Engineering Mat...

Evaluate the iterative steps for solution using Newton-Raphson method for the equation x³ – 23 = 0

The quadratic equation 2x² – 3x + 3 = 0 is to be solved numerically starting with an initial guess as x₀ = 0. The new estimate of x after the first iteration using Newton Raphson method is ______

Considering four subintervals, the value of \(\mathop \smallint \limits_0^1 \frac{1}{{1 + x}}dx\) by Trapezoidal rule is:

Compute the root of function f(x) = x² – 3 after two iterations using bisection method in the interval [0, 2]

Solve the differential equation \(\frac{{dy}}{{dx}} - \left( {xy + {x^2}} \right) = 0\) with the initial condition y(0) = 0 using Euler’s first order method with step size of 1.

Apply Gauss-seidel method to solve the equations: 20x + y – 2z = 17; 3x + 20y – z = -18; 2x – 3y + 20z = 25. Assume initial guess x₀ = y₀ = z₀ = 0, then value of ‘z’ after first iteration is ______

The area of seven horizontal cross-sections of a water reservoir at the interval of 9m is 210, 250, 320, 350, 290, 230 and 170 m². The estimated volume of the reservoir in m³ using Simpson (1/3)rd rule is ________. 

An equation \(\sin x = \frac{1}{x}\) is required to be solved by the Bisection method where x lies between 1 and 1.5 (x is in radian). The approximate root of the fourth iteration will be: (Correct upto four decimal places)

Consider the first order initial value problem y’ + y = 0, y(0) = 1 for x = 0.1, the solution obtained using a single iteration  of the third order Runge Kutta method with step-size h = 0.1 is _________.

Find the positive root of x⁴ – x = 10 after 1^st iteration and 2^nd iteration (x₂) with initial value x₀ = 2. Using Newton-Raphson method