Derive the Runge-Kutta 2nd Order formula for the differential equation
Consider the first-order ordinary differential equation dy/dx = f(x, y), with the initial condition y(x₀) = y₀. The aim of the Runge–Kutta second-order method is to obtain an approximation for y at xₙ₊₁ = xₙ + h that is accurate up to terms of order h². Expand __________ __________ ______ ___ ________ _________ __________ _________ _________.
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Using the linear congruential random number generator determine the first five random numbers given that: ,
,
, and
. Transform these random numbers to lie between 0 and
Explain why a polynomial is a useful choice for an interpolating function.
Derive the Runge-Kutta 2nd Order formula for the differential equation
Use the Secant method with initial guesses of and
to calculate the root/s of the equation
up to three iterations. Calculate the absolute relative approximate error at each stage.
Evaluate the integral using the 2-point Gaussian Quadrature Rule with the following values:
| Points | Weights | Arguments |
| 1 | 1.0 | 0.577 |
| 2 | 1.0 |
|
Explain round-off error and truncation error, with one example of each.
What would be the maximum possible period of this random number sequence?