Numerical Methods (CE)

Consider the differential equation given below. Using the Euler method with the step size (h) of 0.5 , the value of $y$ at $x=1.0$ is equal to _________ (rounded off to 1 decimal place). $$ \frac{d y}{d x}=y+2 x-x^2 ; y(...

The smallest positive root of the equation $$x^5 - 5 x^4 - 10 x^3 + 50 x^2 + 9 x - 45 = 0$$ lies in the range

The second derivative of a function $f$ is computed using the fourth-order Central Divided Difference method with a step length $h$. The CORRECT expression for the second derivative is

Consider the data of $f(x)$ given in the table. $i$ $0$ $1$ $2$ $x_i$ $1$ $2$ $3$ $f(x_i)$ $0$ $0.3010$ $0.4771$ The value of $f(1.5)$ estimated using second-order Newton’s interpolation formula is ________________ (roun...

The differential equation, $\rm \frac{du}{dt}+2tu^2=1,$ is solved by employing a backward difference scheme within the finite difference framework. The value of 𝑢 at the (𝑛 − 1) th time-step, for some 𝑛, is 1.75. The...

Consider the differential equation $${{dy} \over {dx}} = 4(x + 2) - y$$ For the initial condition y = 3 at x = 1, the value of y at x = 1.4 obtained using Euler's method with a step-size of 0.2 is ________. (round off to...

Consider the following recursive iteration scheme for different values of variable P with the initial guess x 1 = 1: $${x_{n + 1}} = {1 \over 2}\left( {{x_n} + {P \over {{x_n}}}} \right)$$, n = 1, 2, 3, 4, 5 For P = 2, x...

Consider the equation $${{du} \over {dt}} = 3{t^2} + 1$$ with $$u=0$$ at $$t=0.$$ This is numerically solved by using the forward Euler method with a step size. $$\,\Delta t = 2.$$ The absolute error in the solution at t...

Newton-Raphson method is to be used to find root of equation $$\,3x - {e^x} + \sin \,x = 0.\,\,$$ If the initial trial value for the root is taken as $$0.333,$$ the next approximation for the root would be _________ (not...

The integral $$\,\int_{{x_1}}^{{x_2}} {{x^2}dx\,\,} $$ with $${x_2} > {x_1} > 0$$ is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $${\rm I}$$ is the exact value of...

The quadratic equation $${x^2} - 4x + 4 = 0$$ is to be solved numerically, starting with the initial guess $${x_0} = 3.$$ The Newton- Raphson method is applied once to get a new estimate and then the Secant method is app...

In Newton-Raphson iterative method, the initial guess value $$\left( {{x_{ini}}} \right)$$ is considered as zero while finding the roots of the equation: $$\,f\left( x \right) = - 2 + 6x - 4{x^2} + 0.5{x^3}.\,\,\,$$ The...

For step-size, $$\Delta x = 0.4,$$ the value of following integral using Simpson's $$1/3$$ rule is ______

The estimate of $$\int\limits_{0.5}^{1.5} {{{dx} \over x}} \,\,$$ obtained using Simpson's rule with three-point function evaluation exceeds the exact value by

The square root of a number $$N$$ is to be obtained by applying the Newton $$-$$ Raphson iteration to the equation $$\,{x^2} - N = 0.\,\,$$ If $$i$$ denotes the iteration index, the correct iterative scheme will be

In the solution of the following set of linear equations by Gauss-elimination using partial pivoting $$$5x+y+2z=34,$$$ $$$4y-3z=12$$$ and $$$10x-2y+z=-4.$$$ The pivots for elimination of $$x$$ and $$y$$ are

The Newton-Raphson iteration $${x_{n + 1}} = {1 \over 2}\left( {{x_n} + {R \over {{x_n}}}} \right)$$ can be used to compute

If the interval of integration is divided into two equal intervals of width $$1.0,$$ the value of the definite integral $$\,\,\int\limits_1^3 {\log _e^x\,\,dx\,\,\,\,} $$ using simpson's one $$-$$ third rule will be

Three values of $$x$$ and $$y$$ are to be fitted in a straight line in the form $$y=a+bx$$ by the method of least squares. Given $$\,\,\,\sum x = 6,\,\,\sum y = 21,\,\,\sum {{x^2} = 14,\,\,\sum {xy} = 46,\,\,\,\,} $$ the...

The following equation needs to be numerically solved using the Newton $$-$$ Raphson method $${x^3} + 4x - 9 = 0.\,\,$$ The iterative equation for this purpose is ($$k$$ indicates the iteration level)

Given $$a>0,$$ we wish to calculate its reciprocal value $${1 \over a}$$ by using Newton - Raphson method for $$f(x)=0.$$ For $$a=7$$ and starting with $${x_0} = 0.2\,\,$$ the first two iterations will be

Given $$a>0,$$ we wish to calculate it reciprocal value $${1 \over a}$$ by using Newton - Raphson method for $$f(x)=0.$$ The Newton - Raphson algorithm for the function will be

Let $$\,\,f\left( x \right) = x - \cos \,x.\,\,\,$$ Using Newton-Raphson method at the $$\,{\left( {n + 1} \right)^{th}}$$ iteration, the point $$\,{x_{n + 1}}$$ is computed from $${x_n}$$ as

Given the differential equation $${y^1} = x - y$$ with initial condition $$y(0)=0.$$ The value of $$y(0.1)$$ calculated numerically upto the third place of decimal by the $${2^{nd}}$$ order Runge-Kutta method with step s...