MuPAD

The default order of a power series is 6. This can be changed either by changing the value of ORDER (analogous to DIGITS above), or by including the order in the series command. This last inclusion is optional.

>> tx:=series(tan(x),x,12);
                  3      5       7       9         11
                 x    2 x    17 x    62 x    1382 x        12
             x + -- + ---- + ----- + ----- + -------- + O(x  )
                 3     15     315    2835     155925
>> cx:=series(cos(x),x,12);
                    2    4    6      8        10
                   x    x    x      x        x          12
               1 - -- + -- - --- + ----- - ------- + O(x  )
                   2    24   720   40320   3628800
>> tx*cx;
                  3    5      7       9        11
                 x    x      x       x        x           12
             x - -- + --- - ---- + ------ - -------- + O(x  )
                 6    120   5040   362880   39916800

This certainly looks like the series for sin(x), but let's see if MuPAD recognizes it as such.

>> sx:=series(sin(x),x,12):
>> is(tx*cx=sx);
                                   TRUE
>> type(tx*cx);
                                  Puiseux

This means that the result of the series product is recognized by MuPAD as an object of type “Puiseux”; that is, a series possibly containing fractional powers.

MuPAD can also deal with polynomials.

>> p1:=x^8-3*x^5+11*x^4-x^2+17;
                            4    2      5    8
                        11 x  - x  - 3 x  + x  + 17
>> p2:=x^3-23*x^2-4*x+11;
                            3             2
                           x  - 4 x - 23 x  + 11
>> divide(p1,p2);
                  2        3       4    5
285641 x + 12337 x  + 533 x  + 23 x  + x  + 6613228,
                           2
   23310861 x + 153111100 x  - 72745491

The result of the last command consists of two terms: the quotient, and the remainder. MuPAD also has commands for extracting coefficients from polynomials, evaluating polynomials using Horner's algorithm, and lots more.

We shall first create a matrix domain:

>> M:=Dom::Matrix();
               Dom::Matrix(Dom::ExpressionField(id, iszero))

The result returned here indicates that MuPAD expects that the elements of matrices will be members of a field for which no normalization is performed (the function id just returns elements as given), and for which 0 is recognized as the zero value.

Now we shall make all the commands in the library linalg available to us:

>> export(linalg);

Now we shall create a few matrices and play with them. First, a matrix with given elements.

>> A:=M([[1,2,3],[-1,3,-2],[4,-5,2]]);
                             +-            -+
                             |   1,  2,  3  |
                             |              |
                             |  -1,  3, -2  |
                             |              |
                             |   4, -5,  2  |
                             +-            -+

We have entered the matrix elements as a list of lists (a list, in MuPAD, is delimited by square brackets).

Next, a matrix with elements randomly chosen to be between -9 and 9. We do this by applying the function returned by random to each of the elements.

>> B:=M(3,3,func(random(-9..9)(),i,j));
                              +-           -+
                              |  -7, -5, 8  |
                              |             |
                              |   3, -1, 7  |
                              |             |
                              |   3, -5, 6  |
                              +-           -+

Clearly this approach can be used to generate any matrix whose elements are functions of their row and column values. There is a randomMatrix command in the linalg library, but it requires the elements to be members of a coefficient ring. For our purposes, it is as easy to roll our own.

>> A*B;
                            +-              -+
                            |   8,  -22, 40  |
                            |                |
                            |   10,  12,  1  |
                            |                |
                            |  -37, -25,  9  |
                            +-              -+
>> det(A);
                                    -37
>> 1/A;
                         +-                     -+
                         |  4/37,  19/37, 13/37  |
                         |                       |
                         |  6/37,  10/37,  1/37  |
                         |                       |
                         |  7/37, -13/37, -5/37  |
                         +-                     -+

As we have seen above, MuPAD supports operator overloading, which means that since A is a matrix, 1/A is interpreted as the inverse of A.

>> A^10;
                   +-                                 -+
                   |   19897010, -20429930,  22281963  |
                   |                                   |
                   |  -42711893,  43857348, -47862790  |
                   |                                   |
                   |   64993856, -66730117,  72852811  |
                   +-                                 -+
>> b:=M(3,1,[7,9,-21]);
                                 +-     -+
                                 |   7   |
                                 |       |
                                 |   9   |
                                 |       |
                                 |  -21  |
                                 +-     -+

Here the first two (optional) values give the number of rows and columns of the matrix, the matrix elements are then given in a single list. If the list isn't long enough, the remaining values will default to zero.

>> linearSolve(A,b);
                                 +-    -+
                                 |  -2  |
                                 |      |
                                 |   3  |
                                 |      |
                                 |   1  |
                                 +-    -+
>> AM:=A.b;
                           +-                 -+
                           |   1,  2,  3,  7   |
                           |                   |
                           |  -1,  3, -2,  9   |
                           |                   |
                           |   4, -5,  2, -21  |
                           +-                 -+

The . operator is concatenation. Again, this is an overloaded operator, as it will work for other data types as well.

>> gaussJordan(AM);
                             +-             -+
                             |  1, 0, 0, -2  |
                             |               |
                             |  0, 1, 0,  3  |
                             |               |
                             |  0, 0, 1,  1  |
                             +-             -+

The linalg library is very full-featured, and contains plenty of commands for operating on matrices and vectors: row and column operations; matrix factorization and decomposition; commands for dealing with matrix polynomials and eigensystems; and so on.