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--- gibson:teaching:spring-2016:math445:lecture:forloop [2016/02/15 09:45]
gibson [User-defined functions]
+++ gibson:teaching:spring-2016:math445:lecture:forloop [2016/02/17 07:38] (current)
gibson
@@ Line 1: / Line 1: @@
-====== Math 445 lecture 7: the "for" loop, fprintf ======
+====== Math 445 lecture 7: fprintf, for loops, functions ======
-===== Formatted printing with fprintf =====
+===== fprintf: formatted printing =====
 **fprintf** is Matlab's **formatted print function**. It's used for printing variables to the screen (or files) in a format that you specify. The basic syntax is
 <code matlab>
-fprintf("format-string", variable, variable, variable, ...)
+fprintf('format-string', variable, variable, variable, ...)
 </code>
@@ Line 18: / Line 18: @@
 </code>
-===== for loops =====
+The format string contains two //slots// ''%s'' and ''%w''. The percent sign is a special marker that signals a slot, and the character following it specifies the type of variable that goes in the slot, ''%s'' for string and ''%d'' for decimal. The values of the variables $w$ and $n$ are placed in those slots when the fprintf function is executed. The ''\n'' is a special //control character// for "new line."
+The most important slot types are
+  * ''%d'' decimal (integer), e.g. 5
+  * ''%f'' floating-point, e.g. 2.75
+  * ''%e'' floating-point in scientific (exponential) notation, e.g. 2.75e+00 for 2.75
+  * ''%s'' string (a sequence of characters), e.g. '' 'banana'''.
+  * ''%c'' character, e.g. '' 'q' ''
+The decimal and floating-point slots can be specialized to print numbers in particular ways, for example,
+<code matlab>
+>> fprintf('The value of pi is %8.3f\n', pi)
+The value of pi is    3.142
+</code>
+This prints $\pi$ with 3 digits after the decimal in a fixed-width field of 8 characters.
+You can put as many slots in a format string as you like. Just provide as many variables as slots, and make sure the types of the slots match the variables.
+===== for loops: repeating sequences of commands=====
+**for** loops are used to repeat a sequence of commands a specified number of times. A **for** loop has an index variable (often $n$ or $i$) whose value changes over every iteration of the loop. For example, this Matlab **for** loop
+<code matlab>
+for n=1:4
+   fprintf('The value of n is %d.\n',n);
+end
+</code>
+produces the output
+<code>
+The value of n is 1.
+The value of n is 2.
+The value of n is 3.
+The value of n is 4.
+</code>
+Note how the //body// of the loop (the fprintf) statement is executed four times, once for each value of $n$ from 1 to 4.
+===== functions =====
+The Matlab **function** command lets you define your own functions. For example, putting
+the following code in a file named ''quadratic.m''
+<file matlab quadratic.m>
+function y = quadratic(x)
+  y = x^2 + 3*x -7;
+end
+</file>
+defines a function named ''quadratic'' that computes the polynomial $y(x) = x^2 +3x -7$. The filename must match the name of the function. If the ''quadratic.m'' function is in Matlab's current working directory, you'll be able to execute the function just like a regular Matlab built-in function, for example
+<code>
+>> quadratic(0)
+ans =
+    -7
+>> quadratic(1)
+ans =
+    -3
+>> quadratic(12.74)
+ans =
+.5276
+</code>
+===== example functions =====
+==== sum the components of a vector ====
+This ''mysum'' function computes the sum of the components in the input vector $x$. There are a
+number of extraneous print statements which show how the calculation works, step-by-step.
+<file matlab mysum.m>
+function s = mysum(x);
+% compute the sum of the components of vector x
+  s = 0;           % variable to hold partial sum
+  n = length(x);   % get length of vector x
+  fprintf('Will need to do %d additions.\n\n', n);
+  for i=1:n
+    fprintf('The current partial sum is s = %f.\n\n', s);
+    pause
+    fprintf('Add %f.\n', x(i));
+    s = s + x(i);
+  end
+  fprintf('The total sum is s = %f.\n', s);
+  pause
+end
+</file>
+You can execute this function with on the input vector ''[4 2 9 1]]'' like this
+<code>
+>> mysum([4 2 9 1])
+Will need to do 4 additions.
+The current partial sum is s = 0.000000.
+Add 4.000000.
+The current partial sum is s = 4.000000.
+Add 2.000000.
+The current partial sum is s = 6.000000.
+Add 9.000000.
+The current partial sum is s = 15.000000.
+Add 1.000000.
+The total sum is s = 16.000000.
+ans =
+>>
+</code>
+The function is a lot easier to read if you take out the print statements
+<file matlab mysum.m>
+function s = mysum(x);
+% compute the sum of the components of vector x
+  s = 0;           % variable to hold partial sum
+  n = length(x);   % get length of vector x
+  for i=1:n        % for each number i from 1 to n
+    s = s + x(i);  %    add x(i) to the partial sum
+  end
+end
+</file>
+==== compute the factorial ====
+This function computes $n!$, the factorial of $n$, according to the formula
+\begin{eqnarray*}
+n! = \prod_{i=1}^n i = 1\cdot 2 \cdot 3 \cdot \ldots \cdot (n-2) \cdot (n-1) \cdot n
+\end{eqnarray*}
+<file matlab myfactorial.m>
+function p = myfactorial(n);
+% compute factorial of n = 1 * 2 * ... (n-2)*(n-1)*n
+  p = 1;   % variable to store partial products
+  % each step through the loop does one more multiplication
+  % in the sequence of products 1, 1*2, 1*2*3, etc.
+  for i=1:n
+    p = p * i;
+  end
+end
+</file>

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