MTH603 Short Questions and Answers

Absolute

Error :

The absolute error is used to denote the actual value of a quantity less it’s rounded value if x and x* are respectively the rounded and actual values of a quantity, then absolute error is defined by AE=|x-x*|

Algebraic equation

The equation f(x)=0 is known as algebraic equation if it is purely a polynomial in x. Like f(x)=3 ,f(x)=x3-3x-4 are some examples of algebraic equations and the if variable is changed to y x or any one then under the same conditions it will be algebraic.

Bisection method :

It is a bracketing method which is used to locate the root of an equation and it make use of intermediate value property to locate the interval in which the root lies , formula which is used is (a+b)/2 where a and b are points such that f(a)f(b)<0.

Bracketing Method :

The iterative methods which require two initial approximation for it’s first iteration are known as bracketing methods. Bisection method is the example of the bracketing method as it requires a interval for the approximation of the root.

rout’s Method :

This method is used to solve the system of the equations by decomposing the system into two matrices L and U where L is a lower triangular matrix and U is an upper triangular matrix. In U all the elements in the main diagonal are 1.

Descartes rule of

signs :

The no of positive roots of an algebraic equation f(x)=0 with real coefficients cannot exceed the no of changes in the in sign of the coefficients in the polynomial f(x)=0, similarly the no of negative roots of f(x)=0 can not exceed the no of changes in the sign of the coefficients of f(-x)=0

Direct methods :

These are the methods which do not need the knowledge of the initial approximation are known as direct methods.

Gauss Elimination

Method :

It is a direct method which is used to solve a system of equations

Gauss seidel iterative

method :

It is an iterative method in which an initial approximation is given. First of all system should be checked either it is diagonally dominant or not if it is not then it is made diagonally dominant. Secondly the first variable is calculated in terms of other variables from first equation and Second variable from second equation and so on .the previous value of the variable is replaced by new value instantly as it is obtained. This is the difference in the gauss seidel and jacobie iterative method.

Graeffee’s root

squaring method :

This method is used to find all the roots of the polynomial equation.

Intermediate value

property :

If for an equation f(x)=0 for two values a and b we have such that f(a)f(b)<0 then there must exist a root between a and b in the interval [a,b].

Inverse of a matrix :

The matrix B is said to be inverse of a matrix A if the product of and B is I identity matrix.

Iterative methods :

Iterative methods are those type of methods which always require an initial

Accuracy

The Extent of the closeness between the actual value and estimated value is known as accuracy. Suppose you have taken readings like 2.1234,2.1236and 2.1238 and 2.45,2.52,2.63 Now if the actual root is 2.65 then the second values are more accurate and not precise but first set of values is precise but not accurate as it differs from the actual value. approximation to start an iteration

Jacobie’s iterative

method: :

It is an iterative method in which an initial approximation is given. First of all system should be checked either it is diagonally dominant or not if it is not then it is made diagonally dominant. Secondly the first variable is calculated in terms of other variables from first equation and Second variable from second equation and so on after first iteration in the second iteration all the variables are replaced by the previous one.

Muller’s Method :

In Muller’s method f(x)=0 is approximated by a second degree polynomial, that is by quadratic equation that fits through three points in the vicinity of a root .Then roots of this quadratic equation are then approximated to the roots of equation f(x)=0

Newton Raphson

Method. :

It is an open method which is used to locate a root of the equation, it needs only one initial approximation for it’s first iteration. xn+1=xn-(fxn/f’xn)

Non singular matrix :

A matrix is said to be non singular if the determinant of the matrix is non zero. |A| is not equal to zero and the inverse of non singular exists.

Open methods :

The methods which require only one initial approximation to start the first iteration, for example the Newton’s Raphsom method is known as the open method as it requires only one initial approximation.

Pivoting :

If any of the pivot elements in gauss elimination become zero then this method fails so to Avoid such type of situation equation are rearranged to get rid of zero pivot element, this procedure is known as pivoting.

Precision :

The extent of closeness of different measurement taken to estimate a certain value. Suppose you have done different iterations to measure the root of an equation and take different values as 2.1234, 2.1236, and 2.1238 these all the values are very close to each other so these are very precise. Regula –Falsi method : It is also an iterative method and is a bracketing method and use intermediate value property to get it’s initial guess. The formula used for this is xn+1=xn-(xn-xn-1/f(xn)-f(xn-1))

Relative Error :

It is the ratio of the absolute error to the actual value of the quantity. Thus RE=AE=AE\|x*|

Relaxation method :

It is also an iterative method and in this method you solve the system of equation by making the greatest residual to zero.

Root :

If you have an equation f(x)=0 then the no a is said to be the root of the equation if f(a)=0 Suppose you have an equation f(x)=x2-4 the 2 is a root of the equation as f(2)=4-4=0

Secant Method :

The secant method is also an open method and it takes two initial values for it’s first approximation, the formula used for this is known as xn+1={xn-1f(xn)-xnf(xn-1)}/{f(xn)-f(xn-1)}

Significant digits :

A significant digit in an approximate no is a digit, which gives reliable information about the size of the number. In other words a significant digit is used to express accuracy, that Is how many digits in the no have meaning.

Singular matrix :

A matrix s said to be a singular matrix if the determinant of the matrix ix zero |A|=0,the inverse of the singular matrix do not exist.

Transcendental

equation :

An equation f(x) =0 is said to be transcendental equation if it contains trigonometric, logarithmic and exponential functions

Truncation Error :

It is defined as the replacement of one series by another series with fewer terms.

SHORT QUESTION ANSWERS

Question:

What is an Open method?

Answer: In the Newton-Raphson method, the root is not bracketed. Only one initial guess of the to get the iterative process started to find the root of an equation. Hence, the method falls in the category of open methods.

Question:

Explain Muller’s method briefly.

Answer: In Muller’s method, f (x) = 0 is approximated by a second degree polynomial; that is b equation that fits through three points in the vicinity of a root. The roots of this quadra are then approximated to the roots of the equation f (x) 0.This method is iterative in n not require the evaluation of derivatives as in Newton-Raphson method. This method used to determine both real and complex roots of f (x) = 0.

Question:

Explain the difference between the linear and non-linear equations.

Answer: Linear Equation An algebraic equation is said to be linear in which each term is either or the product of a constant and the first power of a single variable. One or more variab involved in the linear equations. e.g. x+3y+z=0 2x-y+4z=7 etc. Non-Linear Equation A is said to Non-Linear equation if it is not linear. Equations involving the power of the or higher, transcendental, logarithmic and trigonometric equations etc lie in the catego Non-Linear equations. e.g. x2+5x+3=0 sinx+3y+9=0 xlogx-7x+4y=2 etc.

Question:

Explain which value is to be choosed as X0 in N-R method.

Answer: If, for a given function, f(a)*f(b)<0, then any value between a and b inclusive can be c

Question:

Define iterative method of solving linear equations with two examples.

Answer: Under iterative methods, the initial approximate solution is assumed to be known and towards the exact solution in an iterative way. We consider Jacobi, Gauss-Seidel and methods under iterative methods.

Question:

Define Pivoting.

Answer: The Gaussian elimination method fails if any one of the pivot elements becomes zero. situation, we rewrite the equations in a different order to avoid zero pivots. Changing equations is called pivoting.

Question:

Write the two steps of solving the linear equations using Gaussian Elimination method.

Answer: In this method, the solution to the system of equations is obtained in two stages. i) the of equations is reduced to an equivalent upper triangular form using elementary transfo the upper triangular system is solved using back substitution procedure

Question:

Describe Gauss-Jordan elimination method briefly

Answer: This method is a variation of Gaussian elimination method. In this method, the elemen below the diagonal are simultaneously made zero. That is a given system is reduced to diagonal form using elementary transformations. Then the solution of the resulting dia

Question:

What is Bracketing method?

Answer: Methods such as bisection method and the false position method of finding roots of a nonlinear equation f(x) = 0 require bracketing of the root by two guesses. Such methods are called bracketing methods. These methods are always convergent since they are based on reducing the interval between the two guesses to zero in on the root.