====== Math 445 lab 5: functions and for loops ======


Matlab vocabulary for this lab
<code>
  for
  fprintf
  function
  mean
  std
  *
  length
  size
  rand
  randn
</code>

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

<code>
1 times pi is 3.1416
2 times pi is 6.2832
</code>

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 

<code>
-10 Farenheit is -23.3333 Celsius.
0 Farenheit is -17.7778 Celsius.
</code>

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 

<latex>
y_i = \sum_{j=1}^n A_{ij} x_j
</latex>

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

<latex>
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,
</latex>

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.