numerical methods

A fifth-degree polynomial in $x$ is defined for $x > 0$. All coefficients of the polynomial are positive. The first derivative of the polynomial is obtained numerically at a point...

Starting with the first approximation as x = 0.5, the second approximation for the root of the following function by the Newton-Raphson method is ________ (rounded off to two decim...

Values of y for different values of x are tabulated below. If a second-degree interpolating polynomial P_2(x) is used to represent y, the value of P_2(0) is ________ (rounded off t...

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...

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 p...

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...

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 interpo...

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 derivativ...

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

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)...

Consider the following recursive iteration scheme for different values of variable P with the initial guess $x_1 = 1$: $x_{n+1} = \frac{1}{2} (x_n + \frac{P}{x_n})$, $n = 1, 2, 3,...

Consider the differential equation $\frac{dy}{dx} = 4(x+2)-y$ 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-...

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}}}} \...

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 ste...

The true value of $\ln(2)$ is 0.69. If the value of $\ln(2)$ is obtained by linear interpolation between $\ln(1)$ and $\ln(6)$, the percentage of absolute error (round off to the n...

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

Variation of water depth (y) in a gradually varied open channel flow is given by the first order differential equation $\frac{dy}{dx} = \frac{1 - e^{-\frac{10}{3}\ln(y)}}{250 - 45e...

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. $$\,\Delta t = 2.$$...

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 approximat...

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) = -...

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 es...

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, th...

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 eliminat...

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

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...

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 firs...

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 $...

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}...