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Lesson 1

Subject: Brackets

Respected Sir, Aslam u alikum, Sir in handouts i can't understand the line, the decimal equivalent of binary number 10011001 is... Sir, the coding is 1001001, how we can determine that we should take powers in -ve order? Q#2: in the question, (0.11.....11(0011))2 why we use brackets to write the result e.g (0011) , Respected Sir, kindly guide me in this regards.. Your's obidiently,

Dear student, indefinitely repeated pattern of numbers has been enclosed in brackets with base as subscript.

Subject: Query

Respected Sir

Asalam o alakium

Plz explain example discussed in the lecture decimal equivalent of binary number 10011001 is

1  × 20 + 0* 2 -1 + 0 × 2-2  + 1 × 2-3 + 1× 2-4 + 0 × 2-5+  0 ×2-6  + 1×2-7

Sir plz explain how u come to that binary number 10011001 in decimal is in fraction .......... that u start from 1 × 20 ............ What is the clue

Thanks

Dear student, in daily life we make use of numbers 0,1,2,…,9 which is decimal system. We need to express this decimal system into other forms too. In computers  different number systems are used

The number systems commonly used in computers are

 Base, N             Number 2                        Binary 8                       Octal 10                      Decimal 16                      Hexadecimal

so we should also know that how a number in one system can be expressed into another system.

You want to get decimal equivalent of binary number 10011001. For this you express given number in form

amNm + am-1Nm-1 +...+a1N1+a0+a-1N-1+...a-mN-m
As you convert from binary so take base m =2 , this above expression takes form as

1  × 20 + 0* 2 -1 + 0 × 2-2  + 1 × 2-3 + 1× 2-4 + 0 × 2-5+  0 ×2-6  + 1×2-7

= 1+0+0+1/8+1/16+0+1/128

=1+1/8+1/16+1/128

=1+0.125+0.0625+0.0078125

=1.1953125

As now you have converted into decimal so enclose this obtained result into brackets and put ‘10’ as base

=(1.1953125)10

Subject: Binary equal to decimal fraction

Respected sir,
(0.7625)10=(0.11...11(0011))2 this also write as (0.7625)10 = (011000011)2

Dear student, you are converting a decimal fraction into binary system. Integers you getting in each step are arranged as: The integral parts from top to bottom are arranged from left to right after the fractional point to represent the binary form.
Let's make an important observation here. Notice that the step 9 to be performed (multiply 0.2 x 2) is exactly the same action we had in step 5. We are then bound to repeat steps 5-9, then return to Step 5 again indefinitely. In other words, we will never get a 0 as the decimal fraction part of our result. Instead we will just cycle through steps 5-9 forever. This means we will obtain the sequence of digits generated in steps 5-9, namely 0011, over and over. Hence, the final binary representation will be. 0.1100001100110011… So write it as (0.1100(0011)2)

Student's Message:

Msg No. 546844

 Subject: equation sir aslamo alikum im saria i want to ask about the equation that is AmNm + AM-1NM-1 +...+A1N1+A0+A-1N-1+...A-MN-M  THAT IS CONFUSING and what is that a and n and m thanx. Post Your Comments Other Students' Comments: 0 Instructor's Reply: Dear student  amNm + am-1Nm-1 +...+a1N1+a0+a-1N-1+...a-mN-m  is representation of arbitrary eal number .in which m is base and ‘a’ are the digits of that given number you want to convert in any other format. Please check from examples in lecture 1 . Student's Message: Msg No. 546773

Lesson 2

 dear sir i can't understand Local truncation error. and 2nd question. rational numbers which is written as p/q what is p and q  as we write a/b or x/y thank yo. plz explain. Post Your Comments Other Students' Comments: 0 http://vustudents.ning.com/   Instructor's Reply: Dear Student,  Truncation error is the error made by cutting an infinite sum and approximating it by a finite sum. For instance, the trigonometric function “the cosine function” as written in series form on page 1 of lecture 2 in topic LOCAL TRUNCATION ERROR. This series is called Taylor series. if we approximate this cos function by the first three non-zero term of its Taylor series, and find its sum. Of course this answer will be different from the sum of infinite terms in this Taylor series, so an error has occurred. the resulting error is a truncation error.     A rational number is any number that can be expressed as the fraction a/b of two integers, with the neumerator 'a' and denominator 'b' such that 'b' is not equal to zero. The set of all rational numbers is usually denoted by a boldface Q. DEAR SIR: AOA:  sir Absolute error and relative error are the examples of Local roundoff error? regards Post Your Comments Other Students' Comments: 0 Instructor's Reply: Dear student, yes you are right.

Lesson 3

 Subject: Root Of Equation Respected Sir, I could not understand the concept of Root Of an Equation. What does mean of by root of an equation and why its root is f(x)=0? Regards, Post Your Comments Other Students' Comments: 0 Instructor's Reply: Dear student, a real number x will be called a solution or a root if it satisfies the equation, The root of a  equation is the value of x for which the equation y=f(x)=0 Example: The roots of x2 – x – 2 = 0 are x = 2 and x = –1. Observe you have equation x2 – x – 2 = 0, in this  f(x) = x2 – x – 2  so take y = f(x) = 0.  The equation is satisfied if we substitute either x = 2 or x = –1 into the equation.

Lesson 4

Lesson no 6

Lecture n0 7

Exact solution of 2/3 is not exists.

TRUE

FALSE

The Jacobi’s method is

A method of solving a matrix equation on a matrix that has ____ zeros along its main diagonal.

No

At least one

A 3 x 3 identity matrix have three and __________eigen values.

Same

Different

Eigenvalues of a symmetric matrix are all _______

Real

Complex

Zero

Positive

The Jacobi iteration converges, if A is strictly diagonally dominant

TRUE

FALSE

Below are all the finite difference methods EXCEPT _________.

Jacobi’s method

Newton’s backward difference method

Stirlling formula

Forward difference method

If n x n matrices A and B are similar, then they have the same eigenvalues (with the same multiplicities).

TRUE

FALSE

If A is a nxn triangular matrix (upper triangular, lower triangular) or diagonal matrix, the eigenvalues of A are the diagonal entries of A.

TRUE

FALSE

The characteristics polynomial of a 3x 3

Identity matrix is __________, if x is the Eigen values of the given 3 x 3 identity matrix. Where symbol ^ shows power.

(X-1)^3

(x+1)^3

X^3-1

X^3+1

Two matrices with the same characteristic polynomial need not be similar.

TRUE

FALSE

Bisection method is a

Bracketing method

Open method

Regula Falsi means

Method of Correct position

Method of unknown position

Method of false position

Method of known position

Eigenvalues of a symmetric matrix are all _________.

Select correct option:

Real

Zero

Positive

Negative

An eigenvector V is said to be normalized if the coordinate of largest magnitude is equal to zero.

Select correct option:

TRUE

FALSE

Exact solution of 2/3 is not exists.

Select correct option:

TRUE

FALSE

The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric ________ definite matrices A.

Select correct option:

Positive

Negative

Differences methods find the ________ solution of the system.

Select correct option:

Numerical

Analytical

The Power method can be used only to find the eigenvalue of A that is largest in absolute  value—we call this Eigenvalue the dominant eigenvalue of A.

Select correct option:

TRUE

FALSE

The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its ________.

Select correct option:

Main diagonal

Last column

Last row

First row

If A is a nxn triangular matrix (upper triangular, lower triangular) or diagonal matrix , the eigenvalues of A are the diagonal entries of A.

Select correct option:

TRUE

FALSE

A 3 x 3 identity matrix have three and different Eigen values.

Select correct option:

TRUE

FALSE

Newton Raphson method falls in the category of

Bracketing method

Open Method

Iterative Method

Indirect Method

Newton Raphson method is also known as

Tangent Method

Root method

Open Method

Iterative Method

Secant Method uses values for approximation

1

3

2

4

Secant Method is than bisection method for finding root

Slow

Faster

In Newton Raphson method

Root is bracketed

Root is not bracketed

Regula falsi method and bisection method are both

Convergent

Divergent

In bisection method the two points between which the root lies are

Similar to each other

Different

Not defined

Opposite

In which methods we do not need initial approximation to start

Indirect Method

Open Method

Direct Method

Iterative Method

Root may be

Complex

Real

Complex or real

None

In Regula falsi method we choose points that have signs

2 points opposite signs

3 points opposite signs

2 points similar signs

None of the given

In a bounded function values lie between

1 and -1

1 and 2

0 and 1

0 and -2

Newton Raphson method is a method which when it leads to division of number close to zero

Diverges

Converges

Which of the following method  is modified form of Newton Raphson Method?

Regula falsi method

Bisection method

Secant method

Jacobi’s Method

Which 1 of the following is generalization of Secant method?

Muller’s Method

Jacobi’s Method

Bisection Method

N-R Method

Secant Method needs starting points

2

3

4

1

Near a simple root Muller’s Method converges than the secant method

Faster

Slower

If   S is an identity    matrix, then

All are true

If we retain r+1 terms in Newton’s forward difference formula, we obtain a polynomial of degree ---- agreeing with  at

r+2

r+1

R

R-1

P in Newton’s forward difference formula is defined as

Octal numbers has the base

10

2

8

16

Newton’s divided difference interpolation formula is used when the values of the independent variable are

Equally spaced

Not equally spaced

Constant

None of the above

Given the following data

 0 1 2 4 1 1 2 5

Value of is

1.5

3

2

1

If is approximated by a polynomial  of degree n then the error is given by

Let  denotes the closed interval spanned by . Then vanishes ------times in the interval .

N-1

N+2

N

N+1

Differential operator in terms of forward difference operator is given by

Finding the first derivative of at =0.4 from the following table:

 0.1 0.2 0.3 0.4 1.10517 1.2214 1.34986 1.49182

Differential operator in terms of ----------------will be used.

Forward difference operator

Backward difference operator

Central difference operator

All of the given choices

For the given table of values

 0.1 0.2 0.3 0.4 0.5 0.6 0.425 0.475 0.4 0.452 0.525 0.575

, using two-point equation will be calculated as.............

-0.5

0.5

0.75

-0.75

In Simpson’s 1/3 rule, is of the form

►

►

►

While integrating, , width of the interval, is found by the formula-----.

None of the given choices

To apply Simpson’s 1/3 rule, valid number of intervals are.....

7

8

5

3

For the given table of values

 0.1 0.2 0.3 0.4 0.5 0.6 0.425 0.475 0.4 0.452 0.525 0.575

, using three-point equation will be calculated as ……

17.5

12.5

7.5

-12.5

To apply Simpson’s 1/3 rule, the  number of intervals in the following must be

2

3

5

7

To apply Simpson’s 3/8 rule, the number of intervals in the following must be

10

11

12

13

If the root of the given equation lies between a and b, then the first approximation to the root of the equation by bisection method is ……

None of the given choices

............lies in the category of iterative method.

Bisection Method

Regula Falsi Method

Secant Method

All of the given choices

For the equation, the root of the equation lies in the interval......

(1, 3)

(1, 2)

(0, 1)

(1, 2)

Rate of change of any quantity with respect to another can be modeled by

An ordinary differential equation

A partial differential equation

A polynomial equation

None of the given choices

If

Then the integral of this equation is a curve in

None of the given choices

Xt-plane

Yt-plane

Xy-plane

In solving the differential equation

,   By Euler’s method  is calculated as

1.44

1.11

1.22

1.33

In second order Runge-Kutta method

is given by

None of the given choices

In fourth order Runge-Kutta method,   is given by

In fourth order Runge-Kutta method,  is given by

None of the given choices

Adam-Moulton P-C method is derived by employing

Newton’s backward difference interpolation formula

Newton’s forward difference interpolation formula

Newton’s divided difference interpolation formula

None of the given choices

The need of numerical integration arises for evaluating the definite integral of a function that has no explicit ____________ or whose antiderivative is not easy to obtain

Derivatives

Antiderivative

If then system will have a

Definite solution

Unique solution

Correct solution

No solution

If  then

There is a unique solution

There exists a complete solution

There exists no solution

None of the above options

Direct method consists of method

2

3

5

4

We consider Jacobi’s method Gauss Seidel Method and relaxation method as

Direct method

Iterative method

Open method

All of the above

In Gauss Elimination method Solution of equation is obtained in

3 stages

2 stages

4 stages

5 stages

Gauss Elimination method fails if any one of the pivot values becomes

Greater

Small

Zero

None of the given

Changing the order of the equation is known as

Pivoting

Interpretation

Full pivoting is than partial pivoting

Easy

More complicated

The following is the variation of Gauss Elimination method

Jacobi’s method

Gauss Jordan Elimination method

Courts reduction method is also known as Cholesky Reduction method

True

False

Jacobi’s method is also known as method of Simultaneous displacement

True

False

Gauss Seidel method is also  known as method of Successive displacement

False

True

In Jacobi’s method approximation calculated is used for

Nothing

Calculating the next approximation

Replaced by previous one

All above

In Gauss Seidel method approximation calculated is replaced by previous one

True

False

Relaxation method is derived by

South well

Not defined

Power method is applicable for only

Real metrics

Symmetric

Unsymmetrical

Both symmetric and real

The process of eliminating value of y for intermediate value of x is know as interpolation

True

False

20 MCQS

2 Questions of 2 marks

1)    1 question was to find the value of p from the given table using
Interpolation Newton's forward Diffrence Formula

2)    we have to calculate the values of x, y and z for the given system of
equations using Gauss Siede iterative method taking initial solution vector
as (0, 0, 0)t

2 Questions of 3 marks

both of the questions were to find the value of theta for the given matrices
using Jacobi's method

2 Questions of 5 marks

1)   Construct backward difference table for the given values

2)   Find the solution of following system of Equations using Jacobi's
iterative method upto three decimal places

83x+11y-4z=95
7x+52y+13z=104
3x+8y+29z=71

taking the initial solution vector as (0, 0, 0)T

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Exact solution of 2/3 is not exists.

TRUE

FALSE

The Jacobi’s method is

a method of solving a matrix equation on a matrix that has ____ zeros along its main diagonal.

no

atleast one

A 3 x 3 identity matrix have three and __________eigen values.

same

different

Eigenvalues of a symmetric matrix are all _______ .

real

complex

zero

positive

The Jacobi iteration converges, if A is strictly diagonally dominant.

TRUE

FALSE

Below are all the finite difference methods EXCEPT _________.

jacobi’s method

newton's backward difference method

Stirlling formula

Forward difference method

If n x n matrices A and B are similar, then they have the same eigenvalues (with the same multiplicities).

TRUE

FALSE

If A is a nxn triangular matrix (upper triangular, lower triangular) or diagonal matrix , the eigenvalues of A are the diagonal entries of A.

TRUE

FALSE

The characteristics polynomial of a 3x 3

identity matrix is __________, if x is the eigen values of the given 3 x 3 identity matrix. where symbol ^ shows power.

(x-1)^3

(x+1)^3

x^3-1

x^3+1

Two matrices with the same characteristic polynomial need not be similar.

TRUE

FALSE