Math 445 lab 5: functions and for loops

Matlab vocabulary for this lab

  for
  fprintf
  function
  mean
  std
  *
  length
  size
  rand
  randn

Problem 1: Write a for loop that will print statements of the form

1 times pi is 3.1416
2 times pi is 6.2832

for the numbers 1 through 10.

Problem 2: If $F$ is a temperature in Farenheit, then $C = \frac{5}{9}(F-32)$ is the same temperature in Celsius. Write a function farenheit2celsius that takes a Farenheit temperature as input, converts it to Celsius, prints a statement of the form 20 Farenheit is -6.6667 Celsius., and returns the Celsius value as its output.

Problem 3: Write a for loop that uses the above function to print a list of statements

-10 Farenheit is -23.3333 Celsius.
0 Farenheit is -17.7778 Celsius.

for Farenheit temperatures from -10 to 100, in steps of 10. Challenge: Get the text to line up nicely using field-width specifiers in fprintf!

Problem 4: Write a function average that takes an input vector $x$ and returns as output the average (mean) value of its elements, according to the formula

$\begin{equation*} \text{average}(x) = \frac{1}{N} \sum_{i=1}^N x_i \end{equation*}$

where $N$ is the number of elements in $x$ . Test your function by comparing its output to the output of the built-in Matlab function mean on a random vector of length 100 whose elements are uniformly distributed between 0 and 1.

Problem 5: Write a function deviation that takes an input vector $x$ and returns the standard deviation of its elements, according to the formula

$\begin{equation*} \text{deviation}(x) = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{X})^2} \end{equation*}$

where $\bar{x}$ is the average the elements of $x$ , and $N$ is the number of elements in $x$ . Test your function by comparing its output to the output of the built-in Matlab function std on a random vector of length 100 whose elements are in a normal (Gaussian) distribution about $x=0$ .

Problem 6: The formula for matrix-vector multiplication $y = Ax$ is

$y_i = \sum_{j=1}^n A_{ij} x_j$

In this formula, $A$ is an $m \times n$ matrix, $x$ is an $n$ -dimensional column vector, and $y$ is an $m$ -dimensional column vector.

Write a function matvecmult that takes a matrix $A$ and a vector $x$ as inputs, computes $y = Ax$ according to that formula, and returns the vector $y$ . Compare your matvecmult to Matlab's built-in matrix-vector multiplication operator * on a random $4 \times 4$ matrix and a random 4d column vector.

Problem 7: The formula for matrix-matrix multiplication $C = AB$ of an $m \times n$ matrix $A$ and an, $n \times p$ matrix $B$ is

$C_{i,j} = \sum_{k=1}^n A_{ik} B_{k,j} \quad \text{ for } i = 1,\dots,m, \text{ and } j = 1,\dots,p,$

The product $C = AB$ is an $m \times p$ matrix.

Write a function matmatmult that takes a matrix $A$ and a vector $x$ as inputs, computes $C = AB$ according to that formula, and returns the matrix $C$ . Compare your matmatmult to Matlab's built-in matrix-vector multiplication operator * on a random $4 \times 3$ matrix $A$ and a random $3 \times 5$ matrix B.

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Math 445 lab 5: functions and for loops

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