Numerical Methods Flashcards
What is a root of a function?
A solution to the function. For the function f(x), the roots are found when x = 0
What numerical method can we use to find the root of a function between two specified limits?
Checking if there is a sign change for ther function betwen the two limits
How does the sign change method work?
If the two limits, a and b, are specified for a function f(x), we find f(a) and f(b), if there is a difference in the sign of the two values, there is a root in their interval
What is the function of the Trapezium and Simpson rules?
They are forms of finding the area under a curve by estimation
Trapezium rule formula?
h/2[(y0 + yn) +2(y1… y(n-1))]
the area under the curve between two limits a and b is equal to…
where n is the no. of strips and h is the strip thickness;** h = (b-a)/n**
When does the trapezium rule give an over estimate?
When the curve of the line is found to slope downwards, hence, the tops of the trapeziums are above the line
When does the trapezium rule give an under estimate?
When the curve of the line is found to slope upwards, hence, the tops of the trapeziums are below the line
How can we determine if the trapezium rule gives an under or over estimate?
By sketching the line
Simpson’s rule formula?
h/3[(y0 + yn) +2(sum of yeven numbers) + 4(sum of yodd numbers)]
the area under the curve between two limits a and b is equal to…
where n is the no. of strips and h is the strip thickness;** h = (b-a)/n** In this rule you do not count the first and last strips twice. Note the emboldened differences betwene the two formulae
What is the Newton-Raphson numerical method?
It is a method that workd on the principle of iteration, that is the repetition of a process with small changes until a relatively constant value is reached
What is iteration?
Iteration is the repeated application of a function or process in which the output of each step is used as the input for the next iteration
such is the case with the N-R method
What does the Newton-Raphson method do ?
It is used to find (successive) roots of an equation
or to at least estimate them
Newton-Raphson method formula?
x_(n+1) = x_(n) - [f(x_(n))/f’(x_(n))]
Where x(n+1) is the next root to be found.
Note that all the things in brackets are subscripts. So using iteration principles, you start off with x0, then you use that to get a value for x1, then you plug that value in until you get a value that looks relatively the same across board, that is now you actual value for your first root.
