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Front 1

Name the two Bracketing methods for finding roots

Back 1

Bisection Method and Method of false position

Front 2

PROS: Bracketing methods vs locally convergent

Back 2

Always convergent once initial interval [a,b] is found where f(a) and f(b) has opposite signs.

(GLOBALLY CONVERGENT)

Needs no analytical derivative (like Newtons method this is not always possible or ez)

Front 3

CONS: Bracketing Method

Back 3

- Slow convergence speed (MOFP has fastest of the two)

- If f(x) has several roots in [a,b] need to find seperate intervals for each root

Front 4

What does it mean if a root method is Locally Convergent?

Back 4

Local Convergencns: Requires a close a approximation to the root as first initial guess to guarantee convergence.

Front 5

Name the locally convergent root methods

Back 5

Newtons Method and Secant method

Front 6

Name the Bracketing Methods to minimize a function

Back 6

Ratio Method and Fibonacci

Front 7

What conditions are there to use bracketing method to minimize a function (of one variable)

Back 7

f(x) is decreasing on [a, p]

f(x) is increasing on [p, b]

Front 8

Golden search method internal points c and d are given as

Back 8

c = a + (1 - r)(b - a)

d = b - (1 - r)(b - a)

Front 9

r in Golden Ratio method is

Back 9

r = ( -1 + sqrt(5) ) / 2

Front 10

PROS/CONS Golden Ratio

Back 10

PROS: Although might be slow, can be appled in cases where f(x) is not differentiable

CONS: If function is very flat near minimum, it has limited accuracy

Front 11

Golden ratio vs Fibonacci method

Back 11

Fib. more efficient than GR for small n (Fibonacci numbers), else identical

Front 12

Explain the difference between local and global error

Back 12

The local error is the truncation error from approximating using finite differences (one iteration).

The global error is due to the accumulation of the truncation error of previous pointswhen calculating the current point.

Front 13

Hvordan vil du effektivt løse matrisesystemet Ax=b

Back 13

• Jakobimetoden

• Gauss-Seidel

• Steepest descent

• Konjugerte gradienters metode

Front 14

Jacobi vs Gauss Seidel

Back 14

Gauss Seidel converts faster than Jacobi (utilizes new values)

But there are cases where Gauss Seidel wont converge but Jacobi will.

Front 15

Cons: Newtons method to find root f(x) = 0

Back 15

Division by zero

X may be far away (slope of f'(X) is far away)

Need to know analytical solution of f'(x) (not always possible)

Front 16

Requirement for Steepest descent and CG?

Back 16

Positive Definite mateix , symmetric

Front 17

Jacobi and Gauss Seidel are methods that finds...

Back 17

solution to Ax=b by iteration.

Front 18

What requirements are there for Jacobi method to converge?

Back 18

The spectral radius (maximum eigenvalue) of A must be less than 1

If A is Strictly Diagonally Dominant - (the max spectral radius is less than one) - the method will converge.

OBS (the method might still also converge if not Strictly Diagonally Dominant)

Front 19

What requirements are there for Gauss Seidel method to converge?

Back 19

The spectral radius (maximum eigenvalue) of A must be less than 1

If A is Strictly Diagonally Dominant - (the max spectral radius is less than one) - the method will converge.

OBS (the method might still also converge if not Strictly Diagonally Dominant)

Front 20

Pros cons Jacobi and Gauss seidel

+ which is faster

Back 20

GS is fastest to converge

Pros: Fast, simple to implement

Cons: Not exact, convergence issues

Front 21

Seidel and Newtons are methods to solve

Back 21

Nonlinear Systems of equations

ex.

f1(x,y)=0

f2(x,y)=0

rewrite to

x=g1(x,y)

y=g2(x,y)

Front 22

What is the requirement of convergence for Seidel and Newtons method to solve non linear equations

Back 22

The rows of the jacobi matrix must be less than one.

dg1/dx + dg1/dy < 1

dg2/dx + dg2/dy < 1

Front 23

Taylor polynomial, Lagrange, Divided Differences and Chebyshev are...

Back 23

Different ways to find polynomials P to use for interpolation

Front 24

Taylor Polynomial interpolation, what is good and bad about this method?

Back 24

Taylor Polynomials only use information about f around a point Xo.

Hence, TP Interpolation is very accurate close to xo but innacurate away from xo

Taylorpolynomials is not good for interpolation

Front 25

Lagrange polynomials are useful for interpolation, what are some things that are not so good about the method?

Back 25

The error term is not usable in practice, since we do not know f. The Error is f''''''''''(x) something (n+1) something

Thus, it is diffucult to know the order of Pn to use.--> May need to try different points. to interpolate.

Front 26

What are the n distinct roots of a chebyshev polynomial Tn(x) ?

Back 26

xk = cos((2k+1)*pi / (2N))

Front 27

What are the roots used for?

Back 27

They can be used as collocation points in Lagrange Polynomial interpolation to minimize the error of lagrange

Using Chebyshev nodes removes the Runge phenomenomnea

Front 28

A good way of reducing error of a differential approximation (finite differences) is to use ...

Back 28

Richardson theorem

Front 29

Explain briefly how richardson theorem works

Back 29

Two Differential approximaions (must be of even order O(h^2k) one with step length h and one with steplength 2h are combined to remove the largest error term.

Can also be used in numerical integration, use on Trapes, Simpsons.. This is the theorem Romberg Integration is built upon.

Front 30

Eulers, Heuns, RK methods are methods used to...

Back 30

Solve ODEs

Front 31

What is the order of the error done in Eulers, Heuns and RK4?

Back 31

Eulers O(h)

Heuns O(h^2)

RK4 O(h^4)

Front 32

What is the difference between Predictor Corrector methods compared to regular numerical solutions of ODEs (euler, heuns, RK)

Back 32

Predictor corrector methods use multistep. (that is they use information from a set of previous points, k, k-1, k-2, ... etc)

Regular methods are singlestep.

Can have implicit and explicit multistep methods.

Use AB(explicit) to estimate next step, use AM(implcit) to correct

Front 33

What are the two ways to solve PDE systems numerically?

Back 33

Depends on how the finite difference method is. If explicit then they are very simple straight forward. If they are Implict you need to solve a system of eq to find the next value. This can be done by

- Iterative:

Jacobi, Gauss seidel

- Gradient search:

Steepest descent

(Newtons)

CG

Front 34

What are the advantages of Spectral Element Method over Finite Differences

Back 34

- High order ---> Accurate solutions

- You can have an irregular mesh

- All boundary conditions are automatically implemented in the method

- Based on weak solution

Front 35

What are the disadvantages of Spectral Element Method over Finite Differences

Back 35

Difficult to implement

- Node numbering

- Mesh generation

- Paralellization

Meshes are tedious to create