Solution of Equations Flashcards
What determines if two points have a root between them?
Plug in the values and look for a sign change as it means the graph crosses the x axis and therefore has a root
What assumptions does the substitution method for determining roots make?
Relies on the graph being continuous and it not being a repeated root
How many roots are there between two points above or below the x axis?
Even number of roots
What are straddle points?
Straddle points are two points between which lies a root
How should a table be laid out for iterative interval bisection?
Iteration
Lower Bound
Sign
Upper Bound
Sign
Median
Sign
When does an iterative interval bisection table stop?
When a value for the root can be estimated to the given decimal places
What is a recurrence relation?
Recurrence relation is a type of iteration when the next term is generated by using the previous term
What is the difference between a convergent and divergent sequence?
Convergent sequences tend towards a limit
Divergent sequences do not
What is a fixed point?
A root of a function is a fixed point since it always gives a fixed value
How can recurrence relations be used to estimate a root?
When x = g(x) and x0 is chosen to create a convergent recurrence relation, the iteration approaches the root of the function
What determines whether a function is convergent?
-1 < g’(x0) < 1
What determines whether a function is divergent?
g’(x0) < -1
1 < g’(x0)
What determines whether a function is cobweb?
g’(x0) < 0
What determines whether a function is staircase?
0 < g’(x0)
What other name are cobweb functions known by? Why?
Cobweb functions are known as oscillating functions as they jump past the root and then behind it
What must be the case for a recurrence relation to estimate the root?
Gradient of function between -1 and 1
x0 is close to the root
What is relaxation?
Relaxation is used on fixed point iteration to accelerate convergence or convert divergent sequences to convergent sequences
What is the formula for relaxation?
x is a root of g(x)
y is given
(1-y)x + y g(x)
Why is x = g(x) used?
x = g(x) is used to allow comparison to y = x
What is the iterative formula for Newton Raphson’s Process?
xn+1 = xn - f(xn) / f ‘ (xn)
Why may Newton Raphson’s Process fail?
Initial value not close to root
Roots are close to stationary point
Graph is not defined for all real numbers
Graph is discontinuous
What method is used if the Newton Raphson’s Process fails?
Secant Method
What is the formula for the method of false position?
c = af(b) - bf(a) / f(b) - f(a)
Where a is the value with a lower ‘x’
b is the value with a higher ‘x’
How does the Secant and Newton-Raphson method differ?
Secant method uses lines that intersect last 2 points and x intercept
Newton-Raphson uses tangent to curve and x intercept
How does Secant and False Position Method differ?
False position requires sign change
False position always converges
Secant is faster
Secant requires two previous values
How does interval bisection work?
Find median of two straddle points and narrow down straddle points by replacing one with the median depending on function value’s sign
How does fixed point iteration work?
Starting point is x value
Plugged into function to produce y value
y=x means y value becomes x value
X value is plugged into the function
This narrows down the straddle points
How does Newton-Raphson Method work?
Tangent to function is drawn for given x value
The x intercept becomes the new x value
How does Secant Method work?
Line is drawn between the current and past point
X intercept value becomes x value for next point on the function
