FirstSteps.sagews

#Getting help
#with specific command: type the command name, append the question mark. E like solve?
#Example
solve?
#Help for specific topic
#Example
search_doc("numerical approximation")

reset

<function reset at 0x7f91e82fe050>

# SageMath as a calculator

15*0.7^3

5.14500000000000

32^15

37778931862957161709568

sqrt(8);sqrt(8.0)

2*sqrt(2) 2.82842712474619

38/46

19/23

pi; N(pi);pi.n(digits=20)

pi 3.14159265358979 3.1415926535897932385

#Exercise 1 (ws) Evaluate the number e with 15 decimal digits
e; N(e);e.n(digits=15)

e 2.71828182845905 2.71828182845905

#Demo2:Functions and expressions. Derivative
#Pre-defined functions
N(log(100));log(100,10);val=log(100.,2)

4.60517018598809 2

val;2^val

6.64385618977472 100.000000000000

#Exercise 2 (ws) (a)Evaluate three built-in functions of your choice for inputs of your choice.
# (b) Explain why SageMath refuses to give the result for exp(3) and force it to evaluate the function
exp(3.);

20.0855369231877

#Simple user defined functions
f(x)=x^2-3*x+4; f(5)

14

exp=sin(x)+x^2;exp(1.1); #You will see error message;Get used to this. Learn by practicing.

2.10120736006144

p=plot(f(x),-2,2,figsize=[3,3],color="blue",thickness=3)

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> NameError: name 'f' is not defined

plot(x^2-3*x+4,-2,2,figsize=[4,3])

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> TypeError: 'tuple' object is not callable

#Demo3: Basic plotting
plot(sin(x),-2,2,figsize=[4,3])

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> TypeError: 'tuple' object is not callable

#Saving a plot in a file:
#give a plot a name (say, p ) and type the command
#p.save('[file name].png')

#Exercise 3 (ws)
#Remove the pound sign before the command "parametric_plot", and create a parametric plot using the #command (choose your favorite color and specify the size of the plot).
#Save the plot in a file with name 'my_first_plot.png'
parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),color="lightblue",figsize=[5 ,5 ])

#Declearing variables
var('a b c')

(a, b, c)

#Solving equations exactly
solve(x^2+b*x-5==0,x)

[x == -1/2*b - 1/2*sqrt(b^2 + 20), x == -1/2*b + 1/2*sqrt(b^2 + 20)]

#find_root?

find_root?

File: /ext/sage/sage-8.8_1804/local/lib/python2.7/site-packages/sage/misc/lazy_import.pyx
Signature : find_root(f, a, b, xtol=1e-12, rtol=8.881784197001252e-16, maxiter=100, full_output=False)
Docstring :
Numerically find a root of "f" on the closed interval [a,b] (or
[b,a]) if possible, where "f" is a function in the one variable.
Note: this function only works in fixed (machine) precision, it is
not possible to get arbitrary precision approximations with it.

INPUT:

* "f" -- a function of one variable or symbolic equality

* "a", "b" -- endpoints of the interval

* "xtol", "rtol" -- the routine converges when a root is known to
  lie within "xtol" of the value return. Should be >= 0. The
  routine modifies this to take into account the relative precision
  of doubles. By default, rtol is "4*numpy.finfo(float).eps", the
  minimum allowed value for "scipy.optimize.brentq", which is what
  this method uses underneath. This value is equal to "2.0**-50"
  for IEEE-754 double precision floats as used by Python.

* "maxiter" -- integer; if convergence is not achieved in
  "maxiter" iterations, an error is raised. Must be >= 0.

* "full_output" -- bool (default: "False"), if "True", also
  return object that contains information about convergence.

EXAMPLES:

An example involving an algebraic polynomial function:

   sage: R.<x> = QQ[]
   sage: f = (x+17)*(x-3)*(x-1/8)^3
   sage: find_root(f, 0,4)
   2.999999999999995
   sage: find_root(f, 0,1)  # abs tol 1e-6 (note -- precision of answer isn't very good on some machines)
   0.124999
   sage: find_root(f, -20,-10)
   -17.0

In Pomerance's book on primes he asserts that the famous Riemann
Hypothesis is equivalent to the statement that the function f(x)
defined below is positive for all x >= 2.01:

   sage: def f(x):
   ....:     return sqrt(x) * log(x) - abs(Li(x) - prime_pi(x))

We find where f equals, i.e., what value that is slightly smaller
than 2.01 that could have been used in the formulation of the
Riemann Hypothesis:

   sage: find_root(f, 2, 4, rtol=0.0001)
   2.0082...

This agrees with the plot:

   sage: plot(f,2,2.01)
   Graphics object consisting of 1 graphics primitive

The following example was added due to
https://trac.sagemath.org/4942 and demonstrates that the function
need not be defined at the endpoints:

   sage: find_root(x^2*log(x,2)-1,0, 2)  # abs tol 1e-6
   1.41421356237

The following is an example, again from
https://trac.sagemath.org/4942 where Brent's method fails.
Currently no other method is implemented, but at least we
acknowledge the fact that the algorithm fails:

   sage: find_root(1/(x-1)+1,0, 2)
   0.0
   sage: find_root(1/(x-1)+1,0.00001, 2)
   Traceback (most recent call last):
   ...
   NotImplementedError: Brent's method failed to find a zero for f on the interval

An example of a function which evaluates to NaN on the entire
interval:

   sage: f(x) = 0.0 / max(0, x)
   sage: find_root(f, -1, 0)
   Traceback (most recent call last):
   ...
   RuntimeError: f appears to have no zero on the interval

#Finding roots approximately
find_root(cos(t)-t,0,3)

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> NameError: name 't' is not defined

# Syntax of user defined function syntax
def char_fcn01(x):
    if x<=0 or x>=1:
        return 0
    else:
        return 1

char_fcn01(0.5);char_fcn(1.5)

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> NameError: name 'char_fcn01' is not defined

#Exercise 4. Define a function that finds the minimal element of the list. Test your function on example of your choice.

def minimal_el(L):
    m=L[0]
    for j in range(1,len(L)):
        if L[j]<m:
            m=L[j]
    return m       
c=range(1,10)
minimal_el(c)
a=[i^2 for i in range (2,4)]
minimal_el(a)

1 4

#Exercise 5. Experiment with commands in the file sage-quickref-BasicMath.pdf. Include two examples using commands of your choice. Your commands should be different from arithmetic operations and the commands used in this worksheedef
def func_f(x)
    if x<=0 or x>=10
        return -1
    else
        return 11

func_f(x^3)

Error in lines 1-5 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags), namespace, locals) File "<string>", line 1 def func_f(x) ^ SyntaxError: invalid syntax