Numerical Methods · Engineering Mathematics · GATE ME

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 emp...

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...

Using the trapezoidal rule, and dividing the interval of integration into three equal sub-intervals, the definite integral $$\,\,\int\limits_{ - 1}^{...

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

Match the CORRECT pairs. ...

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...

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

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 pla...

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

$$\,\,P\,\,\,\left( {0,3} \right),\,\,Q\,\,\,\left( {0.5,4} \right),\,\,$$ and $$\,\,R\,\,\,\left( {1,5} \right)\,\,\,$$ are three points on the curv...

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 ...

Gauss-Seidel method is used to solve the following equations (as per the given order). $$${x_1} + 2{x_2} + 3{x_3} = 5$$$ $$$2{x_1} + 3{x_2} + {x_3} =...

The error in numerically computing the integral $$\,\int\limits_0^\pi {\left( {\sin \,x + \cos \,x} \right)dx\,\,\,} $$ using the trapezoidal rule wi...

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 ...

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

The values of function $$(x)$$ at $$5$$ discrete points are given below: Using Trapezodial rule with step size of $$0.1,$$ the value of $$\,\int\lim...

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$...

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

Consider an ordinary differential equation $${{dx} \over {dt}} = 4t + 4.\,\,$$ If $$x = {x_0}$$ at $$t=0,$$ the increment in $$x$$ calculated using R...

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