NOUN TMA Solutions: PHY 314 - Numerical Computations

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LadyB
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NOUN TMA Solutions: PHY 314 - Numerical Computations

Postby LadyB » Fri Jul 27, 2018 1:18 pm

NOUN TMA Solutions: PHY 314 - Numerical Computations

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Q21 What is the value of $$y_{2}$$ in (19) above

-0.021

Q22 With a step length of $$\frac{1}{10}$$, find the value of $$ y_{1}$$ at $$x = \frac{1}{5}$$ given the ordinary differential equation $$\frac{dy}{dx} - y + x = 0$$; $$y(0) = 0$$, using the second order Runge-Kutta method

-0.005

Q23 In any one step numerical scheme, $$K_{1}$$ is expressed as

$$hf(x_{0} , y_{0})$ $

Q24 The formulae for the third order Runge-Kutta method is

$$y_{n+1} = y_{n} + \frac{1}{6}(K_{1} + 4K_{2} + K_{3})$$

Q25 Using the third order Runger Kuttamethod, find the value of $$K_{2}$$ given that $$2y^{1} = 2x - 2y, y(0) = 2$$ with step length 0.1

-0.085

Q26 Which is the correct expression for second order Runger - Kutta method

$$y_{n+1} = y_{n} + \frac{1}{2}(K_{1} + K_{2})$$

Q27 Evaluate the integral of the function $$x(t) = \frac{1}{2}t^{2} + \frac{5}{2}t + 2; 0 \leq t \leq 0.6$$ with six interval, using Simpson's one-third rule

1.686

Q28 The error in the trapezoidal rule in a subinterval $$h$$ is given as

$$E = \frac{(b-a)h^{2}}{12}f^{\prime \prime}(x )$$

Q29 Evaluate the integral of the function in (11) using Simpson's one third rule

25.5

Q30 Integrate the function below using the trapezoidal rule $$f(x) = 6x^{2} + 10x - 2$$; $$1 \leq x \leq 4$$; with step size $$\frac{1}{2}$$

26.62 5

Q31 The expression for first central difference

$$\frac{1}{2}(f_{j+1} - f_{j})$$

Q32 The expression for first backward difference is

$$f_{j-1} - f_{j}$$

Q33 What is the value of $$\Delta^{5} y_{0}$$?

32

Q34 What is the value of $$\Delta^{3} y_{1}$$?

24

Q35 What is the value of $$\Delta y_{3}$$?

54

Q36 Construct a table of forward difference scheme for the following data and use it for questions 5-8 $$\begin{matrix} x: & 0 & 1 & 2 & 3 & 4 & 5\\ 3^x:& 1.0 & 3.0 & 9.0 & 27.0 & 81.0 & .... \end{matrix}$$. What is the missing value in the table?

243

Q37 What is the degree of the polynomial function from which the table was obtained?

2

Q38 What is the value of $$\Delta^{3} y_{3}$$

0

Q39 What is the value of $$\Delta y_{1}$$

-5

Q40 From the table below form a table of forward difference scheme and use it for questions 1-4. $$\begin{matrix} x: & -2 & -1 & 0 & 1 & 2 & 3\\ y: & 14 & 7 & 2 & -1 & -2 & -1 \end{matrix}$$. What is $$y(-1)$$

7






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LadyB
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Re: NOUN TMA Solutions: PHY 314 - Numerical Computations

Postby LadyB » Fri Jul 27, 2018 1:24 pm

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Q41 Newton Raphson method will fail if

$$f^{\prime \prime}(x) = 0$$

Q42 Find the upper bound of the error you are likely to incur in using the bisection method in finding the root of an equation if the two starting points are $$1.4$$ and $$2.5$$ and you need 8 steps to achieve the required toleration

$$4.297 \times 10^{-3}$$

Q43 Perform four iteration of the Newton â􀳦?􀳦􀳦?? Raphson method to obtain the approximate value of the cube root of $$17$$ staring with $$x_{0} = 2$$.

2.75

Q44 Refer to questio 12. $$x_{4}$$

0.2016

Q45 Refer to questio 12. $$x_{3}$$

0.2016

Q46 Refer to questio 12. $$x_{2}$$

0.2016

Q47 Perform four iteration of the Newton - Raphson method to find the root of the equation $$f(x) = x^{3} - 5x + 1$$ staring with $$x_{0} = 0.5$$. What is the value of $$x_{1}$$

0.1765

Q48 Refer to questio 9. After the first bisection, the interval shifts

$$(m,b_{0})$$

Q49 Refer to questio 9. Evaluate $$f(m)$$

0.0532

Q50 Refer to questio 9. What is the bisection point $$(m)$$ of $$x$$ ?

0.5

Q51 Refer to questio 9. Evaluate $$f(1)$$

-1.718

Q52 Giving the function $$f(x) = \cos x - x e^{x}$$ with $$ x \in (0,1)$$. Evaluate $$f(0)$$

1

Q53 If we know that a root of $$f(x) = 0 $$ lies in the interval $$(a_{0},b_{0})$$, we bisect the interval at the point

$$\frac{(a_{0} + b_{0})}{2}$$

Q54 Which of the following iterative scheme is based on the repeated application of the intermediate value theorem

Bisection m ethod

Q55 In a numerical experiment, the data gathered shows that the exact solution (x) and numerical solution ($$\bar{x}$$) are 20 and 18 respectively. What is the the relative error

0.28

Q56 The following are demerits of the method of bisection of solving algebraic and transcendental equations EXCERPT

We can trach the er rors

Q57 An equation involving expressions such as square roots, trigonometric and exponetial functions is said to be

transcend ental

Q58 If we use the Jacobi method to solve the equation $$\begin{matrix} 5x & +2y & +z & = & -12\\ -x & +4y & +2z & = & 20\\ 2z & -3y & +10z & = & 3 \end{matrix}$$, after the first improvement we have

$$(-4.556, 2.745, 2.467)$$

Q59 Using Gaus-Jordan method, find inverse of the matrix $$\begin{bmatrix} 1 & 1 & 3\\ 1 & 3 & -3\\ -2 & -4 & -4 \end{bmatrix}$$

$$\begin{bmatrix} 3 & 1 & 3/2\\ -5/4 & -1/4 & -3/4\\ -1/4 & -1/4 & -1/4 \end{bmatrix}$$

Q60 The latent heat of vaporiztion of steam r, is given is given in the following table at different temperatures t: $$\begin{matrix} t: & 40 & 50 & 60 & 70 & 80 & 90 & 100 & 110\\ r: & 1069.1 & 1063.6 & 1058.2 & 1052.7 & 1049.3 & 1041.8 &1036.3 & 1030.8 \end{matrix}$$. For this range of temperature, a relation of the form $$r=a+bt$$, where a and b are constants, is known to fit the data. Find the values of a and b by the method of group averages.

a = 1090.26, b = -0.534
LadyB
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Re: NOUN TMA Solutions: PHY 314 - Numerical Computations

Postby LadyB » Fri Jul 27, 2018 1:25 pm

Q1 If $$a=4b^{2}c^{3}/d^{4}$$ and errors in a,b,c be 0.001, compute the relative maximum error in a when b=c=d=1

0.009

Q2 Use Romberg's method to compute $$ \int_{0}^{1}\frac{dx}{1+x^{2}} $$ correct to 4 places

0.7855

Q3 Use Simpson's 3/8 rule to approximate the following definite integral, $$ \int_{0}^{1}\frac{dx} {\sqrt{(x^{2}+1)(3x^{2}+4)}} $$ with ð􀳦??􀳦?􀳦=6:

0.402181

Q4 Integrate $$ f(x, y)=4e^{0.8x}-0.5y $$ by fourth-order Runge-Kutta method using h = 0.5 with y(0) = 2 from x = 0 to 0.5

3.751699

Q5 Use the fourth-order Runge-Kutta method to integrate $$ f(x, y)=â􀳦?􀳦??2x^{3} + 12x^{2}â􀳦?􀳦?? 20x + 8.5 $$using a step size of h = 0.5 and an initial condition of y = 1 at x = 0

3.2187 5

Q6 Using Euler's modified method, obtain a solution of the equation $$ \frac{dv}{du}= u+| \sqrt{v}|$$ , with initial condition $$v=1$$ at $$u=0$$ for the range $$ 0\leq u\leq 0.6$$ in steps of 0.2

1.8861

Q7 Given that $$ \frac{dv}{du}= 2+\sqrt{uv}$$ and $$v=1$$ when $$u=1. Find approximate value of v at $$u=2$$ in steps of 0.2, using Eulerâ􀳦?􀳦􀳦?􀳦s modified method

5.051

Q8 Use Runge's method to approximate v when u=1.1, given that v=1.2 when u=1 and $$ \frac{dv}{du}= 3u+v^{2}$$

1.7278

Q9 Use Taylor's method to get the approximate value of y at x=0.2 for the differential equation $$ \frac{dy}{dx}=2y+3e^{x}$$ , $$ y(0)=0 $$

0.811

Q10 Employ Picard's method to obtain, correct to four places of decimal, solution of the differential equation $$ \frac{dy}{dx}=x^{2}+y^{2}$$ for $$ x=0.4$$, given that $$ y=0$$ when $$ x=0$$

0.0214

Q11 If $$a=4b^{2}c^{3}/d^{4}$$ and errors in a,b,c be 0.001, compute the relative maximum error in a when b=c=d=1

0.009

Q12 Use Romberg's method to compute $$ \int_{0}^{1}\frac{dx}{1+x^{2}} $$ correct to 4 places

0.7855

Q13 Use Simpson's 3/8 rule to approximate the following definite integral, $$ \int_{0}^{1}\frac{dx} {\sqrt{(x^{2}+1)(3x^{2}+4)}} $$ with ð􀳦??􀳦?􀳦=6:

0.402181

Q14 Evaluate the integral $$ \int_{0}^{\frac{\pi }{2}}xsinxdx $$ (where is in radians) with a step-size of $$ \Delta x=\frac{\pi }{16} $$, using Simpsonâ􀳦?􀳦􀳦?􀳦s one-third rule

0.99997

Q15 Approximate the area under the curve $$ y=\sqrt{x} $$ on the interval $$ 2\leq x\leq 4$$ with n=5 subintervals using the Trapezoidal rule.

3.447715

Q16 Calculate the error in using Trapezoid rule to solve for $$ \int_{0}^{2}e^{x^{2}}dx using n=4 subintervals

4.19193
Q17 Find the true error E in solving $$ x= \int_{8}^{30}\left (2000ln\left [\frac{140000}{140000-2100t} \right ]-9.8t\right ) dt $$ using Simpsonâ􀳦?􀳦􀳦?􀳦s $$ \frac{1}{3}rd $$ rule

-4.38

Q18 Use the Simpsonâ􀳦?􀳦􀳦?􀳦s rule to solve for $$ \int_{0}^{2}e^{x^{2}}dx $$

17.353626

Q19 Use the trapezoid rule to solve for $$ \int_{0}^{2}e^{x^{2}}dx $$ $$, using n=4 subintervals

20.644559

Q20 One entry in the following table is incorrect and y is a cubic polynomial in x. Construct a foeard difference table and use it to locate and correct the error. $$\begin{matrix} x: & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ y: & 25 & 21 & 18 & 18 & 27 & 45 & 76 & 123 \end{matrix}$$.

The entry corresponding to x = 3 is in error and the true value of y = 19


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LadyB
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Re: NOUN TMA Solutions: PHY 314 - Numerical Computations

Postby LadyB » Fri Jul 27, 2018 1:26 pm

Q41 Newton Raphson method will fail if

$$f^{\prime \prime}(x) = 0$$

Q42 Find the upper bound of the error you are likely to incur in using the bisection method in finding the root of an equation if the two starting points are $$1.4$$ and $$2.5$$ and you need 8 steps to achieve the required toleration

$$4.297 \times 10^{-3}$$

Q43 Perform four iteration of the Newton â􀳦?􀳦􀳦?? Raphson method to obtain the approximate value of the cube root of $$17$$ staring with $$x_{0} = 2$$.

2.75

Q44 Refer to questio 12. $$x_{4}$$

0.2016

Q45 Refer to questio 12. $$x_{3}$$

0.2016

Q46 Refer to questio 12. $$x_{2}$$

0.2016

Q47 Perform four iteration of the Newton - Raphson method to find the root of the equation $$f(x) = x^{3} - 5x + 1$$ staring with $$x_{0} = 0.5$$. What is the value of $$x_{1}$$

0.1765

Q48 Refer to questio 9. After the first bisection, the interval shifts

$$(m,b_{0})$$

Q49 Refer to questio 9. Evaluate $$f(m)$$

0.0532

Q50 Refer to questio 9. What is the bisection point $$(m)$$ of $$x$$ ?

0.5

Q51 Refer to questio 9. Evaluate $$f(1)$$

-1.718

Q52 Giving the function $$f(x) = \cos x - x e^{x}$$ with $$ x \in (0,1)$$. Evaluate $$f(0)$$

1

Q53 If we know that a root of $$f(x) = 0 $$ lies in the interval $$(a_{0},b_{0})$$, we bisect the interval at the point

$$\frac{(a_{0} + b_{0})}{2}$$

Q54 Which of the following iterative scheme is based on the repeated application of the intermediate value theorem

Bisection m ethod

Q55 In a numerical experiment, the data gathered shows that the exact solution (x) and numerical solution ($$\bar{x}$$) are 20 and 18 respectively. What is the the relative error

0.28

Q56 The following are demerits of the method of bisection of solving algebraic and transcendental equations EXCERPT

We can trach the er rors

Q57 An equation involving expressions such as square roots, trigonometric and exponetial functions is said to be

transcend ental

Q58 If we use the Jacobi method to solve the equation $$\begin{matrix} 5x & +2y & +z & = & -12\\ -x & +4y & +2z & = & 20\\ 2z & -3y & +10z & = & 3 \end{matrix}$$, after the first improvement we have

$$(-4.556, 2.745, 2.467)$$

Q59 Using Gaus-Jordan method, find inverse of the matrix $$\begin{bmatrix} 1 & 1 & 3\\ 1 & 3 & -3\\ -2 & -4 & -4 \end{bmatrix}$$

$$\begin{bmatrix} 3 & 1 & 3/2\\ -5/4 & -1/4 & -3/4\\ -1/4 & -1/4 & -1/4 \end{bmatrix}$$

Q60 The latent heat of vaporiztion of steam r, is given is given in the following table at different temperatures t: $$\begin{matrix} t: & 40 & 50 & 60 & 70 & 80 & 90 & 100 & 110\\ r: & 1069.1 & 1063.6 & 1058.2 & 1052.7 & 1049.3 & 1041.8 &1036.3 & 1030.8 \end{matrix}$$. For this range of temperature, a relation of the form $$r=a+bt$$, where a and b are constants, is known to fit the data. Find the values of a and b by the method of group averages.

a = 1090.26, b = -0.534





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LadyB
Posts: 283
Joined: Sun Nov 05, 2017 7:32 pm
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Re: NOUN TMA Solutions: PHY 314 - Numerical Computations

Postby LadyB » Fri Jul 27, 2018 1:26 pm

Q41 Newton Raphson method will fail if

$$f^{\prime \prime}(x) = 0$$

Q42 Find the upper bound of the error you are likely to incur in using the bisection method in finding the root of an equation if the two starting points are $$1.4$$ and $$2.5$$ and you need 8 steps to achieve the required toleration

$$4.297 \times 10^{-3}$$

Q43 Perform four iteration of the Newton â􀳦?􀳦􀳦?? Raphson method to obtain the approximate value of the cube root of $$17$$ staring with $$x_{0} = 2$$.

2.75

Q44 Refer to questio 12. $$x_{4}$$

0.2016

Q45 Refer to questio 12. $$x_{3}$$

0.2016

Q46 Refer to questio 12. $$x_{2}$$

0.2016

Q47 Perform four iteration of the Newton - Raphson method to find the root of the equation $$f(x) = x^{3} - 5x + 1$$ staring with $$x_{0} = 0.5$$. What is the value of $$x_{1}$$

0.1765

Q48 Refer to questio 9. After the first bisection, the interval shifts

$$(m,b_{0})$$

Q49 Refer to questio 9. Evaluate $$f(m)$$

0.0532

Q50 Refer to questio 9. What is the bisection point $$(m)$$ of $$x$$ ?

0.5

Q51 Refer to questio 9. Evaluate $$f(1)$$

-1.718

Q52 Giving the function $$f(x) = \cos x - x e^{x}$$ with $$ x \in (0,1)$$. Evaluate $$f(0)$$

1

Q53 If we know that a root of $$f(x) = 0 $$ lies in the interval $$(a_{0},b_{0})$$, we bisect the interval at the point

$$\frac{(a_{0} + b_{0})}{2}$$

Q54 Which of the following iterative scheme is based on the repeated application of the intermediate value theorem

Bisection m ethod

Q55 In a numerical experiment, the data gathered shows that the exact solution (x) and numerical solution ($$\bar{x}$$) are 20 and 18 respectively. What is the the relative error

0.28

Q56 The following are demerits of the method of bisection of solving algebraic and transcendental equations EXCERPT

We can trach the er rors

Q57 An equation involving expressions such as square roots, trigonometric and exponetial functions is said to be

transcend ental

Q58 If we use the Jacobi method to solve the equation $$\begin{matrix} 5x & +2y & +z & = & -12\\ -x & +4y & +2z & = & 20\\ 2z & -3y & +10z & = & 3 \end{matrix}$$, after the first improvement we have

$$(-4.556, 2.745, 2.467)$$

Q59 Using Gaus-Jordan method, find inverse of the matrix $$\begin{bmatrix} 1 & 1 & 3\\ 1 & 3 & -3\\ -2 & -4 & -4 \end{bmatrix}$$

$$\begin{bmatrix} 3 & 1 & 3/2\\ -5/4 & -1/4 & -3/4\\ -1/4 & -1/4 & -1/4 \end{bmatrix}$$

Q60 The latent heat of vaporiztion of steam r, is given is given in the following table at different temperatures t: $$\begin{matrix} t: & 40 & 50 & 60 & 70 & 80 & 90 & 100 & 110\\ r: & 1069.1 & 1063.6 & 1058.2 & 1052.7 & 1049.3 & 1041.8 &1036.3 & 1030.8 \end{matrix}$$. For this range of temperature, a relation of the form $$r=a+bt$$, where a and b are constants, is known to fit the data. Find the values of a and b by the method of group averages.

a = 1090.26, b = -0.534





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LadyB
Posts: 283
Joined: Sun Nov 05, 2017 7:32 pm
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Re: NOUN TMA Solutions: PHY 314 - Numerical Computations

Postby LadyB » Fri Jul 27, 2018 1:29 pm

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Q61 An experiment gave the following values: $$\begin{matrix} v(m/min): & 350 & 400 & 500 & 600\\ t(min): & 61 & 26 & 7 & 2.6\end{matrix}$$. It is known that $$v$$ and $$t$$ are connected by the relation $$v=at^{b}$$. Find the best possible values of a and b.

a = 699.8, b = -0.1679

Q62 An experiment gave the following values: $$\begin{matrix} v(m/min): & 350 & 400 & 500 & 600\\ t(min): & 61 & 26 & 7 & 2.6\end{matrix}$$. It is known that $$v$$ and $$t$$ are connected by the relation $$v=at^{b}$$. Linearization of this relation took the form $$Y=A+bY$$, (where logarithms were taken to base 10). What are the missing values in the following table: $$\begin{matrix} v(m/min) & t(mun) & X & Y\\ 350& 61 & 1.7853 & 2.5441\\ 400 & 26 & ...... & 2.6021\\ 500 & 7 & 0.8451 & ..... \\ 600 &2.6 & 0.4150 & 2.7782 \end{matrix}$$ ?

$$1.4150$$, $$2.6990$$

Q63 P is the pull required to lift a load W using a pulley system. The linear law connecting P and W is of the form $$P=mW+c$$. Using the following data: $$\begin{matrix} P(N): & 12 & 15 & 21 & 25\\ W(N): & 50 & 70 & 100 & 120 \end{matrix}$$, where P and W are in units of newtons (N), calculate P when W = 150 N.

30.4635 kg

Q64 If P is the pull required to lift a load W using a pulley system, find a linear law of the form $$P=mW+c$$ connecting P and W, using the following data: $$\begin{matrix} P(N): & 12 & 15 & 21 & 25\\ W(N): & 50 & 70 & 100 & 120 \end{matrix}$$, where P and W are in units of newtons (N).

$$P=2.2759+0.1879W$$

Q65 The value of x and y in an experiment are expressed hypothetically by the law $$y=ae^{bx}$$. Which of the following correctly gives the linearised plot of the data obtained?

$$x$$ on the abscissa and $$log_{10}y$$ on the ordinate

Q66 R is the resistance to the motion of a train at speed V. Find, by plotting appropriate values in a graph, a law of the type $$R=a+bV^{2}$$ (i.e. determine the approximate values of a and b) connecting R and V using the following data: $$\begin{matrix} V(km/h): & 10 &20 &30 &40 & 50\\ R(kg/ton): & 8 & 10 & 15 &21 & 30 \end{matrix}$$

a = 7.35, b = 0.0085

Q67 Round off the number 37.46235 to four significant figures and clculate the percentage error

$$6.27\times{10^{-3}}$$

Q68 Whch of the following is an error arising from the replacement of an infinite or iterative process by a finite process?

truncation erro r

Q69 Which of the following is NOT one of the errors in numerical computation?

real error

Q70 What is 6.4356 rounded off to 3 significant figures?

$$6.44$$

Q71 Which of the following is an exact number

$$\frac{1}{4}$$

Q72 Round the number 0.0045829 to 3 significant figure

0.00458

Q73 Round the number 329.5 to 3 significant figure

329

Q74 In a numerical experiment, the data gathered shows that the exact solution (x) and numerical solution ($$\bar{x}$$) are 20 and 18 respectively. What is the the percentage error

10

Q75 In a numerical experiment, the data gathered shows that the exact solution (x) and numerical solution ($$\bar{x}$$) are 20 and 18 respectively. What is the the relative error

0.1

Q76 In a numerical experiment, the data gathered shows that the exact solution (x) and numerical solution ($$\bar{x}$$) are 20 and 18 respectively. What is the absolute error in the information above

2

Q77 An error in numerical computation that is present in the statement of the problem before solution is known as

inhere nt error

Q78 The error that occur as a result of cutting off of a significant part of an unending decimal number is called

Tru ncation Error

Q79 Round the number 3.142857143 to 5 significant figure

3.142 8

Q80 The difference between the numerical solution and exact solution is called

Error


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