A4templates.sagews

reset

<function reset at 0x7f5cab46af50>

#Exercise 1:Synthetic division
# Make a function long_div(L, c)  that takes a list L=[a_n,_(n-1),...,a_0] of length n+1 of coefficients of a polynomial p  and a number c and returns the list B  of length n+1. The first  n  elements of B are the coefficients of the quotient (p(x))/(x-c). The last element on the list S is the remainder. Test your function on example of your choice.

#//Input:
#//Output:
def long_div(L, c):
    S=[L[0]]
    for j in range(1,len(L)):
        temp=L[j]+c*L[j-1] #See Lecture 4 for the formula that defines coefficient of the quotient
        S.append(temp)
    return S
L=[3,2,-5,1];# p=3x^3+2x^2-5x+1
long_div(L, 2)

[3, 8, -1, -9]

#Answer: quotient: 3x^2-4x-9; remainder: 11. Check:
simplify(expand((3*x^2+8*x+11)*(x-2)+23))

3*x^3 + 2*x^2 - 5*x + 1

#################
#Exercise 2. Finding integer roots of monic polynomials
#(a) Make a function find_int_roots(L) that takes a list L of coefficients of a monic polynomial p with integer constant term and returns the list S  of integer roots of the polynomial. If the polynomial does not have integer roots, the function returns the empty list. Test your function on the polynomial p(x)=x^4+2 x^3-x^2-8 x-12. Find all integer roots.
#Your code will be more readable if you define first a function that evaluates a polynomial
def poly_val(A,c):# See pseudocode in Lecture 1
    n=len(A)
    val=A[0]
    for j in range(1,n):
        val=(val*c+A[j])
    return val

def find_int_roots(L): #L=[1,a_(n-1),...,a_0],a_0 must be an integer
    divs_pos=divisors(L[-1])# L[-1] is the last element on the list
    divs_neg=[-c for c in divs_pos]
    divs=divs_pos + divs_neg
    int_roots=[]
    for c in divs:
        if poly_val(L,c)==0:
            int_roots.append(c)
    return int_roots

poly_val([1,2,-1,-8,-12],1)

-18

L=[1,2,-1,-8,-12];find_int_roots(L)
#Answer to example in (a):integer zeros of p(x)=x^4+2*x^3-x^2-8*x-12 are

[2, -2]

#(b) Use your function constructed in Exercise 1 to find the polynomial p(x)/((x-2)(x+2)).
#I use the function: p(x)=x^4+2 x^3-x^2-8 x-12
L=[1,2,-1,-8,-12]
L1=long_div(L,2);L1

[1, 4, 3, -10, -28]

L2=long_div(L1,-2);L2
#Answer to (b): p(x)/(x^2-4)=?.

[1, 2, -5, -16, -8]

#Exercise 3 (manual). Describe a set of points in the pq-plane that does not lie on any tangent to the envelope. What does it mean for the corresponding quadratic equations?
#There is no real root? 
#It has infinity quadratic equations
reset
var('t c')
p=x^3+3*x^2-2*x+4
q=p.sub(x=t-c);q

<function reset at 0x7fef23eeef50> (t, c)

Error in lines 4-4 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1191, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> File "sage/structure/element.pyx", line 489, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4599) return self.getattr_from_category(name) File "sage/structure/element.pyx", line 502, in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4708) return getattr_from_other_class(self, cls, name) File "sage/cpython/getattr.pyx", line 394, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2607) raise AttributeError(dummy_error_message) AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'sub'

r=q.expand();r

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1191, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> NameError: name 'q' is not defined

#########################

##REMARK.
#You can ask SageMath to construct the list of polynomial coefficients using more involved structure called polynomial ring
var('x')
_.<x> = PolynomialRing(RR)#This line says that you will make polynomials with real coefficients in powers of the variable x
p=x^3-2*x^2+3*x-7
L=p.coefficients();L# This command returns the list of coefficients of p

x [-7.00000000000000, 3.00000000000000, -2.00000000000000, 1.00000000000000]

L.reverse();L# This command reverses the order of the list elements

[1.00000000000000, -2.00000000000000, 3.00000000000000, -7.00000000000000]

#With these commands you can modify the codes of several exercises in assignments. For example, synthetic_div.
//Input: polynomial q with real coefficients and a real number c
//Output: coefficients (or approximations) of the polynomial q/(x-c) and the remainder
def long_div2(q, c):
    L=q.?();L.reverse()
    S=[L[0]]
    for j in ?:
        temp=?
        S.append(temp)
    return S
synthetic_div2(p,2)

[1.00000000000000, 0.000000000000000, 3.00000000000000, -1.00000000000000]

#Result: (x^3-2*x^2+3*x-7)/(x-2)=x^2+3,remainder -1
#Check (with SageMath "black box" command)
f(x)=x^3-2*x^2+3*x-7
g(x)=x-2
f.maxima_methods().divide(g)

[x^2 + 3, -1]

#You do not need poly_val function when you work with polynomial instead of the list of its coefficients. Here is p(2):
p.subs(x=2)

-1.00000000000000

#########################

#Exercise 4. Make a function remove_xsquared(L) that takes a polynomial with real coefficients and returns the number c such that substitution x = t-c eliminates the quadratic term in p. Check your code on the polynomial p(x)=x^3+3.5 x^(2)-2.1*x+4.0.
#I experimented with SageMath commands for this task interactively:
var('t c')
q=p.subs(x=t-c);r=q.expand();r.simplify()

(t, c) -1.0*c^3 + 3.0*c^2*t - 3.0*c*t^2 + 1.0*t^3 - 2.0*c^2 + 4.0*c*t - 2.0*t^2 - 3.0*c + 3.0*t - 7.0

r.coefficient(t,n=2)# What does this comman do?

-3.00000000000000*c - 2.00000000000000

c=-2/3;r.subs(c=-2/3)# No term with t^2. It works.

1.00000000000000*t^3 + 1.66666666666667*t - 5.59259259259259

#Now it is your turn to combine the commands into the function remove_xsquared(q)
#//Input:
#//Output:
def remove_xsquared(q):
    var('t c')
    r=q.subs(x=t-c)
    ...

##########################

#Exercise 5. Make a function roots_of_unity(n) that takes a natural number n and returns a list of all n^th  roots of unity. Use your function for n = 7  and plot corresponding regular heptagon inscribed into the unit circle. Note that a heptagon is not constructible polygon, that is, it cannot be constructed with compass and straightedge.
#//
#//
def roots_of_unity(n):
    L=[N(cos(2*pi*k/n)+I*sin(2*pi*k/n)) for k in range(n)]#Encode the formula for roots of unity from Lecture 4
    return L
roots_of_unity(7);L

[1.00000000000000, 0.623489801858734 + 0.781831482468030*I, -0.222520933956314 + 0.974927912181824*I, -0.900968867902419 + 0.433883739117558*I, -0.900968867902419 - 0.433883739117558*I, -0.222520933956314 - 0.974927912181824*I, 0.623489801858734 - 0.781831482468030*I] [1.00000000000000, 0.623489801858734 + 0.781831482468030*I, -0.222520933956314 + 0.974927912181824*I, -0.900968867902419 + 0.433883739117558*I, -0.900968867902419 - 0.433883739117558*I, -0.222520933956314 - 0.974927912181824*I, 0.623489801858734 - 0.781831482468030*I]

[(L[j].real(),L[j].imag())for j in range(7)];S

[(1.00000000000000, 0), (0.623489801858734, 0.781831482468030), (-0.222520933956314, 0.974927912181824), (-0.900968867902419, 0.433883739117558), (-0.900968867902419, -0.433883739117558), (-0.222520933956314, -0.974927912181824), (0.623489801858734, -0.781831482468030)]

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1191, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> NameError: name 'S' is not defined

point(S,aspect_ratio=1,size=30)

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1191, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> NameError: name 'S' is not defined

#Use the function for k=7 and plot the heptagon using the roots of unity to specify the coordinates of the vertices.

############################

#Mini project
#Follow the directions in Lecture 4.

︠6ec9afc3-e934-42bb-89af-d803aa451cef︠

︠27622957-6ee4-4455-9efe-9a2c71996845︠