diff --git a/changelog b/changelog index da0e041..d32578a 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20100418 tpd src/axiom-website/patches.html 20100418.02.tpd.patch +20100418 tpd src/input/Makefile add liska.input +20100418 tpd add src/input/liska.input add liska test suite 20100418 tpd src/axiom-website/patches.html 20100418.01.tpd.patch 20100418 tpd src/algebra/Makefile add BLAS1 dcopy help and regression test 20100418 tpd books/bookvol10.5 add BLAS1 dcopy diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index b25a835..8dfdcbb 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -2629,5 +2629,7 @@ books/bookvol10.5 add BLAS1 dasum function
books/bookvol10.5 add BLAS1 daxpy
20100418.01.tpd.patch books/bookvol10.5 add BLAS1 dcopy
+20100418.02.tpd.patch +add src/input/liska.input add liska test suite
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index bcacc33..e80186e 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -341,7 +341,7 @@ REGRES= ackermann.regress \ kamke4.regress kamke5.regress kamke6.regress kamke7.regress \ kernel.regress knot2.regress kovacic.regress kuipers.regress \ laplace.regress leg.regress limit.regress linalg.regress \ - lindep.regress list.regress liu.regress \ + lindep.regress liska.regress list.regress liu.regress \ lode.regress lodesys.regress lodo1.regress \ lodo2.regress lodo3.regress lodof.regress lodo.regress \ log.regress \ @@ -635,7 +635,8 @@ FILES= ${OUT}/ackermann.input \ ${OUT}/laplace.input ${OUT}/leg.input \ ${OUT}/lextripk.input ${OUT}/lib.input ${OUT}/limit.input \ ${OUT}/linalg.input \ - ${OUT}/lindep.input ${OUT}/liss1.input ${OUT}/liss2.input \ + ${OUT}/lindep.input ${OUT}/liska.input \ + ${OUT}/liss1.input ${OUT}/liss2.input \ ${OUT}/list.input ${OUT}/liu.input \ ${OUT}/lode.input ${OUT}/lodesys.input \ ${OUT}/lodo1.input ${OUT}/lodo2.input ${OUT}/lodof.input \ @@ -958,6 +959,7 @@ DOCFILES= \ ${DOC}/lextripk.input.dvi \ ${DOC}/lib.input.dvi ${DOC}/limit.input.dvi \ ${DOC}/linalg.input.dvi ${DOC}/lindep.input.dvi \ + ${DOC}/liska.input.dvi \ ${DOC}/liss1.input.dvi ${DOC}/liss2.input.dvi \ ${DOC}/list.input.dvi ${DOC}/liu.input.dvi \ ${DOC}/lode.input.dvi \ diff --git a/src/input/liska.input.pamphlet b/src/input/liska.input.pamphlet new file mode 100644 index 0000000..770304a --- /dev/null +++ b/src/input/liska.input.pamphlet @@ -0,0 +1,1262 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input liska.input} +\author{Timothy Daly} +\maketitle +\begin{abstract} +Richard Liska, Ladislav Drska, Jiri Limpouch, Milan Sinor, Michael Wester, +Franz Winkler "Computer Algebra - algorithms, systems and applications'' +June 2, 1997 +\verb|kfe.fjfi.cvut.cz/~liska/ca/all.html| +\end{abstract} +\eject +\tableofcontents +\eject +\begin{chunk}{*} +)set break resume +)sys rm -f liska.output +)spool liska.output +)set message test on +)set message auto off +)clear all +\end{chunk} +\section{Number Domains} +\subsection{Big integers} +\begin{chunk}{*} +)clear all + +--S 1 of 94 +23^12 +--R +--R +--R (1) 21914624432020321 +--R Type: PositiveInteger +--E 1 + +--S 2 of 94 +factorial 60 +--R +--R +--R (2) +--R 8320987112741390144276341183223364380754172606361245952449277696409600000000_ +--R 000000 +--R Type: PositiveInteger +--E 2 + +--S 3 of 94 +bi:=23^4*37*59*101 +--R +--R +--R (3) 61700183203 +--R Type: PositiveInteger +--E 3 + +--S 4 of 94 +factor bi +--R +--R +--R 4 +--R (4) 23 37 59 101 +--R Type: Factored Integer +--E 4 + +--S 5 of 94 +bia:=23*11^6 +--R +--R +--R (5) 40745903 +--R Type: PositiveInteger +--E 5 + +--S 6 of 94 +gcd(bi,bia) +--R +--R +--R (6) 23 +--R Type: PositiveInteger +--E 6 + +\end{chunk} +\subsection{Rational numbers} +\begin{chunk}{*} +)clear all + +--S 7 of 94 +1234567890/98765432 +--R +--R +--R 617283945 +--R (1) --------- +--R 49382716 +--R Type: Fraction Integer +--E 7 + +--S 8 of 94 +rn:=1/2+2/15-64/47 +--R +--R +--R 1027 +--R (2) - ---- +--R 1410 +--R Type: Fraction Integer +--E 8 +\end{chunk} +\subsection{Complex numbers} +\begin{chunk}{*} +)clear all + +--S 9 of 94 +cn:=(2+3*%i)*(15-6*%i)+2/(2-4*%i) +--R +--R +--R 63 + 115%i +--R (1) ---------- +--R 2 + %i +--R Type: Fraction Complex Integer +--E 9 + +--S 10 of 94 +real(cn)+%i*imag(cn) +--R +--R +--R 63 + 115%i +--R (2) ---------- +--R 2 + %i +--R Type: Expression Complex Integer +--E 10 +\end{chunk} +\subsection{Algebraic numbers} +\begin{chunk}{*} +)clear all + +--S 11 of 94 +sqrt2:=rootOf(sqrt2^2-2) +--R +--R +--R (1) sqrt2 +--R Type: AlgebraicNumber +--E 11 + +--S 12 of 94 +1/(sqrt2+1) +--R +--R +--R (2) sqrt2 - 1 +--R Type: AlgebraicNumber +--E 12 + +--S 13 of 94 +(x^2+2*sqrt2*x+2)/(x+sqrt2) +--R +--R +--R (3) x + sqrt2 +--R Type: Fraction Polynomial AlgebraicNumber +--E 13 + +--S 14 of 94 +t1:=(x^3+(sqrt2-2)*x^2-(2*sqrt2+3)*x-3*sqrt2)/(x^2-2) +--R +--R +--R 2 +--R x - 2x - 3 +--R (4) ----------- +--R x - sqrt2 +--R Type: Fraction Polynomial AlgebraicNumber +--E 14 + +--S 15 of 94 +normalize t1 +--R +--R +--R 2 +--R - x + 2x + 3 +--R (5) ------------- +--R sqrt2 - x +--R Type: Expression Integer +--E 15 + +--S 16 of 94 +sqrt(x^2-2*sqrt2*x*y+2*y^2) +--R +--R +--R +-----------------------+ +--R | 2 2 +--R (6) \|- 2x y sqrt2 + 2y + x +--R Type: Expression Integer +--E 16 + +--S 17 of 94 +sqrt5:=rootOf(sqrt5^2-5) +--R +--R +--R (7) sqrt5 +--R Type: AlgebraicNumber +--E 17 + +--S 18 of 94 +cbrt3:=rootOf(cbrt3^3-3) +--R +--R +--R (8) cbrt3 +--R Type: AlgebraicNumber +--E 18 + +--S 19 of 94 +cbrt3^3 +--R +--R +--R (9) 3 +--R Type: AlgebraicNumber +--E 19 + +--S 20 of 94 +sqrt5^2 +--R +--R +--R (10) 5 +--R Type: AlgebraicNumber +--E 20 + +--S 21 of 94 +cbrt +--R +--R +--R (11) cbrt +--R Type: Variable cbrt +--E 21 + +--S 22 of 94 +sqrt(x^2+2*(sqrt5-cbrt3)*x+5-2*sqrt5*cbrt3+cbrt3^2) +--R +--R +--R +-------------------------------------------------+ +--R | 2 2 +--R (12) \|(- 2cbrt3 + 2x)sqrt5 + cbrt3 - 2x cbrt3 + x + 5 +--R Type: Expression Integer +--E 22 +\end{chunk} +\subsection{Big floating point numbers} +\begin{chunk}{*} +)clear all + +--S 23 of 94 +rn:=-1027/1410 +--R +--R +--R 1027 +--R (1) - ---- +--R 1410 +--R Type: Fraction Integer +--E 23 + +--S 24 of 94 +cn:=(167*%i+241)/5 +--R +--R +--R 63 + 115%i +--R (2) ---------- +--R 2 + %i +--R Type: Fraction Complex Integer +--E 24 + +--S 25 of 94 +rn::Float +--R +--R +--R (3) - 0.7283687943 2624113475 +--R Type: Float +--E 25 + +--S 26 of 94 +cn::Complex Float +--R +--R +--R (4) 48.2 + 33.4 %i +--R Type: Complex Float +--E 26 + +--S 27 of 94 +%pi::Float +--R +--R +--R (5) 3.1415926535 897932385 +--R Type: Float +--E 27 + +--S 28 of 94 +cos(%pi::Float) +--R +--R +--R (6) - 1.0 +--R Type: Float +--E 28 + +--S 29 of 94 +sin 1.0 +--R +--R +--R (7) 0.8414709848 0789650665 +--R Type: Float +--E 29 + +--S 30 of 94 +digits 50 +--R +--R +--R (8) 20 +--R Type: PositiveInteger +--E 30 + +--S 31 of 94 +%pi::Float +--R +--R +--R (9) 3.1415926535 8979323846 2643383279 5028841971 693993751 +--R Type: Float +--E 31 + +--S 32 of 94 +cos %pi::Float +--R +--R +--R (10) - 1.0 +--R Type: Float +--E 32 + +--S 33 of 94 +cos(%pi/6::Float) +--R +--R +--R (11) 0.8660254037 8443864676 3723170752 9361834714 0262690519 +--R Type: Float +--E 33 + +--S 34 of 94 +digits 20 +--R +--R +--R (12) 50 +--R Type: PositiveInteger +--E 34 + +--S 35 of 94 +exp(-100000.1^2) +--R +--R +--R (13) 0.1184406313 2021703038 E -4342953504 +--R Type: Float +--E 35 + +--S 36 of 94 +tan(1.0+1.0*%i) +--R +--R +--R (14) 0.2717525853 1951171653 + 1.0839233273 386945435 %i +--R Type: Complex Float +--E 36 + +--S 37 of 94 +log(1.0+1.0*%i) +--R +--R +--R (15) 0.3465735902 7997265471 + 0.7853981633 9744830961 %i +--R Type: Complex Float +--E 37 +\end{chunk} +\section{Polynomials} +\subsection{Basic operations} +\begin{chunk}{*} +)clear all + +--S 38 of 94 +pol:=(a+b+c)^4 +--R +--R +--R (1) +--R 4 3 2 2 2 3 2 2 3 +--R c + (4b + 4a)c + (6b + 12a b + 6a )c + (4b + 12a b + 12a b + 4a )c +--R + +--R 4 3 2 2 3 4 +--R b + 4a b + 6a b + 4a b + a +--R Type: Polynomial Integer +--E 38 + +--S 39 of 94 +dpol:=D(pol,a) +--R +--R +--R (2) +--R 3 2 2 2 3 2 2 3 +--R 4c + (12b + 12a)c + (12b + 24a b + 12a )c + 4b + 12a b + 12a b + 4a +--R Type: Polynomial Integer +--E 39 + +--S 40 of 94 +D(D(pol,a),b,2) +--R +--R +--R (3) 24c + 24b + 24a +--R Type: Polynomial Integer +--E 40 + +--S 41 of 94 +t1:=integrate(dpol,a) +--R +--R +--R (4) +--R 3 2 2 2 2 3 3 2 2 3 4 +--R 4a c + (12a b + 6a )c + (12a b + 12a b + 4a )c + 4a b + 6a b + 4a b + a +--R Type: Polynomial Fraction Integer +--E 41 + +--S 42 of 94 +pol-t1 +--R +--R +--R 4 3 2 2 3 4 +--R (5) c + 4b c + 6b c + 4b c + b +--R Type: Polynomial Fraction Integer +--E 42 + +--S 43 of 94 +(a^2-b^2)/(a^2-2*a*b+b^2) +--R +--R +--R - b - a +--R (6) ------- +--R b - a +--R Type: Fraction Polynomial Integer +--E 43 + +--S 44 of 94 +g:=34*x^19-91*x+70*x^7-25*x^16+20*x^3-86 +--R +--R +--R 19 16 7 3 +--R (7) 34x - 25x + 70x + 20x - 91x - 86 +--R Type: Polynomial Integer +--E 44 + +--S 45 of 94 +f1:=g*(64*x^34-21*x^47-126*x^8-46*x^5-16*x^60-81) +--R +--R +--R (8) +--R 79 76 67 66 63 61 60 54 +--R - 544x + 400x - 1120x - 714x + 205x + 1456x + 1376x - 1470x +--R + +--R 53 50 48 47 41 37 35 +--R 2176x - 2020x + 1911x + 1806x + 4480x + 1280x - 5824x +--R + +--R 34 27 24 21 19 16 15 +--R - 5504x - 4284x + 1586x + 1150x - 2754x + 2025x - 8820x +--R + +--R 12 11 9 8 7 6 5 3 +--R - 3220x - 2520x + 11466x + 9916x - 5670x + 4186x + 3956x - 1620x +--R + +--R 7371x + 6966 +--R Type: Polynomial Integer +--E 45 + +--S 46 of 94 +f2:=g*(72*x^60-25*x^25-19*x^23-22*x^39-83*x^52+54*x^10+81) +--R +--R +--R (9) +--R 79 76 71 68 67 63 61 +--R 2448x - 1800x - 2822x + 2075x + 5040x + 1440x - 6552x +--R + +--R 60 59 58 55 53 52 46 +--R - 6192x - 5810x - 748x - 1110x + 7553x + 7138x - 1540x +--R + +--R 44 42 41 40 39 32 30 +--R - 850x - 1086x + 625x + 2002x + 2367x - 1750x - 1330x +--R + +--R 29 28 26 25 24 23 19 17 +--R 1836x - 500x + 545x + 2150x + 1729x + 1634x + 2754x + 3780x +--R + +--R 16 13 11 10 7 3 +--R - 2025x + 1080x - 4914x - 4644x + 5670x + 1620x - 7371x - 6966 +--R Type: Polynomial Integer +--E 46 + +--S 47 of 94 +gcd(f1,f2) +--R +--R +--R 19 16 7 3 +--R (10) 34x - 25x + 70x + 20x - 91x - 86 +--R Type: Polynomial Integer +--E 47 + +\end{chunk} +\subsection{Factorization} +\begin{chunk}{*} +)clear all + +--S 48 of 94 +factor(a^2-b^2) +--R +--R +--R (1) - (b - a)(b + a) +--R Type: Factored Polynomial Integer +--E 48 + +--S 49 of 94 +factor(a^2+b^2,[rootOf(i^2+1)]) +--R +--R +--R (2) (b - i a)(b + i a) +--R Type: Factored Polynomial AlgebraicNumber +--E 49 + +--S 50 of 94 +fa:=(x^2*z+y^4*z^2+5)*(x*y^3+z^2)*(-x^3*y+z^2+3)*(x^3*y^4+z^2) +--R +--R +--R (3) +--R 4 8 2 7 3 8 7 3 5 4 6 +--R y z + x z + (x y + x y - x y + 3y + 5)z +--R + +--R 5 4 3 3 5 2 5 +--R (x y + x y - x y + 3x )z +--R + +--R 4 11 6 9 4 3 8 7 3 4 3 3 4 +--R (x y - x y + (- x + 3x )y + 3x y + 5x y + 5x y - 5x y + 15)z +--R + +--R 6 7 8 5 6 5 4 3 3 3 +--R (x y - x y + (- x + 3x )y + 3x y )z +--R + +--R 7 12 4 11 4 7 6 5 4 3 4 3 2 +--R (- x y + 3x y + 5x y - 5x y + (- 5x + 15x )y + 15x y )z +--R + +--R 9 8 6 7 7 8 4 7 +--R (- x y + 3x y )z - 5x y + 15x y +--R Type: Polynomial Integer +--E 50 + +--S 51 of 94 +factor fa +--R +--R +--R 2 3 2 3 2 3 4 4 2 2 +--R (4) (z - x y + 3)(z + x y )(z + x y )(y z + x z + 5) +--R Type: Factored Polynomial Integer +--E 51 +\end{chunk} +\subsection{Groebner bases} +\begin{chunk}{*} +)clear all + +--S 52 of 94 +polys:=[45*p+35*s-165*b-36, _ + 35*p+40*z+25*t-27*s, _ + 15*w+25*p*s+30*z-18*t-165*b^2, _ + -9*w+15*p*t+20*z*s, _ + w*p+2*z*t-11*b^3, _ + 99*w-11*s*b+3*b^2, _ + b^2+33/50*b+2673/10000] +--R +--R +--R (1) +--R [35s + 45p - 165b - 36, 40z + 25t - 27s + 35p, +--R 2 3 +--R 30z + 15w - 18t + 25p s - 165b , 20s z - 9w + 15p t, 2t z + p w - 11b , +--R 2 2 33 2673 +--R 99w - 11b s + 3b , b + -- b + -----] +--R 50 10000 +--R Type: List Polynomial Fraction Integer +--E 52 + +--S 53 of 94 +vars:=[w,p,z,t,s,b] +--R +--R +--R (2) [w,p,z,t,s,b] +--R Type: List OrderedVariableList [w,p,z,t,s,b] +--E 53 + +--S 54 of 94 +groebner(polys) +--R +--R +--R (3) +--R 49 1143 19 1323 37 27 5 9 +--R [z + -- b + ----, w + --- b + -----, t - -- b + ---, s - - b - ---, +--R 36 2000 120 20000 15 250 2 200 +--R 31 153 2 33 2673 +--R p - -- b - ---, b + -- b + -----] +--R 18 200 50 10000 +--R Type: List Polynomial Fraction Integer +--E 54 + +--S 55 of 94 +solve(polys,vars) +--R +--R +--R (4) +--R [ +--R - 9500b - 3969 3100b + 1377 - 24500b - 10287 1850b - 81 +--R [w= --------------, p= ------------, z= ----------------, t= ----------, +--R 60000 1800 18000 750 +--R 500b + 9 2 +--R s= --------, 10000b + 6600b + 2673= 0] +--R 200 +--R ] +--R Type: List List Equation Fraction Polynomial Integer +--E 55 + +\end{chunk} +\section{Rational functions} +\begin{chunk}{*} + +--S 56 of 94 +rf:=(3*a*b^2-5*a^2*b)/(a^4-2) +--R +--R +--R 2 2 +--R 3a b - 5a b +--R (5) ------------ +--R 4 +--R a - 2 +--R Type: Fraction Polynomial Integer +--E 56 + +--S 57 of 94 +integrate(rf,a) +--R +--R +--R (6) +--R +--------------------------------------+ +--R | 2 2 4 +--I (\|- 12%%J1 - 8%%J0 %%J1 - 12%%J0 + 9b - 2%%J1 - 2%%J0) +--R * +--R log +--R 4 4 4 +--I ((800%%J0 - 216b )%%J1 - 216b %%J0 - 225b ) +--R * +--R +--------------------------------------+ +--R | 2 2 4 +--I \|- 12%%J1 - 8%%J0 %%J1 - 12%%J0 + 9b +--R + +--R 4 2 2 4 4 2 +--I (1600%%J0 - 432b )%%J1 + (1600%%J0 + 450b )%%J1 - 432b %%J0 +--R + +--R 4 8 7 4 3 +--I 450b %%J0 + 243b + 810a b + 1875b + 3125a b +--R + +--R +--------------------------------------+ +--R | 2 2 4 +--I (- \|- 12%%J1 - 8%%J0 %%J1 - 12%%J0 + 9b - 2%%J1 - 2%%J0) +--R * +--R log +--R 4 4 4 +--I ((- 800%%J0 + 216b )%%J1 + 216b %%J0 + 225b ) +--R * +--R +--------------------------------------+ +--R | 2 2 4 +--I \|- 12%%J1 - 8%%J0 %%J1 - 12%%J0 + 9b +--R + +--R 4 2 2 4 4 2 +--I (1600%%J0 - 432b )%%J1 + (1600%%J0 + 450b )%%J1 - 432b %%J0 +--R + +--R 4 8 7 4 3 +--I 450b %%J0 + 243b + 810a b + 1875b + 3125a b +--R + +--I 4%%J1 +--R * +--R log +--R 4 2 2 4 3 +--I (- 3200%%J0 + 864b )%%J1 + (- 3200%%J0 - 900b )%%J1 - 3200%%J0 +--R + +--R 4 8 7 4 3 +--I 1800b %%J0 - 243b + 810a b + 1875b + 3125a b +--R + +--I 4%%J0 +--R * +--R log +--R 3 4 2 4 8 7 4 +--I 3200%%J0 + 864b %%J0 - 2700b %%J0 - 243b + 810a b - 5625b +--R + +--R 3 +--R 3125a b +--R / +--R 4 +--R Type: Union(Expression Integer,...) +--E 57 + +--S 58 of 94 +t1:=(10*x^2-11*x-6)/(x^3-x^2-2*x) +--R +--R +--R 2 +--R 10x - 11x - 6 +--R (7) -------------- +--R 3 2 +--R x - x - 2x +--R Type: Fraction Polynomial Integer +--E 58 + +--S 59 of 94 +FACU:=Factored UnivariatePolynomial(x,Fraction Integer) +--R +--R +--R (8) Factored UnivariatePolynomial(x,Fraction Integer) +--R Type: Domain +--E 59 + +--S 60 of 94 +partialFraction(numerator(t1)::FACU,denominator(t1)::FACU) +--R +--R +--R 2 5 3 +--R (9) ----- + ----- + - +--R x - 2 x + 1 x +--R Type: PartialFraction UnivariatePolynomial(x,Fraction Integer) +--E 60 +\end{chunk} +\section{Solving Equations} +\subsection{Linear systems} +\begin{chunk}{*} +)clear all + +--S 61 of 94 +solve([2*x1+x2+3*x3-9, x1-2*x2+x3+2, 3*x1+2*x2+2*x3-7],[x1,x2,x3]) +--R +--R +--R (1) [[x1= - 1,x2= 2,x3= 3]] +--R Type: List List Equation Fraction Polynomial Integer +--E 61 +\end{chunk} +\subsection{Nonlinear equations} +\begin{chunk}{*} +)clear all + +--S 62 of 94 +solve(x^8-8*x^7+34*x^6-92*x^5-175*x^4-236*x^3+226*x^2-140*x+46,x) +--R +--R +--R 8 7 6 5 4 3 2 +--R (1) [x - 8x + 34x - 92x - 175x - 236x + 226x - 140x + 46= 0] +--R Type: List Equation Fraction Polynomial Integer +--E 62 + +--S 63 of 94 +solve(log(acos(asin(x^(2/3)-b)-1))+2,x) +--R +--R +--R (2) +--R +---------------------+ +--R 1 | 1 +--R [x= (- sin(cos(---) + 1) - b) |sin(cos(---) + 1) + b , +--R 2 | 2 +--R %e \| %e +--R +---------------------+ +--R 1 | 1 +--R x= (sin(cos(---) + 1) + b) |sin(cos(---) + 1) + b ] +--R 2 | 2 +--R %e \| %e +--R Type: List Equation Expression Integer +--E 63 +\end{chunk} +\subsection{Nonlinear systems} +\begin{chunk}{*} +)clear all + +-- Maple claims: +-- c1 = c1 +-- c2 = -(c1*(- epsilon*alpha+c1*epsilon*beta-theta*alpha+c1*theta*beta)/ +-- (-eta*epsilon+c1*theta*gamma) +-- c3 = -(c1*(- alpha*eta+beta*c1*eta-gamma*c1*alpha+gamma*c1^2*beta)/ +-- (-eta*epsilon+c1*theta*gamma) +-- +-- MMA claims: +-- c1 is an arbitrary complex number +-- c2 = (c1*(alpha-beta*c1)*(epsilon+theta))/(-epsilon*eta+c1*gamma*theta) +-- c3 = (c1*(-alpha+beta*c1)*(eta+c1*gamma))/(epsilon*eta-c1*gamma*theta) +-- +--S 64 of 94 +solve([alpha*c1-beta*c1^2-gamma*c1*c2+epsilon*c3,_ + -gamma*c1*c2+(epsilon+theta)*c3-eta*c2,_ + gamma*c1*c2+eta*c2-(epsilon+theta)*c3], + [c3,c2,c1]) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R system does not have a finite number of solutions +--R +--R Continuing to read the file... +--R +--E 64 +\end{chunk} +\section{Analytical Operations} +\subsection{Limits} +\begin{chunk}{*} +)clear all + +--S 65 of 94 +limit(sin(x)/x,x=0) +--R +--R +--R (1) 1 +--R Type: Union(OrderedCompletion Expression Integer,...) +--E 65 + +--S 66 of 94 +limit((3*sin(%pi*x)-sin(3*%pi*x))/x^3,x=0) +--R +--R +--R 3 +--R (2) 4%pi +--R Type: Union(OrderedCompletion Expression Integer,...) +--E 66 + +--S 67 of 94 +limit((2*x+5)/(3*x-2),x=%plusInfinity) +--R +--R +--R 2 +--R (3) - +--R 3 +--R Type: Union(OrderedCompletion Fraction Polynomial Integer,...) +--E 67 +\end{chunk} +\subsection{Taylor series} +\begin{chunk}{*} +)clear all + +)set streams calculate 4 + +--S 68 of 94 +series(%e^x,x=0) +--R +--R +--R 1 2 1 3 1 4 5 +--R (1) 1 + x + - x + - x + -- x + O(x ) +--R 2 6 24 +--R Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +--E 68 +\end{chunk} +\subsection{Summation and Products} +\begin{chunk}{*} +)clear all + +--S 69 of 94 +sum(n^2*x^n,n) +--R +--R +--R 2 3 2 2 2 n - 1 +--R ((n - 2n + 1)x + (- 2n + 2n + 1)x + n x)x +--R (1) -------------------------------------------------- +--R 3 2 +--R x - 3x + 3x - 1 +--R Type: Expression Integer +--E 69 + +--S 70 of 94 +sum(cos((2*r-1)*%pi/(2*n+1)),r) +--R +--R +--R --+ 2%pi r - %pi +--R (2) > cos(------------) +--R --+ 2n + 1 +--R r +--R Type: Expression Integer +--E 70 + +--S 71 of 94 +product(%e^(sin(n*x)),n) +--R +--R +--R ++-++ sin(n x) +--R (3) | | %e +--R | | +--R n +--R Type: Expression Integer +--E 71 + +--S 72 of 94 +sum(n*2^n/factorial(n+2),n) +--R +--R +--R n - 1 +--R 2 2 +--R (4) - -------- +--R (n + 1)! +--R Type: Expression Integer +--E 72 +\end{chunk} +\subsection{Integration} +\begin{chunk}{*} +)clear all + +--S 73 of 94 +integrate(x^2*(a+b*x)^p,x) +--R +--R +--R (1) +--R 3 2 3 3 3 2 2 2 2 2 3 p log(b x + a) +--R ((b p + 3b p + 2b )x + (a b p + a b p)x - 2a b p x + 2a )%e +--R ----------------------------------------------------------------------------- +--R 3 3 3 2 3 3 +--R b p + 6b p + 11b p + 6b +--R Type: Union(Expression Integer,...) +--E 73 + +--S 74 of 94 +integrate(x^2*log(x^2+a^2),x) +--R +--R +--R 3 2 2 3 x 3 2 +--R 3x log(x + a ) - 6a atan(-) - 2x + 6a x +--R a +--R (2) ----------------------------------------- +--R 9 +--R Type: Union(Expression Integer,...) +--E 74 + +--S 75 of 94 +integrate(x*d^x*sin(x),x) +--R +--R +--R (3) +--R 3 2 2 +--R (x log(d) - log(d) + x log(d) + 1)sin(x) - x cos(x)log(d) +--R + +--R 2cos(x)log(d) - x cos(x) +--R * +--R x log(d) +--R %e +--R / +--R 4 2 +--R log(d) + 2log(d) + 1 +--R Type: Union(Expression Integer,...) +--E 75 + +--S 76 of 94 +integrate(x*sqrt(a+b*x)^p,x) +--R +--R +--R +-------+ +--R 2 2 2 2 p log(\|b x + a ) +--R ((2b p + 4b )x + 2a b p x - 4a )%e +--R (4) ---------------------------------------------------- +--R 2 2 2 2 +--R b p + 6b p + 8b +--R Type: Union(Expression Integer,...) +--E 76 + +--S 77 of 94 +integrate(2*x*%e^(x^2)*log(x)+%e^(x^2)/x+(log(x)-2)/(log(x)^2+x)^2+_ + ((2/x)*log(x)+(1/x)+1)/(log(x)^2+x),x) +--R +--R +--R 2 2 +--R 2 2 x 3 x +--R (log(x) + x)log(log(x) + x) + %e log(x) + (x %e - 1)log(x) +--R (5) ---------------------------------------------------------------- +--R 2 +--R log(x) + x +--R Type: Union(Expression Integer,...) +--E 77 +\end{chunk} +\subsection{Ordinary differential equations} +\begin{chunk}{*} +)clear all + +--S 78 of 94 +y:=operator 'y +--R +--R +--R (1) y +--R Type: BasicOperator +--E 78 + +--S 79 of 94 +solve(D(y(x),x)+y(x)*sin(x)/cos(x)-1/cos(x)=0,y,x) +--R +--R +--R (2) [particular= sin(x),basis= [cos(x)]] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 79 + +--S 80 of 94 +solve(x*(1-x^2)*D(y(x),x)+(2*x^2-1)*y(x)-x^3*y(x)^3=0,y,x) +--R +--R +--R 5 2 4 2 +--R (- 2x + 4)y(x) + 5x - 5x +--R (3) ---------------------------- +--R 2 +--R 5y(x) +--R Type: Union(Expression Integer,...) +--E 80 + +--S 81 of 94 +solve(D(y(x),x,2)+4*D(y(x),x)+4*y(x)-x*exp(x)=0,y,x) +--R +--R +--R - 2x x 3 +--R (3x - 2)%e (%e ) - 2x - 2x +--R (4) [particular= --------------------,basis= [%e ,x %e ]] +--R 27 +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 81 +\end{chunk} +\subsection{Substitutions -- pattern matching} +\begin{chunk}{*} +)clear all + +--S 82 of 94 +sincosRules:=rule + cos(x)*cos(y) == (cos(x+y)+cos(x-y))/2 + cos(x)*sin(y) == (cin(x+y)-sin(x-y))/2 + sin(x)*sin(y) == (cos(x-y)-cos(x+y))/2 + cos(x)^2 == (1+cos(2*x))/2 + sin(x)^2 == (1-cos(2*x))/2 +--R +--R +--R (1) +--I %QQ cos(y + x) + %QQ cos(y - x) +--I {%QQ cos(x)cos(y) == -------------------------------, +--R 2 +--I %QR sin(y - x) + %QR'cin(y + x) +--I %QR cos(x)sin(y) == -------------------------------, +--R 2 +--I - %QS cos(y + x) + %QS cos(y - x) +--I %QS sin(x)sin(y) == ---------------------------------, +--R 2 +--R 2 cos(2x) + 1 2 - cos(2x) + 1 +--R cos(x) == -----------, sin(x) == -------------} +--R 2 2 +--R Type: Ruleset(Integer,Integer,Expression Integer) +--E 82 + +--S 83 of 94 +sincosRules (a1*cos(wt)+a3*cos(3*wt)+b1*sin(wt)+b3*sin(3*wt))^3 +--R +--R +--R (2) +--R 3 3 2 2 2 2 +--R b3 sin(3wt) + (3b1 b3 sin(wt) + 3a3 b3 cos(3wt) + 3a1 b3 cos(wt))sin(3wt) +--R + +--R 2 2 +--R 3b1 b3 sin(wt) + (6a3 b1 b3 cos(3wt) + 6a1 b1 b3 cos(wt))sin(wt) +--R + +--R 2 2 2 2 +--R 3a3 b3 cos(3wt) + 6a1 a3 b3 cos(wt)cos(3wt) + 3a1 b3 cos(wt) +--R * +--R sin(3wt) +--R + +--R 3 3 2 2 2 +--R b1 sin(wt) + (3a3 b1 cos(3wt) + 3a1 b1 cos(wt))sin(wt) +--R + +--R 2 2 2 2 +--R (3a3 b1 cos(3wt) + 6a1 a3 b1 cos(wt)cos(3wt) + 3a1 b1 cos(wt) )sin(wt) +--R + +--R 3 3 2 2 2 2 +--R a3 cos(3wt) + 3a1 a3 cos(wt)cos(3wt) + 3a1 a3 cos(wt) cos(3wt) +--R + +--R 3 3 +--R a1 cos(wt) +--R Type: Expression Integer +--E 83 + +--S 84 of 94 +int:=operator('int) +--R +--R +--R (3) int +--R Type: BasicOperator +--E 84 + +-- bug in this rule form? int(x+ :y,z) == int(x,z)+int(y,z) +-- and also this one? int(x+y,z) == int(x,z)+int(y,z) +--S 85 of 94 +intRules:=rule + int(x+y,z) == int(x,z)+int(y,z) + int(k*x | freeOf?(k,z),z) == k*int(x,z) + int(y | integer? y, z) == y*z + int(x^(?p | D(p,x)=0),x) == x^(p+1)/(p+1) +--R +--R +--R (4) +--R {int(y + x,z) == 'int(y,z) + 'int(x,z), int(k x,z) == k'int(x,z), +--R p + 1 +--R p x +--R int(y,z) == y z, int(x ,x) == ------} +--R p + 1 +--R Type: Ruleset(Integer,Integer,Expression Integer) +--E 85 + +-- +-- should be: +-- b+1 2 2 2 2 +-- a + (a b + a ) int(b,a) + (3a - 5a)b + 3a - 5a +-- -------------------------------------------------- +-- b + 1 +-- +--S 86 of 94 +intRules int(a^2*b+a^b+3*a-5,a) +--R +--R There are 9 exposed and 0 unexposed library operations named elt +--R having 3 argument(s) but none was determined to be applicable. +--R Use HyperDoc Browse, or issue +--R )display op elt +--R to learn more about the available operations. Perhaps +--R package-calling the operation or using coercions on the arguments +--R will allow you to apply the operation. +--R +--RDaly Bug +--R Cannot find a definition or applicable library operation named elt +--R with argument type(s) +--R Polynomial Integer +--R Variable a +--R +--R Perhaps you should use "@" to indicate the required return type, +--R or "$" to specify which version of the function you need. +--E 86 + +--S 87 of 94 +intRules int(a^(a+1),a) +--R +--R +--R a + 2 +--R a +--R (5) ------ +--R a + 2 +--R Type: Expression Integer +--E 87 +\end{chunk} +\section{Matrices} +\begin{chunk}{*} + +--S 88 of 94 +xx:=matrix([[a11,a12],[a21,a22]]) +--R +--R +--R +a11 a12+ +--R (6) | | +--R +a21 a22+ +--R Type: Matrix Polynomial Integer +--E 88 + +--S 89 of 94 +yy:=matrix([[y1],[y2]]) +--R +--R +--R +y1+ +--R (7) | | +--R +y2+ +--R Type: Matrix Polynomial Integer +--E 89 + +--S 90 of 94 +determinant xx +--R +--R +--R (8) a11 a22 - a12 a21 +--R Type: Polynomial Integer +--E 90 + +--S 91 of 94 +zz:=inverse(xx)*yy +--R +--R +--R +- a12 y2 + a22 y1+ +--R |-----------------| +--R |a11 a22 - a12 a21| +--R (9) | | +--R | a11 y2 - a21 y1 | +--R |-----------------| +--R +a11 a22 - a12 a21+ +--R Type: Matrix Fraction Polynomial Integer +--E 91 + +--S 92 of 94 +inverse(xx)^2 +--R +--R +--R (10) +--R [ +--R 2 +--R a22 + a12 a21 +--R [--------------------------------------, +--R 2 2 2 2 +--R a11 a22 - 2a11 a12 a21 a22 + a12 a21 +--R - a12 a22 - a11 a12 +--R --------------------------------------] +--R 2 2 2 2 +--R a11 a22 - 2a11 a12 a21 a22 + a12 a21 +--R , +--R +--R - a21 a22 - a11 a21 +--R [--------------------------------------, +--R 2 2 2 2 +--R a11 a22 - 2a11 a12 a21 a22 + a12 a21 +--R 2 +--R a12 a21 + a11 +--R --------------------------------------] +--R 2 2 2 2 +--R a11 a22 - 2a11 a12 a21 a22 + a12 a21 +--R ] +--R Type: Matrix Fraction Polynomial Integer +--E 92 + +--S 93 of 94 +v:=matrix([[2,-1,1],[0,1,1],[-1,1,1]]) +--R +--R +--R + 2 - 1 1+ +--R | | +--R (11) | 0 1 1| +--R | | +--R +- 1 1 1+ +--R Type: Matrix Integer +--E 93 + +--S 94 of 94 +eigenvectors v +--R +--R +--R (12) +--R +1+ +0+ +--R | | | | +--R [[eigval= 1,eigmult= 2,eigvec= [|1|]],[eigval= 2,eigmult= 1,eigvec= [|1|]]] +--R | | | | +--R +0+ +1+ +--RType: List Record(eigval: Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)),eigmult: NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer) +--E 94 + +\end{chunk} +\section{Graphics} +\subsection{2D Graphics} +\begin{verbatim} + +draw(sin(%e*x),x=0..%pi) +draw([besselJ(0,x),besselJ(2,x),besselJ(5,x)],x=0..10,_ + title=="Bessel Functions BesselJ(n,x)") +\end{verbatim} +\subsection{3D Graphics} +\begin{verbatim} +draw(sin(%pi*sin(x+y)),x=-3..3,y=-3..3) +draw(tan(x*y),x=-2/3*%pi..2/3*%pi,y=-2/3*%pi..2/3*%pi) +\end{verbatim} +\subsection{Parametric plots} +\begin{verbatim} +draw(surface(sin(v),sin(2*v)*sin(u),sin(2*v)*cos(u)),_ + u=0..2*%pi,v=-%pi/2..%pi/2) +\end{verbatim} +\begin{chunk}{*} +)spool +)lisp (bye) + +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} +