The following code:

print( 1 + ( 1/2 + i*sqrt(3)/2 )^3 )
print( simplify( 1 + ( 1/2 + i*sqrt(3)/2 )^3) )
print( N(1 + ( 1/2 + i*sqrt(3)/2 )^3) )

Results in the output:

(1/2*I*sqrt(3) + 1/2)^3 + 1
(1/2*I*sqrt(3) + 1/2)^3 + 1
2.22044604925031e-16 + 1.11022302462516e-16*I

but I'd expect the middle one, and perhaps even the first one, to be just "0." I think this means that there is something which I don't understand about how these expressions are handled.

The following code is from the 1st edition of my textbook, Sage for Undergraduates. It is producing some extremely unexpected errors (technically, deprication warnings). Something about an invalid escape sequence, \i, which I'm not using.

Can someone explain how I should change my code, so that these warnings do not appear?

Code:

P1= text('$\int_0^2 \sqrt{x} \, dx$', (0.5, 1.1), fontsize=20)

P2= plot( sqrt(x), 0, 2, fill=True)

P= P1 + P2

P.show()

Warning: <input>:1: DeprecationWarning: invalid escape sequence \i

Quick Link: https://sagecell.sagemath.org/?z=eJwV...

I was wondering if anyone has made a video to help students transition from using the Sage Cell Server (for simple tasks) to the Sage Cloud system?

I'm thinking of something very short, to give flavor and "the basics." Perhaps something 5-10 minutes long suitable for 200-level math, physics and engineering majors.

I'm guessing that you need either the Taylor expansion or some competitor of it, like the Chebychev polynomials. The main issue is that you have both sin x and a linear polynomial in x. Therefore, I really can't see how you would "solve for x." I'm about 98% sure but I don't want to say 100%, because mathematics is always full of surprises.

One option, and this might be suboptimal, is to replace sin(x) with its taylor series.

Because

sin(x) = x - (1/6)x^3 + (1/120)x^5 - (1/5040)x^7 + (1/362880)x^9 - O(x^11)

we would change

Asin(x) = BC - B*x

into instead

Ax - (A/6)x^3 + (A/120)x^5 - (A/5040)x^7 + (A/362880)x^9 - O(A x^11) = BC - Bx

giving you then

-BC + (A+B)x - (A/6)x^3 + (A/120)x^5 - (A/5040)x^7 + (A/362880)x^9 - O(A x^11) = 0

or if you want a few more terms

-BC + (A+B)x - (A/3!)x^3 + (A/5!)x^5 - (A/7!)x^7 + (A/9!)x^9 - (A/11!)x^11 + (A/13!)x^13 + O(A x^15) = 0

This does reveal that varying C will only affect the answer by a constant.

If A and B were fixed constants, known in advance, then you could use "for" and "find_root" to find a value of x for a long sequence of C values, forming a table of values.

For example:

http://sagecell.sagemath.org/?z=eJxtk...

Quick Question

...Did you intend I_L and i_L to be distinct? did you intend D and d to be distinct? how about v_C and V_C? In Sage, the capitalization matter, and these would be treated as distinct.

Solution

Assuming that you wanted them to be distinct, the following will work:

var("D I_L Ron d i_L V_g v_g V_C n v_C")

f = -DI_LRon - I_LRond - DRoni_L - Rondi_L + DV_g + V_gd + Dv_g + dv_g + DV_C/n + V_Cd/n + Dv_C/n + dv_C/n - V_C/n - v_C/n

print "Before:"

print f

print "After:"

print f(i_L=0, d=0, v_g=0, v_C=0)

Output

Before:

-DI_LRon - I_LRond - DRoni_L - Rondi_L + DV_g + V_gd + Dv_g + dv_g + DV_C/n + V_Cd/n + Dv_C/n + dv_C/n - V_C/n - v_C/n

After:

-DI_LRon + DV_g + DV_C/n - V_C/n

The author of this question intends for you to plug 5 in for x and k in for y. If you do this, you will obtain

2(5) + k - 7 = 0

10 + k - 7 = 0

3 + k = 0

k = -3

We can check our work by confirming that (5, -3) is a solution to 2x + y - 7 = 0.

2(5) + (-3) - 7 = 10 - 3 - 7 = 7 - 7 = 0.

Maybe you might try looking at the answer that I posted to your previous question titled, "Please Help!"

Using what you find there, try to solve this one yourself.

To do this in Sage, I tried the following, which worked:

var("x y") solve( [11x+15y+23 ==0, 7x-2y-20 == 0], [x,y] )

However, I'm wondering if what you really want to know is how to do this by hand... If so, please consider reading

• Ch 1, Sec 2 "Intersecting Two Lines, Part One"
• Ch 1, Sec 3 "Intersecting Two Lines, Part Two"

of my free electronic online textbook-in-progress for the freshmen math course that business majors take: "Finite & Financial Mathematics for University Freshmen." All you have to do is:

• Go to www.gregorybard.com
• Click on "Finite Math" on the left hand side of the page
• Each section is its own pdf file.

The function you are looking for is contour_plot. The following works:

var("y")
contour_plot( y^2 - x^3 - x == 0, (x,-5,5), (y,-5,5), fill=False, axes=true )

I tried the following:

var("x y p")

solve( [ 4x + 6/y == 15, 6x - 8/y == 14, y == p*x - 2 ], [p, x, y] )

that produces the response

[[p == (4/3), x == 3, y == 2]]

which appears to be correct.

Personally, I think it is a serious mistake to install Sage locally on a personal computer. The Sage Cloud interface is far easier to use, requires no installation (it works through the web browser), and it brings you all the advantages of cloud computing---for example, your data lives forever on a server even if your laptop suffers some grave mishap. There are many features for sharing your work with others, and so forth. There's just no reason to bother with a local install.

https://cloud.sagemath.com/

Even simpler is the Sage Single-Cell Server which is great for small Sage tasks.

https://sagecell.sagemath.org/

calc314, I'm sorry, but that's not correct. See where it says "Current function value: 2.032982"? That should be zero. Also, it does not match the symbolic solution at all. Perhaps this is a bug, and minimize should give an error message, since the minimum wasn't reached?

rws, I'm not sure where the plus signs went. They definitely belong. It must have been an error during cut-and-paste. I was using the Sage Single-Cell Server.

%% step one: var("y z L")

F(x, y, z, L) = (x-3)^2 + (y-1)^2 + (z+1)^2 + L*(x^2 + y^2 + z^2 - 4)

derivative(F)

%% step two:

eqn1= 2Lx + 2*x - 6 == 0

eqn2= 2Ly + 2*y - 2 == 0

eqn3= 2Lz + 2*z + 2 == 0

eqn4= x^2 + y^2 + z^2 - 4 == 0

solve( [eqn1, eqn2, eqn3, eqn4], [x, y, z, L] )

%% correct answer obtained: [[x == -6/11sqrt(11), y == -2/11sqrt(11), z == 2/11sqrt(11), L == -1/2sqrt(11) - 1],

[x == 6/11sqrt(11), y == 2/11sqrt(11), z == -2/11sqrt(11), L == 1/2sqrt(11) - 1]]

%% moral of the story: if symbolic methods work fine, why bother with numerical methods?! ;-)

if f(x1,x2,..,xn) is convex (or concave) then minimizing or maximizing subject to the linear constraint g(x1,x2,..,xn)=0 should be the same as minimizing or maximizing

F(x1,x2,...,xn,L) = f(x1,x2,...xn) + L*g(x1,x2,...,xn)

in the unconstrained sense. See "minimize?" for help. Convex functions (or concave) functions will have a unique minimum (or unique maximum), so everything should work out.

I've tried all sorts of initial conditions but this just won't converge. It is an easy Lagrange multipliers problem.

var("y z L") 
F(x, y, z, L) = (x-3)^2 +  (y-1)^2 +  (z + 1)^2 +  L*(x^2 +  y^2 +  z^2 - 4)  
minimize(F, [4, 3, 2, 1], algorithm='ncg' )

Any thoughts? Are there ways of forcing it to use more iterations, or something??