Numerical Methods Flashcards
What is the equation for the numerical integration using the trapezium rule?
Where n represents the number of strips, or number of ordinates - 1
What is the equation for the numerical integration using the mid-ordinate rule?
Where n represents the number of strips, or number of ordinates
When do the trapezium rule and mid-ordinate rule give underestimates and overestimates?
The trapezium rule underestimates when the gradient of the curve is decreasing and overestimates when the gradient of the curve is increasing.
The mid-ordinate rule overestimates when the gradient of the curve is decreasing and underestimates when the gradient of the curve is increasing.
What is Simpson’s rule and what is the equation?
Taking two strips at a time, finding a parabola which passes through the three points at the top of these strips, and integrating this quadratic function to find the area under the parabola.
Where n is even and represents the number of strips, or number of ordinates - 1
What is Euler’s method for the numerical solution of differential equations?
Where h represents a small increment in the value of x
What is the Improved Euler method for the numerical solution of differential equations?
For dy/dx = f(x,y), where h represents a small increment in the value of x
What is the change of sign method for locating the interval at which a root is located for f(x) = 0?
This method must be used for f(x) being a continuous function and f(x) intersects the x-axis. Use a table of values for x and f(x). When there is a change of sign for f(x), for values of x=a1, x=a2, a root of f(x) is present in the interval a1 < x < a2
What are the failures of the change of sign method?
- where the graph of f(x) = 0 touches the x-axis, without crossing it. In this case a
repeated root occurs and there is no change of sign of f(x) on either side of the root. - if there is a discontinuity in the graph. In this case, there may be a change of sign
without a root, so a change of sign method will find a non-existent root. - if the initial interval is too large it may include more than one root.
How are roots of f(x) = 0 found using fixed point iteration, and how is the process represented graphically?
f(x) = 0 is rearranged into the form x = g(x). Roots of the equation x = g(x) are therefore roots of the equation f(x) = 0. This gives rise to the iterative formula of:
Graphically, the method works as follows:
1) Starting with the initial approximation x_0 , you find g(x_0) , which gives you a point on the curve y = g(x).
2) Take this value of g(x_0) as x_1, so you are moving horizontally to the line y = x.
3) Find the value of g(x_1), so you are moving vertically to a new point on the curve y = g(x), nearer to the root
4) Repeat this process.
What is the restriction of the gradient of g(x) in the vicinity of the root?
The curve needs to be less steep than the line y = x, so the gradient needs to be between -1 and 1.
Give two diagrams, representing the two main examples of staircase diagrams, in which all the iterations are on the same side of the root.
The diagram on the left shows the gradient of the curve as positive and less than 1. The iterations coverage towards the root.
The diagram on the right shows the gradient of the curve as positive and is greater than 1. The iterations diverge away from the root.
Give two diagrams, representing the two main examples of cobweb diagrams, in which the iterations oscillate about the root.
The diagram on the left shows the gradient of the curve as negative and greater than -1. The iterations oscillate about the root and converge towards it.
The diagram on the right shows the gradient of the curve as negative and less than -1. The iterations oscillate about the root and diverge away from ti.
What are the failures of fixed-point iteration?
The success or failure depends on the rearrangement being used and on the starting point. Iterations may diverge away from the root, or they may converge to a root other than the intended one.
What is the equation for the newton-raphson method?
What are the failures of the newton-raphson method?
If the starting approximation is near to a turning point, the iterations may diverge, or coverage to a different root, as this means that f’(x) is small. It can also fail if there is a discontinuity in the graph.
