Summary
Summary OCR MEI Mathematics: Year 2 Pure - Numerical Methods Cheat Sheet
- Course
- Institution
This document briefly summarises the key points in the Year 2 'Numerical Methods' topic of OCR's Mathematics (MEI) course.
[Show more]
1 review
By: tsmith3553 • 1 year ago
Seller
Follow
JodbyBerundi
Reviews received
Content preview
Numerical MethodsChange of Sign Methods
● Sometimes there is no easy way of finding the roots of an equation by
factorising (e.g x3 − 7x + 3 = 0 )
● Alternative is to look at graph to find the interval where the roots lie (e.g
between 2 and 3)
● Then try plugging in values of x between that interval and wait for the
answer to change sign (from + to - or visa versa)
● Repeat until gradually you gain appropriate levels of accuracy (enough
decimal places)
● Can go wrong:
○ If a repeated root occurs (so it touches the x-axis and never goes
below)
○ If there is a discontinuity in the graph (if there’s a change of sign
without a root)
Fixed Point Iteration
● Rearrange f (x) = 0 into x = g(x) and solve to find roots (if done incorrectly
can mean converging to different root or not converging at all)
● This is essentially splitting f (x) into the lines y = x and x = g (x)
● This means the point on the graph where y = x and x = g (x) meet is the
same x-value as the root of the equation f (x) = 0
● You can do this by gradually gaining more accuracy from a start point:
○ xn+1 = g(xn ), n = 0, 1, 2, 3...
○ Start with an initial approximation, x0 , and find g (x0 )
○ Take this value of g (x0 ) as a new value, x1 , and find g (x1 )
○ Repeat until particular decimal point is constant for two or three
iterations
● The sequence converges on a root of the equation, providing x0 is a close
enough approximation and the curve is not too steep close to the root
● The gradient of the curve close to the root must be between -1 and 1
Staircase Diagrams
Iterations are on the same side of the root
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller JodbyBerundi. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.90. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75323 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now