Numerical Methods (ME)

$$ \text { The values of a function } f \text { obtained for different values of } x \text { are shown in the table below. } $$ $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 0.25 & 0.5 & 0.75 & 1.0 \\ \hline f(x) & 0.9...

In order to numerically solve the ordinary differential equation dy/dt = -y for t > 0 , with an initial condition y(0) = 1 , the following scheme is employed: $\frac{y_{n+1} - y_{n}}{\Delta t} = -\frac{1}{2}(y_{n+1} + y_...

The initial value problem $\rm \frac{dy}{dt}+2y=0, y(0)=1$ is solved numerically using the forward Euler’s method with a constant and positive time step of Δt. Let 𝑦 𝑛 represent the numerical solution obtained after 𝑛...

Consider the definite integral $\int^2_1(4x^2+2x+6)dx$ Let I e be the exact value of the integral. If the same integral is estimated using Simpson’s rule with 10 equal subintervals, the value is I s . The percentage erro...

$$\,\,P\,\,\,\left( {0,3} \right),\,\,Q\,\,\,\left( {0.5,4} \right),\,\,$$ and $$\,\,R\,\,\,\left( {1,5} \right)\,\,\,$$ are three points on the curve defined by $$\,\,f\left( x \right),\,\,$$ Numerical integration is ca...

The error in numerically computing the integral $$\,\int\limits_0^\pi {\left( {\sin \,x + \cos \,x} \right)dx\,\,\,} $$ using the trapezoidal rule with three intervals of equal length between $$0$$ and $$\pi $$ is ______...

Numerical integration using trapezoidal rule gives the best result for a single variable function, which is

The root of the function $$f\left( x \right) = {x^3} + x - 1$$ obtained after first iteration on application of Newton-Raphson scheme using an initial guess of $${x_0} = 1$$ is

Solve the equation $$x = 10\,\cos \,\left( x \right)$$ using the Newton-Raphson method. The initial guess is $$x = {\pi \over 4}.$$ The value of the predicted root after the first iteration, up to second decimal, is ____...

Simpson's $${1 \over 3}$$ rule is used to integrate the function $$f\left( x \right) = {3 \over 5}{x^2} + {9 \over 5}\,\,$$ between $$x=0$$ and $$x=1$$ using the least number of equal sub-intervals. The value of the inte...

Newton-Raphson method is used to find the roots of the equation, $${\,{x^3} + 2{x^2} + 3x - 1 = 0}$$ If the initial guess is $${x_0} = 1,$$ then the value of $$x$$ after $${2^{nd}}$$ iteration is ___________.

Using a unit step size, the value of integral $$\int\limits_1^2 {x\,\ln \,xdx\,\,\,} $$ by trapezoidal rule is ___________.

Using the trapezoidal rule, and dividing the interval of integration into three equal sub-intervals, the definite integral $$\,\,\int\limits_{ - 1}^{ + 1} {\left| x \right|dx\,\,} $$ is

The value of $$\int\limits_{2.5}^4 {\ln \left( x \right)} $$ calculated using the Trapezoidal rule with five sub-intervals is

The definite integral $$\,\int\limits_1^3 {{1 \over x}} dx\,\,$$ is evaluated using Trapezoidal rule with a step size of $$1.$$ The correct answer is

The real root of the equation $$5x-2cosx=0$$ (up to two decimal accuracy) is

Consider an ordinary differential equation $${{dx} \over {dt}} = 4t + 4.\,\,$$ If $$x = {x_0}$$ at $$t=0,$$ the increment in $$x$$ calculated using Runge-Kutta fourth order multi-step method with a step size of $$\Delta...

The integral $$\,\int\limits_1^3 {{1 \over x}\,\,dx\,\,\,} $$ when evaluated by using simpson's $$1/{3^{rd}}$$ rule on two equal sub intervals each of length $$1,$$ equals to

Starting from $$\,{x_0} = 1,\,\,$$ one step of Newton - Raphson method in solving the equation $${x^3} + 3x - 7 = 0$$ gives the next value $${x_1}$$ as

The order of error in the simpson's rule for numerical integration with a step size $$h$$ is

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...

Simpson's rule for integration gives exact result when $$f(x)$$ is a polynomial function of degree less than or equal to ________.