Fri Jan 9 15:07:24 MST 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 % ---------- Programming and Miscellaneous ---------- % How easy is it to substitute x for a + b in the following expression? % => (x + c)^2 + (d - x)^2 off exp; Time: 10 ms xpr:= (a + b + c)^2 + (d - a - b)^2; 2 2 xpr := (b + c + a) + (b - d + a) Time: 0 ms sub(a + b = x, xpr); ***** a + b invalid as kernel Cont? (Y or N) ?y Time: 10 ms sub(b = x - a, xpr); 2 2 (c + x) + (d - x) Time: 0 ms clear xpr; Time: 0 ms on exp; Time: 0 ms % How easy is it to substitute r for sqrt(x^2 + y^2) in the following % expression? => x/r x/sqrt(x^2 + y^2); x --------------- 2 2 sqrt(x + y ) Time: 10 ms sub(sqrt(x^2 + y^2) = r, ws); x --- r Time: 10 ms % Change variables so that the following transcendental expression is % converted into a rational expression [Vernor Vinge] % => (r - 1)^4 (u^4 - r u^3 - r^3 u + r u + r^4)/[u^4 (2 r - 1)^2] off exp; Time: 0 ms q:= (1/r^4 + 1/(r^2 - 2*r*cos(t) + 1)^2 - 2*(r - cos(t))/(r^2 * (r^2 - 2*r*cos(t) + 1)^(3/2))) / (1/r^4 + 1/(r - 1)^4 - 2*(r - 1)/(r^2 * (r^2 - 2*r + 1)^(3/2))); 2 2 2 q := (((2*(r + 1 - 2*cos(t)*r)*(cos(t) - r) + sqrt( - 2*cos(t)*r + r + 1)*r ) 2 2 2 2 6 *r + sqrt( - 2*cos(t)*r + r + 1)*(r + 1 - 2*cos(t)*r) )*(r - 1) 2 *abs(r - 1))/(sqrt( - 2*cos(t)*r + r + 1)*( 2 2 5 2 ((r - 2*r + 1)*abs(r - 1)*r - 2*(r - 1) )*r 2 4 2 2 + (r - 2*r + 1)*(r - 1) *abs(r - 1))*(r + 1 - 2*cos(t)*r) ) Time: 150 ms sub(cos(t) = (r^2 - u^2 + 1)/(2*r), q); 2 2 2 3 4 6 ( - (((u - 1 + r )*u - abs(u)*r )*r - abs(u)*u )*(r - 1) *abs(r - 1))/(( 2 2 5 2 ((r - 2*r + 1)*abs(r - 1)*r - 2*(r - 1) )*r 2 4 4 + (r - 2*r + 1)*(r - 1) *abs(r - 1))*abs(u)*u ) Time: 20 ms ws where abs(~e) => e; 2 2 3 4 5 - (((u - 1 + r )*u - r )*r - u )*(r - 1) --------------------------------------------- 3 2 4 (4*r - 8*r + 5*r - 1)*u Time: 10 ms on factor; Time: 0 ms ws; 2 2 3 4 4 - (((r + u - 1)*u - r )*r - u )*(r - 1) --------------------------------------------- 2 4 (2*r - 1) *u Time: 50 ms off factor; Time: 0 ms on exp; Time: 10 ms % Establish a rule to symmetrize a differential operator: [Stanly Steinberg] % f g'' + f' g' -> (f g')' operator f, g, !~f, !~g; Time: 0 ms symmetrize:= ~f(~x)*df(~g(~x), ~x, 2) + df(~f(~x), ~x)*df(~g(~x), ~x) => df(f(x)*df(g(x), x), x); symmetrize := ~(f (~ x))*df(~(g (~ x)),~x,2) + df(~(f (~ x)),~x)*df(~(g (~ x)),~x) => df(f(x)*df(g(x),x),x) Time: 0 ms q:= f(x)*df(g(x), x, 2) + df(f(x), x)*df(g(x), x); q := df(f(x),x)*df(g(x),x) + df(g(x),x,2)*f(x) Time: 0 ms q where symmetrize; ***** df(f(x)*df(g(x),x),x) invalid as rule Cont? (Y or N) ?y Time: 0 ms % => 2 (f g')' + f g 2*q + f(x)*g(x) where symmetrize; ***** df(f(x)*df(g(x),x),x) invalid as rule Cont? (Y or N) ?y Time: 0 ms clear f, g, q; Time: 0 ms % Infinite lists: [1 2 3 4 5 ...] * [1 3 5 7 9 ...] % => [1 6 15 28 45 66 91 ...] l1:= {1, 2, 3, 4, 5}$ Time: 10 ms l2:= {1, 3, 5, 7, 9}$ Time: 0 ms for i:= 1:length(l1) collect part(l1, i)*part(l2, i); {1,6,15,28,45} Time: 0 ms clear l1, l2; Time: 0 ms % Write a simple program to compute Legendre polynomials procedure p(n, x); 1/(2^n*factorial(n)) * df((x^2 - 1)^n, x, n)$ Time: 10 ms % p[0](x) = 1, p[1](x) = x, p[2](x) = (3 x^2 - 1)/2, % p[3](x) = (5 x^3 - 3 x)/2, p[4](x) = (35 x^4 - 30 x^2 + 3)/8 for i:= 0:4 do write("p(", i, ", x) = ", p(i, x)); p(0, x) = 1 p(1, x) = x 2 3*x - 1 p(2, x) = ---------- 2 2 x*(5*x - 3) p(3, x) = -------------- 2 4 2 35*x - 30*x + 3 p(4, x) = ------------------- 8 Time: 20 ms % p[4](1) = 1 sub(x = 1, p(4, x)); 1 Time: 0 ms % Now, perform the same computation using a recursive definition procedure pp(n, x); if n = 0 then 1 else if n = 1 then x else ((2*n - 1)*x*pp(n - 1, x) - (n - 1)*pp(n - 2, x))/n$ *** Function `pp' has been redefined Time: 10 ms for i:= 0:4 do write("pp(", i, ", x) = ", pp(i, x)); pp(0, x) = 1 pp(1, x) = x 2 3*x - 1 pp(2, x) = ---------- 2 2 x*(5*x - 3) pp(3, x) = -------------- 2 4 2 35*x - 30*x + 3 pp(4, x) = ------------------- 8 Time: 10 ms pp(4, 1); 1 Time: 0 ms clear p, pp; Time: 0 ms % Iterative computation of Fibonacci numbers on fort; Time: 0 ms procedure myfib(n); begin scalar i, j, k, f; if n < 0 then error('undefined) else if n < 2 then return(n) else <>; return(f)>> end$ Time: 0 ms % Convert the function into FORTRAN syntax off fort; Time: 0 ms % Create a list of the first 11 values of the function. for i:= 0:10 collect myfib(i); {0,1,1,2,3,5,8,13,21,34,55} Time: 20 ms clear myfib; Time: 0 ms % Define the function p(x) = x^2 - 4 x + 7 such that p(lambda) = 0 for % lambda = 2 +- i sqrt(3) and p(A) = [[0 0], [0 0]] for A = [[1 -2], [2 3]] % (the lambda are the eigenvalues and p(x) is the characteristic polynomial of % A) [Johnson and Reiss, p. 184] procedure p(x); x^2 - 4*x + 7$ Time: 10 ms p(2 + i*sqrt(3)); 0 Time: 0 ms p(mat((1, -2), (2, 3))); ***** Matrix mismatch Cont? (Y or N) ?y Time: 20 ms clear p; Time: 0 ms % Define a function to be the result of a calculation operator f; Time: 0 ms -log(x^2 - 2^(1/3)*x + 2^(2/3))/(6 * 2^(2/3)) + atan((2*x - 2^(1/3))/(2^(1/3) * sqrt(3))) / (2^(2/3) * sqrt(3)) + log(x + 2^(1/3))/(3 * 2^(2/3)); 1/3 2 - 2*x 2/3 1/3 2 ( - 6*atan(--------------) - sqrt(3)*log(2 - 2 *x + x ) 1/3 2 *sqrt(3) 1/3 2/3 + 2*sqrt(3)*log(2 + x))/(6*2 *sqrt(3)) Time: 20 ms for all x let f(x) = ws; Time: 10 ms xpr:= f(y); 1/3 2 - 2*y 2/3 1/3 2 xpr := ( - 6*atan(--------------) - sqrt(3)*log(2 - 2 *y + y ) 1/3 2 *sqrt(3) 1/3 2/3 + 2*sqrt(3)*log(2 + y))/(6*2 *sqrt(3)) Time: 30 ms % Display the top-level structure of a nasty expression, hiding the % lower-level details. structr(xpr); - 6*ans1 - ans2*ans3 + 2*ans2*ans4 ------------------------------------- 2 6*ans2*ans5 where ans4 := log(ans5 + y) 2/3 2 ans3 := log(2 - ans5*y + y ) 1/2 ans2 := 3 ans5 - 2*y ans1 := atan(--------------) ans5*sqrt(3) 1/3 ans5 := 2 Time: 10 ms clear xpr, f; Time: 0 ms % Convert the following expression into TeX or LaTeX sqrt((exp(x^2) + exp(-x^2))/(sqrt(3)*x - sqrt(2))); 2 2*x e + 1 sqrt(-----------------------------) 2 2 x x e *sqrt(3)*x - e *sqrt(2) Time: 20 ms load_package(rlfi)$ Time: 10 ms on latex; \documentstyle{article} \begin{document} Time: 0 ms sqrt((exp(x^2) + exp(-x^2))/(sqrt(3)*x - sqrt(2))); \begin{displaymath} \sqrt {\left(e^{2 x^{2}}+1\right)/\left(e^{x^{2}} \sqrt {3} x-e^{x^{2}} \sqrt {2}\right)} \end{displaymath} Time: 20 ms off lasimp; Time: 0 ms sqrt((exp(x^2) + exp(-x^2))/(sqrt(3)*x - sqrt(2))); \begin{displaymath} \sqrt {\left(\exp \left(x^{2}\right)+\exp \left(-x^{2}\right)\right)/\left( \sqrt {3} x-\sqrt {2}\right)} \end{displaymath} Time: 0 ms off latex; \end{document} Time: 0 ms load_package(tri)$ % TeX-REDUCE-Interface 0.50 % set greek asserted % set lowercase asserted % \tolerance 10 % \hsize=150mm Time: 20 ms on TeXIndent; Time: 0 ms y = sqrt((exp(x^2) + exp(-x^2))/(sqrt(3)*x - sqrt(2))); $$\displaylines{\qdd y= \[\sqrt{ \frac{e^{2\cdot x^{2}} +1}{ e^{x^{2}}\cdot \sqrt{3}\cdot x -e^{x^{2}}\cdot \sqrt{2}}} \] \cr}$$ Time: 40 ms off TeXIndent; Time: 0 ms % ---------- Quit ---------- quit; Quitting real 10.37 user 0.68 sys 1.26