BCA MATHS

5th semester

Solutions to Scientific Computing Questions
1. Imagine you can only store and use integers with 10 digits. What results when you
add 9999999999 + 0000000001?
(0000000000)
2. Using a factorial method which returns N! as an int, you get the following results:
12! = 479001600
13! = 1932053504
14! = 1278945280
15! = 2004310016
16! = 2004189184
17! = -288522240
What happened? Where did things start going wrong?
(Overflow. Data type int can only store from -2147483648 to +2147483647. 13! is over
six billion, so it is too big for an int, the leftmost bits fall off and the 32 bits left in
memory are used as the answer.)
3. What will the following java fragment print?
double x1 = 0.3;
double x2 = 0.1 + 0.1 + 0.1;
StdOut.println(x1 == x2);
double z1 = 0.5;
double z2 = 0.1 + 0.1 + 0.1 + 0.1 + 0.1;
StdOut.println(z1 == z2);
(First prints false, second prints true. But the point is that this issue is inherent
in floating point representation of number, so don’t test float and double for strict
equality.)
4. Is the previous result a consequence of Round Off Error or Catastrophic Cancellation?
(round off error, but this will lead to ”What is catastrophic cancellation?”)
5. Will the following java fragment print 0.0?
System.out.println( (.3 -.1 -.1-.1)*1e15);
(No, it prints -0.027755575615628914)
6. Is the previous result a consequence of Round Off Error or Catastrophic Cancellation?
(Both. The result that (.3-.1-.1-.1) is not 0.0 is a result of round off error. When we
multiply that erroneous result by a large number, or repeatedly use it in a calculation,
we get catastrophic cancellation - something that should have cancelled out and been
zero is now big enough to cause a problem.)
1
7. Give an example when java will give you NaN.
(Math.sqrt(-1))
8. Give an example when java will give you Infinity.
( 1.0/0.0 )
9. What will the following java fragment print?
System.out.println( 1/0 );
(Trick question. It won’t print any answer. This is integer division and will cause a
runtime error: ArithmeticException / 0)
10. What will the following java fragment print?
System.out.println( 1000000000000. + .00001);
(1.0E12 Another round off error. The lesson here is don’t add or subtract very differently
sized floating point numbers.)
11. Why is an ill-conditioned problem worse than an unstable algorithm?
(With an ill-conditioned problem, computing a reasonable answer is doomed from the
start. Any slight variation in the starting condition results in a huge deviation from
the desired answer. No algorithm can compensate for this.
With an unstable algorithm, you might be able to rearrange the terms or steps so that
you can avoid the problem area and be able to compute a close enough answer, as long
as the problem is well-conditioned.)

Bit patna 4th semester syllabus

Calculas problem

BCA THIRD SEMESTER

DISCRETE METHAMATICS STRUCTURES

Module -I
Sets and Subsets, Operations on Sets, Characteristics functions, Properties of Integers.

Module -II
Propositions and Logical Operations, Conditional Statements, Methods of Proof, Mathematical Induction.

Module -III
Permutations, Combinations, Pigeonhole Principles, Recurrence Relations.

Module -IV
Product Sets and Partitions, Relations and Digraphs, Paths in Relations and Digraphs, Properties of Relations, Equivalence Relations, Operations of Relations.

Module -V
Functions, Growth of Functions, Permutation Functions.

Module -VI
Partially Ordered Sets, External Elements of Partially Ordered Sets, Lattices.

Module -VII
Binary Operations, Semigroups, Groups, Other Mathematical Structures – Rings, Fields.

BCA SECOND SEMESTER

Module -I
Set Theory: Introduction, Venn Diagram, Algebra of sets, Cardinal number of set, Cartesian product.

Module -II
Relations: Relation, Inverse Relation, Composition of relation, Equivalence of relations and Classes.

Module -III
Functions and Binary Operations: Functions as a set, Representation of function, Types of Functions, Functions applicable in Computer Science: Integer function, Factorial function, Absolute, Characteristic, Floor, Ceiling, Remainder, Boolean, Exponential, Logarithmic, Hashing functions and Binary operation on a set.

Module -IV
Abstract Algebra: Group, Subgroup, Order of group, Cyclic group and Simple theorems, Ring, Integral domain, Field and Simple theorem related to Integral domain and field.

Module -V
Linear Algebra: Vector spaces, Subspaces, Span of a set, Linear Dependence, Independence, Dimension and Basis, Linear Transformation: Definition and example, Range and Kernel of a Linear map, Rank and Nullity.

Module -VI
Matrix Algebra: Elementary Transformation, Elementary matrices, Inverse of a matrix by Row operation, Rank of matrix by Echelon form, Solution of a System of Linear Simultaneous equation by matrix method, Eigen valve and Eigen vector, Caley Hamilton theorem (without proof), to find the inverse of a non singular matrix using Caley-Hamilton theorem.

Module -VII
3D Coordinate Geometry: Cylindrical, Spherical Co-ordinate Systems, Distance formula, Direction Cosines, Direction rations, Projections, Plane, Different form of planes with simple treatment, Shortest distance between two skew lines, Equation of sphere, Tangent plane at a point on the sphere.

BCA FIRST SEMESTER

Module I
Successive differentiation, Leibnitz Theorem, Partial derivative, Partial derivative of Higher orders, Homogeneous function, Euler’s Theorem on Homogeneous functions.

Module II
Total differentiation, Jacobians, Properties of Jacobians, Jacobians of Implicit functions, Approximation of Error.

Module III
Expansion of functions of One variable and two variable by Taylor’s and Maclaurin’s Theorem.

Module IV
Maxima and Minima of functions of One and two variables with simple applications, Table of Standard Curves including Cardioid, Astroid, Lemniscate.

Module V
Double Integration, Eualuation of Double Integration, Change of Order of Integration, Area by Double Integration, Triple Integrals.

Module VI
Definition of Differential equation, Order and degree of a differential equation, Solution of differential equation, Differential equation of first order and first degree: variables separable, Linear differential equation and Equation of Higher order Linear differential equation with constant coefficients.

Module VII
Partial differential equations, Order and degree of partial differential equation, Lagrange’s Linear Equations, Linear Homogeneous partial differential equation, Non Homogeneous partial differential equations.