strings, dictionaries, and files

Transcription

1 strings, dictionaries, and files 1 Polynomials in Several Variables representing and storing polynomials polynomials in dictionaries and in files 2 Two Modules in Python the module randpoly the module filepoly 3 Real Algebraic Geometry surf to visualize algebraic curves and surfaces visualization with surfex MCS 507 Lecture 18 Mathematical, Statistical and Scientific Software Jan Verschelde, 8 October 2012 Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

2 strings, dictionaries, and files 1 Polynomials in Several Variables representing and storing polynomials polynomials in dictionaries and in files 2 Two Modules in Python the module randpoly the module filepoly 3 Real Algebraic Geometry surf to visualize algebraic curves and surfaces visualization with surfex Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

3 data structures Python glues with strings, files, and dictionaries: String representations of objects serve well to pass data between different programs. On files we store objects permanently. A dictionary associates keys to values and scales well to organize large data sets. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

4 multivariate polynomials Our mathematical object of today is a polynomial in several variables with complex coefficients. The basic data structure is a tuple of two lists: a list of coefficients of type complex, a list of tuples, exponents are natural numbers. The monomial ( j)*x1**2*x2**4 is the kth element in the exponent list: (2, 4) and kth coefficient: ( j). Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

5 random polynomials We generate random polynomials as follows: 1 Generate a list C of random coefficients: c = cos(θ) + I sin(θ) for angles θ uniformly distributed in [0, 2π]. 2 Generate a list E of random exponents, the exponents of a monomial are stored in a tuple t: 1 for a number of variables n choose an index k {0, 1,...,n 1}, 2 do a coin flip e [0, 1]: t k = t k + e, and repeat d times for a monomial of degree d. So a polynomial p is then the tuple (C, E). Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

6 string representations Defining a string representation for (C, E): instead of tuple of lists, we see p as a polynomial, we verify (C, E) as a sympy expression with eval. For n variables, we have n symbols in the list S. p = "" for all c C and e E do t = str(c) for all k {0, 1,...,n} do if e k > 0 then t = t + * + S k if e k > 1 then t = t + ** + str(e k ) p = p + t Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

7 strings, dictionaries, and files 1 Polynomials in Several Variables representing and storing polynomials polynomials in dictionaries and in files 2 Two Modules in Python the module randpoly the module filepoly 3 Real Algebraic Geometry surf to visualize algebraic curves and surfaces visualization with surfex Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

8 duplicate monomials If we generate many monomials of low degree, then duplicate exponents are very likely. Goal: add coefficients of monomials with same exponents. Store (C, E) into a dictionary D: D[e] = c for all pairs (c, e) with c C, e E. Every key in the dictionary is unique: if D.has_key(e), then D[e] = D[e] + c. Then p is described as D = {e : c, e E, c C}. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

9 a polynomial on file Five monomials in three unknowns: Every line has one monomial: real and complex part of the coefficient, followed by n natural numbers. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

10 files in Python Opening and closing files: file = open(name, w ) for writing, file = open(name, r ) for reading. file.close() sends buffer to file when w. Writing and reading lines to file: s = str(data); file.write(s + \n ) line = file.readline() L = line.split( ) The result of line.split( ) is a list L of items which are separated by spaces in the string line. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

11 split and join To convert lines separated by spaces to a comma separated value (csv) file, we convert lines with split and join: >>> line = " " >>> L = line.split( ) >>> L [ , ,, 3, 0, 1 ] >>> s =,.join(l) >>> s ,0.0841,,3,0,1 >>> line.split() [ , , 3, 0, 1 ] Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

12 two Python modules We develop the code in two files: 1 A module to generate random polynomials: generate coefficients C and exponents E, turn (C, E) into string representation and then into sympy expression. 2 A module to make a dictionary representation and write/read a random polynomial to/from file. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

13 strings, dictionaries, and files 1 Polynomials in Several Variables representing and storing polynomials polynomials in dictionaries and in files 2 Two Modules in Python the module randpoly the module filepoly 3 Real Algebraic Geometry surf to visualize algebraic curves and surfaces visualization with surfex Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

14 random coefficients def random_coefficient(): Returns a random complex coefficient, uniformly distributed on the unit circle. from math import cos, sin, pi from random import uniform u = uniform(0,2*pi) return complex(cos(u),sin(u)) Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

15 random exponents def random_exponent(n,d): Returns an n-tuple obtained after d coin flips. from random import randint L = [0 for i in xrange(n)] for i in xrange(d): L[randint(0,n-1)] += randint(0,1) return tuple(l) Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

16 a random polynomial def randpoly(n,d,m): A random polynomial in n variables of degree at most d and m terms is returns as a tuple of two lists: coefficients and exponents. C = [random_coefficient() for i in xrange(m)] E = [random_exponent(n,d) for i in xrange(m)] return (C,E) Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

17 the main function def main(): Prompts the user for number of variables, largest degree, number of monomials, and then generates a random polynomial. n = input( give number of variables : ) d = input( give the largest degree : ) m = input( give number of monomials : ) (C,E) = randpoly(n,d,m) print coefficients & exponents :, (C,E) S = [ x + str(i) for i in xrange(1,n+1)] strp = strpoly(s,c,e) print string representation :, strp Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

18 monomials as strings def strmon(s,c,e): Returns a string representation of a monomial using the list of symbols in S, the coefficient c, and the exponents e. r = ( + + str(c)) for i in xrange(len(e)): if e[i] > 0: r += ( * + S[i]) if e[i] > 1: r += ( ** + str(e[i])) return r Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

19 polynomials as strings def strpoly(s,c,e): Returns a string representation of a polynomial in the variables in S with coefficients in C and exponents in E. p = "" for i in xrange(len(c)): p += strmon(s,c[i],e[i]) return p Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

20 sympy expressions import sympy as sp def sympypoly(s,c,e): Turns the polynomial with coefficients in C, exponents in E and symbols in S into a sympy expression. strp = strpoly(s,c,e) sp.var(s) return eval(strp) Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

21 strings, dictionaries, and files 1 Polynomials in Several Variables representing and storing polynomials polynomials in dictionaries and in files 2 Two Modules in Python the module randpoly the module filepoly 3 Real Algebraic Geometry surf to visualize algebraic curves and surfaces visualization with surfex Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

22 polynomials in sympy >>> import sympy as sp >>> x,y = sp.var( x,y ) >>> s = 2*x + y - 8 >>> p = eval(s) >>> p 2*x + y - 8 >>> type(p) <class sympy.core.add.add > >>> q = sp.poly(p) >>> type(q) <class sympy.polys.polytools.poly > >>> q Poly(2*x + y - 8, x, y, domain= ZZ ) >>> q.terms() [((1, 0), 2), ((0, 1), 1), ((0, 0), -8)] Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

23 the main function import randpoly as rp import sympy as sp def main(): Tests manipulation of a random polynomial as a sympy polynomial. (C,E) = rp.randpoly(3,7,5) S = [ x, y, z ] p = rp.sympypoly(s,c,e) q = sp.poly(p) print a random polynomial :, q Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

24 the I in sympy $ python filepoly.py a random polynomial : Poly( *x**2*z**2*I *x**2*z** *x*y**2*z*I *x*y**2*z *x*y*z*I *x*y*z *y*z**3*I *y*z** *y*z**2*I *y*z**2, x, y, z, I, domain= RR ) Problem: I is another variable in sympy, exponent tuples are 4-tuples in the example above. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

25 sympy Poly dict def dictpoly(p): Returns a dictionary representation of the polynomial p. The imaginary unit I is the last power of every exponent. R = {} T = p.terms() for t in T: (e,c) = t a = e[0:-1] Ideg = e[len(e)-1] cf = (c if Ideg == 0 else complex(0,c)) if not R.has_key(a): R[a] = cf else: R[a] += cf return R Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

26 tableau format def strexp(e): Returns a string representation of the exponent in the tuple e. s = "" for d in e: s += ( + str(d)) return s def dictprint(n,d): Prints the tableau format of the dictionary D, n = #variables. print len(d), n for k in D.keys(): c = D[k] print sp.re(c), sp.im(c), strexp(k) Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

27 writing to file def dictwrite(name,n,d): Writes the dictionary D in tableau format to file with the given name. file = open(name, w ) s = str(len(d)) + + str(n) + \n file.write(s) for k in D.keys(): c = D[k] s = str(sp.re(c)) s += + str(sp.im(c)) s += + strexp(k) + \n file.write(s) file.close() Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

28 extracting monomials def cffexp(s): Returns the tuple of coefficient and exponent stored in s. L = s.split( ) c = complex(eval(l[0]),eval(l[1])) e = [] for k in xrange(2,len(l)): if L[k]!= : e.append(eval(l[k])) return (c, tuple(e)) Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

29 reading from file def dictread(name): Opens the file with name, reads it and returns the dictionary representation of the polynomial stored in the file. file = open(name, r ) line = file.readline() L = line.split( ); n = eval(l[1]) s = reading + L[0] + monomials s += in + str(n) + variables... print s; D = {} while True: s = file.readline() if s == : break (c,e) = cffexp(s); D[e] = c file.close() return D Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

30 strings, dictionaries, and files 1 Polynomials in Several Variables representing and storing polynomials polynomials in dictionaries and in files 2 Two Modules in Python the module randpoly the module filepoly 3 Real Algebraic Geometry surf to visualize algebraic curves and surfaces visualization with surfex Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

31 surf surf is a tool to visualize algebraic curves and surfaces. Free software distributed under the GNU General Public License with homepage at Developed since 1996 (version dates from 2006), the authors are Stephan Endrass, Hans Huelf, Ruediger Oertel, Ralf Schmitt, Kai Schneider and Johannes Beigel. It aims to visualize real algebraic geometry: 1 plane curves defined by one polynomial in two variables 2 space curves defined by two polynomials in three variables 3 surface defined by one polynomial in three variables with stable enough algorithms to handle curve/surface singularities (points with multiplicities), although it is best if the equations are reduced (free of multiple components). Real root finding is at the heart of the algorithms in surf. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

32 27 lines on a cubic surface Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

33 using surf The distribution comes with many examples. The picture on the previous slide is the surface defined in the script clebsch27lines.pic. Scripts have the suffix.pic and can be loaded into surf or surf can be started with the name of the script at the command line. $ surf clebsch27lines.pic Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

34 strings, dictionaries, and files 1 Polynomials in Several Variables representing and storing polynomials polynomials in dictionaries and in files 2 Two Modules in Python the module randpoly the module filepoly 3 Real Algebraic Geometry surf to visualize algebraic curves and surfaces visualization with surfex Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

35 surfex surfex is distributed under the GNU General Public License. surfex is written by by Oliver Labs and Stephan Holzer It needs surf, Javaview, and convert (for the animations). The PhD thesis of Oliver Labs (Mainz, 2005) on Hypersurfaces with Many Singularities. History Constructions Algorithms Visualization provides a good background for the software. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

36 ray tracing Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

37 the raytraced picture Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

38 Summary + Exercises Strings, dictionaries, and files help glue programs. Exercises: 1 Suppose our multivariate polynomials would be products of linear equations. Describe how you would change the data structures. Explain why expanding the polynomials with sympy is a very bad idea. 2 Change the module randpoly so that dense polynomials are generated. A polynomial of degree d is dense if all monomials of degree d appear with nonzero coefficient. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39

39 homework & midterm The fourth homework is due on Friday 19 October, 10AM: exercises 3 and 6 of Lecture 13; exercises 2 and 4 of Lecture 14; exercises 1 and 3 of Lecture 15; exercises 2 and 5 of Lecture 16; exercises 3 and 4 of Lecture 17. Friday 12 October is our midterm exam, which could be either or An in-class conventional exam, open book and notes, but without computer; A take-home exam due on Monday 15 October at 10AM which must be solved individually, but will need the use of software. Scientific Software (MCS 507) Strings, Dictionaries, and Files 8 Oct / 39