Ch1.1 Numerical Analysis

Numerical Analysis: Widely Used

Numerical analysis underlies many fields which require numerical answers to mathematical questions.
From video games to climate science, engineering & physics to psychology, numerical analysis performs the calculations we use daily.
Medical professionals use numerical analysis to process advanced imagery.
Numerical analysis supports almost any field when computers are used to solve mathematical problems.

Numerical Analysis Constraints

Limited amount of computer memory, for storing and processing data.
Computers only represent a finite collection of numbers.
Most numbers on the number line \( (-\infty, \infty) \), or even just \( [0,1] \), are absent from computer's number line. Thus most computed results are estimates of the actual solution.
Another constraint is computational time. Running algorithms for numerical methods, data storage and access, and so on, takes time.
Fast and accurate numerical methods are not always readily available, and may take time for numerical analysts and programmers to create.

Numerical View of Functions

In precalculus, we learn that a function can be represented in terms of a formula, a graph and/or a table of values (x-y table).
The table of values is considered a numerical representation of the function.
Numerical values for a function can be found by data collection or by using formulas (if one is available).
In computational science, numerical representations are often the only way solutions to equations can be found.

x	y
0	1
1	3
2	5

Example: Representations of a Function

For \( f(x) = 2x+1 \):

x <- c(0,1,2)
f <- 2*x+1
f

[1] 1 3 5

x	y
0	1
1	3
2	5

title

Numerical Approach

Using analytical methods (algebra, formulas from calculus and other math classes) to solve for exact solutions are preferred.
However, numerical methods are often the only way to solve problems in science, engineering, business, etc.

https://calcworkshop.com title

x <- c(0,1,2)
f <- 2*x+1
f

[1] 1 3 5

Numerical Methods vs Numerical Analysis

Numerical methods is concerned with the different formulas available for common problems in computational science.
Numerical analysis is concerned with analzying how fast, efficient and accurate the different numerical methods are.

title

Example: Square Root of 2

In figure below, the Babylonian method for approximating \( \sqrt{2} \) is stable, fast and accurate (but not Method X…).

\[ \begin{align*} x_{k+1} & = \frac{x_{k}}{2} + \frac{1}{x_k} \\ x_{k+1} & = \left(x_k^2 -2\right)^2 + x_k \end{align*} \]

https://en.wikipedia.org/wiki/Numerical_analysis title

R Code for Babylonian Method

babylonian <- function(x0,m) {
  # x0 = initial value
  # m = number of iterations

  # Iteration function:
  g <- function(x){x/2 + 1/x}

  # Start by displaying initial value
  cat("n = ", 0,",", "x(n) = ", x0, "\n")

  # Iteration loop
  for(n in 1:m) {
    x1 <- g(x0)
    x0 <- x1
    cat("n = ", n,",", "x(n) = ", x1, "\n")  }
  }

Example: Babylonian Method

title

\[ \small{x_{k+1} = \frac{x_{k}}{2} + \frac{1}{x_k}} \]

sqrt(2)

[1] 1.414214

babylonian(1.4,3)

n =  0 , x(n) =  1.4 
n =  1 , x(n) =  1.414286 
n =  2 , x(n) =  1.414214 
n =  3 , x(n) =  1.414214

babylonian(1.42,3)

n =  0 , x(n) =  1.42 
n =  1 , x(n) =  1.414225 
n =  2 , x(n) =  1.414214 
n =  3 , x(n) =  1.414214

R Code for Method X

methodX <- function(x0,m) {
  # x0 = initial value
  # m = number of iterations

  # Iteration function:
  g <- function(x){(x^2 -2)^2 + x}

  # Start by displaying initial value
  cat("n = ", 0,",", "x(n) = ", x0, "\n")

  # Iteration loop
  for(n in 1:m) {
    x1 <- g(x0)
    x0 <- x1
    cat("n = ", n,",", "x(n) = ", x1, "\n")  }
  }

Example: Method X

title

\[ \small{x_{k+1} = \left(x_k^2 -2\right)^2 + x_k} \]

sqrt(2)

[1] 1.414214

methodX(1.4,8)

n =  0 , x(n) =  1.4 
n =  1 , x(n) =  1.4016 
n =  2 , x(n) =  1.402861 
n =  3 , x(n) =  1.403884 
n =  4 , x(n) =  1.404732 
n =  5 , x(n) =  1.405446 
n =  6 , x(n) =  1.406057 
n =  7 , x(n) =  1.406586 
n =  8 , x(n) =  1.407049

Example: Method X

title

\[ \small{x_{k+1} = \left(x_k^2 -2\right)^2 + x_k} \]

sqrt(2)

[1] 1.414214

methodX(1.42,8)

n =  0 , x(n) =  1.42 
n =  1 , x(n) =  1.420269 
n =  2 , x(n) =  1.420564 
n =  3 , x(n) =  1.420888 
n =  4 , x(n) =  1.421246 
n =  5 , x(n) =  1.421643 
n =  6 , x(n) =  1.422087 
n =  7 , x(n) =  1.422586 
n =  8 , x(n) =  1.42315

Numerical Computation & Error Analysis

Numerical analysis is more than the search for numerical solutions to mathematical problems.
It is also the search for usable estimates of those solutions.
Sometimes an error analysis is more extensive than coding the numerical method that produces the solution.
From there, we can evaluate how usable our estimate is.