Sat Jul 11 23:29:25 MDT 1998 aquarius% reduce REDUCE 3.6, 15-Jul-95, patched to 15 Apr 96 ... 1: % ----------[ R e d u c e ]---------- % ---------- Initialization ---------- on time; Time: 0 ms % ---------- Series ---------- % Taylor series---this first example comes from special relativity % => 1 + 1/2 (v/c)^2 + 3/8 (v/c)^4 + 5/16 (v/c)^6 + O((v/c)^8) load_package(taylor)$ Time: 110 ms taylorprintterms:= all$ Time: 0 ms 1/sqrt(1 - (v/c)^2); abs(c) --------------- 2 2 sqrt(c - v ) Time: 10 ms taylor(ws, v, 0, 7); 1 2 3 4 5 6 8 1 + ------*v + ------*v + -------*v + O(v ) 2 4 6 2*c 8*c 16*c Time: 10 ms taylorcombine(1/ws^2); 1 2 8 1 - ----*v + O(v ) 2 c Time: 0 ms % Note: sin(x) = x - x^3/6 + x^5/120 - x^7/5040 + O(x^9) % cos(x) = 1 - x^2/2 + x^4/24 - x^6/720 + O(x^8) % tan(x) = x + x^3/3 + 2/15 x^5 + 17/315 x^7 + O(x^9) tsin:= taylor(sin(x), x, 0, 7); 1 3 1 5 1 7 8 tsin := x - ---*x + -----*x - ------*x + O(x ) 6 120 5040 Time: 20 ms tcos:= taylor(cos(x), x, 0, 7); 1 2 1 4 1 6 8 tcos := 1 - ---*x + ----*x - -----*x + O(x ) 2 24 720 Time: 30 ms % Note that additional terms will be computed as needed taylorcombine(tsin/tcos); 1 3 2 5 17 7 8 x + ---*x + ----*x + -----*x + O(x ) 3 15 315 Time: 0 ms taylor(tan(x), x, 0, 7); 1 3 2 5 17 7 8 x + ---*x + ----*x + -----*x + O(x ) 3 15 315 Time: 10 ms clear tsin, tcos; Time: 0 ms % => -x^2/6 - x^4/180 - x^6/2835 - O(x^8) taylor(log(sin(x)/x), x, 0, 7); 1 2 1 4 1 6 8 - ---*x - -----*x - ------*x + O(x ) 6 180 2835 Time: 30 ms taylor(sin(x)/x, x, 0, 7); 1 2 1 4 1 6 8 1 - ---*x + -----*x - ------*x + O(x ) 6 120 5040 Time: 10 ms taylorcombine(log(ws)); 1 2 1 4 1 6 8 - ---*x - -----*x - ------*x + O(x ) 6 180 2835 Time: 10 ms % => [a f'(a d) + g(b d) + integrate(h(c y), y = 0..d)] % + [a^2 f''(a d) + b g'(b d) + h(c d)] (x - d) operator f, g, h; Time: 0 ms df(f(a*x), x) + g(b*x) + int(h(c*y), y, 0, x); df(f(a*x),x) + g(b*x) + int(h(c*y),y,0,x) Time: 140 ms taylor(ws, x, d, 1); g(b*d) + sub(x=d,df(f(a*x),x)) + sub(x=d,int(h(c*y),y,0,x)) + ( sub(x=d,df(f(a*x),x,2)) + sub(x=d,df(g(b*x),x)) 2 + sub(x=d,df(int(h(c*y),y,0,x),x)))*(x - d) + O((x - d) ) Time: 20 ms % Taylor series of nonscalar objects (noncommutative multiplication) % => (B A - A B) t^2/2 + O(t^3) [Stanly Steinberg] operator A, B; Time: 10 ms noncom A, B; Time: 0 ms e^((A() + B())*t) - e^(A()*t) * e^(B()*t); 0 Time: 0 ms taylor(e^((A() + B())*t) - e^(A()*t) * e^(B()*t), t, 0, 4); 0 Time: 10 ms clear A, B; Time: 0 ms % Laurent series: % => sum( Bernoulli[k]/k! x^(k - 2), k = 1..infinity ) % = 1/x^2 - 1/(2 x) + 1/12 - x^2/720 + x^4/30240 + O(x^6) % [Levinson and Redheffer, p. 173] taylor(1/(x*(exp(x) - 1)), x, 0, 8); -2 1 -1 1 1 2 1 4 1 6 1 8 x - ---*x + ---- - -----*x + -------*x - ---------*x + ----------*x 2 12 720 30240 1209600 47900160 9 + O(x ) Time: 20 ms % Puiseux series (terms with fractional degree): % => 1/sqrt(x - 3/2 pi) + (x - 3/2 pi)^(3/2) / 12 + O([x - 3/2 pi]^(7/2)) taylor(sqrt(sec(x)), x, 3/2*pi, 3); 3*pi - 1/2 1 3*pi 3/2 3*pi 7/2 (x - ------) + ----*(x - ------) + O((x - ------) ) 2 12 2 2 Time: 40 ms % Generalized Taylor series => sum( [x log x]^k/k!, k = 0..infinity ) taylor(x^x, x, 0, 3); 4 x + O(x ) x Time: 10 ms % Compare the generalized Taylor series of two different formulations of a % function => log(z) + log(cosh(w)) + tanh(w) z + O(z^2) s1:= taylor(log(sinh(z)) + log(cosh(z + w)), z, 0, 1); sinh(w) 2 s1 := log(z) + (log(cosh(w)) + ---------*z + O(z )) cosh(w) Time: 30 ms s2:= taylor(log(sinh(z) * cosh(z + w)), z, 0, 1); s2 := taylor(log(cosh(w + z)*sinh(z)),z,0,1) Time: 20 ms taylorcombine(s1 - s2); log(z) - taylor(log(cosh(w + z)*sinh(z)),z,0,1) sinh(w) 2 + (log(cosh(w)) + ---------*z + O(z )) cosh(w) Time: 10 ms clear s1, s2; Time: 0 ms % Look at the generalized Taylor series around x = 1 % => (x - 1)^a/e^b [1 - (a + 2 b) (x - 1) / 2 + O((x - 1)^2)] log(x)^a*exp(-b*x); a log(x) --------- b*x e Time: 10 ms taylor(ws, x, 1, 2); a 1 - a - 2*b 2 (x - 1) *(---- + ------------*(x - 1) + O((x - 1) )) b b e 2*e Time: 20 ms % Asymptotic expansions => sqrt(2) x + O(1/x) taylor(sqrt(2*x^2 + 1), x, infinity, 0); 1 1 sqrt(2)*----- + O(---) -1 x x Time: 10 ms % Wallis' product => 1/sqrt(pi n) + ... [Knopp, p. 385] load_package(specfn)$ *** psi already defined as operator *** ci already defined as operator *** si already defined as operator Time: 1460 ms plus GC time: 40 ms taylor(1/2^(2*n) * Binomial(2*n, n), n, infinity, 0); binomial(2*n,n) taylor(-----------------,n,infinity,0) 2*n 2 Time: 30 ms % => 0!/x - 1!/x^2 + 2!/x^3 - 3!/x^4 + O(1/x^5) [Knopp, p. 544] exp(x) * int(exp(-t)/t, t, x, infinity); x exp( - t) e *int(-----------,t,x,infinity) t Time: 0 ms taylor(ws, x, infinity, 5); ***** Zero divisor ***** Error during expansion (possible singularity!) Cont? (Y or N) ?y Time: 10 ms % Multivariate Taylor series expansion => 1 - (x^2 + 2 x y + y^2)/2 + O(x^4) taylor(cos(x + y), {x, y}, 0, 3); 1 2 1 2 4 1 - ---*y - y*x - ---*x + O({x,y} ) 2 2 Time: 50 ms % Power series (compute the general formula) load_package(fps)$ Time: 250 ms FPS(log(sin(x)/x), x); sin(x) fps(log(--------),x,0) x Time: 5260 ms plus GC time: 690 ms FPS(exp(-x)*sin(x), x); k k/2 k*pi - x *2 *sin(------) 4 infsum(------------------------,k,0,infinity) factorial(k) Time: 910 ms plus GC time: 70 ms taylor(ws, x, 0, 7); ***** Syntax error: 0()()(infsum(0,k,0,infinity)( ***** Bus Error in inprint Cont? (Y or N) ?y Time: 60 ms % Derive an explicit Taylor series solution of y as a function of x from the % following implicit relation: % y = x - 1 + (x - 1)^2/2 + 2/3 (x - 1)^3 + (x - 1)^4 + 17/10 (x - 1)^5 + ... x = sin(y) + cos(y); x=cos(y) + sin(y)$ Time: 10 ms taylor(rhs(ws), y, 0, 7); taylor(1 + y - 1/2*y**2 - 1/6*y**3 + 1/24*y**4 + 1/120*y**5 - 1/720*y**6 - 1/ 5040*y**7,y,0,7)$ Time: 60 ms y = taylorrevert(ws, y, x); y=taylor(x - 1 + 1/2*(x - 1)**2 + 2/3*(x - 1)**3 + (x - 1)**4 + 17/10*(x - 1)**5 + 37/12*(x - 1)**6 + 41/7*(x - 1)**7,x,1,7)$ Time: 50 ms % Pade (rational function) approximation => (2 - x)/(2 + x) load_package(rataprx)$ Time: 70 ms pade(exp(-x), x, 0, 1, 1); ( - x + 2)/(x + 2)$ Time: 60 ms % Fourier series of f(x) of period 2 p over the interval [-p, p] load_package(camal)$ Time: 30 ms % => - (2 p / pi) sum( (-1)^n sin(n pi x / p) / n, n = 1..infinity ) %fourier(x, x, p); fourier(x); [(x)]$ Time: 0 ms % => p / 2 % - (2 p / pi^2) sum( [1 - (-1)^n] cos(n pi x / p) / n^2, n = 1..infinity ) %fourier(abs(x), x, p); fourier(abs(x)); ***** Unknown function ***** Bus Error in calling prin2 Cont? (Y or N) ?y Time: 0 ms % ---------- Quit ---------- quit; Quitting real 38.76 user 9.80 sys 3.01 ------------------------------------------------------------------------------- Sat Jul 11 23:38:30 MDT 1998 aquarius% reduce REDUCE 3.6, 15-Jul-95, patched to 15 Apr 96 ... 1: % ----------[ R e d u c e ]---------- % ---------- Initialization ---------- on time; Time: 0 ms % ---------- Series ---------- load_package(taylor)$ Time: 110 ms taylorprintterms:= all$ Time: 0 ms % Derive an explicit Taylor series solution of y as a function of x from the % following implicit relation: % y = x - 1 + (x - 1)^2/2 + 2/3 (x - 1)^3 + (x - 1)^4 + 17/10 (x - 1)^5 + ... x = sin(y) + cos(y); x=cos(y) + sin(y) Time: 0 ms taylor(rhs(ws), y, 0, 7); 1 2 1 3 1 4 1 5 1 6 1 7 8 1 + y - ---*y - ---*y + ----*y + -----*y - -----*y - ------*y + O(y ) 2 6 24 120 720 5040 Time: 40 ms y = taylorrevert(ws, y, x); 1 2 2 3 4 17 5 37 6 y=x - 1 + ---*(x - 1) + ---*(x - 1) + (x - 1) + ----*(x - 1) + ----*(x - 1) 2 3 10 12 41 7 8 + ----*(x - 1) + O((x - 1) ) 7 Time: 20 ms % Pade (rational function) approximation => (2 - x)/(2 + x) load_package(rataprx)$ Time: 230 ms pade(exp(-x), x, 0, 1, 1); - x + 2 ---------- x + 2 Time: 40 ms % Fourier series of f(x) of period 2 p over the interval [-p, p] load_package(camal)$ Time: 30 ms % => - (2 p / pi) sum( (-1)^n sin(n pi x / p) / n, n = 1..infinity ) %fourier(x, x, p); fourier(x); [(x)] Time: 10 ms % => p / 2 % - (2 p / pi^2) sum( [1 - (-1)^n] cos(n pi x / p) / n^2, n = 1..infinity ) %fourier(abs(x), x, p); fourier(abs(x)); ***** Unknown function ***** Bus Error in calling prin2 Cont? (Y or N) ?y Time: 0 ms % ---------- Quit ---------- quit; Quitting real 11.21 user 0.56 sys 1.15