# 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.

As Linux continues to play an ever increasing role in corporate data centers and institutions, ensuring the integrity and protection of these systems must be a priority. With 60% of the world's websites and an increasing share of organization's mission-critical workloads running on Linux, failing to stop malware and other advanced threats on Linux can increasingly impact an organization's reputation and bottom line.

Sponsored by Bit9

Most companies incorporate backup procedures for critical data, which can be restored quickly if a loss occurs. However, fewer companies are prepared for catastrophic system failures, in which they lose all data, the entire operating system, applications, settings, patches and more, reducing their system(s) to “bare metal.” After all, before data can be restored to a system, there must be a system to restore it to.

In this one hour webinar, learn how to enhance your existing backup strategies for better disaster recovery preparedness using Storix System Backup Administrator (SBAdmin), a highly flexible bare-metal recovery solution for UNIX and Linux systems.

Sponsored by Storix

## Trending Topics

Nmap—Not Just for Evil! | Mar 05, 2015 |

Resurrecting the Armadillo | Mar 04, 2015 |

March 2015 Issue of Linux Journal: System Administration | Mar 02, 2015 |

March 2015 Video Preview | Mar 02, 2015 |

High-Availability Storage with HA-LVM | Feb 26, 2015 |

DNSMasq, the Pint-Sized Super Dæmon! | Feb 24, 2015 |

- Nmap—Not Just for Evil!
- Resurrecting the Armadillo
- High-Availability Storage with HA-LVM
- March 2015 Issue of Linux Journal: System Administration
- Real-Time Rogue Wireless Access Point Detection with the Raspberry Pi
- DNSMasq, the Pint-Sized Super Dæmon!
- Localhost DNS Cache
- Days Between Dates: the Counting
- The Usability of GNOME
- Linux for Astronomers

## 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.