Numerical Methods Flashcards
Triangle inequalities
|x + y| <= |x| + |y||x - y| >= | |x| - |y| |
Taylor series expansion
Jacobian matrix
Matrix inverse (2D)
Decomposition method for a matrix A
let A = LU
=> LUx = b
where Ux = y
Then solve Ly = b, Ux = y
L - lower triangular matrix
U - upper triangular matrix
Properties of a vector norm
A normed vector space over a field F, is a vector field V equipped with a map, the norm, ||-||: V -> F satisfying:
* for all x in V, ||x|| >= 0, and ||x|| = 0 if and only if x = 0 is in V
* for all a in F, x in V, ||a x|| = |a| ||x||
* for all x,y in V, ||x + y|| <= ||x|| + ||y|| (triangle inequality)
Properties of a matrix norm
Provides a measure of the size of a square (nxn) matrix. The norm, ||-|| satisfies:
* for all nxn A, ||A|| >= 0, and ||A|| = 0 if and only if A = 0
* for all scalar a, nxn A, ||a A|| = |a| ||A||
* for all nxn A, B, ||A + B|| <= ||A|| + ||B|| (triangle inequality 1)
* for all nxn A, B, ||A . B|| <= ||A|| . ||B|| (triangle inequality 2)
Matrix and Vector norm compatibility result
A matrix norm ||-||q is compatible with a vector norm ||-||p if:
* ||A x||p <= ||A||q . ||x||p
* for all n-vectors x, nxn matricies A
||-||1,n,inf for vectors x,y
||-||1,inf for matrix A
Jacobi’s method iteration scheme
A = I - (A_L + A_U)
Newton’s method iteration scheme
Gauss-Seidel method iteration scheme
Secant method iteration scheme
Newton’s method from Taylor series iteration scheme
Centred difference formulas for finite differencing
– for a linear problem
First order difference formula for finite differencing
Centred difference formulas for finite differencing
– for a PDE
Define and provide the equations for FTCS and BTCS
FTCS - Forward Time Central Space
BTCS - Backward Time Central Space
FTCS algorithm for the heat equation
Explain how to use a shooting method to solve a BVP using an IVP
Given y'(1) = a
- Create an IVP (using the given equation,
y(0;z) = b, andy'(0;z) = z) - Determine
y'(1) - Define an equation
phi(z) = y'(1;z) - a - Use root finding routine to determine a root
z_cs.t.phi(z_c) = 0 - The solution of the IVP is the solution to the original BVP
Explain the application of the contraction mapping theorem to a mapping function g(x), differentiable inside the interval I
- If
g(x)is a contraction mapping inI, then there is a unique fixed pointx = g(x)inI, - and the continuous map
x_n+1 = g(x_n)converges to the fixed point if: -
g(I) c= Iand|g'(x)| < 1for allxinI
Define the convergence matrix M, and how to compute it
M = N^(-1) . P- let
A = N - P N = coefficient for x^(n+1)P = coefficient for x^(n)
Explain “explicit” and “implicit” in the context of finite difference schemes to solve linear partial differential equations
- explicit: FD schemes give an explicit formula for
U^(n+1)_iat each new time step - implicit: discretisation results in a system of equations for
U^(n+1)_i, which need to be solved to find each of theU^(n+1)_i
Explain “consistency”, “stability”, and “convergence” in the context of finite difference algorithms to solve partial differential equations
- A scheme is consistent if local error tends to zero as grid spacing tends to zero
- A scheme is stable if errors in solution decay in time
- A scheme is convergent if in limit of zero grid spacing it converges to an exact solution of the PDE
State the Lax theorem, in the context of finite difference algorithms to solve partial differential equations
It is necessary and sufficient for a well-posed linear IVP to be stable and consistent, in order for it to be convergent
