 1 and
x = 2 by Newtons forward difference formula using the following
table:
x 1 2 3 4 5 6
y 3.9183 4.5212 5.2535 6.1523 7.2498 8.5892
Use 5-digit floating point arithmetic with rounding.
9. The distance y(t) traversed in time t by a point moving in a
straight line is given below:
t (sec) 0 0.01 0.02 0.03 0.04 0.05 0.06
y(t) 0.00 1.53 6.04 13.41 23.42 35.74 50.12
Find the approximate velocity dydt
and acceleration d2ydt2
at t = 0.01 using Newtons
forward interpolation. Use 5-digit floating point arithmetic
with chopping.
10. Fit a curve of the type y = aebx for the following data
set:
x 77 100 185 239 285
y 2.4 3.4 7.0 1.1 19.6
Use 5-digit floating point arithmetic with rounding.
18
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-10
Note: All the problems use 5-digit floating point arithmetic
with rounding.
1. Use the composite Trapezoidal rule to approximate the
following integrals
(i)
21
x ln(x) dx, (ii)
22
x3 ex dx
with n = 4 equal parts.
2. Approximate 20
x2 ex2
dx
using h = 0.25,
(a) Use the composite Trapezoidal rule.
(b) Use the composite Simpsons rule.
3. The Trapezoidal rule applied to 20f(x) dx gives the value 4
and Simpsons rule gives
the value 2. What is f(1)?
4. Determine the number of subintervals N so that the composite
Trapezoidal rule give
the value of the integral 10
1
1 + x2dx,
correct up to 4-decimal digits.
5. Use Simpsons 13rd rule to approximate the integral
10
1
1 + xdx,
with 4 equal subintervals.
19
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6. Given the function f at the following values
x 1.8 2.0 2.2 2.4
f(x) 3.1213 4.4214 6.0424 8.0302
approximate 2.41.8
f(x) dx using Simpsons 38th rule.
7. The quadrature formula 11
f(x) dx = c0 f(1) + c1 f(0) + c2 f(1)
is exact for all polynomial of degree less than or equal to 2.
Determine c0, c1 and c2.
8. Find a, b and in the integration rule 10
f(x) dx = a f() + bf(1),
so that it is exact for polynomial of as high degree as
possible.
9. Determine values for the coefficients A0, A1 and A2 so that
the quadrature formula 11
f(x) dx = A0f
(12
)+ A1f(0) + A2f
(1
2
),
has degree of precision atleast 2.
10. Evaluate the integral 41
xe2x
1 + x2dx,
by using 3-point Gauss-Legendre quadrature formula.
11. Use three-point Gauss-Legendre quadrature formula to
evaluate the integral 32
cos 2x
1 + sin xdx.
20
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-11
Note: All the problems use 5-digit floating point arithmetic
with rounding.
1. Find the general solutions of the following difference
equations
(a)
yn+2 2yn+1 + yn = 0.
(b)
yn+2 + yn+1 6yn = 0.
(c)
yn+2 + yn = 0.
2. Use Eulers method to find the approximate solution y(0.5) of
the initial value problem
y = te3t 2y,y(0) = 0 with h = 0.5.
The exact solution is given by
y(t) =1
5te3t 1
25e3t +
1
25e2t.
Compute the relative error between the exact solution and the
approximate solution.
3. Use Eulers method to find the approximate solution y(0.25) of
the initial value problem
y = cos 2t+ sin 3t,
y(0) = 1 with h = 0.25.
The exact solution is given by
y(t) =1
2sin 2t 1
3cos 3t+
4
3.
Compute the relative error between the exact solution and the
approximate solution.
21
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4. Use Taylors method of order 2 to find the approximate value
of y(1.1), y(1.2) and
y(1.3) as a solution of the equation
dy
dx= x2 + y3
with y(1) = 2 and spacing h = 0.1.
5. Use modified Euler and second-order Taylors method to find
x(1.5) of the following
initial value problem (IVP)
dx
dt= 1 +
x
t, 1 t 2
with
x(1) = 1, step size h = 0.5.
6. Find the approximate solution y(0.1) of the initial value
problem
dy
dx= 2y + x,
y(0) = 1.75,
by using forward Euler, backward Euler and modified Euler method
with step size
h = 0.1. Hence, compare the solutions obtained with the exact
solution
y = 2e2x x2 1
4.
22
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Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-12
Note: All the problems use 5-digit floating point arithmetic
with rounding.
1. Find the approximate solution y(0.2) of the initial value
problem
y + 2xy2 = 0,
y(0) = 1 with h = 0.2
using second-order Runge-Kutta method.
2. Find the approximate solution x(1.1) of the initial value
problem
dx
dt= 3t x
t,
x(1) = 2 with h = 0.1
using second-order Runge-Kutta method. Compare the approximate
solution with the
exact solution
x(t) = t2 +1
t.
3. Compute the approximate solution (0.1) and (0.1) of the
catalyzed reaction problem
= (1 + ), = ( ),
(0) = 1, (0) = 1,
using 4th order Runge-Kutta method with step size h = 0.1.
4. Consider the mass-spring-damper system with nonlinear
damping
u1 = u2, u1(0) = 0.75,
u2 = (1 u21
)u2 u1, u2(0) = 0,
where = 4. Compute u1(0.5) and u2(0.5) by using Runge-Kutta 4th
order method
with h = 0.5.
23
-
5. Use 4th order Runge-Kutta (RK) method to find the approximate
value y(0.1) of the
following initial value problem (IVP)
y 2y + 2y = e2t sin t,y(0) = 0.4, y(0) = 0.6.
6. Use two-step Adams-Bashforth method to approximate the
solution x(0.2) of the initial
value problem
dx
dt= t x3 x,
x(0) = 1 with h = 0.1.
Use any second-order one-step method to determine x(0.1).
7. Find y(1.4) by Adams-Moulton 4th order predictor-corrector
pair with modifier as a
solution of
y = x3 + x y,
y(1) = 2, y(1.1) = 1.6, y(1.2) = 0.34, and y(1.3) = 0.594 with
spacing h = 0.1.
24
-
Birla Institute of Technology and Science, Pilani-K. K. Birla
Goa Campus
Second Semester 2011-2012
Numerical Analysis
(AAOC C341)
Tutorial Sheet-13
Note: All the problems use 5-digit floating point arithmetic
with rounding.
1. Find the approximate solution of the boundary value problem
(BVP)
y + xy + y = x2,
y(0) = 0, y(1) = 1,
using the finite difference method. Use uniform partition of [0,
1] with N = 4 sub
intervals and replace y and y by second order accuracy finite
differences.
2. Find the approximate solution of the boundary value problem
(BVP)
y = 3y + 2y + 2x+ 3, 0 < x < 1y(0) = 2, y(1) = 1,
using the finite difference method. Use uniform partition of [0,
1] with N = 4 sub
intervals and replace y and y by second order accuracy finite
differences.
3. Use central finite difference method to solve the following
boundary value problem
(BVP)
y + y = 0, 0 < x 0 is
(A) 1
(B) does not exist (C) 2 (D) none of these.
13. The asymptotic error constant in the secant method is:
14. One wants to compute the positive root of the equation x = a
b x2 (a, b > 0) by usingthe iterative method xn+1 = a b x2n.
What is the condition for convergence?(A) ab > 3/4 (B) a + b = 0
(C) ab < 4/3
(D) none of these.
31
-
15. Find the solution of the system of linear algebraic
equations using Gaussian elimination
with partial pivoting
0.03 x1 + 58.9 x2 = 59.2, 5.31 x1 6.10 x2 = 47.0.
16. Suppose p approximates p = 150 with relative error at most
103 then the largest
interval in which p must lie
(A) (149.85, 150.51) (B) (148.85, 150.15) (C) (149.85,
150.15)
(D) none of these.
17. The equation f(x) = x2 2 = 0 has a zero on the interval [1,
2]. Choose p0 = 1 andp1 = 2 then find a root correct upto
2-significant digits using secant method:
18. The most accurate value of f(x) = cos2 x sin2 x at x =
0.78530 is(A) 0.00019633 (B) 0.99999 (C) 0.00020 (D) none of
these.
19. Do one iteration of the method of false position to find the
root of f(x) = 3x4 4x3 +3x 2 = 0 on the interval (0, 1.5):
20. Evaluate the polynomial f(x) = 1.1071 x3+0.3129 x20.0172
x+1.1075, for x = 0.1234in nested form.
21. One root of the quadratic equation x2 + 62.10x + 1 = 0 is x
= 62.085, then otherroot is:
22. The most accurate value of ex at x = 0.99999 (under the
freedom that any first degree
Taylors polynomial can be used) is:
23. Let the equation f(x) = 0 that has a root of multiplicity 5.
To obtain quadratic
convergence which iteration formula has to be used:
32
-
(A) xn+1 = xn5 f(xn)f(xn)
(B) xn+1 = xn2 f(xn)f (xn) (C) xn+1 = 5xnf(xn)f (xn)
(D) none of these.
24. The relative propagated error in the product2 pi is nearly
equal to
(A) 1.1928 107 (B) 1.1928 106 (C) 1.1928 105(D) none of
these.
25. State the intermediate value theorem (IVT):
**********THE END**********
33
-
Birla Institute of Technology and Science, Pilani-K. K. Birla,
GOA Campus
FIRST SEMESTER 2011-2012
23rd October, 2011
Test-2 (Closed Book)
Course Title: NUMERICAL ANALYSIS Max. Marks: 75
Course No: AAOC C341 Time: 1 hour
Instructions
(i) Answer all the four questions. Start a new question in a new
page and answer
all its parts in the same place.
(ii) Write all the steps clearly and give explanations for the
complete credit.
(iii) Make an index on the front page of the main answer sheet.
Incomplete
index costs you 5 marks.
NOTE: Use 5-digit floating point arithmetic with rounding
wherever nec-
essary.
1. (a) Let xe be the solution of Ax = b, assuming that det(A) 6=
0 and x be the solutionof Ax = b+ b. Then prove that (5 + 12
marks)
xe xxe K(A)
bb ,
where K(A) is the condition number of matrix A.(b) Solve the
following system of linear equations by Gauss-Seidel iterative
method.
6x+ 3y + 12z = 36,
4x+ 11y z = 33,8x 3y + 2z = 20.
Use (x(0), y(0), z(0)) = (0, 0, 0) and perform two
iterations.
2. (a) For what postive values of and , the given matrix (3 + 3
+ 3 + 12 marks)
A =
3 2
5
2 1
is strictly diagonally dominant?
34
-
(b) Suppose A and B are symmetric positive definite n n
matrices. Is A + B sym-metric positive definite? Justify your
answer.
(c) Find the minimum value of condition number for any n n
matrix.
(d) Use Doolittle decomposition to solve the following system of
linear equations
3x+ 2y z = 7,6x+ 8y + z = 3,
4x+ 2y + 7z = 33.
3. (a) Suppose that f is continuous and has continuous first and
second derivatives on the
interval [x0, x1]. Then prove the following bound on the error
due to linear interpolation
of f (8 + 9 marks)
|f(x) P1(x)| h2
8max
x[x0,x1]|f (x)|, where h = x1 x0.
(b) A bus traveling along a straight road is clocked at a number
of points. The data
from the observations are given in the following table, where
the time is in seconds and
the distance is in feet:
Time 0 3 5 8 13
Distance 0 225 383 623 993
Construct the piecewise linear interpolation and hence predicit
the position of the bus
when t = 10 seconds.
4. (a) The table below gives the values of tan x: (10 + 10
marks)
x 0.10 0.15 0.20 0.25 0.30
y = tanx 0.1003 0.1511 0.2027 0.2553 0.3093
Use Newtons forward interpolating polynomial to find the value
of tan(0.12).
(b) Fit a curve of the type y = a bx for the following data:
x 4 6 8 10 12
y 13.72 12.90 12.01 11.14 10.31
using method of least squares.
**********THE END**********
35
-
Birla Institute of Technology and Science, Pilani-K. K. Birla,
GOA Campus
FIRST SEMESTER 2011-2012
12th December, 2011
Comprehensive Examination (PART-A)
(Closed Book)
Course Title: NUMERICAL ANALYSIS Max. Marks: 60
Course No: AAOC C341 Time: 1 hour 15 min
Instructions
(i) Answer all the questions. There are 20 questions, each worth
3 marks.
(ii) Round the circle in your choice of the correct answer for
multiple questions
and write the correct answer by filling up the blanks.
Answer only in the given blank space. No rough work in anywhere
of this sheet will be allowed.
NAME: Id No.
Section No. Instructor Name:
Note: Use 5-digit floating point arithmetic with rounding
wherever nec-
essary.
1. Evaluate the polynomial
f(x) = 2.752 x3 2.957 x2 + 3.273 x 4.765,
in nested form at x = 1.077.
2. The following difference equation is
un+2 + un+1 + un 3 = 0.
(A) homogeneous with order 2 (B) inhomogeneous with order 1
(C) inhomogeneous with order 3 (D) none of these
3. The convergence of the sequence generated by the formula
pn+1 =p3n + 3pna
3p2n + a,
towarda is third order. Then the asymptotic error constant
is:
36
-
4. Consider the bisection method starting with the interval
[1.5, 3.5]. What is the
maximum distance possible between the root p and the mid point
of this interval?
(A) 2n (B) 2n+2 (C) 2n+1 (D) none of these
5. The l-norm of the vector X , where
X =
[4
(k + 1),
2
k2, k2 ek
]T,
for a fixed positive integer k is:
6. An interpolating polynomial P (x) of degree at most 2 such
that P (0) = 1, P (1) = 1
and P (2) = 1 is
(A) P (x) = 2x2 2x + 1 (B) P (x) = x2 2x + 1 (C) P (x) = 1(D)
none of these
7. How many equal subintervals would be required to approximate
10
41+x2
dx to within
0.0001 by the composite Trapezoidal rule?
8. Use Newtons method to approximate 39. Start with p0 = 2 then
find p1:
9. Use first order Taylors method to find the approximate
solution x(1.1) of the initial
value problem (IVP):
x = (t x)3 (xt
)2,
x(1) = 0.5, h = 0.1
x(1.1) =
10. Consider the table:xi 1 2 3 4 5
ui = u(xi) 2 5 10 20 30where i = 1, , 5
then the value of 2u4 is:
11. The Trapezoidal rule applied to 20f(x) dx gives the value 8
and 1
3rd Simpsons rule
gives the value 6. What is f(1)?
(A) 2.55 (B) 3.0 (C) 2.0 (D) none of these
12. Let g(x) = f [x0, x1, x2, x] then g(x) =
37
-
13. Perform one iteration of Newtons method for the system
4x21 x22 = 0,4x1x
22 x1 = 1
with[x(0)1 , x
(0)2
]= [0, 1].
14. Determine the LU -factorization of the matrix
1 5
3 16
in which both L and U have
unit diagonal elements.
15. For what value(s) of , the integration formula
11
f(x) dx f() + f(),
is exact for all quadratic polynomials?:
16. Use 2nd-order Runge-Kutta method to find the approximate
solution x(2.5) of the
initial value problem (IVP)
x(t) = t (x+ 1)2,
x(2) = 1/2, h = 0.5.
x(2.5) =
17. Let A and B be nn matrices and let k(A) is the condition
number of A in l-norm,then
(A) k(AB) = k(A) k(B) (B) k(AB) k(A) k(B)(C) k(AB) k(A) k(B) (D)
none of these.
18. Find the first approximation for the eigenvector
corresponding to the dominant eigen-
value of the matrix using Power method:
3 2 23 1 31 2 0
, (0) =
0.1
0.2
0.3
:
38
-
19. Find l2-norm of the matrixA, where A =
2 11 2
.
20. Which of the following method has order of convergence
1.
(A) Secant method (B) Method of false position (C) Newtons
method
(D) Mullers method
***************THE END**************
39
-
Birla Institute of Technology and Science, Pilani-K. K. Birla,
GOA Campus
FIRST SEMESTER 2011-2012
12th December, 2011
Comprehensive Examination: Part-B
(Closed Book)
Course Title: NUMERICAL ANALYSIS Max. Marks: 70
Course No: AAOC C341 Time: 1 hour 45 min
Instructions
(i) Answer all the four questions. Start a new question in a new
page and answer
all its parts in the same place.
(ii) Write all the steps clearly and give explanations for the
complete credit.
(iii) Make an index on the front page of the main answer sheet.
Incomplete
index costs you 5 marks.
Note: Use 5-digit floating point arithmetic with rounding
wherever nec-
essary.
1. (a) The distance y(t) traversed in time t by a point moving
in a straight line is given
below: (10 + 8 M)
t (sec) 0 0.01 0.02 0.03 0.04 0.05 0.06
y(t) 0.00 1.53 6.04 13.41 23.42 35.74 50.12
Find an approximate velocity at t = 0.015 by using Newtons
forward interpolation.
(b) Use Crout decomposition to solve the following system of
linear equations
x1 + x2 + 2x3 = 2,x1 + 2x3 = 1,
3x1 + 2x2 x3 = 0.
2. (a) Find the values of , and such that the quadrature rule
(10 + 8 M) 10
f(x)x(1 x) dx = f(0) + f
(1
2
)+ f(1),
40
-
is exact for polynomials of highest possible degree and use the
formula to evaluate 10
1x x3 dx.
(b) Derive the 2-point Gaussian-Legendre quadrature formula and
use it to find the
approximate value of the integral 10
x2 ex dx.
3. (a) Neglecting the effect of air resistance, the motion of a
pendulum can be modeled
by the second-order initial value problem (IVP) (12 + 10 M)
L + g sin = 0,
(0) = 0, (0) = 0,
where denotes the angle which the pendulum rod makes with the
vertical, L is the
length of the pendulum rod and g is the acceleration due to
gravity. Take L = 1 meter,
g = 9.8m/s2 and 0 = 1.5 radians then compute (0.5) by using 4th
order Runge-Kutta
method with h = 0.5.
(b) Find y(0.4) by Adams-Bashforth-Moulton 4th order
predictor-corrector pair with
modifier as a solution of
dy
dx= x y +
y,
y(0) = 1,
with y(0.1) = 1.1079, y(0.2) = 1.2337, y(0.3) = 1.3807 and
spacing h = 0.1.
4. Use second order finite difference method to solve the
following boundary value problem
(BVP): (12 M)
y =2y 4(1 + x)2
, 0 < x 1,y(0) = 0, y(1) 2 y(1) = 0.
Use uniform partition of [0, 1] with two subintervals.
*************The End*************
41
-
Numerical Analysis(AAOC C341)Formula Sheet
1. Secant Method:
pn+1 = pn f(pn) pn pn1f(pn) f(pn1) , n = 1, 2, 3, ,
2. Newtons Method:
pn+1 = pn f(pn)f (pn)
, n = 0, 1, 2, ,
3. Newtons Method for Multiple Roots:
xn+1 = xn m f(xn)f (xn)
, n = 0, 1, 2, , for f(x) = (x r)m h(x), h(r) 6= 0.
4. Fixed Point Iteration:
pn+1 = g (pn) , n = 0, 1, 2, .
5. Newtons Method for system of equations: f1(x) = 0, f2(x) = 0,
, fn(x) = 0:
x(n+1) = x(n)+x, n = 0, 1, 2, , where
f1x1
f1x2
f1xn
f2x1
f2x2
f2xn
......
. . ....
fnx1
fnx2
fnxn
x =
f1(x)
f2(x)...
fn(x)
.
6. Norms for x
-
9. Condition Number:
k(A) = A A1.
10. Lagrange Interpolation for points (xi, fi), i = 0, 1, 2, ,
n;
li(x) = nj=0, j 6=i
x xjxi xj , pn(x) =
ni=0
li(x) fi.
11. Divided differences: example of f [x0, x1, x2] =f [x0,x1]f
[x1,x2]
x0x2:
pn(x) = f(x0) + f [x0, x1](x x0) + f [x0, x1, x2](x x0)(x x1) +
+ f [x0, x1, x2, , xn] (x x0) (x x1) (x x2) (x xn1).
12. Interpolation Error for pn(x):
E(x) =f (n+1)()
(n+ 1)!(x x0) (x x1) (x xn).
13.
u(x) u(x)h
=u(x+ h) u(x)
h, (forward difference formula)
u(x) u(x)h
=u(x) u(x h)
h, (backward difference formula)
u(x) u(x)2h
=u(x+ h) u(x h)
2h, (central difference formula)
14. Central difference formula:
u(x) =u(x+ h) 2u(x) + u(x h)
h2+O(h2), h > 0.
15. Trapezoidal rule: x1x0
f(x) dx =h
2[f(x0) + f(x1)] h
3
12f (), where x0 < < x1.
16. Simpsons 13rd rule: x2
x0
f(x) dx =h
3[f(x0) + 4f(x1) + f(x2)] h
5
90f (4)(), where x0 < < x2.
17. Simpsons 38th rule: x3
x0
f(x) dx =3h
8[f(x0) + 3f(x1) + 3f(x2) + f(x3)]3h
5
80f (4)(), where x0 < < x3.
43
-
18. Gaussian Quadrature (n-point) rule: 11
f(x) dx n
j=1
wj f(j),
where wj are weights and j are Gaussian points.
19. Single step methods to solve the initial value problem
(IVP):
dy
dx= f(x, y(x)),
y(x0) = y0.
(a) Forward Eulers method:
yn+1 = yn + h f(xn, yn), n = 0, 1, 2,
(b) Backward Eulers method:
yn+1 = yn + h f(xn+1, yn+1), n = 0, 1, 2,
(c) Modified Eulers method:
yn+1 = yn +h
2
[f(xn, yn) + f(xn+1, y
n+1)
], n = 0, 1, 2, ,
here yn+1 = yn + h f(xn, yn).
(d) A second-order Runge-Kutta Method:
yn+1 = yn +h
2(k1 + k2) , n = 0, 1, 2,
where k1 = f(xn, yn) and k2 = f(xn + h, yn + h k1).
(e) A fourth-order Runge-Kutta method:
yn+1 = yn +h
6(k1 + 2 k2 + 2 k3 + k4) , n = 0, 1, 2,
where
k1 = f(xn, yn), k2 = f(xn +h
2, yn +
k12),
k3 = f(xn +h
2, yn +
k22), k4 = f(xn + h, yn + k3).
20. Multi step methods to solve the initial value problem
(IVP):
dy
dx= f(x, y(x)),
y(x0) = y0.
44
-
(a) Two step Adams-Bashforth method:
yn+1 = yn +h
2[3f(xn, yn) f(xn1, yn1)] , n = 1, 2, 3,
(b) Four step Adams-Bashforth method:
yn+1 = yn+h
24[55f(xn, yn) 59f(xn1, yn1) + 37f(xn2, yn2) 9f(xn3, yn3)] ,
where n = 3, 4, 5, (c) Three step Adams-Moulton method:
yn+1 = yn +h
24[9f(xn+1, yn+1) + 19f(xn, yn) 5f(xn1, yn1) + f(xn2, yn2)]
,
where n = 2, 3, 4, (d) Adams-Moulton Predictor-Corrector
formula:
yn+1 = yn +h
24[55f(xn, yn) 59f(xn1, yn1) + 37f(xn2, yn2) 9f(xn3, yn3)] ,
yn+1 = yn +h
24
[9f(xn+1, y
n+1) + 19f(xn, yn) 5f(xn1, yn1) + f(xn2, yn2)
].
45
-
Bibliography
[1] R. L. Burden and J. D. Faires, Numerical Analysis; Theory
and Applications, India
Edition Cengage Learning (2010).
[2] Brian Bradie, A friendly introduction to Numerical Analysis,
Pearson Education, 2007.
[3] S. D. Conte and Carl de Boor, Elementary Numerical Analysis:
An Algorithmic Ap-
proach, International Series in Pure and Applied Mathematics,
3rd Edition, 1980.
[4] Curtis F. Gerald, Patrick O. Wheatley, Applied Numerical
Analysis, Pearson Education,
7th Edn., 2009.
[5] Joe D. Hoffman, Numerical Methods for Engineers and
Scientists, CRC Press, Second
Edition, 2010.
[6] Kendall E Atkinson, An Introduction to Numerical Analysis,
John Wiley & Sons, 2001.
[7] Srimanta Pal, Numerical Methods Principles, Analyses and
Algorithms, Oxford Univer-
sity Press, 2009.
[8] Saumyen Guha and Rajesh Srivastava, Numerical Methods for
Engineering and Science,
Oxford University Press, 2010.
[9] Steven C Chapra, Applied Numerical Methods with MATLAB for
Engineers and Scien-
tists, Tata McGraw-Hill, Second Edition, 2007.
[10] Victor S. Ryabenkii and Semyon V. Tsynkov, A theoretical
Introduction to Numerical
Analysis, Chapman & Hall/CRC, 2007.
46
-
1
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI-K. K. Birla
GOA CAMPUS
INSTRUCTION DIVISION SECOND SEMESTER 2011-2012
Course Handout (Part II) Date: 06/01/2012
In addition to part I (General Handout for all courses appended
to the time table) this portion gives further specific details
regarding the course.
Course No. : AAOC C341 Course Title : Numerical Analysis
Instructor-in-charge : P. DHANUMJAYA Instructors : Sangeeta
Jaiswal, Muslim, Balchand Prajapati, Samanta Gauranga
1. Scope and Objective of the Course: This course enables one to
devise algorithms for numerical solutions of different mathematical
problems and also discuss the error analysis of different
algorithms. 2. Text Books: T1. Numerical Analysis; Theory and
Applications, R. L. Burden and J. D. Faires, Cengage Learning,
India Edition (2010). T2. Applied Numerical Analysis, Curtis F.
Gerald, Patrick O. Wheatley, Pearson Education, 7th Edn., 2009. 3.
Reference Books: R1. A friendly introduction to Numerical Analysis,
Brian Bradie, Pearson Education, 2007. R2. An Introduction to
Numerical Analysis, Kendall E Atkinson, John Wiley & Sons,
2001. R3. Numerical Methods Principles, Analyses and Algorithms,
Srimanta Pal, Oxford University Press, 2009. 4. Course Plan:
Lect. No. Learning Objective Topics to be Covered (Refer to T1,
T2)
1-3 To understand the potential and pitfall of the numerical
computation.
Computer arithmetic, Kinds of Errors in Numerical Computation,
Significant digits, Error bounds and Evaluation of polynomials.
4-10 To find the roots of nonlinear equations and understand the
relative strengths and weaknesses of each method.
Bisection method, Fixed-point iteration method, Newtons method,
Secant method, Falseposition method, Mullers method, Newtons method
for multiple roots, Order of convergences of all the above methods.
Newtons method and fixed-point iteration method for the system of
non-linear equations.
11-16
To solve the system of linear algebraic equations by using
direct methods and iterative methods. Compute the determinant of a
matrix, matrix inverse and understand the relative strengths and
weakness of each method.
The Gaussian elimination method, Pathology in linear
systems-singular matrices, Determinants and matrix inversions,
Doolittle and Crout decompositions, Tridiagonal and positive
definite matrices, Norms, Condition numbers and errors in
solutions; Iterative methods: Jacobi, Gauss-Seidel and SOR
Methods.
-
2
17-22 To construct an interpolating polynomial and evaluate at
unknown points
Lagrange interpolation, Existence and uniqueness of
interpolating polynomial, Divided differences, Newton's forward and
backward interpolations, Errors of interpolations, Piecewise
linear, Piecewise quadratic interpolations, Cubic spline
construction and Least-Square Regression.
23-29
To compute numerical derivatives and integrations using discrete
data points and learn how to integrate functions containing
singularities
Numerical differentiation, Newton-Cotes integration formulae,
Composite rules, Error terms for Newton-Cotes formulae and
composite rules, Method of undetermined coefficients, Two point and
Three point Gaussian-Legendre quadrature rules.
30-36
To compute the numerical solution of initial value problems
(IVPs)
Difference Equations, Forward Euler, Backward Euler and Modified
Euler methods, Taylor Series methods, 2nd order and 4th order
Runge-Kutta methods, System of ODEs and Higher order ODEs.
Multistep methods: Adams-Bashforth methods, Adams-Moulton methods,
Adams-Moulton Predictor-Corrector Method.
37-39 To solve two point boundary value problems (BVP) Finite
difference methods and Shooting methods
40-42 Eigenvalues and eigenvectors of matrices Power method,
Inverse power method, QR methods of finding eigenvalues and
eigenvectors of matrices. 5. Self Learning Component (SLC):
(i) Implementation of Bisection, Secant, False-position,
Newton's method, Fixed-Point method and verifying order of
convergence of each method by using MATLAB.
(ii) Implementation of Gaussian elimination method and iterative
methods (Jacobi, Gauss-Seidel and SOR methods) using MATLAB.
(iii) Finding numerical integration using different quadrature
rules. (iv) Solving initial value problems (IVPs) and boundary
value problems (BVPs) using MATLAB.
6. Evaluation Scheme:
EC No. Evaluation Component Duration
Weightage
(%) Date, Time Remarks
1 Test 1 60 Min. 25 22/02/2012, 8.30-9.30 AM CB 2 Test 2 60 Min.
25 30/03/2012, 8.30-9.30 AM CB
3 Tutorial Test/Quiz/
Self Learning Component/ Assignments/Lab Exam
*** 10 ***
4 Comprehensive Exam 3 Hours 40 03/05/2012, 9:00-12:00 Noon
CB
*** To be announced latter. 7. Problems: Students are strongly
advised to workout all the problems in the text-books (T1, T2) and
do similar problems from the reference books (R1, R2, R3, R4). It
is also strongly recommended that the students should implement all
the algorithms on computers to get a better understanding of the
subject.
8. Chamber Consultation Hours: To be announced by the respective
instructor.
9. Make-up: Make up for any component of evaluation will be
given only in the genuine cases.
10. Notices: All the notices regarding this course will be put
up only in the course ftp. Instructor-In-Charge
AAOC C341
-
Numerical Analysis(AAOC C341)
Important Dates
S. No Evaluation Component Date Time
1 Test-1 22-02-2012 8:30 - 9:30 AM
2 Test-2 30-03-2012 8:30 - 9:30 AM
3 Comprehensive Exam 03-05-2012 9:00 - 12:00 Noon
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Table of Contents.pdfroot_bits.pdfNA-Handout.pdf