# 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); PuiseuxThis 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.

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Thank you to our 2014 Editorial Advisors!

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## Comments

## Sellout

Mupad has been bought out by mathworks and all code is now under matlab (junk) licence.

any and all open source work is now dead.

## Thankyou for a well written a

Thankyou for a well written article. TeXmacs acts as an excellent interface to mupad. I assume that the TeXmacs screen display generated by TeX. The graphics is generated by javaview. The combination of TeXmacs and javaview greatly enhance the mupad experience.