Modeling quizzes

Quiz 1

Intro concepts, Iteration, convergence, residual, FD,TE,RO

Iteration (Loop

x₁ =1/2 (x₀+ r/x₀)

Residual

f(x_i)=ans or |f(x_i+1)|=TOL

Fixed point

Integer

Floating point

Real

Round off error

Finite accuracy which stores real numbers (storage error) Finite accuracy of arithmetic inherent error (arithmetic error)

Iterative convergence

|x₁-x₀|=ans

E_machine

smallest number you can add or subtract and still find a difference

Forward difference example

Truncation error

Due to approximations used to represent exact mathematical operations and quantities. We cant actually use an infinite number of terms.

Quiz 2

Data representation and bisection

Bisection

departs from premise that [a,b] brackets a simple root in between a and b which is p of f(x)

Bisection steps

chk midterm by p₁=a+b/2

if |f(p₁)<TOL

yes then stop; no keep going

if root is less than p₁ then [a,p]

If its greater than p₁ then [p,b]

+ then x>0, 0 then x=0, - then x<0

sign f(a)*sign(f(p₁) < 0

IEEE (Institute for electrical and electronics engineers)

Sign bit, characteristic, mantissa, bias is set by the computer when precision is declared.
c>b t is pos

c<b t is neg

t=c-b

Single Precision integer

16 bits

Largest possible integer +- 32, 767

Double Precision integer

32 bits
Largest possible integer +- 2,147,483,647

Double precision real

64 bits

Largest number is 2047 and bias is 1023

smallest number 10^-308 which is zero in double precision corresponds to zero in c and f

largest number 10³⁰⁸ corresponds to 1 in c and f

any attempt to make any smaller than +-10^-308 it just resets to that and its not fatal (real underflow error)

any attempt to make any larger than +- 10³⁰⁸ it is a fatal error (real overflow error) NAN

Single precision real

32 bits
Largest number is 255 and bias is 127

smallest number corresponds to zero +-10^-38

largest number corresponds to one +-10³⁸

any attempt to make any smaller than +-10^-38 it just resets to that and its not fatal (real underflow error)

any attempt to make any larger than +- 10³⁸ it is a fatal error (real overflow error) NAN

ex

x=-39.9
(-39.9)₁₀=-1.00111111100110011×2⁵

b=127

t=c-b

c=t+b

=5+127

=132

(132)₁₀=(10000100)₂

Putting in binary

1 is odd and 0 is even

For just a number like 25 just divide by 2 till you get to 1

and for a decimal like 0.2 just multiply by 2 till you get to the amount needed (most likely given in the question)

Ex

Error abs and relative

Horner’s Method

It can be used for both 1st and 2nd derivatives, It’s used to reduce round off error, and it can be applied to polynomials with complex arguments.

Quiz 3

fixed point, Newton-Raphson,Gauss elimination

Fixed point

Used for iterative solution of linear system of equations to solve f(x)=0 and then rearrange somehow in the form of x=g(x)

You’ll start with an initial guess then iterate

x₁= g(x₀)

x₂= g(x₁)

x₃= g(x₂)

…

x_n+1=g(x_n)

ex

You can rearrange it other ways too

To find when fixed point iteration converges

You need to use intermediate value theorem for derivatives.

Fixed Point Theorem

A sufficient but not necessary condition for the fixed point iteration. Provides a convergence criteria basically.
x=g(x)

to converge is that

|g’(x_n)|<1 for n=0,1,2,…

We can use it to check the initial guess

Newton-Raphson

Method requires: f(x) and f’(x)

converges “very fast” when it converges

diverges “very fast” when it diverges

a general way to find a fixed point form for any f(x) or geometrically

NR example

Only one initial guess

Do residual, do the 1st update

Then the residual of that |f₁(x₂)|=ans, then iterative convergence |x₂-x₁|=ans

Geometric interpretations

Secant Method

Approximates f’(x_n) in the NR by a backward finite difference generate iteration.

Secant Method ex

Needs two initial guesses, do residuals for both guesses, then do the 1st update, residual from that answer then the iterative convergence.

Muller’s method

Extension of secant method

Good method for complex roots

Rate of convergence is K=1.84

Requires 3 initial guesses
Assume a quadratic and use this formula to find x₃ and solve the quadratic

To avoid RO error if b²»4ac express result in an alt manner
Chk residual, update solution and we use them to find x₄ and etc

Finding roots of

initial guess

count the number of iterations to convergence

replace the location of the initial guess with the number of iterations

plot the result

Ex of the roots

End of 1st exam material

yippee

Small system of equations

N < 2,000

Use direct methods and N is limited by R.O

Large system of equations

N > 2,000
Use iterative methods. No limits on N

Gauss elimination

Its how we solve Ax=b

rearrange exactly using ERO’s Ax=b into an upper triangular form where we can find easily

det(A)

solution of Ax=b

FVM and FEM

Finite Volume Methods and Finite Element Methods

We write the equations as Ax=b

Vector norms and properties

Measures of size of vectors

||x||

a[NxN] matrix operates on an [Nx1] vector and produces another [Nx1] vector

a linear transformation

To be able to solve a₁ and a₂ must be linearly independent

Matrix Operations

Example

Transpose

Example

Inverse

Extra example

Useful Property

(AB)^-1= B^-1A^-1

Can you always solve Ax=b

No, you can only solve if A^-1 exists and this is assured if the determinant is not zero

|A|≠0 or det(A)≠0

Determinant refresher

Ex:

Cramer’s Rule

Legit take the det(A)
For larger N, Cramer’s rule isnt practical it takes N! floating point operations

Ex using cramer’s rule

ERO’s

Used to rearrange in a form where the determinant can be calculated

1. multiply row by non-zero constant (scaling which scales determinant)

2. add multiple of one row to another row (rotation about root)

3. switch two rows (reorder the way equations are written change the sign of the determinant by -1)