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REDUCE 3.4.1, 15-Jul-92 ... 1: (NUMERIC) on errcont; bounds (x,x=(1 .. 2)); 1 .. 2 bounds (2*x,x=(1 .. 2)); 2 .. 4 bounds (x**3,x=(1 .. 2)); - 1 ------ .. 8 8 bounds (x*y,x=(1 .. 2),y=(-1 .. 0)); 1 - 2 .. --- 2 bounds (x**3+y,x=(1 .. 2),y=(-1 .. 0)); - 9 ------ .. 8 8 bounds (x**3/y,{x=(1 .. 2),y=(-1 .. -0.5)}); 1 - 16 .. --- 4 bounds (x**3/y,x=(1 .. 2),y=(-1 .. -0.5)); 1 - 16 .. --- 4 % unbounded expression (pole at y=0) bounds (x**3/y,x=(1 .. 2),y=(-1 .. 0.5)); ***** unbounded in range on rounded; bounds(e**x,x=(1 .. 2)); 2.71828182846 .. 7.38905609893 bounds((1/2)**x,x=(1 .. 2)); 0.5 .. 0.25 off rounded; bounds(abs x,x=(1 .. 2)); 1 .. 2 bounds(abs x,x=(-3 .. 2)); 0 .. 3 bounds(abs x,x=(-3 .. -2)); 2 .. 3 bounds(sin x,x=(1 .. 2)); - 1 .. 1 on rounded; bounds(sin x,x=(1 .. 2)); 0.841470984808 .. 1 bounds(sin x,x=(1 .. 10)); - 1 .. 1 bounds(sin x,x=(1001 .. 1002)); 0.167266541974 .. 0.919990597586 bounds(log x,x=(1 .. 10)); 0 .. 2.30258509299 bounds(tan x,x=(1 .. 1.1)); 1.55740772465 .. 1.96475965725 bounds(cot x,x=(1 .. 1.1)); 0.508968105239 .. 0.642092615934 bounds(asin x,x=(-0.6 .. 0.6)); - 0.643501108793 .. 0.643501108793 bounds(acos x,x=(-0.6 .. 0.6)); 0.927295218002 .. 2.21429743559 bounds(sqrt(x),x=(1 .. 1.1)); 1 .. 1.04880884817 bounds(x**(7/3),x=(1 .. 1.1)); 1 .. 1.2490589397 bounds(x**y,x=(1 .. 1.1),y=(2 .. 4)); 0.870336363636 .. 1.4641 off rounded; % MINIMA (steepest descent) % Rosenbrock function (minimum extremely hard to find). fktn := 100*(x1^2-x2)^2 + (1-x1)^2; 4 2 2 2 FKTN := 100*X1 - 200*X1 *X2 + X1 - 2*X1 + 100*X2 + 1 num_min(fktn, x1=-1.2, x2=1, accuracy=6); {0.000000218702240318,{X1=0.999532844964,X2=0.99906807244}} % infinitely many local minima num_min(sin(x)+x/5, x=1); { - 1.33422674663,{X= - 1.77215430279}} % bivariate polynomial num_min(x^4 + 3 x^2 * y + 5 y^2 + x + y, x=0.1, y=0.2); { - 0.832523282274,{X= - 0.889601609042,Y= - 0.33989805551}} % ROOTS (non polynomial: damped Newton) num_solve (cos x -x, x=0,accuracy=6); *** precision increased to 13 {X=0.739085133385} % automatically randomized starting point num_solve (cos x -x,x, accuracy=6); {X=0.739085134394} % syntactical errors: forms do not evaluate to purely % numerical values num_solve (cos x -x, x=a); ***** A invalid as number num_solve (cos x -a, x=0); ***** A invalid as number ***** error during function evaluation (e.g. singularity) num_solve (sin x = 0, x=3); *** precision increased to 13 {X=3.1415926533} % blows up: no real solution exists num_solve(sin x = 2, x=1); ***** Newton method does not converge % solution in complex plane(only fond with complex starting point): on complex; *** Domain mode ROUNDED changed to COMPLEX-ROUNDED num_solve(sin x = 2, x=1+i); {X=1.5707963268 + 1.31695789692*I} off complex; *** Domain mode COMPLEX-ROUNDED changed to ROUNDED % blows up for derivative 0 in starting point num_solve(x^2-1, x=0); ***** zero divisor in quotient ***** error during function evaluation (e.g. singularity) % succeeds because of perturbed starting point num_solve(x^2-1, x=0.1); {X=1.00000000643} % bivariate equation system num_solve({sin x=cos y, x + y = 1},{x=1,y=2}); {X= - 4.99778688329,Y=5.99778688329} on evallhseqp; % So both sides of equation evaluate. sub(ws,{sin x=cos y, x + y = 1}); {0.959549703325=0.959549556645,1=1} % INTEGRALS num_int( x**2,x=(1 .. 2),accuracy=3); 2.33333333333 % critical function: almost flat line with one % high narrow needle. needle := 1/(10**-4 + x**2); 1 NEEDLE := -------------- 2 1 X + ------- 10000 num_int(needle,x=(-1 .. 1),accuracy=3); 312.165724458 % 312.16 num_int(exp(-x**2),x=(-10 .. 10),accuracy=3); 1.7724717458 % 1.772 num_int(exp(-x**2),x=(-10**2 .. 10**2)); *** ROUNDBF turned on to increase accuracy 1.74618555047 % 1.7461 off roundbf; % cases with singularities num_int(1/sqrt x ,x=(0 .. 1)); ***** zero divisor in quotient requested accuracy cannot be reached within iteration limit critical area of function near to {X=0.0} current error estimate: 0.00337429826955 1.99997531551 % 1.999 num_int(1/sqrt abs x ,x=(-1 .. 1)); ***** zero divisor in quotient requested accuracy cannot be reached within iteration limit critical area of function near to {X=0.0} current error estimate: 0.00870788028243 3.99992568283 % 3.999 % simple multidimensional integrals num_int(x+y,x=(0 .. 1),y=(2 .. 3)); 3.0 num_int(sin(x+y),x=(0 .. 1),y=(0 .. 1)); 0.773147731572 % APPROXIMATION %approximate sin x by a cubic polynomial num_fit(sin x,{1,x,x**2,x**3},x=for i:=0:20 collect 0.1*i); *** precision increased to 15 3 2 { - 0.0847539695007*X - 0.134641944761*X + 1.06263064633*X - 0.00519313406389, { - 0.00519313406389,1.06263064633, - 0.134641944761, - 0.0847539695007}} % approximate x**2 by a harmonic series in the interval [0,1] num_fit(x**2,1 . for i:=1:5 join {sin(i*x), cos(i*x)}, x=for i:=0:10 collect i/10); { - 0.196187855271*SIN(5*X) + 0.874456227951*SIN(4*X) - 0.38159923677*SIN(3*X) - 3.3562268088*SIN(2*X) + 5.34063539945*SIN(X) - 0.13327743733*COS(5*X) + 1.56208432179*COS(4*X) - 5.82407625714*COS(3*X) + 9.95627217794*COS(2*X) - 11.0565397235*COS(X) + 5.49553691049, {5.49553691049,5.34063539945, - 11.0565397235, - 3.3562268088,9.9562 7217794, - 0.38159923677, - 5.82407625714,0.874456227951,1.56208432 179, - 0.196187855271, - 0.13327743733}} % approximate a set of points by a polynomial pts:=for i:=1 step 0.1 until 3 collect i$ vals:=for each p in pts collect (p+2)**3$ num_fit(vals,{1,x,x**2,x**3},x=pts); 3 2 {1.0*X + 5.99999999998*X + 12.0*X + 7.99999999998,{7.99999999998,12 .0,5.99999999998,1.0}} first ws - (x+2)**3; 3 2 2.70006239589E-12*X - 0.0000000000166000546642*X + 0.0000000000329993810055*X - 0.0000000000205000461051 % ODE SOLUTION (Runge-Kutta) depend(y,x); % approximate y=y(x) with df(y,x)=2y in interval [0 : 5] num_odesolve(df(y,x)=y,y=2,x=(0 .. 5),iterations=20); {{0.0,2.0}, {0.25,2.56805083336}, {0.5,3.29744254135}, {0.75,4.23400003313}, {1.0,5.43656365675}, {1.25,6.98068591465}, {1.5,8.96337814026}, {1.75,11.5092053514}, {2.0,14.778112197}, {2.25,18.9754716714}, {2.5,24.3649879195}, {2.75,31.2852637657}, {3.0,40.1710738427}, {3.25,51.5806798292}, {3.5,66.2309039102}, {3.75,85.0421639903}, {4.0,109.196300053}, {4.25,140.210824675}, {4.5,180.034262576}, {4.75,231.168569021}, {5.0,296.826318159}} % same with negative direction num_odesolve(df(y,x)=y,y=2,x=(0 .. -5),iterations=20); {{0.0,2.0}, {-0.25,1.55760156616}, {-0.5,1.21306131944}, {-0.75,0.944733105504}, {-1.0,0.735758882366}, {-1.25,0.573009593743}, {-1.5,0.446260320318}, {-1.75,0.34754788692}, {-2.0,0.27067056649}, {-2.25,0.210798449139}, {-2.5,0.164169997261}, {-2.75,0.127855722424}, {-3.0,0.0995741367451}, {-3.25,0.0775484156713}, {-3.5,0.0603947668512}, {-3.75,0.0470354917175}, {-4.0,0.036631277782}, {-4.25,0.0285284678218}, {-4.5,0.0222179930796}, {-4.75,0.0173033904088}, {-5.0,0.0134758940003}} % giving a nice picture when plotted num_odesolve(df(y,x)=1- x*y**2 ,y=0,x=(0 .. 4),iterations=20); {{0.0,0.0}, {0.2,0.199600912185}, {0.4,0.393714914143}, {0.6,0.569482634317}, {0.8,0.710657687321}, {1.0,0.805480021865}, {1.2,0.852604290316}, {1.4,0.860563376499}, {1.6,0.842333333669}, {1.8,0.809992008198}, {2.0,0.772211952477}, {2.2,0.734163639954}, {2.4,0.69843323517}, {2.6,0.66601919664}, {2.8,0.637070047105}, {3.0,0.611341375875}, {3.2,0.588447372816}, {3.4,0.567985133961}, {3.6,0.549587947477}, {3.8,0.53294225579}, {4.0,0.517787833885}} end; Time: 48739 ms plus GC time: 2601 ms Quitting