# MuPAD

There are a few basic number theory functions in the kernel;
others are contained in the **numlib**
library.

>> isprime(997); TRUE >> Factor(2^67-1); 193707721 761838257287 >> nextprime(1000000); 1000003 >> powermod(9382471,322973,1298377); 880825 >> phi(nextprime(2^20)-1); 498400

Here **phi** is Euler's totient function
returning the number of integers less than and relatively prime to
its argument. These functions allow us to perform simple RSA
encryption and decryption. Suppose we choose two primes and compute
their product:

>> p:=nextprime(5678); 5683 >> q:=nextprime(6789); 6791 >> N:=p*q; 38593253Now we have to choose an integer e relatively prime to (p-1)*(q-1); a smaller prime will do; say e:=17.

>> e:=17:The values e and N are our “public key”. Now we find the d, the inverse of e modulo (p-1)*(q-1). This is very easily done using the convenient overloading of the reciprocal function:

>> d:=1/e mod (p-1)*(q-1); 6808373Suppose someone wishes to send us a message M < N; say

>> M:=24367139;They can encrypt it using our public key values:

>> M1:=powermod(M,e,N); 18476508We can now decrypt this using the value d (and N):

>> powermod(M1,d,N); 24367139This is indeed the value of the original message.

We have seen a glimpse of MuPAD's symbolic abilities in the equation solving above. But MuPAD can do much more than this: all manner of algebraic simplification; rewriting in a different form; partial fractions; and so on.

>> expand((x+2*y-3*z)^4); 4 4 4 3 3 3 3 3 x + 16 y + 81 z + 32 x y + 8 x y - 108 x z - 12 x z - 216 y z - 3 2 2 2 2 2 2 2 96 y z + 216 x y z - 144 x y z - 72 x y z + 24 x y + 54 x z + 2 2 216 y z >> Factor(%); 4 (x + 2 y - 3 z) >> sum(1/(k*(k+2)*(k+4)),k=1..n); 2 3 4 310 n + 337 n + 110 n + 11 n ---------------------------------------- 2 3 4 4800 n + 3360 n + 960 n + 96 n + 2304 >> partfrac(%); 1 1 1 1 --------- - --------- - --------- + --------- + 11/96 8 (n + 3) 8 (n + 1) 8 (n + 2) 8 (n + 4) >> normal(%); 2 3 4 310 n + 337 n + 110 n + 11 n ---------------------------------------- 2 3 4 4800 n + 3360 n + 960 n + 96 n + 2304 >> Factor(%); 2 n (n + 5) (55 n + 11 n + 62) ---------------------------------- 96 (n + 1) (n + 2) (n + 3) (n + 4) >> sum(sin(k*x),k=1..n); (I exp(-I x) - I exp(I x) + I exp(-I n x) - I exp(I n x) - I exp(- I x - I n x) + I exp(I x + I n x)) / 4 - 2 exp(-I x) - 2 exp(I x) >> rewrite(%,sincos); (2 sin(x) + 2 sin(n x) + I cos(x + n x) - 2 sin(x + n x) - I cos(- x - n x) ) / 4 - 4 cos(x)

MuPAD's calculus skills are excellent. It implements very powerful algorithms for differentiation, integration, and limits.

>> diff(x^3,x); 2 3 x >> diff(exp(exp(x)),x$4); 2 3 exp(x) exp(exp(x)) + 7 exp(x) exp(exp(x)) + 6 exp(x) exp(exp(x)) + 4 exp(x) exp(exp(x))

The dollar operator, **$**, is MuPAD's
sequencing operator. As with most operators, it is overloaded; in
the context of a derivative it is interpreted as a multiple
derivative. We can of course also perform partial differentiation.

>> int(sec(x),x); ln(2 sin(x) + 2) ln(2 - 2 sin(x)) ---------------- - ---------------- 2 2 >> int(cos(x)^3, x=-PI/4..PI/3); 1/2 1/2 5 2 3 3 ------ + ------ 12 8 >> int(E^(-x^2),x=0..0.5); / 1 \ int| --------, x = 0..0.5 | | 2 | | x | \ exp(1) / >> float(%); 0.461281006412792448755702936740453103083759088964291146680472565934983884\ 2952938567126622486999424745If we require only a numeric result, then we don't want to force MuPAD to attempt a symbolic or exact solution first. In such a case we may use the

**hold**command, which returns the input unevaluated, but “holds” onto it for the purposes of later evaluation. Thus we may enter:

>> hold(int(exp(-x^2),x=0..0.5));followed by the

**float**command.

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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.