Core 3 - Numerical Methods Flashcards
What is Simpson’s Rule? What principal does it work on?
It takes an approximation of the curve using three points. The general method is:
1) Find h by dividing the difference between the 2 limits by the number of strips. NOT FUCKING ORDINATES
2) Find the values of all the y values for the corresponding x values using table mode.
3) Substitute into the formula
A= (h/3)(ends + 4odd + 2even)
What is the mid-ordinate rule? What principal does it work on?
It finds the point in the middle of the strip, and starts the trapezium from there. This is more accurate than Trap. Rule as it under/overestimates by a smaller amount.
1) Find h by dividing the difference between the 2 limits by the number of strips. NOT ORDINATES.
2) Draw up table, then find the outputs for the x inputs in the middle of those you found first. e.g. if x(edge)=1,2,3 x(mid)=1.5,2.5. These are the points we use to find the outputs.
3) sub into equation A=h(sum of outputs)
What is the formula for simpsons rule?
h/3(ends + 4odds + 2even)
What is the formula for mid ordinate rule?
h(sum of ordinates) Hard part is finding the ordinates.
How do you prove there is a root of an equation between two numbers?
Equate them and rearrange to =0.
Substitute the two values in and look for a change in sign.
The graph is continuous and there is a change of sign. A root therefore exists such that 2 (less than) x (less than) 3
How do you remember the difference between the two formulae for mid-ordinate and simpson’s rule?
Mid ordinate requires effort to find the values, then just sub into simple equation (h*sum of values)
Simpson’s is easy to find the values, but a more complicated equation : h/3(ends + 4odds + 2even)
What is the method for finding a root of an equation?
Substitute the two given values into the equation when set to =0, and hope they give +- values. Then concluding statement.
What is the concluding statement for finding a root?
The graph is continuous and there is a change of sign, therefore a root alpha exists such that 1
When drawing cobweb/staircase diagrams, what is the rule?
Up to the Curve,
Across to the Line
When could the mid ordinate rule provide an overestimate?
the part that it overshoots by is greater than the bit it missed off, so gradient decreasing e.g. ln(x)
When could mid ordinate rule provide an underestimate?
Part it overshoots is smaller than the part it missed off, like a decreasing gradient e.g. 1/x
How do you calculate h
largest limit-smallest limit/number of strips
Jan 13 mistake - what must you check for?
Make sure your ordinates only go up to the values stated, NOT BEYOND!!!!
