Symbolic Math Toolbox

Simple Example

Suppose we want to write an M-file that takes two polynomials or two integers and returns their greatest common divisor. For example, the greatest common divisor of 14 and 21 is 7. The greatest common divisor of x^2-y^2 and x^3-y^3 is x - y.

The first thing we need to know is how to call the greatest common divisor function in Maple. We use the mhelp function to bring up the Maple online help for the greatest common divisor (gcd).

Let's try the gcd function

• mhelp gcd
  

which returns

• gcd - greatest common divisor of polynomials
  
  lcm - least common multiple of polynomials
  
  Calling Sequence:
       gcd(a,b,'cofa','cofb')
       lcm(a,b,...)
  
  Parameters:
       a, b      - multivariate polynomials over an algebraic number
                   field or an algebraic function field 
       cofa,cofb - (optional) unevaluated names 
  
  Description:
  - The gcd function computes the greatest common divisor of two 
  polynomials a and b. 
  
  - If the coefficients of a and b are integers, then a primitive 
  unit normal greatest common divisor is returned. In other words, 
  the coefficients of the result are relatively prime integers and 
  the leading coefficient is a positive integer. 
  
  - If the coefficients of a or b are rational numbers or belong to 
  an algebraic number or function field, then the monic greatest 
  common divisor of a and b is computed. See type,algnum and 
  type,algfun.
  
  - Algebraic numbers and functions may be represented by radicals 
  (see type,radical) or with the RootOf notation. See evala.
  
  - Names occurring inside a RootOf or a radical are viewed as 
  elements of the coefficient field, provided the RootOf defines an 
  algebraic function. Therefore, they may occur in denominators as 
  well. Other names are not allowed in denominators. 
  
  - If a or b contains objects that are not algebraic numbers or 
  algebraic functions, these objects will be frozen before the 
  computation proceeds. See frontend.
  
  - The RootOf and the radicals defining the algebraic numbers must 
  form an independent set of algebraic quantities, otherwise an 
  error is returned. Note that this condition needs not be satisfied 
  if the expression contains only algebraic numbers in radical 
  notation (i.e. 2^(1/2), 3^(1/2), 6^(1/2)). A basis over Q for the 
  radicals can be computed by Maple in this case. 
  
  - Since the ordering of the variables depends on the session, the 
  result may also depend on the session when a and b have several 
  variables. 
  
  - The lcm function computes the least common multiple of an 
  arbitrary number of polynomials. 
  
  - The optional third argument cofa is assigned the cofactor 
  a/gcd(a,b). 
  
  - The optional fourth argument cofb is assigned the cofactor 
  b/gcd(a,b). 
  .
  .
  .
  

Since we now know the Maple calling syntax for gcd, we can write a simple M-file to calculate the greatest common divisor. First, create the M-file gcd in the @sym directory and include the commands below.

• function g = gcd(a, b)
  g = maple('gcd',a, b);
  

If we run this file

• syms x y
  z = gcd(x^2-y^2,x^3-y^3)
  w = gcd(6, 24)
  

we get

• z = 
  -y+x
  
  w = 
  6
  

Now let's extend our function so that we can take the gcd of two matrices in a pointwise fashion:

• function g = gcd(a,b)
  
  if any(size(a) ~= size(b))
    error('Inputs must have the same size.')
  end
  
  for k = 1: prod(size(a))
    g(k) = maple('gcd',a(k), b(k));
  end
  
  g = reshape(g,size(a));
  

Running this on some test data

• A = sym([2 4 6; 3 5 6; 3 6 4]);
  B = sym([40 30 8; 17 60 20; 6 3 20]);
  gcd(A,B)
  

we get the result

• ans =
  [ 2, 2, 2 ]
  [ 1, 5, 2 ]
  [ 3, 3, 4 ]
  

Using Maple Functions

Vectorized Example