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SCHOOL OF MATHEMATICAL SCIENCES
MTH3051
Introduction to Computational Mathematics
Assignment 1. Long Multiplication.
Due date: Lab class for week 4
This assignment contains just one question worth 25 marks (equal
to 6% of the finalmark for this unit).
Late submissions will be subject to late penalties (see the unit
guide for fulldetails).
1. Multiplication
If you were to ask any primary school student to compute 342
times 837 using longmultiplication you could reasonably expect that
they might produce a table somewhatlike the following
342
8372394
+ 10260
+ 273600
= 286254
The layout is simple. The first two lines are the integers to be
multiplied. The nextthree lines are the results of multiplying each
digit of the first number by successivedigits of the second number.
The final line equals the sum of the previous three lines.All of
the calculations are done from right to left.
Each column in the table is meant to record just one decimal
digit and yet on occasionswe obtain a two digit number (2 by 7 =
14, 6 plus 6 = 12). In such cases we carrythe extra digits to the
next column (2 by 7 = 14, write down 4 and carry the 1). Butthere
is no reason to invoke the carry as soon as it arises, in fact all
of the carries canbe deferred to the very end. This allows us to
write the multiplication of 342 by 837 astwo tables, the first
records the successive pairs of two digit multiplications
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School of Mathematical Sciences Monash University
3 4 2
8 3 721 28 14
+ 9 12 6
+ 24 32 16
= 24 41 49 34 14
followed by the carry process
24 41 49 34 14
= 24 41 49 35 4
= 24 41 52 5 4
= 24 46 2 5 4
= 28 6 2 5 4
= 2 8 6 2 5 4
Your job is to write a single Matlab function called multiply.
Here is a skeleton example
function xy = multiply (x,y)
%
% This function computes the product of two positive integers, x
and y
% The result is returned as another integer xy.
%
% Usage: xy = multiply (x,y);
% Input: x and y positive integers stored as row vectors
% Output: xy the positive integer (row vector) equal to x times
y
... Your code goes here ...
Here is a simple example of how you would use your multiply
function
Matlab >> x = [1,2,3,4,5]; % first integer
Matlab >> y = [6,7,8]; % second integer
Matlab >> z = multiply(x,y); % do the long multiply
Setting up the row vectors for x and y is a bit tedious so I
have written a small Matlabprogram called decimaldigits.m (which
you can find on the Moodle web page). Usingthat program, the above
example would now be just
Matlab >> x = decimaldigits(12345); % first integer
Matlab >> y = decimaldigits(678); % second integer
Matlab >> z = multiply(x,y); % do the long multiply
Note that this decimaldigits function only works for integers
less than about 16 digits.So if you really need to multiply very
long numbers you may have to create the rowvectors for x and y by
hand (as in the first example above).
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School of Mathematical Sciences Monash University
Ive also written another Matlab function factorial.m (also on
Moodle). This functioncalls the multiply function to compute the
factorial of any integer. Youll find that thenumber of digits in
the factorial blows out very quickly and looks really messy
whendisplayed as a row vector on the screen. To make the output
look a bit more tidy Ivewritten one final Matlab function
longprint.m. This prints the integers in a compactformat (and you
can specify how many digits to print per line). Here is an
example
Matlab >> fact = factorial(100); % compute 100!
Matlab >> longprint(fact,30); % print 30 digits per
line
Matlab >> % can combine the functions to compute and print
63! by 47!
Matlab >>
longprint(multiply(factorial(63),factorial(47)),40);
2. What to submit
When submitting the assignment you should include
F A clear explanation of how your algorithm works (this could be
your re-wordingof the introduction given on the assignment sheet,
you do not need to explain thelines in your code that should be
evident from the comments in the code),
F Printouts of your Matlab multiply function.
F Printouts of the answers for 100! and 63! times 47!
Marks will be awarded for correctness as well as elegance (in
particular how well youdocument the working of your Matlab
programs. It should be easy for any body toread and understand your
program. So use well chosen comments, but not so detailedthat it
becomes hard to find the code!). But, and this is important, do not
rely onthe comments in your Matlab code as the primary place in
which you describe youralgorithm. There should be a separate
discussion of the algorithm including any newequations you derived.
This discussion should clearly describe your algorithm.
Marks will be lost for poorly written programs, lack of clear
and meaningfulcomments, and for an inadequate discussion.
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