Sage Workshop (June 5th, 2017)

Transcription

1 Sage Presentation Sage Workshop (June 5th, 2017) Jonathan Tyler About Sage 1. Goal: Create a viable, free, open-source alternative to Maple, MATLAB, Mathematica, and Magma. 2. Emphasis is on openness, community, cooperation, and collaboration. 3. Based upon the Python Language What can Sage do? 1. Simple Computations 2. Computations in Algebra, Geometry, Number Theory, Cryptography, Numerical Analysis, and more! Ways to use Sage 1. Notebook graphical interface 2. Interactive command line 3. By writing interpreted and compiled programs in Sage 4. By writing stand-alone Python scripts Tutorial For the complete tutorial, visit Installation For a guide to installation, visit Basics "=" is used for assignment. Variables are dynamic meaning you can change their types without reseting them. ==, >=, <=, <, and > are used for comparison +,-,*,/ are the standard operations ** and ^ are exponentiation (different from Python as ** is exponentiation) % is mod // returns the integer portion of the quotient Many basic functions are already built-in such as sqrt and all trig and hyperbolic trig functions (trig functions take radians) exp(x) is e^x. The function returns exacxt values. Use the function or method n() to find a decimal approximation. Same with other functions such as trig functions. You can change precision in number of bits or precision in number of digits by adding arguments to the n method or function. exp(2) e^2 show(sin(pi/3)) sin(pi/3).n()

2 show(exp(2).n()) show(exp(2).n(2)) show(exp(2).n(50)) show(exp(2).n(digits=2)) Order of Operations in Sage: Try 3^2*4+2%5 4*(10 // 4) + 10%4 == 10 show(3^2*4+2%5) show(4*(10//4)+10%4) Defining Symbolic Expression in Sage Sage allows for symbolic computation and plotting. To define a symbolic variable, use the syntax: x,y,z = vars('x y z'). x is automatically a symbolic variable without definition. You can list as many variables as you want. Note that the variables are separated by a comma outside the function, and not separated by one inside the function. Then you can define expressions with these symbolic variables. These symbolic variables will play a big role later in solving for roots of functions and in plotting. Let's say you want to define an expression in sage of the form f = y 7 y 2 +4y+12. y=var(' y ') f = (y-7)/(y^2+4*y+12) show(f) y 2 y y + 12 Or perhaps g = 3x+11 x 2 x 6. g = (3*x+11)/(x^2-x-6) show(g) x 2 3 x + 11 x 6 Here, we illustrate the difference between method and function. Say you want to compute the partial fraction of g above. Try partial_fraction(g) and g.partial_fraction(). partial_fraction(g) Traceback (click to the left of this block for traceback)... NameError: name 'partial_fraction' is not defined

3 Use the parametic plot command to display a plot of the curve α(t) = ( t 2 1, t 3 t) for t [ 3, 3]. g.partial_fraction() -1/(x + 2) + 4/(x - 3) Plotting in Sage 2D plotting The most basic Sage command for 2D plotting, used for graphs of the form y = f(x) among other things, is called plot. The format is plot(function,range,options) where function is the function to be plotted the simplest way to specify one is via a formula for f(x) (an expression in Sage) range is the range of x-values you want to see plotted, entered in the form (x, low, high) options can be used to control the form of the plot if desired. No options need be specified. More on this later. Plot the graph of y = x 4 2 x 3 + x 5 sin( x 2 ) for 2 x 1. plot(x^4-2*x^3+x-5*sin(x^2), (x,-2,1)) From the formula y = x 4 2 x 3 + x 5 sin( x 2 ), you might guess that there is at least one other x-intercept for this graph for x > 1 (why?). To see that part of the graph as well, edit your previous command line to change the right hand endpoint of the interval of x values (do not retype the whole command). Click the evaluate link to have Sage execute the command again. Experiment until you are sure that your plot shows all the x-intercepts of this graph. (You can repeat this process of editing a command and re-running it as often as you want; the previous output is replaced by the new output each time.) Parametic plotting in 2D The basic format for parametric curves α(t) = (x(t), y(t)) is parametric_plot([x-comp,y-comp],t-range,options) where x-comp and y-comp are the x- and y-component functions of the curve to be plotted (each of these can be an expression involving the variable/parameter t), t-range is the range of t-values you want to see plotted, and options can be used to control the form of the plot if desired. No options need be specified.

4 t = var('t') parametric_plot([t^2-1,t^3-t], (t,-3,3)) Combining plots Frequently, it's the relationship between two or more different graphs that you want to understand by looking at a plot. Sage has a very straightforward method for combining plots. You can use the plot or parametric plot commands as above to generate the component plots, but you assign the output to names instead of displaying them. Then you can enter a command that combines the plots and displays them together like this: sp1=plot(x**4-3*x,(x,-5,5)) sp2=plot(x**3-5,(x,-5,5)) sp1+sp2 Combine the plots of the following expressions: w = sin(x) v = sin(x + 2) for 0 x 2π with the added conditions: graph of graph of w v should be red. should be a blue and dashed. w = plot(sin(x), (x,0,2*pi), color='red') v = plot(sin(x+2), (x,0,2*pi), color='blue', linestyle='dashed') show(w+v)

5 Example: Figure out a two ways to display the circle x 2 + y 2 = 1. t=var('t') parametric_plot([cos(t), sin(t)], (t,0,2*pi)) There is a built-in function that will plot a circle for you. Type in circle? in the prompt for help on how to use it. circle((0,0),1)

6 Implicit 2D plotting The 2D plots above are all either graphs of functions plot the affine variety y = f(x) or parametric curves, or a combination of these plots. In addition to these, we will also want to be able to V(g(x, y)) = {(x, y) R 2 : g(x, y) = 0} for a general polynomial g(x, y) in two variables. The technical name for this in Sage is implicit curve plotting. To plot these curves, we need a new command called implicit_plot. The format for the implicit_plot command is implicit_plot(expression,x-range,y-range) The expression should be the expression g(x, y) (Sage assumes you mean to plot g(x, y) = 0.) This command generates a plot of the part of the variety V(g(x, y)) in the rectangular box in the plane defined by the x- and y-ranges. Now plot the circle x 2 + y 2 = 1 using implicit plot. (Now you know four different ways to plot a circle in Sage.) implicit_plot(x^2+y^2-1, (x,-1,1), (y,-1,1))

7 Example: Use the implicit plot command to plot the real zero set of f = x 3 3xy 2 3. Some questions to think about as you look at this: If you set x = c (a constant), how many points (c, y) are there on the curve? How many points (x, d) are there for y = d? Do your answers depend on c, d? How? implicit_plot(x^3-3*x*y^2-3, (x,-10,10), (y,-10,10)) 3D curve and surface plotting R 3 Now we move up a dimension and consider plotting curves and surfaces in. The parametric_plot3d command can be used to draw parametric curves and parametric surfaces in. To use it to draw a parametric curve, you would enter a command of the form R 3

8 parametric_plot3d([x-comp,y-comp,z-comp],(t,low,high)) where x-comp = x(t), y-comp = y(t), z-comp = z(t) are the parametric equations of the curve and the range of parameter values to be plotted is low t high. Try plotting the curve α(t) = (t, t 2, t 3 ) for 2 t 2. α(t) is called the twisted cubic. The first plot you see here might be rather uninformative. Fortunately, Sage also lets you look at a 3D plot from different viewpoints. To do this, left-click the rotation symbol to the left of the plot then hold down the left mouse button and move the cursor to rotate the graph. Experiment with this until you feel comfortable. parametric_plot3d([t,t^2,t^3], (t,-2,2)) Parametric surface plotting R 3 The parametric_plot3d command is also used for plotting parametric surfaces in. The differences between this use of the command and that above is that thecomponent functions will depend on two parameters, say u, v, and you will need to (define and) specify plotting ranges for each one. Plotting a graph z = f(x, y) The command for plotting graphs of functions of two variables is called plot3d (naturally enough!) Its format is similar to, but not exactly the same as, the format of plot. To draw a graph with plot3d, you use a command of the format: plot3d(function,xrange,yrange,options) The function is the function f(x, y), entered in the usual Sage syntax for expressions. The xrange and yrange specify a rectangular box in the plane that the plot will be constructed over; the options can be used to specify how the plot is drawn. Look at the online help for plot3d if you want to see what things are possible. You can change viewpoint (rotate the graph); The method is the same as that for the parametric_plot3d command described above. Implicit surface plots To plot an surface defined as a variety V(g(x, y, z)) = {(x, y, z) : g(x, y, z) = 0} you use the implicit_plot3d command. Look up the on-line help listing for this command. Basic Rings You can work in many different rings in Sage. ZZ - integers, RR - reals, QQ - rationals, CC - complex numbers Polynomial rings - realpoly.<z> = PolynomialRing(RR), ratpoly.<t> = PolynomialRing(QQ) GF(p^q) field with p^q elements. If q > 1, you must assign a generator to the field (e.g., GF(27, 'a') Can also work in the p-adic rings. notation: Zp(p) 1/2 in QQ True sqrt(3) in QQ False sqrt(3) in QQbar True

9 QQbar == RR False pi in QQ False pi in RR True i in RR False F_27 = GF(27,'a') F_27 Finite Field in a of size 3^3 F_27.0 a Polynomials Say we want to perform computations with polynomials in the ring R = Q[x] (What does Q[x] mean?). Sage lets us define this structure in several different ways. Here is the most direct: R.<x> = PolynomialRing(QQ) or R = QQ['x'] or R.<x> = QQ['x'] (Other possibilities for the coefficient ring would be ZZ -- the ring of integers, Integers(n) for the integers mod finite field of order p, and many, many others.) (Example 2) Define the ring R = R[x] where R = F 5 in a few different ways. R.<x> = PolynomialRing(RR, 'x') n (fill in the specific integer n, of course!), GF(p) for the If you need to determine whether something (say f) is really a polynomial (for example, if it is causing an error message), you can enter a command like this: f.parent() Define a polynmial in the R defined above. Then use the parent method to determine the parent ring. Does it match what you entered above? f = x^2-2 f.parent() Univariate Polynomial Ring in x over Real Field with 53 bits of precision Operations on polynomials Sage has built-in commands: factor to factor a polynomial. This gives the factorization over the specified coefficient ring. For instance, if we defined Q[x] as above, then the factor command does not know about radicals, complex numbers, etc. Formats: factor(poly) or poly.factor(). This works on polynomials in any number of variables. (Example 5) Define the polynomial x^2-2 in RR[x] and t^2-2 in QQ[t]. Compare the factorizations. Q.<t> = PolynomialRing(QQ,'t') show(factor(x^2-2)) show(factor(t^2-2)) (x ) (x ) ( t 2 2) Factor the following polynomials over the rationals: a.) b.) p = x 2 2x + 1 s = x 3 3 x 2 + 4

10 c.) t = x 2 2. p = t^2-2*t+1 s = t^3-3*t^2+4; r = t^2-2 show(factor(p)) show(factor(s)) show(factor(r)) (t 1) 2 (t + 1) (t 2) 2 ( t 2 2) quo_rem() to compute the quotient and remainder on division of one polynomial f(x) by another g(x). This has a slightly strange format: f.quo_rem(g) Note: the polynomial g in the parentheses is used as the divisor; the polynomial being acted on (the f) is the dividend. If f or g contain other variables besides x, this will fail. The output from this is a Sage list -- the quotient is the first element and the remainder is the second element. If you want to assign names to the outputs, you can do something like this: (q,r) = f.quo_rem(g) Compute the following: a.) Divide x 4 5 x 3 + 8x 2 4x by x 2 (what do you expect to get?) b.) Divide x 2 by x 4 5 x 3 + 9x 2 4x. c.) Divide x 3 4x 5 by x 2 2x 4. p1 = t^4-5*t^3+8*t^2-4*t; p2 = t-2; p3 = t^4-5*t^3+9*t^2-4*t; p4 = t^3-4*t-5; p5 = t^2-2*t-4; [q1,r1] = p1.quo_rem(p2) [q2, r2] = p2.quo_rem(p3) [q3,r3] = p4.quo_rem(p5) show([q1,r1]) show([q2,r2]) show([q3,r3]) [ t 3 3 t 2 + 2t, 0] [0, t 2] [t + 2, 4t + 3] gcd to compute the greatest common divisor of two polynomials f(x) and g(x). Format: f.gcd(g). The greatest common divisor of a set of several polynomials can be computed by nested gcd's (see p. 43 of ``IVA''). For example, gcd(f, g, h) can be computed by f.gcd(g).gcd(h) Compute the gcds of the following pairs of polynomials. Given your results from Example 7, think critically about whether or not these answers make sense. a.) b.) gcd( x 5 2x 4 4 x 3 + 8x 2 2x + 4, x 2) gcd( x 3 4x 5, x 2 2x 4) f = t^5-2*t^4-4*t^3+8*t^2-2*t+4; g = t-2; p = t^3-4*t-5; q = t^2-2*t-4; show(f.gcd(g)) show(p.gcd(q)) t 2 1 diff(f,x) computes the formal derivative of f with respect to x. Example 9 Here is a sample sequence of Sage input lines illustrating some of these commands. Try entering and executing them in sequence. Look carefully at the output and make sure you understand what happened in each case: R.<x> = PolynomialRing(QQ) cube = (x**2-16)**3 print cube s = (x**2-x+1)*(x - 4) print s (q,r) = cube.quo_rem(s) print "q = ", q,", r = ", r q*s + r == cube

11 R.<x> = PolynomialRing(QQ,'x') cube = (x^2-16)^3 print cube s=(x^2-x+1)*(x-4) print s (q,r) = cube.quo_rem(s) print "q = ", q, ",r = ", r q*s+r == cube Lists x^6-48*x^ *x^ x^3-5*x^2 + 5*x - 4 q = x^3 + 5*x^2-28*x - 161,r = 123*x^ *x True To represent points in affine space, we use coordinate vectors. The same idea is also useful to represent any ordered collection of information. In Sage, these ordered collections are called lists, and the elements of a list can be anything. A list is indicated by a pair of square brackets ([,]) enclosing the items in the list, separated by commas. For instance, the input line list1 = ['a','b','c','d','e'] creates a list with five items (the letters a, b, c, d, e, treated as character strings, not as symbolic variables), and assigns the list to the name list1. Create a list of the numbers 1 through 9 and the words dog, cat, emu. list1 = [1,2,3,4,5,6,7,8,9] list2 = ['dog', 'cat', 'emu'] show(list1) show(list2) [1, 2, 3, 4, 5, 6, 7, 8, 9] [dog, cat, emu] What if you wanted a list of the numbers from 1 to 100? Obviously, you wouldn't want to type them all out. You could try a command like range(100). Try the command range(100). list3 = range(100) show(list3) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61 That's not quite what we want. Sage (due to the underlying programming language, Python) starts its index at 0. There are multiple ways to fix this. One way is by creating a new list using range as a tool: [ i+1 for i in range(100)]. Another is to just use range(1,101). Enter the codes into sage to check that they give you the correct list. Also try creating the following lists: a.) n 2 for n from 1 to 20. b.) The even numbers from 16 to 48. list4 = [n^2 for n in range(1,21)] list5 = [2*i for i in range(8,25)] show(list4, list5) [1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400] [16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48] We have said that lists are ordered. What this means, for instance, is that list1 above is different from the list defined by list2 = ['a','d','c','e','b']. You can test this by entering the definition of list2, then the command

12 list1 == list2 A list can have any number of items, including no items at all -- the empty list is written as []. The same item can also occur at several places in a list (unlike the case for a set, which is an unordered collection of information with no repetitions). The items in a list do not all have to be items of the same type. For instance, another perfectly OK Sage list is the following: list3 = [['a','b','c'],'d','e'] Some useful list operations The built-in len function returns the number of items in a list. (If one of the items is a list itself, it is counted as a single item.) To pick out the ith item in a named list (like list1), you can use the notation list1[i]. Caution: Lists are always numbered starting from 0. So list1[1] is actually the second element(!) You can also pick out a ``slice'' or consecutive sublist -- the items in the slots numbered first to last from a list by using the format list1[first:last]. This gives a list as output. The append operator inserts a new entry at the end of a list. Try list1.append('k') to see the result. Sage has an interesting feature called ``tab-completion'' that can be used as a quick reference tool. Try entering list and then press the TAB key. You should see a listing of different ways the partial command could be completed. Tuples and Sets Sage also has data types called tuples and sets. A tuple is similar to a list, but once it is created it cannot be changed (it is immutable). A set is an unordered list. You can find information about these in the online tutorials and manual. Matrices The matrix command creates a matrix. There are a few ways that it can make a matrix. The easiest is matrix([[row1],[row2],...,[last row]]) You can also specify the ring that you want to work in by: matrix(ring, [[row1],[row2],...,[last row]]) or you can specify the ring and the dimensions. matrix(ring, # of rows, # of columns, [elements listed from left to right and top to bottom]) Define a 2x2 matrix with elements 1,2,3,4 in three different ways. M1 = matrix([[1,2],[3,4]]) M2 = matrix(zz, 2,2, [1,2,3,4]) M1 == M2 True Apparently, this is the hardest known sudoku ever published. Sage has a built-in function that will solve sudokus for you! A = matrix(zz, 9, [8,0,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,7,0,0,9,0,2,0,0,0,5,0,0,0,7,0,0,0,0,0,0,0,4,5,7,0,0,0,0,0,1,0,0,0,3,0,0,0,1,0,0 A [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] sudoku(a) [ ] [ ] [ ] [ ] [ ] [ ]

13 [ ] [ ] [ ] m=matrix([[10,3],[5,6]]);m; m.parent() [10 3] [ 5 6] Full MatrixSpace of 2 by 2 dense matrices over Integer Ring newm=matrix(gf(5),[[10,3],[5,6]]); newm; newm.parent() [0 3] [0 1] Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 5 The m.inverse() method finds the matrix inverse of a matrix m. Example: Find the matrix inverses of m and newm. show(m.inverse()) show(newm.inverse()) Traceback (click to the left of this block for traceback)... ZeroDivisionError: input matrix must be nonsingular To create a matrix from a list: Define the space of matrices you'd like to work over: M=MatrixSpace(ZZ,2) Then build your matrix from a list: m1=m([10,3,5,6]) Example: Define m1 via this method. What is the ring M1=MatrixSpace(ZZ,2,3)? M1 = MatrixSpace(ZZ,2) m1 = M1([10,3,5,6]) show(m1) 10 3 ( ) ( ) Sage has many other built-in matrix functions and methods, such as: Example: m.trace() and m.determinant() m.transpose() m.kernel(), m.nullity(), m.row_span() m.minors(k) will give you the k k minors of m m.charpoly(). if you want to use a specific variable for the polynomial (ex: t), you can do m.charpoly('t') m.eigenvectors(), m.eigenspaces() m.echelon_form() What does m.fcp() do? m.fcp() reset() x^2-16*x + 45 Computational Biology Example We use Sage to compute the steady states of a system of ordinary differential equations. This problem reduces to finding the zero set of a system of polynomials. dx/dt = (k1 - k4)*x-k2*x*y dy/dt = -k3*y+k5*z

14 dz/dt = k4*x-k5*z x,y,z,k1,k2,k3,k4,k5 = var('x y z k1 k2 k3 k4 k5') dxdt = (k1-k4)*x-k2*x*y; dydt = -k3*y+k5*z; dzdt = k4*x-k5*z sols = solve([dxdt ==0, dydt == 0, dzdt == 0],[x,y,z], solution_dict = True) show(sols) k 1 k 3 k 3 k 4 k 1 k 3 k 3 k 4 k 1 k 4 [{x : 0, z : 0, y : 0}, {x :, z :, y : }] k 2 k 4 k 2 k 5 k 2 Now we use the matrix features of Sage to compute the eigenvalues of the Jacobian matrix. We also build the Hurwitz matrices and compute the determinants. We call the steady state (x*,y*,z*). x_star = sols[1][x]; y_star = sols[1][y]; z_star = sols[1][z] show(x_star,y_star,z_star) k 1 k 3 k 3 k 4 k 2 k 4 k 1 k 4 k 2 k 1 k 3 k 3 k 4 k 2 k 5 Jac = matrix([[(k1-k4)-k2*y_star, -k2*x_star, 0],[0, -k3, k5],[k4, 0, -k5]]) show(jac) 0 0 k 4 k 1 k 3 k 3 k 4 k 4 k k 5 k 5 Jac [ 0 -(k1*k3 - k3*k4)/k4 0] [ 0 -k3 k5] [ k4 0 -k5] eigs = Jac.eigenvalues() eigs[0] -1/18*((k3 + k5)^2-3*k3*k5)*(-i*sqrt(3) + 1)/(-1/27*(k3 + k5)^3-1/2*k1*k3*k5 + 1/6*(k3 + k5)*k3*k5 + 1/2*k3*k4*k5 + 1/6*sqrt(1/3)*sqrt((4*k1*k3^3-4*k3^3*k4 + (4*k1 - k3-4*k4)*k5^3-2*(3*k1*k3 - k3^2-3*k3*k4)*k5^2 + (27*k1^2*k3-6*k1*k3^2 - k3^3 + 27*k3*k4^2-6*(9*k1*k3 - k3^2)*k4)*k5)*k3*k5))^(1/3) - 1/2*(-1/27*(k3 + k5)^3-1/2*k1*k3*k5 + 1/6*(k3 + k5)*k3*k5 + 1/2*k3*k4*k5 + 1/6*sqrt(1/3)*sqrt((4*k1*k3^3-4*k3^3*k4 + (4*k1 - k3-4*k4)*k5^3-2*(3*k1*k3 - k3^2-3*k3*k4)*k5^2 + (27*k1^2*k3-6*k1*k3^2 - k3^3 + 27*k3*k4^2-6*(9*k1*k3 - k3^2)*k4)*k5)*k3*k5))^(1/3)*(i*sqrt(3) + 1) - 1/3*k3-1/3*k5 As you can see, the above output is very messy. We can compute determinants of much easier matrices to find when the system is stable. cpoly = Jac.charpoly('x') cpoly x^3 + (k3 + k5)*x^2 + k3*k5*x + k1*k3*k5 - k3*k4*k5 We build the Hurwitz matrices now. a0 = 1 a1 = k3+k5 a2 = k3*k5 a3 = k1*k3*k5-k3*k4*k5 a0 = 1; a1 = k3+k5; a2 = k3*k5; a3 = k1*k3*k5-k3*k4*k5 H1 = matrix([a1]); H2 = matrix([[a1, a0],[a3, a2]]); H3 = matrix([[a1, a0, 0], [a3, a2, a1], [0, 0, a3]]) d_h1 = det(h1); d_h2 = det(h2); d_h3 = det(h3) show(d_h1) show(d_h2) show(d_h3) k 3 + k 5 k 1 k 3 k 5 + ( k 3 + k 5 ) k 3 k 5 + k 3 k 4 k 5

15 2 ( k 1 k 3 k 5 k 3 k 4 k 5 )( k 3 + k 5 ) k 3 k 5 ( k 1 k 3 k 5 k 3 k 4 k 5 ) Let's say that we want to plot when the determinant of H1 is positive. We can plot the d_h1 = 0 as a curve in the k3-k5 plane. implicit_plot(d_h1, (k3, -10, 10), (k5, -10, 10)) Let us say that we want to write a program that takes a list and an item and adds the item to the beginning of the list. We can do this easily in Sage. def add_list(l,item): l.reverse() l.append(item) l.reverse() return l add_list(range(3),-1) [-1, 0, 1, 2] Now, let's say we want to create a list containing the entries of a given list, followed by a second copy of that same list. We write a function. Then we load it and execute it. The extend method is really helpful here. def double_list(l): l.extend(l) return l double_list(range(3)) [0, 1, 2, 0, 1, 2] Finally, let's say we want to insert a new item as the third entry from the start in a list (assuming it has at least two entries to begin with). def add_third(l, third): l[2] = third return l add_third([1],1)

16 Traceback (click to the left of this block for traceback)... IndexError: list assignment index out of range add_third(range(3),3) [0, 1, 3]