Numerical Methods

Question 1

Triangle inequalities

Answer 1

|x + y| <= |x| + |y|
|x - y| >= | |x| - |y| |

Question 2

Taylor series expansion

Answer 2

Question 3

Jacobian matrix

Answer 3

Question 4

Matrix inverse (2D)

Answer 4

Question 5

Decomposition method for a matrix A

Answer 5

let A = LU
=> LUx = b
where Ux = y
Then solve Ly = b, Ux = y

L - lower triangular matrix
U - upper triangular matrix

Question 6

Properties of a vector norm

Answer 6

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)

Question 7

Properties of a matrix norm

Answer 7

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)

Question 8

Matrix and Vector norm compatibility result

Answer 8

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

Question 9

||-||1,n,inf for vectors x,y

Answer 9

Question 10

||-||1,inf for matrix A

Answer 10

Question 11

Jacobi’s method iteration scheme

Answer 11

A = I - (A_L + A_U)

Question 12

Newton’s method iteration scheme

Answer 12

Question 13

Gauss-Seidel method iteration scheme

Answer 13

Question 14

Secant method iteration scheme

Answer 14

Question 15

Newton’s method from Taylor series iteration scheme

Answer 15

Question 16

Centred difference formulas for finite differencing
– for a linear problem

Answer 16

Question 17

First order difference formula for finite differencing

Answer 17

Question 18

Centred difference formulas for finite differencing
– for a PDE

Answer 18

Question 19

Define and provide the equations for FTCS and BTCS

Answer 19

FTCS - Forward Time Central Space
BTCS - Backward Time Central Space

Question 20

FTCS algorithm for the heat equation

Answer 20

Question 21

Explain how to use a shooting method to solve a BVP using an IVP

Given y'(1) = a

Answer 21

• Create an IVP (using the given equation, y(0;z) = b, and y'(0;z) = z)
• Determine y'(1)
• Define an equation phi(z) = y'(1;z) - a
• Use root finding routine to determine a root z_c s.t. phi(z_c) = 0
• The solution of the IVP is the solution to the original BVP

Question 22

Explain the application of the contraction mapping theorem to a mapping function g(x), differentiable inside the interval I

Answer 22

• If g(x) is a contraction mapping in I, then there is a unique fixed point x = g(x) in I,
• and the continuous map x_n+1 = g(x_n) converges to the fixed point if:
• g(I) c= I and |g'(x)| < 1 for all x in I

Question 23

Define the convergence matrix M, and how to compute it

Answer 23

• M = N^(-1) . P
• let A = N - P
• N = coefficient for x^(n+1)
• P = coefficient for x^(n)

Question 24

Explain “explicit” and “implicit” in the context of finite difference schemes to solve linear partial differential equations

Answer 24

• explicit: FD schemes give an explicit formula for U^(n+1)_i at 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 the U^(n+1)_i

Question 25

Explain "consistency", "stability", and "convergence" in the context of finite difference algorithms to solve partial differential equations

Answer 25

* 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

Question 26

State the Lax theorem, in the context of finite difference algorithms to solve partial differential equations

Answer 26

It is necessary and sufficient for a well-posed linear IVP to be stable and consistent, in order for it to be convergent