Základy_práce.sagews

%typeset_mode True

Komentář a matematika $x^2+y^2$.

1+1;#komentar

2

 #Příkaz provedeme klávesou Shift+Enter, odřádkování Enter - změna v nastavení účtu

$\displaystyle 2$

1+1;2+2 #Více příkazů na jednom řádku oddělujeme středníkem (;)

factor?

/ext/sage/sage-8.0/local/lib/python2.7/site-packages/urllib3/contrib/pyopenssl.py:46: DeprecationWarning: OpenSSL.rand is deprecated - you should use os.urandom instead import OpenSSL.SSL

File: /ext/sage/sage-8.0/local/lib/python2.7/site-packages/sage/arith/misc.py
Signature : factor(n, proof=None, int_=False, algorithm='pari', verbose=0, **kwds)
Docstring :
Returns the factorization of "n".  The result depends on the type
of "n".

If "n" is an integer, returns the factorization as an object of
type "Factorization".

If n is not an integer, "n.factor(proof=proof, **kwds)" gets
called. See "n.factor??" for more documentation in this case.

Warning: This means that applying "factor" to an integer result
  of a symbolic computation will not factor the integer, because it
  is considered as an element of a larger symbolic ring.EXAMPLES:

     sage: f(n)=n^2
     sage: is_prime(f(3))
     False
     sage: factor(f(3))
     9

INPUT:

* "n" - an nonzero integer

* "proof" - bool or None (default: None)

* "int_" - bool (default: False) whether to return answers as
  Python ints

* "algorithm" - string

  * "'pari'" - (default) use the PARI c library

  * "'kash'" - use KASH computer algebra system (requires the
    optional kash package be installed)

  * "'magma'" - use Magma (requires magma be installed)

* "verbose" - integer (default: 0); PARI's debug variable is set
  to this; e.g., set to 4 or 8 to see lots of output during
  factorization.

OUTPUT:

* factorization of n

The qsieve and ecm commands give access to highly optimized
implementations of algorithms for doing certain integer
factorization problems. These implementations are not used by the
generic factor command, which currently just calls PARI (note that
PARI also implements sieve and ecm algorithms, but they aren't as
optimized). Thus you might consider using them instead for certain
numbers.

The factorization returned is an element of the class
"Factorization"; see Factorization?? for more details, and examples
below for usage. A Factorization contains both the unit factor (+1
or -1) and a sorted list of (prime, exponent) pairs.

The factorization displays in pretty-print format but it is easy to
obtain access to the (prime,exponent) pairs and the unit, to
recover the number from its factorization, and even to multiply two
factorizations. See examples below.

EXAMPLES:

   sage: factor(500)
   2^2 * 5^3
   sage: factor(-20)
   -1 * 2^2 * 5
   sage: f=factor(-20)
   sage: list(f)
   [(2, 2), (5, 1)]
   sage: f.unit()
   -1
   sage: f.value()
   -20
   sage: factor( -next_prime(10^2) * next_prime(10^7) )
   -1 * 101 * 10000019

   sage: factor(-500, algorithm='kash')      # optional - kash
   -1 * 2^2 * 5^3

   sage: factor(-500, algorithm='magma')     # optional - magma
   -1 * 2^2 * 5^3

   sage: factor(0)
   Traceback (most recent call last):
   ...
   ArithmeticError: factorization of 0 is not defined
   sage: factor(1)
   1
   sage: factor(-1)
   -1
   sage: factor(2^(2^7)+1)
   59649589127497217 * 5704689200685129054721

Sage calls PARI's factor, which has proof False by default. Sage
has a global proof flag, set to True by default (see
"sage.structure.proof.proof", or proof.[tab]). To override the
default, call this function with proof=False.

   sage: factor(3^89-1, proof=False)
   2 * 179 * 1611479891519807 * 5042939439565996049162197

   sage: factor(2^197 + 1)  # long time (2s)
   3 * 197002597249 * 1348959352853811313 * 251951573867253012259144010843

Any object which has a factor method can be factored like this:

   sage: K.<i> = QuadraticField(-1)
   sage: factor(122 - 454*i)
   (-3*i - 2) * (-i - 2)^3 * (i + 1)^3 * (i + 4)

To access the data in a factorization:

   sage: f = factor(420); f
   2^2 * 3 * 5 * 7
   sage: [x for x in f]
   [(2, 2), (3, 1), (5, 1), (7, 1)]
   sage: [p for p,e in f]
   [2, 3, 5, 7]
   sage: [e for p,e in f]
   [2, 1, 1, 1]
   sage: [p^e for p,e in f]
   [4, 3, 5, 7]

Sage má automatické doplňování příkazů. Zkuste napsat do následujícího políčka text fac a stisknout tabelátor. Ukážou se všechny příkazy začínající těmito slovy. Vyberte si kterýkoliv z nich, připište za něj otazník  a stiskněte opět tabelátor. Objeví se nápověda k tomutu příkazu.

factor
v1=factorial(20)-factorial(12);v1

<function factor at 0x7f0ae7e6a668>

$\displaystyle 2432902007697638400$

factor(v1);

$\displaystyle 2^{10} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11^{3} \cdot 41976119$

v1.next_prime()

$\displaystyle 2432902007697638557$

(2^30/3^20)*sqrt(2) #Sage pracuje v přesné aritmetice.

$\displaystyle \frac{1073741824}{3486784401} \, \sqrt{2}$

(2^30/3^20)*sqrt(2).n()

$\displaystyle 0.435501618497698$

(2^30/3^20)*sqrt(2).n(digits=100)

$\displaystyle 0.4355016184976975426780034699980734981419044126751400227013960351069068541814735297648401904204298624$

n((2^30/3^20)*sqrt(2))

$\displaystyle 0.435501618497698$

var('i') #Narozdíl od Maplu a Maximy je třeba proměnné (kromě proměnné x) předem deklarovat.

$\displaystyle i$

sum((1+i)/(1+i^4), i,1,20)

$\displaystyle \frac{27531089155139512664184734944799665266675157706349981434584200143}{21653762264974190884124027285387789492160976952723410943046016049}$

N(sum((1+i)/(1+i^4), i,1,20), digits=30)

$\displaystyle 1.27142289724276359903349287003$

sum((1+i)/(1+i^4), i,1,20).n(digits=30)

$\displaystyle 1.27142289724276359903349287003$

var('j')

$\displaystyle j$

sum((1+j)/(1+j^4), j,1,20)

$\displaystyle \frac{27531089155139512664184734944799665266675157706349981434584200143}{21653762264974190884124027285387789492160976952723410943046016049}$

N(pi);n(pi)

$\displaystyle 3.14159265358979$

$\displaystyle 3.14159265358979$

((3+5*I)*(7+4*I)); #Komplexní jednotku zadáváme pomocí I.

$\displaystyle 47 i + 1$

var('y')

$\displaystyle y$

(x+y)^3*(x-y)^2

$\displaystyle {\left(x + y\right)}^{3} {\left(x - y\right)}^{2}$

((x+y)^3*(x-y)^2).expand() #roznásobení závorek

$\displaystyle x^{5} + x^{4} y - 2 \, x^{3} y^{2} - 2 \, x^{2} y^{3} + x y^{4} + y^{5}$

((x+y)^3*(x-y)^2).expand().factor() #rozklad na součin

$\displaystyle {\left(x + y\right)}^{3} {\left(x - y\right)}^{2}$

simplify(x/2+x/5) #zjednoduseni vyrazu

$\displaystyle \frac{7}{10} \, x$

Uložení výrazu do proměnné.

f=(x^3-x)/(x^4-1);f

$\displaystyle \frac{x^{3} - x}{x^{4} - 1}$

f.simplify_full()

$\displaystyle \frac{x}{x^{2} + 1}$

vyraz=sin(x)^2+cos(x)^2;vyraz.simplify_trig()

$\displaystyle 1$

(cos(x)^2).trig_reduce()

$\displaystyle \frac{1}{2} \, \cos\left(2 \, x\right) + \frac{1}{2}$

Jedno znaméno = se v programu Sage používá pro přiřazení hodnoty do proměnné. Pro rovnice se používají dvě rovnítka za sebou.

x^2-7==0 #rovnice

$\displaystyle x^{2} - 7 = 0$

(x^2-7==0).rhs();(x^2-7==0).lhs() #prava a leva strana

$\displaystyle 0$

$\displaystyle x^{2} - 7$

solve(x^2-7==0, x)

[$\displaystyle x = -\sqrt{7}$, $\displaystyle x = \sqrt{7}$]

Získání všech řešení.

solve(sin(3*x+pi/3)==1/2,x) #neziskame vsechna reseni

[$\displaystyle x = -\frac{1}{18} \, \pi$]

solve(sin(3*x+pi/3)==1/2,x,to_poly_solve='force')

[$\displaystyle x = \frac{1}{6} \, \pi + \frac{2}{3} \, \pi z_{72}$, $\displaystyle x = -\frac{1}{18} \, \pi + \frac{2}{3} \, \pi z_{74}$]

eq1=y==3*x+1;eq2=y==x-1;eq1;eq2

$\displaystyle y = 3 \, x + 1$

$\displaystyle y = x - 1$

sol=solve([eq1,eq2],x,y); sol #reseni soustavy rovnic

[[$\displaystyle x = \left(-1\right)$, $\displaystyle y = \left(-2\right)$]]

Práce s $n$-ticemi

A=solve(x^4-2*x^2==0,x);A

[$\displaystyle x = -\sqrt{2}$, $\displaystyle x = \sqrt{2}$, $\displaystyle x = 0$]

A[0] #prvni prvek (index 0)

$\displaystyle x = -\sqrt{2}$

A[1]

$\displaystyle x = \sqrt{2}$

Deklarace funkce

f(x)=x^2

f

$\displaystyle x \ {\mapsto}\ x^{2}$

f(4)

$\displaystyle 16$

f=x^2 #deklarace f jako vyrazu, ne funkce

f

$\displaystyle x^{2}$

f(x=4)

$\displaystyle 16$

f.subs(x=4)

$\displaystyle 16$

Grafika

plot(x * sin(x), (x, -2, 10))

plot(ln(x), (x,0,10))

plot(e^x, (x,-3,5),ymax=100,ymin=0)

plot(tan(x), (x,-3,3), ymax=10, ymin=-10,detect_poles=True)

Popis obrázku

p1=plot(sin(x), x, -pi, pi);show(p1)

p2=text("$y=\sin\, x$", (2.75,0.75), color="black", fontsize="large")

text?

p1+p2

(p1+p2).save("sin.pdf") #Export do PDF formátu

Animace

t = var('t');
a = animate((sin(c*pi*t) for c in sxrange(1,2,.2)));
a.show()

Interaktivní obrázky

@interact
def interactive_function(a = slider(-5, 5, 1, default=4)):
    f(x) = a*x^2
    plot(f, (x, -5, 5), figsize=3).show()

Interact: please open in CoCalc

plot(e^x+ln(abs(4-x)), (x,0,5))

plot(e^x+ln(abs(4-x)), (x,3.99,4.01),detect_poles=True)

rov1=sqrt(x-2)==sqrt(2*x-3);rov1

$\displaystyle \sqrt{x - 2} = \sqrt{2 \, x - 3}$

res1=solve(rov1, x, to_poly_solve='force');res1

[$\displaystyle x = 1$]

p1=plot(sqrt(x-2), (x,2,7));p2=plot(sqrt(2*x-3), (x,1.5,7));p1+p2

rov1.subs(x=(res1[0].rhs()))

$\displaystyle i = i$

$\displaystyle \log\left(x + 4\right) + \log\left(x - 2\right) = \log\left(x + 3\right)$

res2=solve(rov2,x, to_poly_solve='force');res2

[$\displaystyle x = -\frac{3}{2} \, \sqrt{5} - \frac{1}{2}$, $\displaystyle x = \frac{3}{2} \, \sqrt{5} - \frac{1}{2}$]

((res2[0])).rhs().n();((res2[1])).rhs().n()

$\displaystyle -3.85410196624968$

$\displaystyle 2.85410196624968$

p1=plot(log(x+4)+log(x-2), (x,2,4));p2=plot(log(x+3), (x,-3,4));p1+p2

solve(2*(2^x)^2-7*2^x+3==0, x, to_poly_solve='force')

[]

plot(2*(2^x)^2-7*2^x+3, (x,-2,2))

solve(2*(y)^2-7*y+3==0, y, to_poly_solve='force')

[$\displaystyle y = \left(\frac{1}{2}\right)$, $\displaystyle y = 3$]

solve(2^x==3, x);solve(2^x==1/2,x)

[$\displaystyle x = \frac{\log\left(3\right)}{\log\left(2\right)}$]

[$\displaystyle x = \left(-1\right)$]

(solve(2^x==3, x)[0]).rhs().n()

$\displaystyle 1.58496250072116$