Coverage for local/lib/python2.7/site-packages/sage/tests/french_book/nonlinear_doctest.py : 0%

## -*- encoding: utf-8 -*- 

""" 

Doctests from French Sage book 

Test file for chapter "Équations non linéaires" ("Nonlinear Equations") 

Tests extracted from ./nonlinear.tex. 

Sage example in ./nonlinear.tex, line 61:: 

sage: R.<x> = PolynomialRing(RealField(prec=10)) 

sage: p = 2*x^7 - 21*x^6 + 64*x^5 - 67*x^4 + 90*x^3 \ 

....: + 265*x^2 - 900*x + 375 

sage: p.roots() 

[(-1.7, 1), (0.50, 1), (1.7, 1), (5.0, 2)] 

sage: p.roots(ring=ComplexField(10), multiplicities=False) 

[-1.7, 0.50, 1.7, 5.0, -2.2*I, 2.2*I] 

sage: p.roots(ring=RationalField()) 

[(1/2, 1), (5, 2)] 

Sage example in ./nonlinear.tex, line 158:: 

sage: R.<x> = PolynomialRing(QQ, 'x') 

sage: p = x^4 + x^3 + x^2 + x + 1 

sage: K.<alpha> = p.root_field() 

sage: p.roots(ring=K, multiplicities=None) 

[alpha, alpha^2, alpha^3, -alpha^3 - alpha^2 - alpha - 1] 

sage: alpha^5 

1 

Sage example in ./nonlinear.tex, line 202:: 

sage: R.<x> = PolynomialRing(RR, 'x') 

sage: d = ZZ.random_element(1, 15) 

sage: p = R.random_element(d) 

sage: p.degree() == sum(r[1] for r in p.roots(CC)) 

True 

Sage example in ./nonlinear.tex, line 231:: 

sage: from itertools import product 

sage: def build_complex_roots(degree): 

....: R.<x> = PolynomialRing(CDF, 'x') 

....: v = [] 

....: for c in product([-1, 1], repeat=degree+1): 

....: v.extend(R(c).roots(multiplicities=False)) 

....: return v 

sage: data = build_complex_roots(12) # long time 

sage: g = points(data, pointsize=1, aspect_ratio=1) # long time 

Sage example in ./nonlinear.tex, line 275:: 

sage: a, b, c, x = var('a, b, c, x') 

sage: p = a * x^2 + b * x + c 

sage: type(p) 

<type 'sage.symbolic.expression.Expression'> 

sage: p.parent() 

Symbolic Ring 

sage: p.roots(x) 

[(-1/2*(b + sqrt(b^2 - 4*a*c))/a, 1), 

(-1/2*(b - sqrt(b^2 - 4*a*c))/a, 1)] 

Sage example in ./nonlinear.tex, line 299:: 

sage: a, b, c, d, e, f, x = var('a, b, c, d, e, f, x') 

sage: p = a*x^5+b*x^4+c*x^3+d*x^2+e*x+f 

sage: try: 

....: p.roots(x) 

....: except RuntimeError: 

....: print('No explicit roots found') 

No explicit roots found 

Sage example in ./nonlinear.tex, line 315:: 

sage: x, a, b, c, d = var('x, a, b, c, d') 

sage: P = a * x^3 + b * x^2 + c * x + d 

sage: alpha = var('alpha') 

sage: P.subs(x=x + alpha).expand().coefficient(x, 2) 

3*a*alpha + b 

sage: P.subs(x = x - b / (3 * a)).expand().collect(x) 

a*x^3 - 1/3*(b^2/a - 3*c)*x + 2/27*b^3/a^2 - 1/3*b*c/a + d 

Sage example in ./nonlinear.tex, line 328:: 

sage: p, q, u, v = var('p, q, u, v') 

sage: P = x^3 + p * x + q 

sage: P.subs(x = u + v).expand() 

u^3 + 3*u^2*v + 3*u*v^2 + v^3 + p*u + p*v + q 

Sage example in ./nonlinear.tex, line 340:: 

sage: P.subs({x: u + v, q: -u^3 - v^3}).factor() 

(3*u*v + p)*(u + v) 

sage: P.subs({x: u+v, q: -u^3 - v^3, p: -3 * u * v}).expand() 

0 

sage: X = var('X') 

sage: solve([X^2 + q*X - p^3 / 27 == 0], X, solution_dict=True) 

[{X: -1/2*q - 1/18*sqrt(12*p^3 + 81*q^2)}, 

{X: -1/2*q + 1/18*sqrt(12*p^3 + 81*q^2)}] 

Sage example in ./nonlinear.tex, line 367:: 

sage: e = sin(x) * (x^3 + 1) * (x^5 + x^4 + 1) 

sage: roots = e.roots() 

sage: print(len(roots)) 

9 

sage: print(roots) 

[(0, 1), 

(-1/2*(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3)*(I*sqrt(3) + 1) 

- 1/6*(-I*sqrt(3) + 1)/(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3), 1), 

(-1/2*(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3)*(-I*sqrt(3) + 1) 

- 1/6*(I*sqrt(3) + 1)/(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3), 1), 

((1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3) + 1/3/(1/18*sqrt(23)*sqrt(3) 

- 1/2)^(1/3), 1), 

(-1/2*I*sqrt(3) - 1/2, 1), 

(1/2*I*sqrt(3) - 1/2, 1), 

(1/2*I*sqrt(3)*(-1)^(1/3) - 1/2*(-1)^(1/3), 1), 

(-1/2*I*sqrt(3)*(-1)^(1/3) - 1/2*(-1)^(1/3), 1), ((-1)^(1/3), 1)] 

Sage example in ./nonlinear.tex, line 424:: 

sage: alpha, m, x = var('alpha, m, x') 

sage: p = function('p')(x) 

sage: q = function('q')(x) 

sage: p = (x - alpha)^m * q 

sage: p.derivative(x) 

(-alpha + x)^(m - 1)*m*q(x) + (-alpha + x)^m*diff(q(x), x) 

sage: simplify(p.derivative(x)(x=alpha)) 

0 

Sage example in ./nonlinear.tex, line 450:: 

sage: R.<x> = PolynomialRing(QQ, 'x') 

sage: p = 128 * x^13 - 1344 * x^12 + 6048 * x^11 \ 

....: - 15632 * x^10 + 28056 * x^9 - 44604 * x^8 \ 

....: + 71198 * x^7 - 98283 * x^6 + 105840 * x^5 \ 

....: - 101304 * x^4 + 99468 * x^3 - 81648 * x^2 \ 

....: + 40824 * x - 8748 

sage: d = gcd(p, p.derivative()) 

sage: (p // d).degree() 

4 

sage: roots = SR(p // d).roots(multiplicities=False) 

sage: roots 

[1/2*I*sqrt(3)*2^(1/3) - 1/2*2^(1/3), 

-1/2*I*sqrt(3)*2^(1/3) - 1/2*2^(1/3), 

2^(1/3), 3/2] 

sage: [QQbar(p(alpha)).is_zero() for alpha in roots] # long time 

[True, True, True, True] 

Sage example in ./nonlinear.tex, line 504:: 

sage: R.<x> = PolynomialRing(RR, 'x') 

sage: p = x^7 - 131/3*x^6 + 1070/3*x^5 - 2927/3*x^4 \ 

....: + 2435/3*x^3 - 806/3*x^2 + 3188/3*x - 680 

sage: sign_changes = \ 

....: [p[i] * p[i + 1] < 0 \ 

....: for i in range(p.degree())].count(True) 

sage: real_positive_roots = \ 

....: sum([alpha[1] \ 

....: if alpha[0] > 0 else 0 for alpha in p.roots()]) 

sage: sign_changes, real_positive_roots 

(7, 5) 

Sage example in ./nonlinear.tex, line 567:: 

sage: def count_sign_changes(l): 

....: changes = [l[i]*l[i + 1] < 0 \ 

....: for i in range(len(l) - 1)] 

....: return changes.count(True) 

sage: def sturm(p, a, b): 

....: assert p.degree() > 2 

....: assert not (p(a) == 0) 

....: assert not (p(b) == 0) 

....: if a > b: 

....: a, b = b, a 

....: remains = [p, p.derivative()] 

....: for i in range(p.degree()): 

....: remains.append(-(remains[i] % remains[i + 1])) 

....: evals = [[], []] 

....: for q in remains: 

....: evals[0].append(q(a)) 

....: evals[1].append(q(b)) 

....: return count_sign_changes(evals[0]) \ 

....: - count_sign_changes(evals[1]) 

Sage example in ./nonlinear.tex, line 591:: 

sage: R.<x> = PolynomialRing(QQ, 'x') 

sage: p = (x - 34) * (x - 5) * (x - 3) * (x - 2) * (x - 2/3) 

sage: sturm(p, 1, 4) 

2 

sage: sturm(p, 1, 10) 

3 

sage: sturm(p, 1, 200) 

4 

sage: p.roots(multiplicities=False) 

[34, 5, 3, 2, 2/3] 

sage: sturm(p, 1/2, 35) 

5 

Sage example in ./nonlinear.tex, line 651:: 

sage: f(x) = 4 * sin(x) - exp(x) / 2 + 1 

sage: a, b = RR(-pi), RR(pi) 

sage: bool(f(a) * f(b) < 0) 

True 

Sage example in ./nonlinear.tex, line 661:: 

sage: solve(f(x) == 0, x) 

[sin(x) == 1/8*e^x - 1/4] 

Sage example in ./nonlinear.tex, line 666:: 

sage: f.roots() 

Traceback (most recent call last): 

... 

RuntimeError: no explicit roots found 

Sage example in ./nonlinear.tex, line 684:: 

sage: a, b = RR(-pi), RR(pi) 

sage: g = plot(f, a, b, rgbcolor='blue') 

Sage example in ./nonlinear.tex, line 720:: 

sage: def phi(s, t): return (s + t) / 2 

sage: def intervalgen(f, phi, s, t): 

....: msg = 'Wrong arguments: f({0})*f({1})>=0)'.format(s, t) 

....: assert (f(s) * f(t) < 0), msg 

....: yield s 

....: yield t 

....: while True: 

....: u = phi(s, t) 

....: yield u 

....: if f(u) * f(s) < 0: 

....: t = u 

....: else: 

....: s = u 

Sage example in ./nonlinear.tex, line 785:: 

sage: a, b 

(-3.14159265358979, 3.14159265358979) 

sage: bisection = intervalgen(f, phi, a, b) 

sage: next(bisection) 

-3.14159265358979 

sage: next(bisection) 

3.14159265358979 

sage: next(bisection) 

0.000000000000000 

Sage example in ./nonlinear.tex, line 805:: 

sage: from types import GeneratorType, FunctionType 

sage: def checklength(u, v, w, prec): 

....: return abs(v - u) < 2 * prec 

sage: def iterate(series, check=checklength,prec=10^-5, maxit=100): 

....: assert isinstance(series, GeneratorType) 

....: assert isinstance(check, FunctionType) 

....: niter = 2 

....: v, w = next(series), next(series) 

....: while (niter <= maxit): 

....: niter += 1 

....: u, v, w = v, w, next(series) 

....: if check(u, v, w, prec): 

....: print('After {0} iterations: {1}'.format(niter, w)) 

....: return 

....: print('Failed after {0} iterations'.format(maxit)) 

Sage example in ./nonlinear.tex, line 837:: 

sage: bisection = intervalgen(f, phi, a, b) 

sage: iterate(bisection) 

After 22 iterations: 2.15847275559132 

Sage example in ./nonlinear.tex, line 899:: 

sage: phi(s, t) = t - f(t) * (s - t) / (f(s) - f(t)) 

sage: falsepos = intervalgen(f, phi, a, b) 

sage: iterate(falsepos) 

After 8 iterations: -2.89603757331027 

Sage example in ./nonlinear.tex, line 906:: 

sage: a, b = RR(-pi), RR(pi) 

sage: g = plot(f, a, b, rgbcolor='blue') 

sage: phi(s, t) = t - f(t) * (s - t) / (f(s) - f(t)) 

sage: falsepos = intervalgen(f, phi, a, b) 

sage: u, v, w = next(falsepos), next(falsepos), next(falsepos) 

sage: niter = 3 

sage: while niter < 9: 

....: g += line([(u, 0), (u, f(u))], rgbcolor='red', 

....: linestyle=':') 

....: g += line([(u, f(u)), (v, f(v))], rgbcolor='red') 

....: g += line([(v, 0), (v, f(v))], rgbcolor='red', 

....: linestyle=':') 

....: g += point((w, 0), rgbcolor='red') 

....: if (f(u) * f(w)) < 0: 

....: u, v = u, w 

....: else: 

....: u, v = w, v 

....: w = next(falsepos) 

....: niter += 1 

Sage example in ./nonlinear.tex, line 942:: 

sage: a, b = RR(pi/2), RR(pi) 

sage: phi(s, t) = t - f(t) * (s - t) / (f(s) - f(t)) 

sage: falsepos = intervalgen(f, phi, a, b) 

sage: phi(s, t) = (s + t) / 2 

sage: bisection = intervalgen(f, phi, a, b) 

sage: iterate(falsepos) 

After 15 iterations: 2.15846441170219 

sage: iterate(bisection) 

After 20 iterations: 2.15847275559132 

Sage example in ./nonlinear.tex, line 954:: 

sage: a, b = RR(pi/2), RR(pi) 

sage: g = plot(f, a, b, rgbcolor='blue') 

sage: phi(s, t) = t - f(t) * (s - t) / (f(s) - f(t)) 

sage: falsepos = intervalgen(f, phi, a, b) 

sage: u, v, w = next(falsepos), next(falsepos), next(falsepos) 

sage: niter = 3 

sage: while niter < 7: 

....: g += line([(u, 0), (u, f(u))], rgbcolor='red', 

....: linestyle=':') 

....: g += line([(u, f(u)), (v, f(v))], rgbcolor='red') 

....: g += line([(v, 0), (v, f(v))], rgbcolor='red', 

....: linestyle=':') 

....: g += point((w, 0), rgbcolor='red') 

....: if (f(u) * f(w)) < 0: 

....: u, v = u, w 

....: else: 

....: u, v = w, v 

....: w = next(falsepos) 

....: niter += 1 

Sage example in ./nonlinear.tex, line 1025:: 

sage: f.derivative() 

x |--> 4*cos(x) - 1/2*e^x 

sage: a, b = RR(pi/2), RR(pi) 

Sage example in ./nonlinear.tex, line 1042:: 

sage: def newtongen(f, u): 

....: while 1: 

....: yield u 

....: u -= f(u) / (f.derivative()(u)) 

sage: def checkconv(u, v, w, prec): 

....: return abs(w - v) / abs(w) <= prec 

Sage example in ./nonlinear.tex, line 1053:: 

sage: iterate(newtongen(f, a), check=checkconv) 

After 6 iterations: 2.15846852566756 

Sage example in ./nonlinear.tex, line 1058:: 

sage: generator = newtongen(f, a) 

sage: g = plot(f, a, b, rgbcolor='blue') 

sage: u, v = next(generator), next(generator) 

sage: niter = 2 

sage: while niter < 6: 

....: g += point((u, 0), rgbcolor='red') 

....: g += line([(u, 0), (u, f(u))], rgbcolor='red', 

....: linestyle=':') 

....: g += line([(u, f(u)), (v, 0)], rgbcolor='red') 

....: u, v = v, next(generator) 

....: niter += 1 

Sage example in ./nonlinear.tex, line 1109:: 

sage: def secantgen(f, a): 

....: yield a 

....: estimate = f.derivative()(a) 

....: b = a - f(a) / estimate 

....: yield b 

....: while True: 

....: fa, fb = f(a), f(b) 

....: if fa == fb: 

....: estimate = f.derivative()(a) 

....: else: 

....: estimate = (fb - fa) / (b - a) 

....: a = b 

....: b -= fb / estimate 

....: yield b 

Sage example in ./nonlinear.tex, line 1136:: 

sage: iterate(secantgen(f, a), check=checkconv) 

After 8 iterations: 2.15846852557553 

Sage example in ./nonlinear.tex, line 1148:: 

sage: g = plot(f, a, b, rgbcolor='blue') 

sage: sequence = secantgen(f, a) 

sage: u, v = next(sequence), next(sequence) 

sage: niter = 2 

sage: while niter < 6: 

....: g += point((u, 0), rgbcolor='red') 

....: g += line([(u, 0), (u, f(u))], rgbcolor='red', 

....: linestyle=':') 

....: g += line([(u, f(u)), (v, 0)], rgbcolor='red') 

....: u, v = v, next(sequence) 

....: niter += 1 

Sage example in ./nonlinear.tex, line 1198:: 

sage: from collections import deque 

sage: basering = PolynomialRing(CC, 'x') 

sage: def quadraticgen(f, r, s): 

....: t = (r + s) / 2 

....: yield t 

....: points = deque([(r,f(r)), (s,f(s)), (t,f(t))], maxlen=3) 

....: while True: 

....: pol = basering.lagrange_polynomial(points) 

....: roots = pol.roots(ring=CC, multiplicities=False) 

....: u = min(roots, key=lambda x: abs(x - points[2][0])) 

....: points.append((u, f(u))) 

....: yield points[2][0] 

Sage example in ./nonlinear.tex, line 1230:: 

sage: generator = quadraticgen(f, a, b) 

sage: iterate(generator, check=checkconv) 

After 5 iterations: 2.15846852554764 

Sage example in ./nonlinear.tex, line 1287:: 

sage: rings = [ZZ, QQ, QQbar, RDF, RIF, RR, AA, CDF, CIF, CC] 

sage: for ring in rings: 

....: print("{0:50} {1}".format(ring, ring.is_exact())) 

Integer Ring True 

Rational Field True 

Algebraic Field True 

Real Double Field False 

Real Interval Field with 53 bits of precision False 

Real Field with 53 bits of precision False 

Algebraic Real Field True 

Complex Double Field False 

Complex Interval Field with 53 bits of precision False 

Complex Field with 53 bits of precision False 

Sage example in ./nonlinear.tex, line 1403:: 

sage: def steffensen(sequence): 

....: assert isinstance(sequence, GeneratorType) 

....: values = deque(maxlen=3) 

....: for i in range(3): 

....: values.append(next(sequence)) 

....: yield values[i] 

....: while 1: 

....: values.append(next(sequence)) 

....: u, v, w = values 

....: yield u - (v - u)^2 / (w - 2 * v + u) 

Sage example in ./nonlinear.tex, line 1419:: 

sage: g(x) = sin(x^2 - 2) * (x^2 - 2) 

sage: sequence = newtongen(g, RR(0.7)) 

sage: accelseq = steffensen(newtongen(g, RR(0.7))) 

sage: iterate(sequence, check=checkconv) 

After 17 iterations: 1.41422192763287 

sage: iterate(accelseq, check=checkconv) 

After 10 iterations: 1.41421041980166 

Sage example in ./nonlinear.tex, line 1432:: 

sage: sequence = newtongen(f, RR(a)) 

sage: accelseq = steffensen(newtongen(f, RR(a))) 

sage: iterate(sequence, check=checkconv) 

After 6 iterations: 2.15846852566756 

sage: iterate(accelseq, check=checkconv) 

After 7 iterations: 2.15846852554764 

Sage example in ./nonlinear.tex, line 1457:: 

sage: result = (f == 0).find_root(a, b, full_output=True) 

sage: result[0], result[1].iterations 

(2.1584685255476415, 9) 

Sage example in ./nonlinear.tex, line 1494:: 

sage: a, b = pi/2, pi 

sage: generator = newtongen(f, a) 

sage: next(generator) 

1/2*pi 

sage: next(generator) 

1/2*pi - (e^(1/2*pi) - 10)*e^(-1/2*pi) 

"""