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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,

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Instructor's Reply:

Dear student, indefinitely repeated pattern of numbers has been enclosed in brackets with base as subscript.
Please read the other replies I gave in this MDB too

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

 and the answer is (1.1953125)10......

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 

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Instructor's Reply:

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

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Instructor's Reply:

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 AmN+ AM-1NM-1 +...+A1N1+A0+A-1N-1+...A-MN-M  THAT IS CONFUSING and what is that a and n and m thanx.

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Instructor's Reply:

Dear student 
amN+ 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

 

Subject: Hexadecimal over Binary,decimal,Octal And Octal over Hexadecimal ,decimal and Binary

Respected Sir,

What is the benefit of using Hexadecimal over Binary,decimal and Octal?
And what the benefit of using Octal over Hexadecimal ,decimal and Binary?

Regards,
Farhan Ahmed

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Instructor's Reply:

Dear student, The binary numeral system, or base-2 number system, represents numeric values using two symbols: 0 and 1. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices such as mobile phones. 
Hexadecimal numbers are used on web pages to specify colors. In URLs, character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space (blank) character (code value 20 in hex, 32 in decimal). In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation &#xcode;, where code is the 1- to 6-digit hex number assigned to the character in the Unicode standard. Thus ’ represents the curled right single quote (Unicode value 2019 in hex, 8217 in decimal). 
these are few applications of using different number systems, similalrly you can explore more about by your own and don't forget to share with me that whatelse you learned by your own. You can tell me through email.

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.

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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

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Instructor's Reply:

Dear student, yes you are right.

Lesson 3

Kindly sir, explain truncation error with proper numerical method

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Instructor's Reply:

Dear student, this question has been already replied in this MDB. Truncate means chop out or cut off. Please read the reply I already gave and try to grasp this concept that if a function has been expressed in infinte series form. This sum can be found by certain formula developed particulary for those series. As may be you don't know how to find sum of certain infinite series, but just here concentrate on the main concept " truncation error how results." Let this infinite series sum has been found by a certain formula and you get its answer. Now let you cut off this series infinite terms and try to approximate it by its 1st 2 or 3 or 4 etc terms. Ofcourse this finite series sum will be different from the exact sum. this gives rise truncation error.

Dear Sir, 
Q.No.1   in bisection method when we find value of x3 why we use two formulas first x0+x2/2 & second x1+x2/2?????????
Q.No.2   in example one of bisection method solve x3 - 9x+1=0 for the root b/w x=2 and x=4  
we use for x3=x0+x2/2 and x4=x3+x2/2 and x5=x4+x2 and x6=x5+x2
but
in next example carry out five iteration for the function f(x)=2xcos(2x)-(x+1)2  
we use formula for x3= x2+x1/2 and x4 = x3+x1/2 and x5= x3+x4/2

Dear Sir what is difference in these two example? plz clear why we use x4 = x3+x2/2 in first example and x4 = x3+x1 in next example? 

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Instructor's Reply:

Dear student, for roots location in given interval we are making use of intermediate value property, and as we using bisection method, in this method root is taken to be mid point of those two values of ‘x’ for which graph of function crosses x-axis. Let for x0 and x1, we having intermediate value property i.e. f(x0)f(x1)<0, then the next root x2= (x0+x1)/2 . similarly this process continues till the required number of iterations. 

One thing more to notice here that from intermediate value property you check the condition that product of function values is less than zero. The next root is then sum of those two ‘x’ values divided by 2. As you mentioned x4=x3+x2/2. it means that for x3andx2 you are getting intermediate value property satisfied 
f(x3)f(x2)<0 thus you locating root x4=x3+x2/2. Similarly others too. 

In example of function f(x)=2xcos(2x)-(x+1)2 or solving any other function keep one thing in your mind that for locating root whenever you use intermediate value property. For those two values of ‘x’ product of their function values is less than zero. The next root is sum of those two values of x divided by 2. that is why in 1st example for x2 and x3 you had f(x2)f(x3)<0 so x4 = (x3+x2)/2 but in next example for x1 ans x3 you were getting f(x1)f(x3)<0 so next root x4=(x1+x3)/2

plz explain about Local truncation error with proper numerical example i cannot understand about that concept

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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.

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,

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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

I am much confused about bisection method ............ How i came to know in which values root lies ................ Plz explain in simple way with example step by step ...........

 

Thanks 

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Instructor's Reply:

Dear student, use intermediate value property to locate two values a and b such that f(a) and f(b) have opposite signs then we use the fact that the root lie between these points ‘a’ and ‘b’. Tell me did you understand it ? One thing important that when in the statement of question you have been given the interval, means two values i.e. initial and end values (a,b) then you do not use intermediate value property in this 1st step as you see in example  on page 4 by bisection method. So initial guess has been known to lie in this interval, but for getting next guess you now make use of intermediate value property. This is what actually you are to do to know that where next value of root is lying. You read again Intermediate value property and then example on page 4. If you still feel any difficulty then ask me.

Subject: Regula Falsi Method

Respected Sir 

http://vustudents.ning.com/

 

Whenever intervals are given then we take as x0 and x1  and start from n =1 and whenever interval is not given them we take x1 and x2 and start fron n = 2 ............

Am i right ?????

 

Thanks 

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Instructor's Reply:

Dear student, there is no difference that either you name two values as  x0 and x1 OR x1 and x2, either interval is given or not. But if it is mentioned in the statement the values already let suppose x0,x1 then ofcourse you will name next value x2 and so on

Lesson no 6

Subject: newton raphson method

dear sir how we can find f'(x)???? and f"(x)????
in example x3-x-1=0  
f'(x)=3x2-1   and  f"(x) = 6x   sir yh kaha sy aa gy??????

and in example  f(x)= x-0.8-0.2sinx
f'(x)= 1-0.2cos x  ????? kesy aya???? plz explain.
 

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Instructor's Reply:

Dear student, f''(x) means 1st derivative of f(x) and f"(x) means 2nd derivative of f(x).

Subject: under root

how can we use under root formula in MS Excel plz write method 

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Instructor's Reply:

Dear student, use function SQRT in exel, for example you want to compute square root of 16. For this enter formula =SQRT(16) and press enter to get answer.

f(x4)=ln1.3037-cos1.3037=1.24816

plz calculate all step 

i do' not understand this calculation 

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Instructor's Reply:

Dear student, compute values ln1.3037 and cos1.3037 separately and then find their difference

Lecture n0 7

lecture 6 example no ! x3-x2-1

.ess question me f(2) and f''(2)

have the same signs so x0=2 kesy ata hay  f(2)=5 and f"(2)=12 hy  plz define

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Instructor's Reply:

Dear student, if f(a) and f''(a)  are of same sign then 'a' can be chosen as initial approximation. In this example you have f(2)=5 and f''(2) =12 as both values '+5' and '+12' are of same sign so x0=2 is selected as initial approximation.

What is difference in secant Method and Muller's method explain it with easy example

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Instructor's Reply:

Dear student, The secant method is modified form of Newton-Raphson method .

 

In Muller’s method, f (x) = 0 is approximated by a second degree polynomial; that is by a

quadratic equation that fits through three points in the vicinity of a root. The roots of this quadratic equation are then approximated to the roots of the equation f (x)= 0.This method isiterative in nature and does not require the evaluation of derivatives as in Newton- Raphson method.

example   f(x)= 4xcos2x-(x-2)sqare
in this example x3=43.2343 while acutal its value becomes 34.3679
plz comments on it...


thanks

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Instructor's Reply:

Dear student, observe that at end points '0' and '8' , values of  f(0) and f(8) both have same negative sign. No change of sign so surely root can't lie in this interval and thats why as you do more iterations the value get more diverge. Observe x1=33.6801 and x2  = 38.8011 and x3=43.2343. Values not getting closer but diverging. So actually in this interval root has not been located.



sir, with respect to calculations x3=34.7679, as both f(x) and f``(x) are negative... 

i think its value of x3 is wrong on handout.....??

example   f(x)= 4xcos2x-(x-2)sqare
in this example x3=43.2343 while acutal its value becomes 34.3679
plz comments on it...


thanks

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Instructor's Reply:

Dear student, observe that at end points '0' and '8' , values of  f(0) and f(8) both have same negative sign. No change of sign so surely root can't lie in this interval and thats why as you do more iterations the value get more diverge. Observe x1=33.6801 and x2  = 38.8011 and x3=43.2343. Values not getting closer but diverging. So actually in this interval root has not been located.



sir, with respect to calculations x3=34.7679, as both f(x) and f``(x) are negative... 

i think its value of x3 is wrong on handout.....??

what is partial pivoting please explain it with easy way

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Instructor's Reply:

Dear student, You have to find solution of system of equations  by Gaussian elimination method. Let suppose you have system of three equations.You are to make selection for diagonal elements in a special way , it is  1st you have diagonal element at position R1C1 means in 1st row and 1st column. For this check entries in C1. For this you are to look for the entry that has largest absolute value in C1 and make it as  the pivot by interchanging two rows. (if there are more than one one such entries having the same largest absolute value , then select the one which is uppermost)). Next  pivot position will be R2C2. For this you will now check entries in C2  that has largest absolute value and do the same strategy as I told you above. Similarly for position R3C3.This strategy is called partial pivoting.

in this example x3=43.2343 while acutal its value becomes 34.3679
plz comments on it...

more over
in example range given is 0 to 8, and our answer is larger ,
is it true....

thanks

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Instructor's Reply:

Dear student, observe that at end points '0' and '8' , values of  f(0) and f(8) both have same negative sign. No change of sign so surely root can't lie in this interval and thats why as you do more iterations the value get more diverge. Observe x1=33.6801 and x2  = 38.8011 and x3=43.
2343. Values not getting closer but diverging. So actually in this interval root has not been located.

example page No. 37

2coshx sinx=1
here value of cosh0.4 is 1.081
sir, how can we calculate it.. plz tell in simple way...
thanks

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Instructor's Reply:

Dear student, check on your calculator, there will be a button typed on it 'hyp'. To compute the said value press 'hyp' then button named 'cos' and then value 0.4 and then press button'=' . You will get this answer

in example x3-3x+1 second step take x2=0 , xo=0.33333 , x1= 0.5
h1= 0.16667 h2= 0.33333
fo= x03-3x0+1= (0.33333)3 -3(0.33333)+1= 0.037046

c=fo= 0.037046    but in handsout  c=0.33333????????? why we use c=0.333333 instead of 0.037046??????? plz explain.

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Instructor's Reply:

Dear student, the root obtained from 1st iteration is x=0.33333, then for proceeding for 2nd iteration this computed root in 1st iteration will be treated as x0 now and  c=fo=f(x0)=f(0.33333)= 0.037046
Do tell me that you understand it now? or need further discussion.

Salam Respected Sir,

A.A

In Jacobi method i canot understand the Leaner system equation please guide me 

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Instructor's Reply:

Dear student, A system of linear equations (or linear system) is a collection of linear equationsas involving the same set of variables. For example,

is a system of three equations in the three variables x, y, z.

Sir i  hv not studied math at inter level and bachelor level so i m confused about direct solution given in handouts and lectures. dnt know how to find out derivatives and function so pls guide me wat i do for better understanding. last time helping material u upload pls also xplane each step how to do that, 

bundle of thanx for ur coperation 

best regards

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Instructor's Reply:

Dear student, there is still nothing to worry about. You are to put a little more effort to learn these terminologies. As far it seems that basic concepts about functions and differentiation are making you confuse. For this I suggest you very direct and short speedy solution that please listen and learn Lecture # 6, 7, 9 of MTH 101 completely and then come to lectures 16,17 of MTH101. In lecture 16, 17 you just need to learn the formulae given as theorem statements. You are just to learn the formulae NOT their proof. It will make you able to grasp your problems yet you facing. After listening and learning you can discuss it with me in a better way through subject email id. I'd really like to know that you got it well so start from today as I told you and discuss it with me.

Subject: Question

A.A

Honorable Sir

What is finite difference and mue U plz explain it 

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Instructor's Reply:

A finite difference is a mathematical expression of the form f(x + b) − f(x + a).

 

Only three forms are commonly considered: forward, backward, and central differences.

 

Central difference  is  = 1/2[f(x+h/2)+f(x-h/2)] 

Subject: mc120204018

what is the mean to use shift E, operator?

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Instructor's Reply:

Dear student, A SHIFT OPERATOR, denoted by E is the operator which shifts the value at the next point with step h  i.e., Ef(x)=f(x+h).

what method of finite difference operators would me more effective. what reason behind this?

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Instructor's Reply:

Dear student, 1st of all you have been told different types of operators and thier inter relationship . These operators will actually be helpful when in coming lectuires for equally spaced data, the interpolation  and differentiation formulae will make use of backward, forward or central differences . For example if we desire the derivative at a point near the beginning of a tabulated data we use Newton's formula involving forward differences and so on . These all you are going to learn in coming lectures so after that you will be able to get it better.

 mc120204018

what is important one type in finite difference operator?

regards.

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Instructor's Reply:

Dear student, there are different types of operators backward difference operator, forward diference operator, shift operator, central difference operator etc. you are being introduced about these. They all are important at their own. You can be asked to use particular one for a problem or problem data itself decides that which operator should be used . So it is not the question that which one is most important.

                                             

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

1.22140

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

0.400

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

0.400

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

http://vustudents.ning.com/

 

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

 

 

 

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