Numerical methods Flashcards
(29 cards)
Method of Bisection
This method takes two points which straddle the root and then find the midpoint of these values. Then compare this with a and b to find the closer values which straddle the root. Repeat this to a suitable accuracy
False Position
Uses two points which straddle the solution
Equation : C = (af(b) - bf(a)) / (f(b) - f(a))
This equation provides a better approximation to the solution
f(c) is compared to f(a) and f(b) in order to straddle the root more precisely
Fixed Point Iteration
Rearrange f(x) = 0 to x = g(x)
This can then be repeated to approximate the root
Fixed Point Iteration - Cobweb When?
g’(value) < 0
Fixed Point Iteration - Cobweb Converge When?
-1 < g’(value) < 0
Fixed Point Iteration - Cobweb Diverge When?
g’(value) < -1
Fixed Point Iteration - Staircase When?
g’(value) > 0
Fixed Point Iteration - Staircase Converge When?
0 < g’(value) < 1
Fixed Point Iteration - Staircase Diverge When?
g’(value) > 1
Def - Order of Convergence
Describes the pattern of how quickly an iterative method approaches a solution
Def - Rate of Convergence
Describes the actual speed of convergence
Order of Convergence - Bisection
1 (Linear)
Order of Convergence - False Position
1 (Linear)
Order of Convergence - Newton-Raphson
2 (Quadratic)
Order of Convergence - Secant
Approximately 1.618
Order of Convergence - Fixed Point Iteration
1 (Linear)
Rate of Convergence - Bisection
Relatively slow as halves each time
Rate of Convergence - False Position
Similar to Bisection
Rate of Convergence - Newton-Raphson
The error is roughly squared in each iteration so has faster convergence compared to linear methods
Rate of Convergence - Secant
Slower than Newton-Raphson but faster than Linear methods
Rate of Convergence - Fixed Point Iteration
Slow - error decreases by a constant factor each iteration
First Order Convergence
The ratios of differences between successive approximations is equal to k (constant)
Midpoint Rule Conversion and order method
Convergence of approximations is first order
Second order method - error is proportional to h^2 so ratio between successive errors is 1/4
Trapezium Rule Conversion and order method
Convergence of approximations is first order
Second order method - error is proportional to h^2 so ratio between successive errors is 1/4
