Course Title:
Numerical Analysis
Course No.: Math. Ed. 447 Nature of course:
Theoretical
Level: BICTE Credit Hour: 3 hours
Semester: Fourth Teaching Hour: 48
hours
1. Course Description:
This course is designed for the students of BICTE under
Tribhuvan University. It helps
students to fulfill their increasing
desire towards numerical answers to applied problems with the help of methods and techniques of numerical
analysis. Although numerical methods
have always been useful,
their role in the present day scientific research is of
fundamental importance. It deals with numerical methods which give the solution when ordinary analytical methods
fail for the solution of transcendental equations. In addition, it deals those numerical
techniques which can be used for the
solution of system of linear equations through matrix computations along with solution
of non-linear equations through interpolation and iterative method of differentiation and integration. This course
also provides a foundation for the mathematical modeling in the field of research.
2.
General Objectives
· To understand errors and approximation.
·
To use different methods for
solving transcendental and linear simultaneous
equations.
·
To define different types of differences and construct their tables,
and establish the relationship between them.
·
To
be familiar with interpolation
and apply suitable interpolation formula for
numerical problems
· To deal with numerical
approximations of derivatives
· To approximate computation of an integral using numerical techniques
3.
Specific Objectives and contents
|
Unit-I Errors
and computation of roots (8
hrs) |
|
|
· To identify the
types of errors ·
To derive general error formula · To generalize a series
approximation ·
To solve linear equations graphically · To find
solution of equations by bisection method ·
To discuss the method of false position ·
To solve equations by iteration method · To use Newton – Raphson iteration formula · To apply the Muller
‘s method to approximate the
roots of equations |
· Significant digits · Errors · General error formula · Error in a
series approximation · Linear
equations · Graphical
solution of equations · Bisection method · The method of false position · Iteration method · Newton – Raphson method · Generalized Newton’s formula for multiple roots · Muller’s method |
|
Unit
–II Solution of Linear
simultaneous Equations (8 hrs) |
|
|
· To identify
linear- simultaneous equations. · To apply Gauss elimination method in solving simultaneous equations · To solve simultaneous equations by Gauss –Jordan method · To solve
LS equations by using Jacobi’s and Gauss –
Seidel iteration method · To discuss
and use factorization, Iterative and partition methods to solve simultaneous equations. |
· Linear
simultaneous equations (LSE) · Gauss elimination method · Gauss – Jordan method · Jacobi –
iteration method · Gauss – Seidel iteration method · Matrix inversion method · Factorization
method · Iteration method · Partition method |
|
Unit –III |
Differences
of polynomials |
(10
hrs) |
|
|
·
To discuss forward and backward difference operators · To construct difference tables ·
To discuss properties of the
forward difference operator · To establish relationship among the operators E ,D and
D ·
To express a given polynomial in factorial notation ·
To identify the central difference
operator and the mean operator ·
To construct the central
difference table · To establish
relationship between the operators D ,Ñ ,E ,m and d |
· Forward difference operator ·
Forward difference table · The operator E · Relation
between the operators E and D · The operator D · Backward
difference table · Factorial
polynomial · Central difference operator · Central
difference table · Mean operator · Relationship between operators D ,Ñ ,E ,m and d |
||
|
Unit –IV |
Interpolation with
Equal Intervals |
(8 hrs) |
|
|
·
To derive and use Newton – Gregory forward interpolation formula ·
To derive and use Newton
–Gregory backward interpolation formula · To apply
forward and backward interpolation
formulae in solving problems ·
To derive and use Gauss’ forward and backward interpolation formula ·
To apply Bessel’s and Stirling's formula for interpolation |
·
Newton –Gregory forward interpolation formula ·
Newton - Gregory backward interpolation formula · Error in the interpolation formula · Gauss’ forward
interpolation formula ·
Gauss’ s backward interpolation formula · Bessel’s formula · Stirling’s formula |
||
|
Unit -v |
Interpolation with
Unequal Intervals |
(4 hrs ) |
|
|
·
To discuss linear and quadratic interpolations · To find divided differences · To establish
the relationship between divided differences and ordinary differences |
· Linear interpolation · Quadratic interpolation · Divided
differences · Second divided difference ·
Newton ’s divided difference interpolation · Relation
between divided differences and ordinary differences |
||
|
Unit – VI |
Numerical Differentiation and integration |
( 10 hrs ) |
|
|
· To derive
formula for the derivative using
forward and backward difference
formula ·
To derive formula for derivative using central difference formula · To derive
general quadrature formula · To apply
trapezoidal rule , Simpson’s one
–third rule ,three-eighth rule , Bool’s rule and Weddle’s rule for solving
numerical problems · To find
errors in quadrature formula |
· Numerical differentiation ·
Derivative using forward
difference formula ·
Derivative using backward
difference formula · Derivative
using central difference formula ·
General quadrature formula for equidistant ordinates · Trapezoidal rule · Simpson,s One –Third rule · Simpson,s Three
– Eighth rule · Bool,s rule · Weddle ,s rule · Errors in quadrature formula |
||
4.Instructional Techniques
|
Units |
Activity and Instructional Techniques |
|
Unit I |
· Individual and group discussion on calculating errors |
|
Unit II |
·
Individual and group discussion on bisection and iteration methods ·
Group and individual assignments
on problems of getting roots by bisection method |
|
Unit III |
· Group and
individual discussion on different methods of solving linear simultaneous
equations |
|
Unit IV |
· Individual and group assignments on finite differences |
|
Unit V |
·
Presentation and discussion on computer programming in c++ of important methods |
|
Unit VI |
· Individual and
group presentation on divided differences and ordinary differences |
5. Evaluation
Internal evaluation
Internal evaluation will be conducted
by course teacher
based on following activities:
a.
Attendance 5 points
b. Participation in learning activity 5 points
c.
First assessment test 10 points
d. Second ssessment test 10 points
e.
Third ssessment test 10 points
` ………………………………………………..
Total 40 points
NOTE: Internal evaluation
and assignments may include the numerical calculation and computation by using
different computer application like as Matlab,
Geobebra and MS Excel also.
External Evaluation:
Faculty of Education, Examination division will conduct final
examination of weight 60 points at the end of semester. This 60 points is divided in final
examination paper as Objective questions (10 x 1) 10 points
Short answer questions (6 x 5) 30 points
Long answer questions (2x 10) 20 points
………………………………………………………………..
Total 60 points
6.Recommended and Reference Books
6.1 Recommended books
Sastry,
S.S. (1990). Introductory methods of numerical analysis ,New Delhi : Prentice- Hall of India ( Units
I – VI )
Gupta S. and Sharma S.(2014).Numerical
analysis ,New Delhi : S.K .Kataria &
Sons ( Units I – VI )
6.2
Reference books
Conte S.D. (1965) , Elementary
numerical analysis Mc Graw-
Hill Froberge
C.E. (1965)
, Introduction to numerical analysis ,Adison Wesley
Jian , M.K.(1971) , Numerical
analysis for scientists and engineers Delhi:S.B.W . Publishers
Sastry S.S. (1997) , Engineering mathematics , New Delhi :
Prentice-Hall of India Stanton
, R.G. (1967) , Numerical
methods for science and
engineering , New Delhi : Prentice-Hall of India
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