# HG changeset patch # User Nick Alexander # Date 1241921338 25200 # Branch symbolics_switch # Node ID e8f0e91a03eabb520b62c3d3876dc7441d017d33 # Parent 6a4c0c6c6c1c85e5d69a176d203a0b55a78ad42d [mq]: wester-partial.patch diff -r 6a4c0c6c6c1c -r e8f0e91a03ea sage/calculus/wester.py --- a/sage/calculus/wester.py Sat May 09 18:41:45 2009 -0700 +++ b/sage/calculus/wester.py Sat May 09 19:08:58 2009 -0700 @@ -28,9 +28,9 @@ sage: # Evaluate e^(Pi*Sqrt(163)) to 50 decimal digits sage: a = e^(pi*sqrt(163)); a - e^(sqrt(163)*pi) + e^(pi*sqrt(163)) sage: print RealField(150)(a) - 2.6253741264076874399999999999925007259719819e17 + 2.6253741264076874399999999999925007259719820e17 :: @@ -76,12 +76,12 @@ sage: # (not exactly ok) Sqrt(14+3*Sqrt(3+2*Sqrt(5-12*Sqrt(3-2*Sqrt(2)))))=3+Sqrt(2). sage: a = sqrt(14+3*sqrt(3+2*sqrt(5-12*sqrt(3-2*sqrt(2))))) sage: b = 3+sqrt(2) - sage: print a, b - sqrt(3 sqrt(2 sqrt(5 - 12 sqrt(3 - 2 sqrt(2))) + 3) + 14) sqrt(2) + 3 - sage: print bool(a==b) + sage: a, b + (sqrt(3*sqrt(2*sqrt(-12*sqrt(-2*sqrt(2) + 3) + 5) + 3) + 14), sqrt(2) + 3) + sage: bool(a==b) False - sage: print float(a-b) - 1.7763568394e-15 + sage: abs(float(a-b)) < 1e-10 + True sage: # 2*Infinity-3=Infinity. sage: 2*infinity-3 == infinity True @@ -114,12 +114,12 @@ sage: f = (exp(x)-1)/(exp(x/2)+1) sage: g = exp(x/2)-1 sage: f - (e^x - 1)/(e^(x/2) + 1) + (e^x - 1)/(e^(1/2*x) + 1) sage: g - e^(x/2) - 1 - sage: print f(x=10.0), g(x=10.0) - 147.4131591025766 147.4131591025766 - sage: print bool(f == g) + e^(1/2*x) - 1 + sage: f(x=10.0).n(53), g(x=10.0).n(53) + (147.413159102577, 147.413159102577) + sage: bool(f == g) True :: @@ -185,13 +185,8 @@ :: sage: # (YES) Partial fraction decomposition of (x^2+2*x+3)/(x^3+4*x^2+5*x+2) - sage: f = (x^2+2*x+3)/(x^3+4*x^2+5*x+2) - sage: print f - 2 - x + 2 x + 3 - ------------------- - 3 2 - x + 4 x + 5 x + 2 + sage: f = (x^2+2*x+3)/(x^3+4*x^2+5*x+2); f + (x^2 + 2*x + 3)/(x^3 + 4*x^2 + 5*x + 2) :: @@ -238,8 +233,8 @@ sage: # (NO) Solve the inequality (x-1)*...*(x-5)<0. sage: eqn = prod(x-i for i in range(1,5 +1)) < 0 sage: # but don't know how to solve - sage: print eqn - (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) < 0 + sage: eqn + (x - 5)*(x - 4)*(x - 3)*(x - 2)*(x - 1) < 0 :: @@ -264,9 +259,8 @@ :: sage: # (YES) Sqrt(997)-(997^3)^(1/6)=0 - sage: a = sqrt(997) - (997^3)^(1/6) - sage: print a - 0 + sage: a = sqrt(997) - (997^3)^(1/6); a + 0 sage: print bool(a == 0) True @@ -287,14 +281,12 @@ sage: # (YES) (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3))-6 = 0 sage: ## same issue as above -- can only do using number fields - sage: a = (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6 - sage: print a - 1/3 1/3 3 1/3 1/3 - (4 + 2 ) - 6 (4 + 2 ) - 6 - sage: print bool(a==0) + sage: a = (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6; a + (2^(1/3) + 4^(1/3))^3 - 6*2^(1/3) - 6*4^(1/3) - 6 + sage: bool(a==0) True - sage: print float(a) - 3.5527136788e-15 + sage: abs(float(a)) < 1e-10 + True sage: ## but we can do it using number fields. sage: reset('x') sage: k. = NumberField(x^3-2) @@ -323,14 +315,8 @@ True sage: abs(float(g(x=0))) < 1e-10 True - sage: print g - 2 x pi - sec (- + ---) 2 - 2 4 sec (x) - -------------- - ----------------- - x pi 2 - 2 tan(- + ---) sqrt(1 - tan (x)) - 2 4 + sage: g + -(tan(x)^2 + 1)/sqrt(-tan(x)^2 + 1) + 1/2*(tan(1/4*pi + 1/2*x)^2 + 1)/tan(1/4*pi + 1/2*x) :: @@ -338,14 +324,12 @@ sage: var('r') r sage: f = log( (2*sqrt(r) + 1) / sqrt(4*r + 4*sqrt(r) + 1)) - sage: print f - 2 sqrt(r) + 1 - log(-------------------------) - sqrt(4 r + 4 sqrt(r) + 1) - sage: print bool(f == 0) + sage: f + log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1)) + sage: bool(f == 0) False - sage: print [float(f(r=i)) for i in [0.1,0.3,0.5]] - [0.0, 0.0, 0.0] + sage: [abs(float(f(r=i))) < 1e-10 for i in [0.1,0.3,0.5]] + [True, True, True] :: @@ -353,36 +337,26 @@ sage: # (4*r+4*Sqrt(r)+1)^(Sqrt(r)/(2*Sqrt(r)+1))*(2*Sqrt(r)+1)^(2*Sqrt(r)+1)^(-1)-2*Sqrt(r)-1=0, assuming r>0. sage: assume(r>0) sage: f = (4*r+4*sqrt(r)+1)^(sqrt(r)/(2*sqrt(r)+1))*(2*sqrt(r)+1)^(2*sqrt(r)+1)^(-1)-2*sqrt(r)-1 - sage: print f - 1 sqrt(r) - ------------- ------------- - 2 sqrt(r) + 1 2 sqrt(r) + 1 - (2 sqrt(r) + 1) (4 r + 4 sqrt(r) + 1) - - 2 sqrt(r) - 1 + sage: f + (2*sqrt(r) + 1)^((2*sqrt(r) + 1))*(4*r + 4*sqrt(r) + 1)^(sqrt(r)/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1 sage: bool(f == 0) False - sage: [float(f(r=i)) for i in [0.1,0.3,0.5]] - [0.0, 0.0, -2.2204460492503131e-16] + sage: [abs(float(f(r=i))) < 1e-10 for i in [0.1,0.3,0.5]] + [True, True, True] :: sage: # (YES) Obtain real and imaginary parts of Ln(3+4*I). - sage: a = log(3+4*I) - sage: print a - log(4 I + 3) - sage: print a.real() - log(5) - sage: print a.imag() - 4 - arctan(-) - 3 + sage: a = log(3+4*I); a + log(4*I + 3) + sage: a.real() + sage: a.imag() :: sage: # (YES) Obtain real and imaginary parts of Tan(x+I*y) - sage: a = tan(x + I*y) - sage: print a - tan( I y + x) + sage: a = tan(x + I*y); a + tan(x + I*y) sage: print a.real() sin(2 x) -------------------- @@ -436,7 +410,7 @@ sage: # x+y+z==6,2*x+y+2*z==10,x+3*y+z==10 sage: # First symbolically: sage: solve([x+y+z==6, 2*x+y+2*z==10, x+3*y+z==10], x,y,z) - [[x == 4 - r1, y == 2, z == r1]] + [[x == -r1 + 4, y == 2, z == r1]] :: @@ -462,8 +436,8 @@ [ 1 c c^2 c^3] [ 1 d d^2 d^3] sage: d = m.determinant() - sage: print d.factor() - (b - a) (a - c) (b - c) (a - d) (b - d) (d - c) + sage: d.factor() + (c - d)*(b - d)*(b - c)*(a - d)*(a - c)*(a - b) :: @@ -507,12 +481,10 @@ sage: # OK Verify some standard limits found by L'Hopital's rule: sage: # Verify(Limit(x,Infinity) (1+1/x)^x, Exp(1)); sage: # Verify(Limit(x,0) (1-Cos(x))/x^2, 1/2); - sage: print limit( (1+1/x)^x, x = oo) + sage: limit( (1+1/x)^x, x = oo) e - sage: print limit( (1-cos(x))/(x^2), x = 1/2) - 1 - 4 - 4 cos(-) - 2 + sage: limit( (1-cos(x))/(x^2), x = 1/2) + -4*cos((1/2)) + 4 :: @@ -547,7 +519,7 @@ sage: var('v,c') (v, c) sage: taylor(1/sqrt(1-v^2/c^2), v, 0, 7) - 1 + v^2/(2*c^2) + 3*v^4/(8*c^4) + 5*v^6/(16*c^6) + 1/2*v^2/c^2 + 3/8*v^4/c^4 + 5/16*v^6/c^6 + 1 :: @@ -565,7 +537,7 @@ :: - sage: # (YES) Taylor expansion of Ln(x)^a*Exp(-b*x) at x=1. + sage: XXX # (YES) Taylor expansion of Ln(x)^a*Exp(-b*x) at x=1. E = EllipticCurve([1,2,3/4,7,19]); E (x - 1)^a/e^b - ((x - 1)^a*a + 2*b*(x - 1)^a)*(x - 1)/(2*e^b) + (3*(x - 1)^a*a^2 + (12*b + 5)*(x - 1)^a*a + 12*b^2*(x - 1)^a)*(x - 1)^2/(24*e^b) - ((x - 1)^a*a^3 + (6*b + 5)*(x - 1)^a*a^2 + (12*b^2 + 10*b + 6)*(x - 1)^a*a + 8*b^3*(x - 1)^a)*(x - 1)^3/(48*e^b) @@ -573,7 +545,7 @@ sage: # (YES) Taylor expansion of Ln(Sin(x)/x) at x=0. sage: taylor(log(sin(x)/x), x, 0, 10) - -x^2/6 - x^4/180 - x^6/2835 - x^8/37800 - x^10/467775 + -1/467775*x^10 - 1/37800*x^8 - 1/2835*x^6 - 1/180*x^4 - 1/6*x^2 :: @@ -603,28 +575,25 @@ sage: # Deriv(x,n) (x^2-1)^n ); sage: # TestYacas(P(4,x), (35*x^4)/8+(-15*x^2)/4+3/8); sage: P = lambda n, x: simplify(diff((x^2-1)^n,x,n) / (2^n * factorial(n))) - sage: print P(4,x).expand() - 4 2 - 35 x 15 x 3 - ----- - ----- + - - 8 4 8 + sage: P(4,x).expand() + 35/8*x^4 - 15/4*x^2 + 3/8 :: sage: # (YES) Define the polynomial p=Sum(i,1,5,a[i]*x^i). sage: # symbolically - sage: ps = sum(var('a%s'%i)*x^i for i in range(1,6)) - sage: print 'symbolic\n',ps - symbolic - 5 4 3 2 - a5 x + a4 x + a3 x + a2 x + a1 x + sage: ps = sum(var('a%s'%i)*x^i for i in range(1,6)); ps + a5*x^5 + a4*x^4 + a3*x^3 + a2*x^2 + a1*x + sage: ps.parent() + Symbolic Ring + sage: # algebraically sage: R = PolynomialRing(QQ,5,names='a') sage: S. = PolynomialRing(R) - sage: p = S(list(R.gens()))*x - sage: print 'algebraic\n',p - algebraic + sage: p = S(list(R.gens()))*x; p a4*x^5 + a3*x^4 + a2*x^3 + a1*x^2 + a0*x + sage: p.parent() + Univariate Polynomial Ring in x over Multivariate Polynomial Ring in a0, a1, a2, a3, a4 over Rational Field :: @@ -634,7 +603,7 @@ sage: # We use the trick of evaluating the algebraic poly at a symbolic variable: sage: restore('x') sage: p(x) - x*(x*(x*(x*(a4*x + a3) + a2) + a1) + a0) + ((((a4*x + a3)*x + a2)*x + a1)*x + a0)*x ::