diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet index 5ede61c..803dabc 100644 --- a/books/bookvolbib.pamphlet +++ b/books/bookvolbib.pamphlet @@ -7095,7 +7095,9 @@ rational right-hand sides etc." isbn = "9780486661698", pagetotal = "240", publisher = "Dover Pubn Inc", - date = "2003-03-28" + date = "2003-03-28", + comment = "documentation for DeRhamComplex" + } \end{chunk} @@ -11974,6 +11976,28 @@ ISSAC 94 ACM 0-89791-638-7/94/0007 \end{chunk} \index{Wester, Michael J.} +\begin{chunk}{axiom.bib} +@misc{West99a, + author = "Wester, Michael J.", + title = "A Critique of the Mathematical Abilities of CA Systems", + year = "1999", + url = "http://math.unm.edu/~wester/cas/book/Wester.pdf", + url2 = "http://math.unm.edu/~wester/cas_review.html", + paper = "West99a.pdf", + abstract = + "Computer algebra systems (CASs) have become an essential computational + tool in the last decade. General purpose CASs, which are designed to + solve a wide variety of problems, have gained special prominence. In + this chapter, the capabilities of seven major general purpose CASs + (Axiom, Derive, Macsyma, Maple, Mathmatica, MuPAD and Reduce) are + reviewed on 542 short problems covering a broad range of (primarily) + symbolic mathematics." + +} + +\end{chunk} + +\index{Wester, Michael J.} \begin{chunk}{ignore} \bibitem[Wester 99]{Wes99} Wester, Michael J. title = "Computer Algebra Systems", @@ -15132,10 +15156,15 @@ Num. Math. 16 205--223. (1970) \subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \index{Schafer, R.D.} -\begin{chunk}{ignore} -\bibitem[Schafer 66]{Sch66} Schafer, R.D. +\begin{chunk}{axiom.bib} +@book{Scha66, + author = "Schafer, R.D.", title = "An Introduction to Nonassociative Algebras", -Academic Press, New York, 1966 + year = "1966", + publisher = "Academic Press, New York", + comment = "documentation for AlgebraGivenByStructuralConstants" + +} \end{chunk} diff --git a/changelog b/changelog index 8235f6d..68812af 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,9 @@ +20141215 tpd src/axiom-website/patches.html 20141215.05.tpd.patch +20141215 tpd src/axiom-website/CATS/westeralgebra.input.pdf add CATS test suite +20141215 tpd src/axiom-website/CATS/westeralgebra.input add CATS test suite +20141215 tpd src/axiom-website/CATS/index.html add new CATS test suite +20141215 tpd src/input/Makefile add westeralgebra.input +20141215 tpd src/input/westeralgebra.input add new test suite 20141215 tpd src/axiom-website/patches.html 20141215.04.tpd.patch 20141215 tpd buglist: bug 7273: wester algebra radicalSolve bug 20141215 tpd src/axiom-website/patches.html 20141215.03.tpd.patch diff --git a/patch b/patch index 643f3fc..a3d04d4 100644 --- a/patch +++ b/patch @@ -1,3 +1 @@ -buglist: bug 7273: wester algebra radicalSolve bug - -Problem 26, 27 failure in westeralgebra CATS test suite +src/input/westeralgebra.input add new CATS test suite diff --git a/src/axiom-website/CATS/index.html b/src/axiom-website/CATS/index.html index 6ed6845..d6d87eb 100644 --- a/src/axiom-website/CATS/index.html +++ b/src/axiom-website/CATS/index.html @@ -774,6 +774,14 @@ which illustrates many 2D graphs. Also included are graphs from the book "Pasta by Design" ISBN 978-0-500-51580-8 (2011) which illustrates many 3D graphics. +


+

The Wester Test Suite



+This portion of the CATS suite involves Michael Wester Test Suite. +

+ + Algebra + source + pdf
diff --git a/src/axiom-website/CATS/westeralgebra.input.pamphlet b/src/axiom-website/CATS/westeralgebra.input.pamphlet new file mode 100644 index 0000000..e54ebf7 --- /dev/null +++ b/src/axiom-website/CATS/westeralgebra.input.pamphlet @@ -0,0 +1,1986 @@ +\documentclass{article} +\usepackage{axiom} +\setlength{\textwidth}{400pt} +\begin{document} +\title{\$SPAD/src/input westeralgebra.input} +\author{Michael Wester} +\maketitle +\begin{abstract} +These problems come from the web page +\begin{verbatim} +http://math.unm.edu/~wester/cas_review.html +\end{verbatim} +\end{abstract} +\eject +\tableofcontents +\eject +\begin{chunk}{*} +)set break resume +)set messages autoload off +)set streams calculate 7 +)sys rm -f westeralgebra.output +)spool westeralgebra.output +)clear all + +\end{chunk} +\section{Algebra} + +One would think that the simplification $2\ 2^n => 2^{(n + 1)}$ would happen +automatically or at least easily ... +\begin{chunk}{*} +--S 1 of 63 +2*2**n +--R +--R +--R n +--R (1) 2 2 +--R Type: Expression(Integer) +--E 1 + +\end{chunk} +And how about $4\ 2^n => 2^{(n + 2)}$? [Richard Fateman] +\begin{chunk}{*} +--S 2 of 63 +4*2**n +--R +--R +--R n +--R (2) 4 2 +--R Type: Expression(Integer) +--E 2 + +\end{chunk} +$(-1)^{(n(n + 1))} => 1$ for integer $n$ +\begin{chunk}{*} +--S 3 of 63 +(-1)**(n*(n + 1)) +--R +--R +--R 2 +--R n + n +--R (3) (- 1) +--R Type: Expression(Integer) +--E 3 + +\end{chunk} +Also easy $=> 2 (3 x - 5)$ +\begin{chunk}{*} +--S 4 of 63 +factor(6*x - 10) +--R +--R +--R (4) 2(3x - 5) +--R Type: Factored(Polynomial(Integer)) +--E 4 + +\end{chunk} +Univariate gcd: $gcd(p1, p2) => 1$, $gcd(p1 q, p2 q) => q$ [Richard Liska] +\begin{chunk}{*} +--S 5 of 63 +p1:= 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81 +--R +--R +--R 60 47 34 8 5 +--R (5) - 16x - 21x + 64x - 126x - 46x - 81 +--R Type: Polynomial(Integer) +--E 5 + +--S 6 of 63 +p2:= 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81 +--R +--R +--R 60 52 39 25 23 10 +--R (6) 72x - 83x - 22x - 25x - 19x + 54x + 81 +--R Type: Polynomial(Integer) +--E 6 + +--S 7 of 63 +q:= 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86 +--R +--R +--R 19 16 7 3 +--R (7) 34x - 25x + 70x + 20x - 91x - 86 +--R Type: Polynomial(Integer) +--E 7 + +--S 8 of 63 +gcd(p1, p2) +--R +--R +--R (8) 1 +--R Type: Polynomial(Integer) +--E 8 + +--S 9 of 63 +gcd(expand(p1*q), expand(p2*q)) - q +--R +--R +--R (9) 0 +--R Type: Polynomial(Integer) +--E 9 + +\end{chunk} +$resultant(p1 q, p2 q) => 0$ +\begin{chunk}{*} +--S 10 of 63 +resultant(expand(p1*q), expand(p2*q), x) +--R +--R +--R (10) 0 +--R Type: Polynomial(Integer) +--E 10 + +\end{chunk} +How about factorization? $=> p1 * p2$ +\begin{chunk}{*} +--S 11 of 63 +factor(expand(p1 * p2)) +--R +--R +--R (11) +--R - +--R 60 47 34 8 5 +--R (16x + 21x - 64x + 126x + 46x + 81) +--R * +--R 60 52 39 25 23 10 +--R (72x - 83x - 22x - 25x - 19x + 54x + 81) +--R Type: Factored(Polynomial(Integer)) +--E 11 + +)clear properties p1 p2 q + +\end{chunk} +Multivariate gcd: $gcd(p1, p2) => 1, gcd(p1 q, p2 q) => q$ +\begin{chunk}{*} +--S 12 of 63 +p1:= 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5 +--R +--R +--R 19 17 5 8 15 9 2 22 +--R (12) (24x y - 47x y )z + 6x y z - 3x + 5 +--R Type: Polynomial(Integer) +--E 12 + +--S 13 of 63 +p2:= 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z +--R +--R +--R 5 8 13 7 7 7 9 16 4 14 +--R (13) 34x y z + 20x y z + 12x y z + 80y z +--R Type: Polynomial(Integer) +--E 13 + +--S 14 of 63 +q:= 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8 +--R +--R +--R 12 7 13 2 8 10 17 5 8 +--R (14) 11x y z - 23x y z + 47x y z +--R Type: Polynomial(Integer) +--E 14 + +--S 15 of 63 +gcd(p1, p2) +--R +--R +--R (15) 1 +--R Type: Polynomial(Integer) +--E 15 + +--S 16 of 63 +gcd(expand(p1*q), expand(p2*q)) - q +--R +--R +--R (16) 0 +--R Type: Polynomial(Integer) +--E 16 + +\end{chunk} +How about factorization? $=> p1 * p2$ +\begin{chunk}{*} +--S 17 of 63 +factor(expand(p1 * p2)) +--R +--R +--R (17) +--R 7 19 17 5 8 15 9 2 22 +--R 2y z((24x y - 47x y )z + 6x y z - 3x + 5) +--R * +--R 5 12 7 6 9 9 3 7 +--R (17x y z + 10x z + 6x y z + 40y ) +--R Type: Factored(Polynomial(Integer)) +--E 17 + +)clear properties p1 p2 q + +\end{chunk} +$=> x^n {\textrm\ for\ } n > 0$ [Chris Hurlburt] +\begin{chunk}{*} +--S 18 of 63 +gcd(2*x**(n + 4) - x**(n + 2), 4*x**(n + 1) + 3*x**n) +--R +--R +--R (18) 1 +--R Type: Expression(Integer) +--E 18 + +\end{chunk} + +Resultants. If the resultant of two polynomials is zero, this implies they +have a common factor. See Keith O. Geddes, Stephen R. Czapor and George +Labahn, ``Algorithms for Computer Algebra'', Kluwer Academic Publishers, 1992, +p. 286 $=> 0$ +\begin{chunk}{*} +--S 19 of 63 +resultant(3*x**4 + 3*x**3 + x**2 - x - 2, x**3 - 3*x**2 + x + 5, x) +--R +--R +--R (19) 0 +--R Type: Polynomial(Integer) +--E 19 + +\end{chunk} +Numbers are nice, but symbols allow for variability---try some high school +algebra: rational simplification $=> (x - 2)/(x + 2)$ +\begin{chunk}{*} +--S 20 of 63 +(x**2 - 4)/(x**2 + 4*x + 4) +--R +--R +--R x - 2 +--R (20) ----- +--R x + 2 +--R Type: Fraction(Polynomial(Integer)) +--E 20 + +\end{chunk} +This example requires more sophistication $=> e^{(x/2)} - 1$ +\begin{chunk}{*} +--S 21 of 63 +[(%e**x - 1)/(%e**(x/2) + 1), (exp(x) - 1)/(exp(x/2) + 1)] +--R +--R +--R x x +--R %e - 1 %e - 1 +--R (21) [-------,-------] +--R x x +--R - - +--R 2 2 +--R %e + 1 %e + 1 +--R Type: List(Expression(Integer)) +--E 21 + +--S 22 of 63 +map(normalize, %) +--R +--R +--R x x +--R - - +--R 2 2 +--R (22) [%e - 1,%e - 1] +--R Type: List(Expression(Integer)) +--E 22 + +\end{chunk} +Expand and factor polynomials +\begin{chunk}{*} +--S 23 of 63 +(x + 1)**20 +--R +--R +--R (23) +--R 20 19 18 17 16 15 14 13 +--R x + 20x + 190x + 1140x + 4845x + 15504x + 38760x + 77520x +--R + +--R 12 11 10 9 8 7 6 +--R 125970x + 167960x + 184756x + 167960x + 125970x + 77520x + 38760x +--R + +--R 5 4 3 2 +--R 15504x + 4845x + 1140x + 190x + 20x + 1 +--R Type: Polynomial(Integer) +--E 23 + +--S 24 of 63 +D(%, x) +--R +--R +--R (24) +--R 19 18 17 16 15 14 13 +--R 20x + 380x + 3420x + 19380x + 77520x + 232560x + 542640x +--R + +--R 12 11 10 9 8 7 +--R 1007760x + 1511640x + 1847560x + 1847560x + 1511640x + 1007760x +--R + +--R 6 5 4 3 2 +--R 542640x + 232560x + 77520x + 19380x + 3420x + 380x + 20 +--R Type: Polynomial(Integer) +--E 24 + +--S 25 of 63 +factor(%) +--R +--R +--R 19 +--R (25) 20(x + 1) +--R Type: Factored(Polynomial(Integer)) +--E 25 + +\end{chunk} +Completely factor this polynomial, then try to multiply it back together! +\begin{chunk}{*} +--S 26 of 63 +radicalSolve(x**3 + x**2 - 7 = 0, x) +--R +--R +--R (26) +--R [ +--R x = +--R +------------------+2 +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (- 9\|- 3 + 9) |------------------ +--R 3| +-+ +--R \| 54\|3 +--R + +--R +------------------+ +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (- 3\|- 3 - 3) |------------------ - 2 +--R 3| +-+ +--R \| 54\|3 +--R / +--R +------------------+ +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (9\|- 3 + 9) |------------------ +--R 3| +-+ +--R \| 54\|3 +--R , +--R +--R x = +--R +------------------+2 +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (- 9\|- 3 - 9) |------------------ +--R 3| +-+ +--R \| 54\|3 +--R + +--R +------------------+ +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (- 3\|- 3 + 3) |------------------ + 2 +--R 3| +-+ +--R \| 54\|3 +--R / +--R +------------------+ +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (9\|- 3 - 9) |------------------ +--R 3| +-+ +--R \| 54\|3 +--R , +--R +------------------+2 +------------------+ +--R | +----+ +-+ | +----+ +-+ +--R |9\|1295 + 187\|3 |9\|1295 + 187\|3 +--R 9 |------------------ - 3 |------------------ + 1 +--R 3| +-+ 3| +-+ +--R \| 54\|3 \| 54\|3 +--R x= ----------------------------------------------------] +--R +------------------+ +--R | +----+ +-+ +--R |9\|1295 + 187\|3 +--R 9 |------------------ +--R 3| +-+ +--R \| 54\|3 +--R Type: List(Equation(Expression(Integer))) +--E 26 + +--S 27 of 63 +reduce(*, map(e +-> lhs(e) - rhs(e), %)) +--R +--R +--R 3 2 +-+ +----+ 3 2 +--R (9x + 9x - 63)\|3 \|1295 + 561x + 561x - 3927 +--R (27) -------------------------------------------------- +--R +-+ +----+ +-+2 +--R 9\|3 \|1295 + 187\|3 +--R Type: Expression(Integer) +--E 27 + +--S 28 of 63 +x**100 - 1 +--R +--R +--R 100 +--R (28) x - 1 +--R Type: Polynomial(Integer) +--E 28 + +--S 29 of 63 +factor(%) +--R +--R +--R (29) +--R 2 4 3 2 4 3 2 +--R (x - 1)(x + 1)(x + 1)(x - x + x - x + 1)(x + x + x + x + 1) +--R * +--R 8 6 4 2 20 15 10 5 20 15 10 5 +--R (x - x + x - x + 1)(x - x + x - x + 1)(x + x + x + x + 1) +--R * +--R 40 30 20 10 +--R (x - x + x - x + 1) +--R Type: Factored(Polynomial(Integer)) +--E 29 + +\end{chunk} +Factorization over the complex rationals + +$=> (2 x + 3 i) (2 x - 3 i) (x + 1 + 4 i) (x + 1 - 4 i)$ +\begin{chunk}{*} +--S 30 of 63 +factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153, [rootOf(i**2 + 1)]) +--R +--R +--R 3i 3i +--R (30) 4(x - 4i + 1)(x - --)(x + --)(x + 4i + 1) +--R 2 2 +--R Type: Factored(Polynomial(AlgebraicNumber)) +--E 30 + +\end{chunk} +Algebraic extensions +\begin{chunk}{*} +--S 31 of 63 +sqrt2:= rootOf(sqrt2**2 - 2) +--R +--R +--R (31) sqrt2 +--R Type: AlgebraicNumber +--E 31 + +\end{chunk} +$=> sqrt2 + 1$ +\begin{chunk}{*} +--S 32 of 63 +1/(sqrt2 - 1) +--R +--R +--R (32) sqrt2 + 1 +--R Type: AlgebraicNumber +--E 32 + +\end{chunk} +$=> (x^2 - 2 x - 3)/(x - sqrt2) = (x + 1) (x - 3)/(x - sqrt2)$ +[Richard Liska] +\begin{chunk}{*} +--S 33 of 63 +(x**3 + (sqrt2 - 2)*x**2 - (2*sqrt2 + 3)*x - 3*sqrt2)/(x**2 - 2) +--R +--R +--R 2 +--R x - 2x - 3 +--R (33) ----------- +--R x - sqrt2 +--R Type: Fraction(Polynomial(AlgebraicNumber)) +--E 33 + +--S 34 of 63 +numer(%)/ratDenom(denom(%)) +--R +--R +--R 2 +--R - x + 2x + 3 +--R (34) ------------- +--R sqrt2 - x +--R Type: Expression(Integer) +--E 34 + +)clear properties sqrt2 + +\end{chunk} +Multiple algebraic extensions +\begin{chunk}{*} +--S 35 of 63 +sqrt3:= rootOf(sqrt3**2 - 3) +--R +--R +--R (35) sqrt3 +--R Type: AlgebraicNumber +--E 35 + +--S 36 of 63 +cbrt2:= rootOf(cbrt2**3 - 2) +--R +--R +--R (36) cbrt2 +--R Type: AlgebraicNumber +--E 36 + +\end{chunk} +$=> 2 cbrt2 + 8 sqrt3 + 18 cbrt2^2 + 12 cbrt2 sqrt3 + 9$ +\begin{chunk}{*} +--S 37 of 63 +(cbrt2 + sqrt3)**4 +--R +--R +--R 2 +--R (37) (12cbrt2 + 8)sqrt3 + 18cbrt2 + 2cbrt2 + 9 +--R Type: AlgebraicNumber +--E 37 + +)clear properties sqrt3 cbrt2 + +\end{chunk} +Factor polynomials over finite fields and field extensions +\begin{chunk}{*} +--S 38 of 63 +p:= x**4 - 3*x**2 + 1 +--R +--R +--R 4 2 +--R (38) x - 3x + 1 +--R Type: Polynomial(Integer) +--E 38 + +--S 39 of 63 +factor(p) +--R +--R +--R 2 2 +--R (39) (x - x - 1)(x + x - 1) +--R Type: Factored(Polynomial(Integer)) +--E 39 + +\end{chunk} +$=> (x - 2)^2 (x + 2)^2 {\textrm\ mod\ } 5$ +\begin{chunk}{*} +--S 40 of 63 +factor(p :: Polynomial(PrimeField(5))) +--R +--R +--R 2 2 +--R (40) (x + 2) (x + 3) +--R Type: Factored(Polynomial(PrimeField(5))) +--E 40 + +--S 41 of 63 +expand(%) +--R +--R +--R 4 2 +--R (41) x + 2x + 1 +--R Type: Polynomial(PrimeField(5)) +--E 41 + +\end{chunk} +$=> (x^2 + x + 1) (x^9 - x^8 + x^6 - x^5 + x^3 - x^2 + 1){\textrm\ mod\ } 65537$ +[Paul Zimmermann] +\begin{chunk}{*} +--S 42 of 63 +factor(x**11 + x + 1 :: Polynomial(PrimeField(65537))) +--R +--R +--R 2 9 8 6 5 3 2 +--R (42) (x + x + 1)(x + 65536x + x + 65536x + x + 65536x + 1) +--R Type: Factored(Polynomial(PrimeField(65537))) +--E 42 + +\end{chunk} +$=> (x - phi) (x + phi) (x - phi + 1) (x + phi - 1)$ + +where $phi^2 - phi - 1 = 0$ or $phi = (1 \pm sqrt(5))/2$ +\begin{chunk}{*} +--S 43 of 63 +phi:= rootOf(phi**2 - phi - 1) +--R +--R +--R (43) phi +--R Type: AlgebraicNumber +--E 43 + +--S 44 of 63 +factor(p, [phi]) +--R +--R +--R (44) (x - phi)(x - phi + 1)(x + phi - 1)(x + phi) +--R Type: Factored(Polynomial(AlgebraicNumber)) +--E 44 + +)clear properties phi p + +--S 45 of 63 +expand((x - 2*y**2 + 3*z**3)**20) +--R +--R +--R (45) +--R 60 2 57 +--R 3486784401z + (- 46490458680y + 23245229340x)z +--R + +--R 4 2 2 54 +--R (294439571640y - 294439571640x y + 73609892910x )z +--R + +--R 6 4 2 2 +--R - 1177758286560y + 1766637429840x y - 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574665062400cos(y) sin(x) + 1292996390400cos(y) sin(x) +--R + +--R 18 7 20 6 +--R - 2298660249600cos(y) sin(x) + 3218124349440cos(y) sin(x) +--R + +--R 22 5 24 4 +--R - 3510681108480cos(y) sin(x) + 2925567590400cos(y) sin(x) +--R + +--R 26 3 28 2 +--R - 1800349286400cos(y) sin(x) + 771578265600cos(y) sin(x) +--R + +--R 30 32 +--R - 205754204160cos(y) sin(x) + 25719275520cos(y) +--R * +--R 12 +--R tan(z) +--R + +--R 17 2 16 4 15 +--R 30780sin(x) - 1046520cos(y) sin(x) + 16744320cos(y) sin(x) +--R + +--R 6 14 8 13 +--R - 167443200cos(y) sin(x) + 1172102400cos(y) sin(x) +--R + +--R 10 12 12 11 +--R - 6094932480cos(y) sin(x) + 24379729920cos(y) sin(x) +--R + +--R 14 10 16 9 +--R - 76622008320cos(y) sin(x) + 191555020800cos(y) sin(x) +--R + +--R 18 8 20 7 +--R - 383110041600cos(y) sin(x) + 612976066560cos(y) sin(x) +--R + +--R 22 6 24 5 +--R - 780151357440cos(y) sin(x) + 780151357440cos(y) sin(x) +--R + +--R 26 4 28 3 +--R - 600116428800cos(y) sin(x) + 342923673600cos(y) sin(x) +--R + +--R 30 2 32 +--R - 137169469440cos(y) sin(x) + 34292367360cos(y) sin(x) +--R + +--R 34 +--R - 4034396160cos(y) +--R * +--R 9 +--R tan(z) +--R + +--R 18 2 17 4 16 +--R 1710sin(x) - 61560cos(y) sin(x) + 1046520cos(y) sin(x) +--R + +--R 6 15 8 14 +--R - 11162880cos(y) sin(x) + 83721600cos(y) sin(x) +--R + +--R 10 13 12 12 +--R - 468840960cos(y) sin(x) + 2031644160cos(y) sin(x) +--R + +--R 14 11 16 10 +--R - 6965637120cos(y) sin(x) + 19155502080cos(y) sin(x) +--R + +--R 18 9 20 8 +--R - 42567782400cos(y) sin(x) + 76622008320cos(y) sin(x) +--R + +--R 22 7 24 6 +--R - 111450193920cos(y) sin(x) + 130025226240cos(y) sin(x) +--R + +--R 26 5 28 4 +--R - 120023285760cos(y) sin(x) + 85730918400cos(y) sin(x) +--R + +--R 30 3 32 2 +--R - 45723156480cos(y) sin(x) + 17146183680cos(y) sin(x) +--R + +--R 34 36 +--R - 4034396160cos(y) sin(x) + 448266240cos(y) +--R * +--R 6 +--R tan(z) +--R + +--R 19 2 18 4 17 +--R 60sin(x) - 2280cos(y) sin(x) + 41040cos(y) sin(x) +--R + +--R 6 16 8 15 +--R - 465120cos(y) sin(x) + 3720960cos(y) sin(x) +--R + +--R 10 14 12 13 +--R - 22325760cos(y) sin(x) + 104186880cos(y) sin(x) +--R + +--R 14 12 16 11 +--R - 386979840cos(y) sin(x) + 1160939520cos(y) sin(x) +--R + +--R 18 10 20 9 +--R - 2837852160cos(y) sin(x) + 5675704320cos(y) sin(x) +--R + +--R 22 8 24 7 +--R - 9287516160cos(y) sin(x) + 12383354880cos(y) sin(x) +--R + +--R 26 6 28 5 +--R - 13335920640cos(y) sin(x) + 11430789120cos(y) sin(x) +--R + +--R 30 4 32 3 +--R - 7620526080cos(y) sin(x) + 3810263040cos(y) sin(x) +--R + +--R 34 2 36 38 +--R - 1344798720cos(y) sin(x) + 298844160cos(y) sin(x) - 31457280cos(y) +--R * +--R 3 +--R tan(z) +--R + +--R 20 2 19 4 18 6 17 +--R sin(x) - 40cos(y) sin(x) + 760cos(y) sin(x) - 9120cos(y) sin(x) +--R + +--R 8 16 10 15 12 14 +--R 77520cos(y) sin(x) - 496128cos(y) sin(x) + 2480640cos(y) sin(x) +--R + +--R 14 13 16 12 +--R - 9922560cos(y) sin(x) + 32248320cos(y) sin(x) +--R + +--R 18 11 20 10 +--R - 85995520cos(y) sin(x) + 189190144cos(y) sin(x) +--R + +--R 22 9 24 8 +--R - 343982080cos(y) sin(x) + 515973120cos(y) sin(x) +--R + +--R 26 7 28 6 +--R - 635043840cos(y) sin(x) + 635043840cos(y) sin(x) +--R + +--R 30 5 32 4 +--R - 508035072cos(y) sin(x) + 317521920cos(y) sin(x) +--R + +--R 34 3 36 2 +--R - 149422080cos(y) sin(x) + 49807360cos(y) sin(x) +--R + +--R 38 40 +--R - 10485760cos(y) sin(x) + 1048576cos(y) +--R Type: Expression(Integer) +--E 47 + +--S 48 of 63 +factor(%) +--R +--R +--R (48) +--R 60 2 57 +--R 3486784401tan(z) + (23245229340sin(x) - 46490458680cos(y) )tan(z) +--R + +--R 2 2 4 +--R (73609892910sin(x) - 294439571640cos(y) sin(x) + 294439571640cos(y) ) +--R * +--R 54 +--R tan(z) +--R + +--R 3 2 2 +--R 147219785820sin(x) - 883318714920cos(y) sin(x) +--R + +--R 4 6 +--R 1766637429840cos(y) sin(x) - 1177758286560cos(y) +--R * +--R 51 +--R tan(z) +--R + +--R 4 2 3 +--R 208561363245sin(x) - 1668490905960cos(y) sin(x) +--R + +--R 4 2 6 +--R 5005472717880cos(y) sin(x) - 6673963623840cos(y) sin(x) +--R + +--R 8 +--R 3336981811920cos(y) +--R * +--R 48 +--R tan(z) +--R + +--R 5 2 4 +--R 222465454128sin(x) - 2224654541280cos(y) sin(x) +--R + +--R 4 3 6 2 +--R 8898618165120cos(y) sin(x) - 17797236330240cos(y) sin(x) +--R + +--R 8 10 +--R 17797236330240cos(y) sin(x) - 7118894532096cos(y) +--R * +--R 45 +--R tan(z) +--R + +--R 6 2 5 +--R 185387878440sin(x) - 2224654541280cos(y) sin(x) +--R + +--R 4 4 6 3 +--R 11123272706400cos(y) sin(x) - 29662060550400cos(y) sin(x) +--R + +--R 8 2 10 +--R 44493090825600cos(y) sin(x) - 35594472660480cos(y) sin(x) +--R + +--R 12 +--R 11864824220160cos(y) +--R * +--R 42 +--R tan(z) +--R + +--R 7 2 6 +--R 123591918960sin(x) - 1730286865440cos(y) sin(x) +--R + +--R 4 5 6 4 +--R 10381721192640cos(y) sin(x) - 34605737308800cos(y) sin(x) +--R + +--R 8 3 10 2 +--R 69211474617600cos(y) sin(x) - 83053769541120cos(y) sin(x) +--R + +--R 12 14 +--R 55369179694080cos(y) sin(x) - 15819765626880cos(y) +--R * +--R 39 +--R tan(z) +--R + +--R 8 2 7 +--R 66945622770sin(x) - 1071129964320cos(y) sin(x) +--R + +--R 4 6 6 5 +--R 7497909750240cos(y) sin(x) - 29991639000960cos(y) sin(x) +--R + +--R 8 4 10 3 +--R 74979097502400cos(y) sin(x) - 119966556003840cos(y) sin(x) +--R + +--R 12 2 14 +--R 119966556003840cos(y) sin(x) - 68552317716480cos(y) sin(x) +--R + +--R 16 +--R 17138079429120cos(y) +--R * +--R 36 +--R tan(z) +--R + +--R 9 2 8 +--R 29753610120sin(x) - 535564982160cos(y) sin(x) +--R + +--R 4 7 6 6 +--R 4284519857280cos(y) sin(x) - 19994426000640cos(y) sin(x) +--R + +--R 8 5 10 4 +--R 59983278001920cos(y) sin(x) - 119966556003840cos(y) sin(x) +--R + +--R 12 3 14 2 +--R 159955408005120cos(y) sin(x) - 137104635432960cos(y) sin(x) +--R + +--R 16 18 +--R 68552317716480cos(y) sin(x) - 15233848381440cos(y) +--R * +--R 33 +--R tan(z) +--R + +--R 10 2 9 +--R 10909657044sin(x) - 218193140880cos(y) sin(x) +--R + +--R 4 8 6 7 +--R 1963738267920cos(y) sin(x) - 10473270762240cos(y) sin(x) +--R + +--R 8 6 10 5 +--R 36656447667840cos(y) sin(x) - 87975474402816cos(y) sin(x) +--R + +--R 12 4 14 3 +--R 146625790671360cos(y) sin(x) - 167572332195840cos(y) sin(x) +--R + +--R 16 2 18 +--R 125679249146880cos(y) sin(x) - 55857444065280cos(y) sin(x) +--R + +--R 20 +--R 11171488813056cos(y) +--R * +--R 30 +--R tan(z) +--R + +--R 11 2 10 +--R 3305956680sin(x) - 72731046960cos(y) sin(x) +--R + +--R 4 9 6 8 +--R 727310469600cos(y) sin(x) - 4363862817600cos(y) sin(x) +--R + +--R 8 7 10 6 +--R 17455451270400cos(y) sin(x) - 48875263557120cos(y) sin(x) +--R + +--R 12 5 14 4 +--R 97750527114240cos(y) sin(x) - 139643610163200cos(y) sin(x) +--R + +--R 16 3 18 2 +--R 139643610163200cos(y) sin(x) - 93095740108800cos(y) sin(x) +--R + +--R 20 22 +--R 37238296043520cos(y) sin(x) - 6770599280640cos(y) +--R * +--R 27 +--R tan(z) +--R + +--R 12 2 11 +--R 826489170sin(x) - 19835740080cos(y) sin(x) +--R + +--R 4 10 6 9 +--R 218193140880cos(y) sin(x) - 1454620939200cos(y) sin(x) +--R + +--R 8 8 10 7 +--R 6545794226400cos(y) sin(x) - 20946541524480cos(y) sin(x) +--R + +--R 12 6 14 5 +--R 48875263557120cos(y) sin(x) - 83786166097920cos(y) sin(x) +--R + +--R 16 4 18 3 +--R 104732707622400cos(y) sin(x) - 93095740108800cos(y) sin(x) +--R + +--R 20 2 22 +--R 55857444065280cos(y) sin(x) - 20311797841920cos(y) sin(x) +--R + +--R 24 +--R 3385299640320cos(y) +--R * +--R 24 +--R tan(z) +--R + +--R 13 2 12 +--R 169536240sin(x) - 4407942240cos(y) sin(x) +--R + +--R 4 11 6 10 +--R 52895306880cos(y) sin(x) - 387898917120cos(y) sin(x) +--R + +--R 8 9 10 8 +--R 1939494585600cos(y) sin(x) - 6982180508160cos(y) sin(x) +--R + +--R 12 7 14 6 +--R 18619148021760cos(y) sin(x) - 37238296043520cos(y) sin(x) +--R + +--R 16 5 18 4 +--R 55857444065280cos(y) sin(x) - 62063826739200cos(y) sin(x) +--R + +--R 20 3 22 2 +--R 49651061391360cos(y) sin(x) - 27082397122560cos(y) sin(x) +--R + +--R 24 26 +--R 9027465707520cos(y) sin(x) - 1388840878080cos(y) +--R * +--R 21 +--R tan(z) +--R + +--R 14 2 13 +--R 28256040sin(x) - 791169120cos(y) sin(x) +--R + +--R 4 12 6 11 +--R 10285198560cos(y) sin(x) - 82281588480cos(y) sin(x) +--R + +--R 8 10 10 9 +--R 452548736640cos(y) sin(x) - 1810194946560cos(y) sin(x) +--R + +--R 12 8 14 7 +--R 5430584839680cos(y) sin(x) - 12412765347840cos(y) sin(x) +--R + +--R 16 6 18 5 +--R 21722339358720cos(y) sin(x) - 28963119144960cos(y) sin(x) +--R + +--R 20 4 22 3 +--R 28963119144960cos(y) sin(x) - 21064086650880cos(y) sin(x) +--R + +--R 24 2 26 +--R 10532043325440cos(y) sin(x) - 3240628715520cos(y) sin(x) +--R + +--R 28 +--R 462946959360cos(y) +--R * +--R 18 +--R tan(z) +--R + +--R 15 2 14 4 13 +--R 3767472sin(x) - 113024160cos(y) sin(x) + 1582338240cos(y) sin(x) +--R + +--R 6 12 8 11 +--R - 13713598080cos(y) sin(x) + 82281588480cos(y) sin(x) +--R + +--R 10 10 12 9 +--R - 362038989312cos(y) sin(x) + 1206796631040cos(y) sin(x) +--R + +--R 14 8 16 7 +--R - 3103191336960cos(y) sin(x) + 6206382673920cos(y) sin(x) +--R + +--R 18 6 20 5 +--R - 9654373048320cos(y) sin(x) + 11585247657984cos(y) sin(x) +--R + +--R 22 4 24 3 +--R - 10532043325440cos(y) sin(x) + 7021362216960cos(y) sin(x) +--R + +--R 26 2 28 +--R - 3240628715520cos(y) sin(x) + 925893918720cos(y) sin(x) +--R + +--R 30 +--R - 123452522496cos(y) +--R * +--R 15 +--R tan(z) +--R + +--R 16 2 15 4 14 +--R 392445sin(x) - 12558240cos(y) sin(x) + 188373600cos(y) sin(x) +--R + +--R 6 13 8 12 +--R - 1758153600cos(y) sin(x) + 11427998400cos(y) sin(x) +--R + +--R 10 11 12 10 +--R - 54854392320cos(y) sin(x) + 201132771840cos(y) sin(x) +--R + +--R 14 9 16 8 +--R - 574665062400cos(y) sin(x) + 1292996390400cos(y) sin(x) +--R + +--R 18 7 20 6 +--R - 2298660249600cos(y) sin(x) + 3218124349440cos(y) sin(x) +--R + +--R 22 5 24 4 +--R - 3510681108480cos(y) sin(x) + 2925567590400cos(y) sin(x) +--R + +--R 26 3 28 2 +--R - 1800349286400cos(y) sin(x) + 771578265600cos(y) sin(x) +--R + +--R 30 32 +--R - 205754204160cos(y) sin(x) + 25719275520cos(y) +--R * +--R 12 +--R tan(z) +--R + +--R 17 2 16 4 15 +--R 30780sin(x) - 1046520cos(y) sin(x) + 16744320cos(y) sin(x) +--R + +--R 6 14 8 13 +--R - 167443200cos(y) sin(x) + 1172102400cos(y) sin(x) +--R + +--R 10 12 12 11 +--R - 6094932480cos(y) sin(x) + 24379729920cos(y) sin(x) +--R + +--R 14 10 16 9 +--R - 76622008320cos(y) sin(x) + 191555020800cos(y) sin(x) +--R + +--R 18 8 20 7 +--R - 383110041600cos(y) sin(x) + 612976066560cos(y) sin(x) +--R + +--R 22 6 24 5 +--R - 780151357440cos(y) sin(x) + 780151357440cos(y) sin(x) +--R + +--R 26 4 28 3 +--R - 600116428800cos(y) sin(x) + 342923673600cos(y) sin(x) +--R + +--R 30 2 32 +--R - 137169469440cos(y) sin(x) + 34292367360cos(y) sin(x) +--R + +--R 34 +--R - 4034396160cos(y) +--R * +--R 9 +--R tan(z) +--R + +--R 18 2 17 4 16 +--R 1710sin(x) - 61560cos(y) sin(x) + 1046520cos(y) sin(x) +--R + +--R 6 15 8 14 +--R - 11162880cos(y) sin(x) + 83721600cos(y) sin(x) +--R + +--R 10 13 12 12 +--R - 468840960cos(y) sin(x) + 2031644160cos(y) sin(x) +--R + +--R 14 11 16 10 +--R - 6965637120cos(y) sin(x) + 19155502080cos(y) sin(x) +--R + +--R 18 9 20 8 +--R - 42567782400cos(y) sin(x) + 76622008320cos(y) sin(x) +--R + +--R 22 7 24 6 +--R - 111450193920cos(y) sin(x) + 130025226240cos(y) sin(x) +--R + +--R 26 5 28 4 +--R - 120023285760cos(y) sin(x) + 85730918400cos(y) sin(x) +--R + +--R 30 3 32 2 +--R - 45723156480cos(y) sin(x) + 17146183680cos(y) sin(x) +--R + +--R 34 36 +--R - 4034396160cos(y) sin(x) + 448266240cos(y) +--R * +--R 6 +--R tan(z) +--R + +--R 19 2 18 4 17 +--R 60sin(x) - 2280cos(y) sin(x) + 41040cos(y) sin(x) +--R + +--R 6 16 8 15 +--R - 465120cos(y) sin(x) + 3720960cos(y) sin(x) +--R + +--R 10 14 12 13 +--R - 22325760cos(y) sin(x) + 104186880cos(y) sin(x) +--R + +--R 14 12 16 11 +--R - 386979840cos(y) sin(x) + 1160939520cos(y) sin(x) +--R + +--R 18 10 20 9 +--R - 2837852160cos(y) sin(x) + 5675704320cos(y) sin(x) +--R + +--R 22 8 24 7 +--R - 9287516160cos(y) sin(x) + 12383354880cos(y) sin(x) +--R + +--R 26 6 28 5 +--R - 13335920640cos(y) sin(x) + 11430789120cos(y) sin(x) +--R + +--R 30 4 32 3 +--R - 7620526080cos(y) sin(x) + 3810263040cos(y) sin(x) +--R + +--R 34 2 36 38 +--R - 1344798720cos(y) sin(x) + 298844160cos(y) sin(x) - 31457280cos(y) +--R * +--R 3 +--R tan(z) +--R + +--R 20 2 19 4 18 6 17 +--R sin(x) - 40cos(y) sin(x) + 760cos(y) sin(x) - 9120cos(y) sin(x) +--R + +--R 8 16 10 15 12 14 +--R 77520cos(y) sin(x) - 496128cos(y) sin(x) + 2480640cos(y) sin(x) +--R + +--R 14 13 16 12 +--R - 9922560cos(y) sin(x) + 32248320cos(y) sin(x) +--R + +--R 18 11 20 10 +--R - 85995520cos(y) sin(x) + 189190144cos(y) sin(x) +--R + +--R 22 9 24 8 +--R - 343982080cos(y) sin(x) + 515973120cos(y) sin(x) +--R + +--R 26 7 28 6 +--R - 635043840cos(y) sin(x) + 635043840cos(y) sin(x) +--R + +--R 30 5 32 4 +--R - 508035072cos(y) sin(x) + 317521920cos(y) sin(x) +--R + +--R 34 3 36 2 +--R - 149422080cos(y) sin(x) + 49807360cos(y) sin(x) +--R + +--R 38 40 +--R - 10485760cos(y) sin(x) + 1048576cos(y) +--R Type: Factored(Expression(Integer)) +--E 48 + + +\end{chunk} +expand$[(1 - c^2)^5 (1 - s^2)^5 (c^2 + s^2)^{10}] => c^{10} s^{10}$ + +when $c^2 + s^2 = 1$ [modification of a problem due to Richard Liska] +\begin{chunk}{*} +--S 49 of 63 +expand((1 - c**2)**5 * (1 - s**2)**5 * (c**2 + s**2)**10) +--R +--R +--R (49) +--R 10 8 6 4 2 30 +--R (c - 5c + 10c - 10c + 5c - 1)s +--R + +--R 12 10 8 6 4 2 28 +--R (10c - 55c + 125c - 150c + 100c - 35c + 5)s +--R + +--R 14 12 10 8 6 4 2 26 +--R (45c - 275c + 710c - 1000c + 825c - 395c + 100c - 10)s +--R + +--R 16 14 12 10 8 6 4 2 +--R 120c - 825c + 2425c - 3960c + 3900c - 2345c + 825c - 150c +--R + +--R 10 +--R * +--R 24 +--R s +--R + +--R 18 16 14 12 10 8 6 +--R 210c - 1650c + 5550c - 10450c + 12055c - 8735c + 3900c +--R + +--R 4 2 +--R - 1000c + 125c - 5 +--R * +--R 22 +--R s +--R + +--R 20 18 16 14 12 10 8 +--R 252c - 2310c + 8970c - 19470c + 26060c - 22253c + 12055c +--R + +--R 6 4 2 +--R - 3960c + 710c - 55c + 1 +--R * +--R 20 +--R s +--R + +--R 22 20 18 16 14 12 10 +--R 210c - 2310c + 10500c - 26400c + 40875c - 40645c + 26060c +--R + +--R 8 6 4 2 +--R - 10450c + 2425c - 275c + 10c +--R * +--R 18 +--R s +--R + +--R 24 22 20 18 16 14 12 +--R 120c - 1650c + 8970c - 26400c + 47400c - 54615c + 40875c +--R + +--R 10 8 6 4 +--R - 19470c + 5550c - 825c + 45c +--R * +--R 16 +--R s +--R + +--R 26 24 22 20 18 16 14 +--R 45c - 825c + 5550c - 19470c + 40875c - 54615c + 47400c +--R + +--R 12 10 8 6 +--R - 26400c + 8970c - 1650c + 120c +--R * +--R 14 +--R s +--R + +--R 28 26 24 22 20 18 16 +--R 10c - 275c + 2425c - 10450c + 26060c - 40645c + 40875c +--R + +--R 14 12 10 8 +--R - 26400c + 10500c - 2310c + 210c +--R * +--R 12 +--R s +--R + +--R 30 28 26 24 22 20 18 +--R c - 55c + 710c - 3960c + 12055c - 22253c + 26060c +--R + +--R 16 14 12 10 +--R - 19470c + 8970c - 2310c + 252c +--R * +--R 10 +--R s +--R + +--R 30 28 26 24 22 20 18 +--R - 5c + 125c - 1000c + 3900c - 8735c + 12055c - 10450c +--R + +--R 16 14 12 +--R 5550c - 1650c + 210c +--R * +--R 8 +--R s +--R + +--R 30 28 26 24 22 20 18 +--R 10c - 150c + 825c - 2345c + 3900c - 3960c + 2425c +--R + +--R 16 14 +--R - 825c + 120c +--R * +--R 6 +--R s +--R + +--R 30 28 26 24 22 20 18 16 4 +--R (- 10c + 100c - 395c + 825c - 1000c + 710c - 275c + 45c )s +--R + +--R 30 28 26 24 22 20 18 2 30 28 +--R (5c - 35c + 100c - 150c + 125c - 55c + 10c )s - c + 5c +--R + +--R 26 24 22 20 +--R - 10c + 10c - 5c + c +--R Type: Polynomial(Integer) +--E 49 + +--S 50 of 63 +groebner([%, c**2 + s**2 - 1]) +--R +--R +--R 2 2 20 18 16 14 12 10 +--R (50) [s + c - 1,c - 5c + 10c - 10c + 5c - c ] +--R Type: List(Polynomial(Integer)) +--E 50 + +--S 51 of 63 +map(factor, %) +--R +--R +--R 2 2 5 10 5 +--R (51) [s + c - 1,(c - 1) c (c + 1) ] +--R Type: List(Factored(Polynomial(Integer))) +--E 51 + +\end{chunk} +$=> (x + y) (x - y) {\textrm\ mod\ } 3$ +\begin{chunk}{*} +--S 52 of 63 +factor(4*x**2 - 21*x*y + 20*y**2 :: Polynomial(PrimeField(3))) +--R +--R There are 22 exposed and 18 unexposed library operations named ** +--R having 2 argument(s) but none was determined to be applicable. +--R Use HyperDoc Browse, or issue +--R )display op ** +--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 +--R Cannot find a definition or applicable library operation named ** +--R with argument type(s) +--R Variable(y) +--R Polynomial(PrimeField(3)) +--R +--R Perhaps you should use "@" to indicate the required return type, +--R or "$" to specify which version of the function you need. +--E 52 + +\end{chunk} +$=> 1/4 (x + y) (2 x + y [-1 + i sqrt(3)]) (2 x + y [-1 - i sqrt(3)])$ +\begin{chunk}{*} +--S 53 of 63 +factor(x**3 + y**3, [rootOf(isqrt3**2 + 3)]) +--R +--R +--R - isqrt3 - 1 isqrt3 - 1 +--R (52) (y + ------------ x)(y + x)(y + ---------- x) +--R 2 2 +--R Type: Factored(Polynomial(AlgebraicNumber)) +--E 53 + +\end{chunk} +Partial fraction decomposition $=> 3/(x + 2) - 2/(x + 1) + 2/(x + 1)^2$ +\begin{chunk}{*} +--S 54 of 63 +(x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2) +--R +--R +--R 2 +--R x + 2x + 3 +--R (53) ----------------- +--R 3 2 +--R x + 4x + 5x + 2 +--R Type: Fraction(Polynomial(Integer)) +--E 54 + +--S 55 of 63 +fullPartialFraction( _ + % :: Fraction UnivariatePolynomial(x, Fraction Integer)) +--R +--R +--R 2 2 3 +--R (54) - ----- + -------- + ----- +--R x + 1 2 x + 2 +--R (x + 1) +--RType: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer))) +--E 55 + +\end{chunk} +Noncommutative algebra: note that $(A B C)^{(-1)} = C^{(-1)} B^{(-1)} A^{(-1)}$ + +$=> A B C A C B - C^{(-1)} B^{(-1)} C B$ +\begin{chunk}{*} +--S 56 of 63 +A : SquareMatrix(2, Integer) +--R +--R Type: Void +--E 56 + +--S 57 of 63 +B : SquareMatrix(2, Integer) +--R +--R Type: Void +--E 57 + +--S 58 of 63 +C : SquareMatrix(2, Integer) +--R +--R Type: Void +--E 58 + +--S 59 of 63 +(A*B*C - (A*B*C)**(-1)) * A*C*B +--R +--R +--R A is declared as being in SquareMatrix(2,Integer) but has not been +--R given a value. +--E 59 + +\end{chunk} +Jacobi's identity: $[A, B, C] + [B, C, A] + [C, A, B] = 0$ where +$[A, B, C] = [A, [B, C]]$ and $[A, B] = A B - B A$ +is the commutator of $A$ and $B$ +\begin{chunk}{*} +--S 60 of 63 +comm2(A, B) == A * B - B * A +--R +--R Type: Void +--E 60 + +--S 61 of 63 +comm3(A, B, C) == comm2(A, comm2(B, C)) +--R +--R Type: Void +--E 61 + +--S 62 of 63 +comm2(A, B) +--R +--R +--R A is declared as being in SquareMatrix(2,Integer) but has not been +--R given a value. +--E 62 + +--S 63 of 63 +comm3(A, B, C) + comm3(B, C, A) + comm3(C, A, B) +--R +--R +--R A is declared as being in SquareMatrix(2,Integer) but has not been +--R given a value. +--E 63 + +)spool + + +)lisp (bye) +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/westeralgebra.input.pdf b/src/axiom-website/CATS/westeralgebra.input.pdf new file mode 100644 index 0000000..a18e6ac Binary files /dev/null and b/src/axiom-website/CATS/westeralgebra.input.pdf differ diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index e82eb3d..65c7105 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4818,6 +4818,8 @@ books/axiom.sty make \sig write lisp signatures
books/bookheader update the credits list in the books
20141215.04.tpd.patch buglist: bug 7273: wester algebra radicalSolve bug
+20141215.05.tpd.patch +src/axiom-website/CATS/westeralgebra.input add CATS test suite diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index bf0bc4c..b7dfcbe 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -359,7 +359,7 @@ REGRESSTESTS= ackermann.regress \ triglim.regress tsetcatvermeer.regress tutchap1.regress \ typetower.regress void.regress uniseg.regress \ unittest1.regress unittest2.regress unittest3.regress unittest4.regress \ - unit-macro.regress wangeez.regress \ + unit-macro.regress wangeez.regress westeralgebra.regress \ zimmbron.regress zimmer.regress \end{chunk} @@ -375,7 +375,8 @@ CATSTESTS= \ schaum21.regress schaum22.regress schaum23.regress schaum24.regress \ schaum25.regress schaum26.regress schaum27.regress schaum28.regress \ schaum29.regress schaum30.regress schaum31.regress schaum32.regress \ - schaum33.regress schaum34.regress + schaum33.regress schaum34.regress \ + westeralgebra.regress \end{chunk} These long-running tests have been split into a different group @@ -670,7 +671,7 @@ FILES= ${OUT}/ackermann.input \ ${OUT}/intg0.input ${OUT}/intheory.input ${OUT}/int.input \ ${OUT}/intlf.input ${OUT}/intmix.input ${OUT}/intrf.input \ ${OUT}/ipftest.input ${OUT}/is.input ${OUT}/isprime.input \ - ${OUT}/kamke0.input ${OUT}/kamke1.input \ + ${OUT}/kamke0.input ${OUT}/kamke1.input \ ${OUT}/kamke2.input ${OUT}/kamke3.input ${OUT}/kamke4.input \ ${OUT}/kamke5.input ${OUT}/kamke6.input ${OUT}/kamke7.input \ ${OUT}/kernel.input ${OUT}/knot.input \ @@ -828,8 +829,8 @@ FILES= ${OUT}/ackermann.input \ ${OUT}/unittest2.input ${OUT}/unittest3.input ${OUT}/unittest4.input \ ${OUT}/unit-macro.input \ ${OUT}/vector.input ${OUT}/vectors.input ${OUT}/viewdef.input \ - ${OUT}/void.input ${OUT}/wiggle.input \ - ${OUT}/wutset.input \ + ${OUT}/void.input \ + ${OUT}/wiggle.input ${OUT}/wutset.input \ ${OUT}/xpoly.input ${OUT}/xpr.input ${OUT}/wangeez.input \ ${OUT}/zimmbron.input \ ${OUT}/zdsolve.input ${OUT}/zimmer.input ${OUT}/zlindep.input @@ -1275,7 +1276,8 @@ DOCFILES= \ ${DOC}/up.input.dvi \ ${DOC}/vector.input.dvi ${DOC}/vectors.input.dvi \ ${DOC}/viewdef.input.dvi ${DOC}/void.input.dvi \ - ${DOC}/wester.input.dvi ${DOC}/wiggle.input.dvi \ + ${DOC}/wester.input.dvi ${DOC}/westeralgebra.input.dvi \ + ${DOC}/wiggle.input.dvi \ ${DOC}/wutset.input.dvi \ ${DOC}/xpoly.input.dvi ${DOC}/xpr.input.dvi \ ${DOC}/wangeez.input.dvi ${DOC}/zimmbron.input.dvi \ diff --git a/src/input/westeralgebra.input.pamphlet b/src/input/westeralgebra.input.pamphlet new file mode 100644 index 0000000..e54ebf7 --- /dev/null +++ b/src/input/westeralgebra.input.pamphlet @@ -0,0 +1,1986 @@ +\documentclass{article} +\usepackage{axiom} +\setlength{\textwidth}{400pt} +\begin{document} +\title{\$SPAD/src/input westeralgebra.input} +\author{Michael Wester} +\maketitle +\begin{abstract} +These problems come from the web page +\begin{verbatim} +http://math.unm.edu/~wester/cas_review.html +\end{verbatim} +\end{abstract} +\eject +\tableofcontents +\eject +\begin{chunk}{*} +)set break resume +)set messages autoload off +)set streams calculate 7 +)sys rm -f westeralgebra.output +)spool westeralgebra.output +)clear all + +\end{chunk} +\section{Algebra} + +One would think that the simplification $2\ 2^n => 2^{(n + 1)}$ would happen +automatically or at least easily ... +\begin{chunk}{*} +--S 1 of 63 +2*2**n +--R +--R +--R n +--R (1) 2 2 +--R Type: Expression(Integer) +--E 1 + +\end{chunk} +And how about $4\ 2^n => 2^{(n + 2)}$? [Richard Fateman] +\begin{chunk}{*} +--S 2 of 63 +4*2**n +--R +--R +--R n +--R (2) 4 2 +--R Type: Expression(Integer) +--E 2 + +\end{chunk} +$(-1)^{(n(n + 1))} => 1$ for integer $n$ +\begin{chunk}{*} +--S 3 of 63 +(-1)**(n*(n + 1)) +--R +--R +--R 2 +--R n + n +--R (3) (- 1) +--R Type: Expression(Integer) +--E 3 + +\end{chunk} +Also easy $=> 2 (3 x - 5)$ +\begin{chunk}{*} +--S 4 of 63 +factor(6*x - 10) +--R +--R +--R (4) 2(3x - 5) +--R Type: Factored(Polynomial(Integer)) +--E 4 + +\end{chunk} +Univariate gcd: $gcd(p1, p2) => 1$, $gcd(p1 q, p2 q) => q$ [Richard Liska] +\begin{chunk}{*} +--S 5 of 63 +p1:= 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81 +--R +--R +--R 60 47 34 8 5 +--R (5) - 16x - 21x + 64x - 126x - 46x - 81 +--R Type: Polynomial(Integer) +--E 5 + +--S 6 of 63 +p2:= 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81 +--R +--R +--R 60 52 39 25 23 10 +--R (6) 72x - 83x - 22x - 25x - 19x + 54x + 81 +--R Type: Polynomial(Integer) +--E 6 + +--S 7 of 63 +q:= 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86 +--R +--R +--R 19 16 7 3 +--R (7) 34x - 25x + 70x + 20x - 91x - 86 +--R Type: Polynomial(Integer) +--E 7 + +--S 8 of 63 +gcd(p1, p2) +--R +--R +--R (8) 1 +--R Type: Polynomial(Integer) +--E 8 + +--S 9 of 63 +gcd(expand(p1*q), expand(p2*q)) - q +--R +--R +--R (9) 0 +--R Type: Polynomial(Integer) +--E 9 + +\end{chunk} +$resultant(p1 q, p2 q) => 0$ +\begin{chunk}{*} +--S 10 of 63 +resultant(expand(p1*q), expand(p2*q), x) +--R +--R +--R (10) 0 +--R Type: Polynomial(Integer) +--E 10 + +\end{chunk} +How about factorization? $=> p1 * p2$ +\begin{chunk}{*} +--S 11 of 63 +factor(expand(p1 * p2)) +--R +--R +--R (11) +--R - +--R 60 47 34 8 5 +--R (16x + 21x - 64x + 126x + 46x + 81) +--R * +--R 60 52 39 25 23 10 +--R (72x - 83x - 22x - 25x - 19x + 54x + 81) +--R Type: Factored(Polynomial(Integer)) +--E 11 + +)clear properties p1 p2 q + +\end{chunk} +Multivariate gcd: $gcd(p1, p2) => 1, gcd(p1 q, p2 q) => q$ +\begin{chunk}{*} +--S 12 of 63 +p1:= 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5 +--R +--R +--R 19 17 5 8 15 9 2 22 +--R (12) (24x y - 47x y )z + 6x y z - 3x + 5 +--R Type: Polynomial(Integer) +--E 12 + +--S 13 of 63 +p2:= 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z +--R +--R +--R 5 8 13 7 7 7 9 16 4 14 +--R (13) 34x y z + 20x y z + 12x y z + 80y z +--R Type: Polynomial(Integer) +--E 13 + +--S 14 of 63 +q:= 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8 +--R +--R +--R 12 7 13 2 8 10 17 5 8 +--R (14) 11x y z - 23x y z + 47x y z +--R Type: Polynomial(Integer) +--E 14 + +--S 15 of 63 +gcd(p1, p2) +--R +--R +--R (15) 1 +--R Type: Polynomial(Integer) +--E 15 + +--S 16 of 63 +gcd(expand(p1*q), expand(p2*q)) - q +--R +--R +--R (16) 0 +--R Type: Polynomial(Integer) +--E 16 + +\end{chunk} +How about factorization? $=> p1 * p2$ +\begin{chunk}{*} +--S 17 of 63 +factor(expand(p1 * p2)) +--R +--R +--R (17) +--R 7 19 17 5 8 15 9 2 22 +--R 2y z((24x y - 47x y )z + 6x y z - 3x + 5) +--R * +--R 5 12 7 6 9 9 3 7 +--R (17x y z + 10x z + 6x y z + 40y ) +--R Type: Factored(Polynomial(Integer)) +--E 17 + +)clear properties p1 p2 q + +\end{chunk} +$=> x^n {\textrm\ for\ } n > 0$ [Chris Hurlburt] +\begin{chunk}{*} +--S 18 of 63 +gcd(2*x**(n + 4) - x**(n + 2), 4*x**(n + 1) + 3*x**n) +--R +--R +--R (18) 1 +--R Type: Expression(Integer) +--E 18 + +\end{chunk} + +Resultants. If the resultant of two polynomials is zero, this implies they +have a common factor. See Keith O. Geddes, Stephen R. Czapor and George +Labahn, ``Algorithms for Computer Algebra'', Kluwer Academic Publishers, 1992, +p. 286 $=> 0$ +\begin{chunk}{*} +--S 19 of 63 +resultant(3*x**4 + 3*x**3 + x**2 - x - 2, x**3 - 3*x**2 + x + 5, x) +--R +--R +--R (19) 0 +--R Type: Polynomial(Integer) +--E 19 + +\end{chunk} +Numbers are nice, but symbols allow for variability---try some high school +algebra: rational simplification $=> (x - 2)/(x + 2)$ +\begin{chunk}{*} +--S 20 of 63 +(x**2 - 4)/(x**2 + 4*x + 4) +--R +--R +--R x - 2 +--R (20) ----- +--R x + 2 +--R Type: Fraction(Polynomial(Integer)) +--E 20 + +\end{chunk} +This example requires more sophistication $=> e^{(x/2)} - 1$ +\begin{chunk}{*} +--S 21 of 63 +[(%e**x - 1)/(%e**(x/2) + 1), (exp(x) - 1)/(exp(x/2) + 1)] +--R +--R +--R x x +--R %e - 1 %e - 1 +--R (21) [-------,-------] +--R x x +--R - - +--R 2 2 +--R %e + 1 %e + 1 +--R Type: List(Expression(Integer)) +--E 21 + +--S 22 of 63 +map(normalize, %) +--R +--R +--R x x +--R - - +--R 2 2 +--R (22) [%e - 1,%e - 1] +--R Type: List(Expression(Integer)) +--E 22 + +\end{chunk} +Expand and factor polynomials +\begin{chunk}{*} +--S 23 of 63 +(x + 1)**20 +--R +--R +--R (23) +--R 20 19 18 17 16 15 14 13 +--R x + 20x + 190x + 1140x + 4845x + 15504x + 38760x + 77520x +--R + +--R 12 11 10 9 8 7 6 +--R 125970x + 167960x + 184756x + 167960x + 125970x + 77520x + 38760x +--R + +--R 5 4 3 2 +--R 15504x + 4845x + 1140x + 190x + 20x + 1 +--R Type: Polynomial(Integer) +--E 23 + +--S 24 of 63 +D(%, x) +--R +--R +--R (24) +--R 19 18 17 16 15 14 13 +--R 20x + 380x + 3420x + 19380x + 77520x + 232560x + 542640x +--R + +--R 12 11 10 9 8 7 +--R 1007760x + 1511640x + 1847560x + 1847560x + 1511640x + 1007760x +--R + +--R 6 5 4 3 2 +--R 542640x + 232560x + 77520x + 19380x + 3420x + 380x + 20 +--R Type: Polynomial(Integer) +--E 24 + +--S 25 of 63 +factor(%) +--R +--R +--R 19 +--R (25) 20(x + 1) +--R Type: Factored(Polynomial(Integer)) +--E 25 + +\end{chunk} +Completely factor this polynomial, then try to multiply it back together! +\begin{chunk}{*} +--S 26 of 63 +radicalSolve(x**3 + x**2 - 7 = 0, x) +--R +--R +--R (26) +--R [ +--R x = +--R +------------------+2 +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (- 9\|- 3 + 9) |------------------ +--R 3| +-+ +--R \| 54\|3 +--R + +--R +------------------+ +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (- 3\|- 3 - 3) |------------------ - 2 +--R 3| +-+ +--R \| 54\|3 +--R / +--R +------------------+ +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (9\|- 3 + 9) |------------------ +--R 3| +-+ +--R \| 54\|3 +--R , +--R +--R x = +--R +------------------+2 +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (- 9\|- 3 - 9) |------------------ +--R 3| +-+ +--R \| 54\|3 +--R + +--R +------------------+ +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (- 3\|- 3 + 3) |------------------ + 2 +--R 3| +-+ +--R \| 54\|3 +--R / +--R +------------------+ +--R | +----+ +-+ +--R +---+ |9\|1295 + 187\|3 +--R (9\|- 3 - 9) |------------------ +--R 3| +-+ +--R \| 54\|3 +--R , +--R +------------------+2 +------------------+ +--R | +----+ +-+ | +----+ +-+ +--R |9\|1295 + 187\|3 |9\|1295 + 187\|3 +--R 9 |------------------ - 3 |------------------ + 1 +--R 3| +-+ 3| +-+ +--R \| 54\|3 \| 54\|3 +--R x= ----------------------------------------------------] +--R +------------------+ +--R | +----+ +-+ +--R |9\|1295 + 187\|3 +--R 9 |------------------ +--R 3| +-+ +--R \| 54\|3 +--R Type: List(Equation(Expression(Integer))) +--E 26 + +--S 27 of 63 +reduce(*, map(e +-> lhs(e) - rhs(e), %)) +--R +--R +--R 3 2 +-+ +----+ 3 2 +--R (9x + 9x - 63)\|3 \|1295 + 561x + 561x - 3927 +--R (27) -------------------------------------------------- +--R +-+ +----+ +-+2 +--R 9\|3 \|1295 + 187\|3 +--R Type: Expression(Integer) +--E 27 + +--S 28 of 63 +x**100 - 1 +--R +--R +--R 100 +--R (28) x - 1 +--R Type: Polynomial(Integer) +--E 28 + +--S 29 of 63 +factor(%) +--R +--R +--R (29) +--R 2 4 3 2 4 3 2 +--R (x - 1)(x + 1)(x + 1)(x - x + x - x + 1)(x + x + x + x + 1) +--R * +--R 8 6 4 2 20 15 10 5 20 15 10 5 +--R (x - x + x - x + 1)(x - x + x - x + 1)(x + x + x + x + 1) +--R * +--R 40 30 20 10 +--R (x - x + x - x + 1) +--R Type: Factored(Polynomial(Integer)) +--E 29 + +\end{chunk} +Factorization over the complex rationals + +$=> (2 x + 3 i) (2 x - 3 i) (x + 1 + 4 i) (x + 1 - 4 i)$ +\begin{chunk}{*} +--S 30 of 63 +factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153, [rootOf(i**2 + 1)]) +--R +--R +--R 3i 3i +--R (30) 4(x - 4i + 1)(x - --)(x + --)(x + 4i + 1) +--R 2 2 +--R Type: Factored(Polynomial(AlgebraicNumber)) +--E 30 + +\end{chunk} +Algebraic extensions +\begin{chunk}{*} +--S 31 of 63 +sqrt2:= rootOf(sqrt2**2 - 2) +--R +--R +--R (31) sqrt2 +--R Type: AlgebraicNumber +--E 31 + +\end{chunk} +$=> sqrt2 + 1$ +\begin{chunk}{*} +--S 32 of 63 +1/(sqrt2 - 1) +--R +--R +--R (32) sqrt2 + 1 +--R Type: AlgebraicNumber +--E 32 + +\end{chunk} +$=> (x^2 - 2 x - 3)/(x - sqrt2) = (x + 1) (x - 3)/(x - sqrt2)$ +[Richard Liska] +\begin{chunk}{*} +--S 33 of 63 +(x**3 + (sqrt2 - 2)*x**2 - (2*sqrt2 + 3)*x - 3*sqrt2)/(x**2 - 2) +--R +--R +--R 2 +--R x - 2x - 3 +--R (33) ----------- +--R x - sqrt2 +--R Type: Fraction(Polynomial(AlgebraicNumber)) +--E 33 + +--S 34 of 63 +numer(%)/ratDenom(denom(%)) +--R +--R +--R 2 +--R - x + 2x + 3 +--R (34) ------------- +--R sqrt2 - x +--R Type: Expression(Integer) +--E 34 + +)clear properties sqrt2 + +\end{chunk} +Multiple algebraic extensions +\begin{chunk}{*} +--S 35 of 63 +sqrt3:= rootOf(sqrt3**2 - 3) +--R +--R +--R (35) sqrt3 +--R Type: AlgebraicNumber +--E 35 + +--S 36 of 63 +cbrt2:= rootOf(cbrt2**3 - 2) +--R +--R +--R (36) cbrt2 +--R Type: AlgebraicNumber +--E 36 + +\end{chunk} +$=> 2 cbrt2 + 8 sqrt3 + 18 cbrt2^2 + 12 cbrt2 sqrt3 + 9$ +\begin{chunk}{*} +--S 37 of 63 +(cbrt2 + sqrt3)**4 +--R +--R +--R 2 +--R (37) (12cbrt2 + 8)sqrt3 + 18cbrt2 + 2cbrt2 + 9 +--R Type: AlgebraicNumber +--E 37 + +)clear properties sqrt3 cbrt2 + +\end{chunk} +Factor polynomials over finite fields and field extensions +\begin{chunk}{*} +--S 38 of 63 +p:= x**4 - 3*x**2 + 1 +--R +--R +--R 4 2 +--R (38) x - 3x + 1 +--R Type: Polynomial(Integer) +--E 38 + +--S 39 of 63 +factor(p) +--R +--R +--R 2 2 +--R (39) (x - x - 1)(x + x - 1) +--R Type: Factored(Polynomial(Integer)) +--E 39 + +\end{chunk} +$=> (x - 2)^2 (x + 2)^2 {\textrm\ mod\ } 5$ +\begin{chunk}{*} +--S 40 of 63 +factor(p :: Polynomial(PrimeField(5))) +--R +--R +--R 2 2 +--R (40) (x + 2) (x + 3) +--R Type: Factored(Polynomial(PrimeField(5))) +--E 40 + +--S 41 of 63 +expand(%) +--R +--R +--R 4 2 +--R (41) x + 2x + 1 +--R Type: Polynomial(PrimeField(5)) +--E 41 + +\end{chunk} +$=> (x^2 + x + 1) (x^9 - x^8 + x^6 - x^5 + x^3 - x^2 + 1){\textrm\ mod\ } 65537$ +[Paul Zimmermann] +\begin{chunk}{*} +--S 42 of 63 +factor(x**11 + x + 1 :: Polynomial(PrimeField(65537))) +--R +--R +--R 2 9 8 6 5 3 2 +--R (42) (x + x + 1)(x + 65536x + x + 65536x + x + 65536x + 1) +--R Type: Factored(Polynomial(PrimeField(65537))) +--E 42 + +\end{chunk} +$=> (x - phi) (x + phi) (x - phi + 1) (x + phi - 1)$ + +where $phi^2 - phi - 1 = 0$ or $phi = (1 \pm sqrt(5))/2$ +\begin{chunk}{*} +--S 43 of 63 +phi:= rootOf(phi**2 - phi - 1) +--R +--R +--R (43) phi +--R Type: AlgebraicNumber +--E 43 + +--S 44 of 63 +factor(p, [phi]) +--R +--R +--R (44) (x - phi)(x - phi + 1)(x + phi - 1)(x + phi) +--R Type: Factored(Polynomial(AlgebraicNumber)) +--E 44 + +)clear properties phi p + +--S 45 of 63 +expand((x - 2*y**2 + 3*z**3)**20) +--R +--R +--R (45) +--R 60 2 57 +--R 3486784401z + (- 46490458680y + 23245229340x)z +--R + +--R 4 2 2 54 +--R (294439571640y - 294439571640x y + 73609892910x )z +--R + +--R 6 4 2 2 +--R - 1177758286560y + 1766637429840x y - 883318714920x y +--R + +--R 3 +--R 147219785820x +--R * +--R 51 +--R z +--R + +--R 8 6 2 4 +--R 3336981811920y - 6673963623840x y + 5005472717880x y +--R + +--R 3 2 4 +--R - 1668490905960x y + 208561363245x +--R * +--R 48 +--R z +--R + +--R 10 8 2 6 +--R - 7118894532096y + 17797236330240x y - 17797236330240x y +--R + +--R 3 4 4 2 5 +--R 8898618165120x y - 2224654541280x y + 222465454128x +--R * +--R 45 +--R z +--R + +--R 12 10 2 8 +--R 11864824220160y - 35594472660480x y + 44493090825600x y +--R + +--R 3 6 4 4 5 2 +--R - 29662060550400x y + 11123272706400x y - 2224654541280x y +--R + +--R 6 +--R 185387878440x +--R * +--R 42 +--R z +--R + +--R 14 12 2 10 +--R - 15819765626880y + 55369179694080x y - 83053769541120x y +--R + +--R 3 8 4 6 5 4 +--R 69211474617600x y - 34605737308800x y + 10381721192640x y +--R + +--R 6 2 7 +--R - 1730286865440x y + 123591918960x +--R * +--R 39 +--R z +--R + +--R 16 14 2 12 +--R 17138079429120y - 68552317716480x y + 119966556003840x y +--R + +--R 3 10 4 8 5 6 +--R - 119966556003840x y + 74979097502400x y - 29991639000960x y +--R + +--R 6 4 7 2 8 +--R 7497909750240x y - 1071129964320x y + 66945622770x +--R * +--R 36 +--R z +--R + +--R 18 16 2 14 +--R - 15233848381440y + 68552317716480x y - 137104635432960x y +--R + +--R 3 12 4 10 5 8 +--R 159955408005120x y - 119966556003840x y + 59983278001920x y +--R + +--R 6 6 7 4 8 2 +--R - 19994426000640x y + 4284519857280x y - 535564982160x y +--R + +--R 9 +--R 29753610120x +--R * +--R 33 +--R z +--R + +--R 20 18 2 16 +--R 11171488813056y - 55857444065280x y + 125679249146880x y +--R + +--R 3 14 4 12 5 10 +--R - 167572332195840x y + 146625790671360x y - 87975474402816x y +--R + +--R 6 8 7 6 8 4 +--R 36656447667840x y - 10473270762240x y + 1963738267920x y +--R + +--R 9 2 10 +--R - 218193140880x y + 10909657044x +--R * +--R 30 +--R z +--R + +--R 22 20 2 18 +--R - 6770599280640y + 37238296043520x y - 93095740108800x y +--R + +--R 3 16 4 14 5 12 +--R 139643610163200x y - 139643610163200x y + 97750527114240x y +--R + +--R 6 10 7 8 8 6 +--R - 48875263557120x y + 17455451270400x y - 4363862817600x y +--R + +--R 9 4 10 2 11 +--R 727310469600x y - 72731046960x y + 3305956680x +--R * +--R 27 +--R z +--R + +--R 24 22 2 20 +--R 3385299640320y - 20311797841920x y + 55857444065280x y +--R + +--R 3 18 4 16 5 14 +--R - 93095740108800x y + 104732707622400x y - 83786166097920x y +--R + +--R 6 12 7 10 8 8 +--R 48875263557120x y - 20946541524480x y + 6545794226400x y +--R + +--R 9 6 10 4 11 2 12 +--R - 1454620939200x y + 218193140880x y - 19835740080x y + 826489170x +--R * +--R 24 +--R z +--R + +--R 26 24 2 22 +--R - 1388840878080y + 9027465707520x y - 27082397122560x y +--R + +--R 3 20 4 18 5 16 +--R 49651061391360x y - 62063826739200x y + 55857444065280x y +--R + +--R 6 14 7 12 8 10 +--R - 37238296043520x y + 18619148021760x y - 6982180508160x y +--R + +--R 9 8 10 6 11 4 +--R 1939494585600x y - 387898917120x y + 52895306880x y +--R + +--R 12 2 13 +--R - 4407942240x y + 169536240x +--R * +--R 21 +--R z +--R + +--R 28 26 2 24 +--R 462946959360y - 3240628715520x y + 10532043325440x y +--R + +--R 3 22 4 20 5 18 +--R - 21064086650880x y + 28963119144960x y - 28963119144960x y +--R + +--R 6 16 7 14 8 12 +--R 21722339358720x y - 12412765347840x y + 5430584839680x y +--R + +--R 9 10 10 8 11 6 +--R - 1810194946560x y + 452548736640x y - 82281588480x y +--R + +--R 12 4 13 2 14 +--R 10285198560x y - 791169120x y + 28256040x +--R * +--R 18 +--R z +--R + +--R 30 28 2 26 +--R - 123452522496y + 925893918720x y - 3240628715520x y +--R + +--R 3 24 4 22 5 20 +--R 7021362216960x y - 10532043325440x y + 11585247657984x y +--R + +--R 6 18 7 16 8 14 +--R - 9654373048320x y + 6206382673920x y - 3103191336960x y +--R + +--R 9 12 10 10 11 8 +--R 1206796631040x y - 362038989312x y + 82281588480x y +--R + +--R 12 6 13 4 14 2 15 +--R - 13713598080x y + 1582338240x y - 113024160x y + 3767472x +--R * +--R 15 +--R z +--R + +--R 32 30 2 28 +--R 25719275520y - 205754204160x y + 771578265600x y +--R + +--R 3 26 4 24 5 22 +--R - 1800349286400x y + 2925567590400x y - 3510681108480x y +--R + +--R 6 20 7 18 8 16 +--R 3218124349440x y - 2298660249600x y + 1292996390400x y +--R + +--R 9 14 10 12 11 10 +--R - 574665062400x y + 201132771840x y - 54854392320x y +--R + +--R 12 8 13 6 14 4 15 2 +--R 11427998400x y - 1758153600x y + 188373600x y - 12558240x y +--R + +--R 16 +--R 392445x +--R * +--R 12 +--R z +--R + +--R 34 32 2 30 +--R - 4034396160y + 34292367360x y - 137169469440x y +--R + +--R 3 28 4 26 5 24 +--R 342923673600x y - 600116428800x y + 780151357440x y +--R + +--R 6 22 7 20 8 18 +--R - 780151357440x y + 612976066560x y - 383110041600x y +--R + +--R 9 16 10 14 11 12 +--R 191555020800x y - 76622008320x y + 24379729920x y +--R + +--R 12 10 13 8 14 6 15 4 +--R - 6094932480x y + 1172102400x y - 167443200x y + 16744320x y +--R + +--R 16 2 17 +--R - 1046520x y + 30780x +--R * +--R 9 +--R z +--R + +--R 36 34 2 32 3 30 +--R 448266240y - 4034396160x y + 17146183680x y - 45723156480x y +--R + +--R 4 28 5 26 6 24 +--R 85730918400x y - 120023285760x y + 130025226240x y +--R + +--R 7 22 8 20 9 18 +--R - 111450193920x y + 76622008320x y - 42567782400x y +--R + +--R 10 16 11 14 12 12 +--R 19155502080x y - 6965637120x y + 2031644160x y +--R + +--R 13 10 14 8 15 6 16 4 +--R - 468840960x y + 83721600x y - 11162880x y + 1046520x y +--R + +--R 17 2 18 +--R - 61560x y + 1710x +--R * +--R 6 +--R z +--R + +--R 38 36 2 34 3 32 +--R - 31457280y + 298844160x y - 1344798720x y + 3810263040x y +--R + +--R 4 30 5 28 6 26 +--R - 7620526080x y + 11430789120x y - 13335920640x y +--R + +--R 7 24 8 22 9 20 10 18 +--R 12383354880x y - 9287516160x y + 5675704320x y - 2837852160x y +--R + +--R 11 16 12 14 13 12 14 10 +--R 1160939520x y - 386979840x y + 104186880x y - 22325760x y +--R + +--R 15 8 16 6 17 4 18 2 19 +--R 3720960x y - 465120x y + 41040x y - 2280x y + 60x +--R * +--R 3 +--R z +--R + +--R 40 38 2 36 3 34 +--R 1048576y - 10485760x y + 49807360x y - 149422080x y +--R + +--R 4 32 5 30 6 28 7 26 +--R 317521920x y - 508035072x y + 635043840x y - 635043840x y +--R + +--R 8 24 9 22 10 20 11 18 +--R 515973120x y - 343982080x y + 189190144x y - 85995520x y +--R + +--R 12 16 13 14 14 12 15 10 16 8 +--R 32248320x y - 9922560x y + 2480640x y - 496128x y + 77520x y +--R + +--R 17 6 18 4 19 2 20 +--R - 9120x y + 760x y - 40x y + x +--R Type: Polynomial(Integer) +--E 45 + +--S 46 of 63 +factor(%) +--R +--R +--R 3 2 20 +--R (46) (3z - 2y + x) +--R Type: Factored(Polynomial(Integer)) +--E 46 + +--S 47 of 63 +expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20) +--R +--R +--R (47) +--R 60 2 57 +--R 3486784401tan(z) + (23245229340sin(x) - 46490458680cos(y) )tan(z) +--R + +--R 2 2 4 +--R (73609892910sin(x) - 294439571640cos(y) sin(x) + 294439571640cos(y) ) +--R * +--R 54 +--R tan(z) +--R + +--R 3 2 2 +--R 147219785820sin(x) - 883318714920cos(y) sin(x) +--R + +--R 4 6 +--R 1766637429840cos(y) sin(x) - 1177758286560cos(y) +--R * +--R 51 +--R tan(z) +--R + +--R 4 2 3 +--R 208561363245sin(x) - 1668490905960cos(y) sin(x) +--R + +--R 4 2 6 +--R 5005472717880cos(y) sin(x) - 6673963623840cos(y) sin(x) +--R + +--R 8 +--R 3336981811920cos(y) +--R * +--R 48 +--R tan(z) +--R + +--R 5 2 4 +--R 222465454128sin(x) - 2224654541280cos(y) sin(x) +--R + +--R 4 3 6 2 +--R 8898618165120cos(y) sin(x) - 17797236330240cos(y) sin(x) +--R + +--R 8 10 +--R 17797236330240cos(y) sin(x) - 7118894532096cos(y) +--R * +--R 45 +--R tan(z) +--R + +--R 6 2 5 +--R 185387878440sin(x) - 2224654541280cos(y) sin(x) +--R + +--R 4 4 6 3 +--R 11123272706400cos(y) sin(x) - 29662060550400cos(y) sin(x) +--R + +--R 8 2 10 +--R 44493090825600cos(y) sin(x) - 35594472660480cos(y) sin(x) +--R + +--R 12 +--R 11864824220160cos(y) +--R * +--R 42 +--R tan(z) +--R + +--R 7 2 6 +--R 123591918960sin(x) - 1730286865440cos(y) sin(x) +--R + +--R 4 5 6 4 +--R 10381721192640cos(y) sin(x) - 34605737308800cos(y) sin(x) +--R + +--R 8 3 10 2 +--R 69211474617600cos(y) sin(x) - 83053769541120cos(y) sin(x) +--R + +--R 12 14 +--R 55369179694080cos(y) sin(x) - 15819765626880cos(y) +--R * +--R 39 +--R tan(z) +--R + +--R 8 2 7 +--R 66945622770sin(x) - 1071129964320cos(y) sin(x) +--R + +--R 4 6 6 5 +--R 7497909750240cos(y) sin(x) - 29991639000960cos(y) sin(x) +--R + +--R 8 4 10 3 +--R 74979097502400cos(y) sin(x) - 119966556003840cos(y) sin(x) +--R + +--R 12 2 14 +--R 119966556003840cos(y) sin(x) - 68552317716480cos(y) sin(x) +--R + +--R 16 +--R 17138079429120cos(y) +--R * +--R 36 +--R tan(z) +--R + +--R 9 2 8 +--R 29753610120sin(x) - 535564982160cos(y) sin(x) +--R + +--R 4 7 6 6 +--R 4284519857280cos(y) sin(x) - 19994426000640cos(y) sin(x) +--R + +--R 8 5 10 4 +--R 59983278001920cos(y) sin(x) - 119966556003840cos(y) sin(x) +--R + +--R 12 3 14 2 +--R 159955408005120cos(y) sin(x) - 137104635432960cos(y) sin(x) +--R + +--R 16 18 +--R 68552317716480cos(y) sin(x) - 15233848381440cos(y) +--R * +--R 33 +--R tan(z) +--R + +--R 10 2 9 +--R 10909657044sin(x) - 218193140880cos(y) sin(x) +--R + +--R 4 8 6 7 +--R 1963738267920cos(y) sin(x) - 10473270762240cos(y) sin(x) +--R + +--R 8 6 10 5 +--R 36656447667840cos(y) sin(x) - 87975474402816cos(y) sin(x) +--R + +--R 12 4 14 3 +--R 146625790671360cos(y) sin(x) - 167572332195840cos(y) sin(x) +--R + +--R 16 2 18 +--R 125679249146880cos(y) sin(x) - 55857444065280cos(y) sin(x) +--R + +--R 20 +--R 11171488813056cos(y) +--R * +--R 30 +--R tan(z) +--R + +--R 11 2 10 +--R 3305956680sin(x) - 72731046960cos(y) sin(x) +--R + +--R 4 9 6 8 +--R 727310469600cos(y) sin(x) - 4363862817600cos(y) sin(x) +--R + +--R 8 7 10 6 +--R 17455451270400cos(y) sin(x) - 48875263557120cos(y) sin(x) +--R + +--R 12 5 14 4 +--R 97750527114240cos(y) sin(x) - 139643610163200cos(y) sin(x) +--R + +--R 16 3 18 2 +--R 139643610163200cos(y) sin(x) - 93095740108800cos(y) sin(x) +--R + +--R 20 22 +--R 37238296043520cos(y) sin(x) - 6770599280640cos(y) +--R * +--R 27 +--R tan(z) +--R + +--R 12 2 11 +--R 826489170sin(x) - 19835740080cos(y) sin(x) +--R + +--R 4 10 6 9 +--R 218193140880cos(y) sin(x) - 1454620939200cos(y) sin(x) +--R + +--R 8 8 10 7 +--R 6545794226400cos(y) sin(x) - 20946541524480cos(y) sin(x) +--R + +--R 12 6 14 5 +--R 48875263557120cos(y) sin(x) - 83786166097920cos(y) sin(x) +--R + +--R 16 4 18 3 +--R 104732707622400cos(y) sin(x) - 93095740108800cos(y) sin(x) +--R + +--R 20 2 22 +--R 55857444065280cos(y) sin(x) - 20311797841920cos(y) sin(x) +--R + +--R 24 +--R 3385299640320cos(y) +--R * +--R 24 +--R tan(z) +--R + +--R 13 2 12 +--R 169536240sin(x) - 4407942240cos(y) sin(x) +--R + +--R 4 11 6 10 +--R 52895306880cos(y) sin(x) - 387898917120cos(y) sin(x) +--R + +--R 8 9 10 8 +--R 1939494585600cos(y) sin(x) - 6982180508160cos(y) sin(x) +--R + +--R 12 7 14 6 +--R 18619148021760cos(y) sin(x) - 37238296043520cos(y) sin(x) +--R + +--R 16 5 18 4 +--R 55857444065280cos(y) sin(x) - 62063826739200cos(y) sin(x) +--R + +--R 20 3 22 2 +--R 49651061391360cos(y) sin(x) - 27082397122560cos(y) sin(x) +--R + +--R 24 26 +--R 9027465707520cos(y) sin(x) - 1388840878080cos(y) +--R * +--R 21 +--R tan(z) +--R + +--R 14 2 13 +--R 28256040sin(x) - 791169120cos(y) sin(x) +--R + +--R 4 12 6 11 +--R 10285198560cos(y) sin(x) - 82281588480cos(y) sin(x) +--R + +--R 8 10 10 9 +--R 452548736640cos(y) sin(x) - 1810194946560cos(y) sin(x) +--R + +--R 12 8 14 7 +--R 5430584839680cos(y) sin(x) - 12412765347840cos(y) sin(x) +--R + +--R 16 6 18 5 +--R 21722339358720cos(y) sin(x) - 28963119144960cos(y) sin(x) +--R + +--R 20 4 22 3 +--R 28963119144960cos(y) sin(x) - 21064086650880cos(y) sin(x) +--R + +--R 24 2 26 +--R 10532043325440cos(y) sin(x) - 3240628715520cos(y) sin(x) +--R + +--R 28 +--R 462946959360cos(y) +--R * +--R 18 +--R tan(z) +--R + +--R 15 2 14 4 13 +--R 3767472sin(x) - 113024160cos(y) sin(x) + 1582338240cos(y) sin(x) +--R + +--R 6 12 8 11 +--R - 13713598080cos(y) sin(x) + 82281588480cos(y) sin(x) +--R + +--R 10 10 12 9 +--R - 362038989312cos(y) sin(x) + 1206796631040cos(y) sin(x) +--R + +--R 14 8 16 7 +--R - 3103191336960cos(y) sin(x) + 6206382673920cos(y) sin(x) +--R + +--R 18 6 20 5 +--R - 9654373048320cos(y) sin(x) + 11585247657984cos(y) sin(x) +--R + +--R 22 4 24 3 +--R - 10532043325440cos(y) sin(x) + 7021362216960cos(y) sin(x) +--R + +--R 26 2 28 +--R - 3240628715520cos(y) sin(x) + 925893918720cos(y) sin(x) +--R + +--R 30 +--R - 123452522496cos(y) +--R * +--R 15 +--R tan(z) +--R + +--R 16 2 15 4 14 +--R 392445sin(x) - 12558240cos(y) sin(x) + 188373600cos(y) sin(x) +--R + +--R 6 13 8 12 +--R - 1758153600cos(y) sin(x) + 11427998400cos(y) sin(x) +--R + +--R 10 11 12 10 +--R - 54854392320cos(y) sin(x) + 201132771840cos(y) sin(x) +--R + +--R 14 9 16 8 +--R - 574665062400cos(y) sin(x) + 1292996390400cos(y) sin(x) +--R + +--R 18 7 20 6 +--R - 2298660249600cos(y) sin(x) + 3218124349440cos(y) sin(x) +--R + +--R 22 5 24 4 +--R - 3510681108480cos(y) sin(x) + 2925567590400cos(y) sin(x) +--R + +--R 26 3 28 2 +--R - 1800349286400cos(y) sin(x) + 771578265600cos(y) sin(x) +--R + +--R 30 32 +--R - 205754204160cos(y) sin(x) + 25719275520cos(y) +--R * +--R 12 +--R tan(z) +--R + +--R 17 2 16 4 15 +--R 30780sin(x) - 1046520cos(y) sin(x) + 16744320cos(y) sin(x) +--R + +--R 6 14 8 13 +--R - 167443200cos(y) sin(x) + 1172102400cos(y) sin(x) +--R + +--R 10 12 12 11 +--R - 6094932480cos(y) sin(x) + 24379729920cos(y) sin(x) +--R + +--R 14 10 16 9 +--R - 76622008320cos(y) sin(x) + 191555020800cos(y) sin(x) +--R + +--R 18 8 20 7 +--R - 383110041600cos(y) sin(x) + 612976066560cos(y) sin(x) +--R + +--R 22 6 24 5 +--R - 780151357440cos(y) sin(x) + 780151357440cos(y) sin(x) +--R + +--R 26 4 28 3 +--R - 600116428800cos(y) sin(x) + 342923673600cos(y) sin(x) +--R + +--R 30 2 32 +--R - 137169469440cos(y) sin(x) + 34292367360cos(y) sin(x) +--R + +--R 34 +--R - 4034396160cos(y) +--R * +--R 9 +--R tan(z) +--R + +--R 18 2 17 4 16 +--R 1710sin(x) - 61560cos(y) sin(x) + 1046520cos(y) sin(x) +--R + +--R 6 15 8 14 +--R - 11162880cos(y) sin(x) + 83721600cos(y) sin(x) +--R + +--R 10 13 12 12 +--R - 468840960cos(y) sin(x) + 2031644160cos(y) sin(x) +--R + +--R 14 11 16 10 +--R - 6965637120cos(y) sin(x) + 19155502080cos(y) sin(x) +--R + +--R 18 9 20 8 +--R - 42567782400cos(y) sin(x) + 76622008320cos(y) sin(x) +--R + +--R 22 7 24 6 +--R - 111450193920cos(y) sin(x) + 130025226240cos(y) sin(x) +--R + +--R 26 5 28 4 +--R - 120023285760cos(y) sin(x) + 85730918400cos(y) sin(x) +--R + +--R 30 3 32 2 +--R - 45723156480cos(y) sin(x) + 17146183680cos(y) sin(x) +--R + +--R 34 36 +--R - 4034396160cos(y) sin(x) + 448266240cos(y) +--R * +--R 6 +--R tan(z) +--R + +--R 19 2 18 4 17 +--R 60sin(x) - 2280cos(y) sin(x) + 41040cos(y) sin(x) +--R + +--R 6 16 8 15 +--R - 465120cos(y) sin(x) + 3720960cos(y) sin(x) +--R + +--R 10 14 12 13 +--R - 22325760cos(y) sin(x) + 104186880cos(y) sin(x) +--R + +--R 14 12 16 11 +--R - 386979840cos(y) sin(x) + 1160939520cos(y) sin(x) +--R + +--R 18 10 20 9 +--R - 2837852160cos(y) sin(x) + 5675704320cos(y) sin(x) +--R + +--R 22 8 24 7 +--R - 9287516160cos(y) sin(x) + 12383354880cos(y) sin(x) +--R + +--R 26 6 28 5 +--R - 13335920640cos(y) sin(x) + 11430789120cos(y) sin(x) +--R + +--R 30 4 32 3 +--R - 7620526080cos(y) sin(x) + 3810263040cos(y) sin(x) +--R + +--R 34 2 36 38 +--R - 1344798720cos(y) sin(x) + 298844160cos(y) sin(x) - 31457280cos(y) +--R * +--R 3 +--R tan(z) +--R + +--R 20 2 19 4 18 6 17 +--R sin(x) - 40cos(y) sin(x) + 760cos(y) sin(x) - 9120cos(y) sin(x) +--R + +--R 8 16 10 15 12 14 +--R 77520cos(y) sin(x) - 496128cos(y) sin(x) + 2480640cos(y) sin(x) +--R + +--R 14 13 16 12 +--R - 9922560cos(y) sin(x) + 32248320cos(y) sin(x) +--R + +--R 18 11 20 10 +--R - 85995520cos(y) sin(x) + 189190144cos(y) sin(x) +--R + +--R 22 9 24 8 +--R - 343982080cos(y) sin(x) + 515973120cos(y) sin(x) +--R + +--R 26 7 28 6 +--R - 635043840cos(y) sin(x) + 635043840cos(y) sin(x) +--R + +--R 30 5 32 4 +--R - 508035072cos(y) sin(x) + 317521920cos(y) sin(x) +--R + +--R 34 3 36 2 +--R - 149422080cos(y) sin(x) + 49807360cos(y) sin(x) +--R + +--R 38 40 +--R - 10485760cos(y) sin(x) + 1048576cos(y) +--R Type: Expression(Integer) +--E 47 + +--S 48 of 63 +factor(%) +--R +--R +--R (48) +--R 60 2 57 +--R 3486784401tan(z) + (23245229340sin(x) - 46490458680cos(y) )tan(z) +--R + +--R 2 2 4 +--R (73609892910sin(x) - 294439571640cos(y) sin(x) + 294439571640cos(y) ) +--R * +--R 54 +--R tan(z) +--R + +--R 3 2 2 +--R 147219785820sin(x) - 883318714920cos(y) sin(x) +--R + +--R 4 6 +--R 1766637429840cos(y) sin(x) - 1177758286560cos(y) +--R * +--R 51 +--R tan(z) +--R + +--R 4 2 3 +--R 208561363245sin(x) - 1668490905960cos(y) sin(x) +--R + +--R 4 2 6 +--R 5005472717880cos(y) sin(x) - 6673963623840cos(y) sin(x) +--R + +--R 8 +--R 3336981811920cos(y) +--R * +--R 48 +--R tan(z) +--R + +--R 5 2 4 +--R 222465454128sin(x) - 2224654541280cos(y) sin(x) +--R + +--R 4 3 6 2 +--R 8898618165120cos(y) sin(x) - 17797236330240cos(y) sin(x) +--R + +--R 8 10 +--R 17797236330240cos(y) sin(x) - 7118894532096cos(y) +--R * +--R 45 +--R tan(z) +--R + +--R 6 2 5 +--R 185387878440sin(x) - 2224654541280cos(y) sin(x) +--R + +--R 4 4 6 3 +--R 11123272706400cos(y) sin(x) - 29662060550400cos(y) sin(x) +--R + +--R 8 2 10 +--R 44493090825600cos(y) sin(x) - 35594472660480cos(y) sin(x) +--R + +--R 12 +--R 11864824220160cos(y) +--R * +--R 42 +--R tan(z) +--R + +--R 7 2 6 +--R 123591918960sin(x) - 1730286865440cos(y) sin(x) +--R + +--R 4 5 6 4 +--R 10381721192640cos(y) sin(x) - 34605737308800cos(y) sin(x) +--R + +--R 8 3 10 2 +--R 69211474617600cos(y) sin(x) - 83053769541120cos(y) sin(x) +--R + +--R 12 14 +--R 55369179694080cos(y) sin(x) - 15819765626880cos(y) +--R * +--R 39 +--R tan(z) +--R + +--R 8 2 7 +--R 66945622770sin(x) - 1071129964320cos(y) sin(x) +--R + +--R 4 6 6 5 +--R 7497909750240cos(y) sin(x) - 29991639000960cos(y) sin(x) +--R + +--R 8 4 10 3 +--R 74979097502400cos(y) sin(x) - 119966556003840cos(y) sin(x) +--R + +--R 12 2 14 +--R 119966556003840cos(y) sin(x) - 68552317716480cos(y) sin(x) +--R + +--R 16 +--R 17138079429120cos(y) +--R * +--R 36 +--R tan(z) +--R + +--R 9 2 8 +--R 29753610120sin(x) - 535564982160cos(y) sin(x) +--R + +--R 4 7 6 6 +--R 4284519857280cos(y) sin(x) - 19994426000640cos(y) sin(x) +--R + +--R 8 5 10 4 +--R 59983278001920cos(y) sin(x) - 119966556003840cos(y) sin(x) +--R + +--R 12 3 14 2 +--R 159955408005120cos(y) sin(x) - 137104635432960cos(y) sin(x) +--R + +--R 16 18 +--R 68552317716480cos(y) sin(x) - 15233848381440cos(y) +--R * +--R 33 +--R tan(z) +--R + +--R 10 2 9 +--R 10909657044sin(x) - 218193140880cos(y) sin(x) +--R + +--R 4 8 6 7 +--R 1963738267920cos(y) sin(x) - 10473270762240cos(y) sin(x) +--R + +--R 8 6 10 5 +--R 36656447667840cos(y) sin(x) - 87975474402816cos(y) sin(x) +--R + +--R 12 4 14 3 +--R 146625790671360cos(y) sin(x) - 167572332195840cos(y) sin(x) +--R + +--R 16 2 18 +--R 125679249146880cos(y) sin(x) - 55857444065280cos(y) sin(x) +--R + +--R 20 +--R 11171488813056cos(y) +--R * +--R 30 +--R tan(z) +--R + +--R 11 2 10 +--R 3305956680sin(x) - 72731046960cos(y) sin(x) +--R + +--R 4 9 6 8 +--R 727310469600cos(y) sin(x) - 4363862817600cos(y) sin(x) +--R + +--R 8 7 10 6 +--R 17455451270400cos(y) sin(x) - 48875263557120cos(y) sin(x) +--R + +--R 12 5 14 4 +--R 97750527114240cos(y) sin(x) - 139643610163200cos(y) sin(x) +--R + +--R 16 3 18 2 +--R 139643610163200cos(y) sin(x) - 93095740108800cos(y) sin(x) +--R + +--R 20 22 +--R 37238296043520cos(y) sin(x) - 6770599280640cos(y) +--R * +--R 27 +--R tan(z) +--R + +--R 12 2 11 +--R 826489170sin(x) - 19835740080cos(y) sin(x) +--R + +--R 4 10 6 9 +--R 218193140880cos(y) sin(x) - 1454620939200cos(y) sin(x) +--R + +--R 8 8 10 7 +--R 6545794226400cos(y) sin(x) - 20946541524480cos(y) sin(x) +--R + +--R 12 6 14 5 +--R 48875263557120cos(y) sin(x) - 83786166097920cos(y) sin(x) +--R + +--R 16 4 18 3 +--R 104732707622400cos(y) sin(x) - 93095740108800cos(y) sin(x) +--R + +--R 20 2 22 +--R 55857444065280cos(y) sin(x) - 20311797841920cos(y) sin(x) +--R + +--R 24 +--R 3385299640320cos(y) +--R * +--R 24 +--R tan(z) +--R + +--R 13 2 12 +--R 169536240sin(x) - 4407942240cos(y) sin(x) +--R + +--R 4 11 6 10 +--R 52895306880cos(y) sin(x) - 387898917120cos(y) sin(x) +--R + +--R 8 9 10 8 +--R 1939494585600cos(y) sin(x) - 6982180508160cos(y) sin(x) +--R + +--R 12 7 14 6 +--R 18619148021760cos(y) sin(x) - 37238296043520cos(y) sin(x) +--R + +--R 16 5 18 4 +--R 55857444065280cos(y) sin(x) - 62063826739200cos(y) sin(x) +--R + +--R 20 3 22 2 +--R 49651061391360cos(y) sin(x) - 27082397122560cos(y) sin(x) +--R + +--R 24 26 +--R 9027465707520cos(y) sin(x) - 1388840878080cos(y) +--R * +--R 21 +--R tan(z) +--R + +--R 14 2 13 +--R 28256040sin(x) - 791169120cos(y) sin(x) +--R + +--R 4 12 6 11 +--R 10285198560cos(y) sin(x) - 82281588480cos(y) sin(x) +--R + +--R 8 10 10 9 +--R 452548736640cos(y) sin(x) - 1810194946560cos(y) sin(x) +--R + +--R 12 8 14 7 +--R 5430584839680cos(y) sin(x) - 12412765347840cos(y) sin(x) +--R + +--R 16 6 18 5 +--R 21722339358720cos(y) sin(x) - 28963119144960cos(y) sin(x) +--R + +--R 20 4 22 3 +--R 28963119144960cos(y) sin(x) - 21064086650880cos(y) sin(x) +--R + +--R 24 2 26 +--R 10532043325440cos(y) sin(x) - 3240628715520cos(y) sin(x) +--R + +--R 28 +--R 462946959360cos(y) +--R * +--R 18 +--R tan(z) +--R + +--R 15 2 14 4 13 +--R 3767472sin(x) - 113024160cos(y) sin(x) + 1582338240cos(y) sin(x) +--R + +--R 6 12 8 11 +--R - 13713598080cos(y) sin(x) + 82281588480cos(y) sin(x) +--R + +--R 10 10 12 9 +--R - 362038989312cos(y) sin(x) + 1206796631040cos(y) sin(x) +--R + +--R 14 8 16 7 +--R - 3103191336960cos(y) sin(x) + 6206382673920cos(y) sin(x) +--R + +--R 18 6 20 5 +--R - 9654373048320cos(y) sin(x) + 11585247657984cos(y) sin(x) +--R + +--R 22 4 24 3 +--R - 10532043325440cos(y) sin(x) + 7021362216960cos(y) sin(x) +--R + +--R 26 2 28 +--R - 3240628715520cos(y) sin(x) + 925893918720cos(y) sin(x) +--R + +--R 30 +--R - 123452522496cos(y) +--R * +--R 15 +--R tan(z) +--R + +--R 16 2 15 4 14 +--R 392445sin(x) - 12558240cos(y) sin(x) + 188373600cos(y) sin(x) +--R + +--R 6 13 8 12 +--R - 1758153600cos(y) sin(x) + 11427998400cos(y) sin(x) +--R + +--R 10 11 12 10 +--R - 54854392320cos(y) sin(x) + 201132771840cos(y) sin(x) +--R + +--R 14 9 16 8 +--R - 574665062400cos(y) sin(x) + 1292996390400cos(y) sin(x) +--R + +--R 18 7 20 6 +--R - 2298660249600cos(y) sin(x) + 3218124349440cos(y) sin(x) +--R + +--R 22 5 24 4 +--R - 3510681108480cos(y) sin(x) + 2925567590400cos(y) sin(x) +--R + +--R 26 3 28 2 +--R - 1800349286400cos(y) sin(x) + 771578265600cos(y) sin(x) +--R + +--R 30 32 +--R - 205754204160cos(y) sin(x) + 25719275520cos(y) +--R * +--R 12 +--R tan(z) +--R + +--R 17 2 16 4 15 +--R 30780sin(x) - 1046520cos(y) sin(x) + 16744320cos(y) sin(x) +--R + +--R 6 14 8 13 +--R - 167443200cos(y) sin(x) + 1172102400cos(y) sin(x) +--R + +--R 10 12 12 11 +--R - 6094932480cos(y) sin(x) + 24379729920cos(y) sin(x) +--R + +--R 14 10 16 9 +--R - 76622008320cos(y) sin(x) + 191555020800cos(y) sin(x) +--R + +--R 18 8 20 7 +--R - 383110041600cos(y) sin(x) + 612976066560cos(y) sin(x) +--R + +--R 22 6 24 5 +--R - 780151357440cos(y) sin(x) + 780151357440cos(y) sin(x) +--R + +--R 26 4 28 3 +--R - 600116428800cos(y) sin(x) + 342923673600cos(y) sin(x) +--R + +--R 30 2 32 +--R - 137169469440cos(y) sin(x) + 34292367360cos(y) sin(x) +--R + +--R 34 +--R - 4034396160cos(y) +--R * +--R 9 +--R tan(z) +--R + +--R 18 2 17 4 16 +--R 1710sin(x) - 61560cos(y) sin(x) + 1046520cos(y) sin(x) +--R + +--R 6 15 8 14 +--R - 11162880cos(y) sin(x) + 83721600cos(y) sin(x) +--R + +--R 10 13 12 12 +--R - 468840960cos(y) sin(x) + 2031644160cos(y) sin(x) +--R + +--R 14 11 16 10 +--R - 6965637120cos(y) sin(x) + 19155502080cos(y) sin(x) +--R + +--R 18 9 20 8 +--R - 42567782400cos(y) sin(x) + 76622008320cos(y) sin(x) +--R + +--R 22 7 24 6 +--R - 111450193920cos(y) sin(x) + 130025226240cos(y) sin(x) +--R + +--R 26 5 28 4 +--R - 120023285760cos(y) sin(x) + 85730918400cos(y) sin(x) +--R + +--R 30 3 32 2 +--R - 45723156480cos(y) sin(x) + 17146183680cos(y) sin(x) +--R + +--R 34 36 +--R - 4034396160cos(y) sin(x) + 448266240cos(y) +--R * +--R 6 +--R tan(z) +--R + +--R 19 2 18 4 17 +--R 60sin(x) - 2280cos(y) sin(x) + 41040cos(y) sin(x) +--R + +--R 6 16 8 15 +--R - 465120cos(y) sin(x) + 3720960cos(y) sin(x) +--R + +--R 10 14 12 13 +--R - 22325760cos(y) sin(x) + 104186880cos(y) sin(x) +--R + +--R 14 12 16 11 +--R - 386979840cos(y) sin(x) + 1160939520cos(y) sin(x) +--R + +--R 18 10 20 9 +--R - 2837852160cos(y) sin(x) + 5675704320cos(y) sin(x) +--R + +--R 22 8 24 7 +--R - 9287516160cos(y) sin(x) + 12383354880cos(y) sin(x) +--R + +--R 26 6 28 5 +--R - 13335920640cos(y) sin(x) + 11430789120cos(y) sin(x) +--R + +--R 30 4 32 3 +--R - 7620526080cos(y) sin(x) + 3810263040cos(y) sin(x) +--R + +--R 34 2 36 38 +--R - 1344798720cos(y) sin(x) + 298844160cos(y) sin(x) - 31457280cos(y) +--R * +--R 3 +--R tan(z) +--R + +--R 20 2 19 4 18 6 17 +--R sin(x) - 40cos(y) sin(x) + 760cos(y) sin(x) - 9120cos(y) sin(x) +--R + +--R 8 16 10 15 12 14 +--R 77520cos(y) sin(x) - 496128cos(y) sin(x) + 2480640cos(y) sin(x) +--R + +--R 14 13 16 12 +--R - 9922560cos(y) sin(x) + 32248320cos(y) sin(x) +--R + +--R 18 11 20 10 +--R - 85995520cos(y) sin(x) + 189190144cos(y) sin(x) +--R + +--R 22 9 24 8 +--R - 343982080cos(y) sin(x) + 515973120cos(y) sin(x) +--R + +--R 26 7 28 6 +--R - 635043840cos(y) sin(x) + 635043840cos(y) sin(x) +--R + +--R 30 5 32 4 +--R - 508035072cos(y) sin(x) + 317521920cos(y) sin(x) +--R + +--R 34 3 36 2 +--R - 149422080cos(y) sin(x) + 49807360cos(y) sin(x) +--R + +--R 38 40 +--R - 10485760cos(y) sin(x) + 1048576cos(y) +--R Type: Factored(Expression(Integer)) +--E 48 + + +\end{chunk} +expand$[(1 - c^2)^5 (1 - s^2)^5 (c^2 + s^2)^{10}] => c^{10} s^{10}$ + +when $c^2 + s^2 = 1$ [modification of a problem due to Richard Liska] +\begin{chunk}{*} +--S 49 of 63 +expand((1 - c**2)**5 * (1 - s**2)**5 * (c**2 + s**2)**10) +--R +--R +--R (49) +--R 10 8 6 4 2 30 +--R (c - 5c + 10c - 10c + 5c - 1)s +--R + +--R 12 10 8 6 4 2 28 +--R (10c - 55c + 125c - 150c + 100c - 35c + 5)s +--R + +--R 14 12 10 8 6 4 2 26 +--R (45c - 275c + 710c - 1000c + 825c - 395c + 100c - 10)s +--R + +--R 16 14 12 10 8 6 4 2 +--R 120c - 825c + 2425c - 3960c + 3900c - 2345c + 825c - 150c +--R + +--R 10 +--R * +--R 24 +--R s +--R + +--R 18 16 14 12 10 8 6 +--R 210c - 1650c + 5550c - 10450c + 12055c - 8735c + 3900c +--R + +--R 4 2 +--R - 1000c + 125c - 5 +--R * +--R 22 +--R s +--R + +--R 20 18 16 14 12 10 8 +--R 252c - 2310c + 8970c - 19470c + 26060c - 22253c + 12055c +--R + +--R 6 4 2 +--R - 3960c + 710c - 55c + 1 +--R * +--R 20 +--R s +--R + +--R 22 20 18 16 14 12 10 +--R 210c - 2310c + 10500c - 26400c + 40875c - 40645c + 26060c +--R + +--R 8 6 4 2 +--R - 10450c + 2425c - 275c + 10c +--R * +--R 18 +--R s +--R + +--R 24 22 20 18 16 14 12 +--R 120c - 1650c + 8970c - 26400c + 47400c - 54615c + 40875c +--R + +--R 10 8 6 4 +--R - 19470c + 5550c - 825c + 45c +--R * +--R 16 +--R s +--R + +--R 26 24 22 20 18 16 14 +--R 45c - 825c + 5550c - 19470c + 40875c - 54615c + 47400c +--R + +--R 12 10 8 6 +--R - 26400c + 8970c - 1650c + 120c +--R * +--R 14 +--R s +--R + +--R 28 26 24 22 20 18 16 +--R 10c - 275c + 2425c - 10450c + 26060c - 40645c + 40875c +--R + +--R 14 12 10 8 +--R - 26400c + 10500c - 2310c + 210c +--R * +--R 12 +--R s +--R + +--R 30 28 26 24 22 20 18 +--R c - 55c + 710c - 3960c + 12055c - 22253c + 26060c +--R + +--R 16 14 12 10 +--R - 19470c + 8970c - 2310c + 252c +--R * +--R 10 +--R s +--R + +--R 30 28 26 24 22 20 18 +--R - 5c + 125c - 1000c + 3900c - 8735c + 12055c - 10450c +--R + +--R 16 14 12 +--R 5550c - 1650c + 210c +--R * +--R 8 +--R s +--R + +--R 30 28 26 24 22 20 18 +--R 10c - 150c + 825c - 2345c + 3900c - 3960c + 2425c +--R + +--R 16 14 +--R - 825c + 120c +--R * +--R 6 +--R s +--R + +--R 30 28 26 24 22 20 18 16 4 +--R (- 10c + 100c - 395c + 825c - 1000c + 710c - 275c + 45c )s +--R + +--R 30 28 26 24 22 20 18 2 30 28 +--R (5c - 35c + 100c - 150c + 125c - 55c + 10c )s - c + 5c +--R + +--R 26 24 22 20 +--R - 10c + 10c - 5c + c +--R Type: Polynomial(Integer) +--E 49 + +--S 50 of 63 +groebner([%, c**2 + s**2 - 1]) +--R +--R +--R 2 2 20 18 16 14 12 10 +--R (50) [s + c - 1,c - 5c + 10c - 10c + 5c - c ] +--R Type: List(Polynomial(Integer)) +--E 50 + +--S 51 of 63 +map(factor, %) +--R +--R +--R 2 2 5 10 5 +--R (51) [s + c - 1,(c - 1) c (c + 1) ] +--R Type: List(Factored(Polynomial(Integer))) +--E 51 + +\end{chunk} +$=> (x + y) (x - y) {\textrm\ mod\ } 3$ +\begin{chunk}{*} +--S 52 of 63 +factor(4*x**2 - 21*x*y + 20*y**2 :: Polynomial(PrimeField(3))) +--R +--R There are 22 exposed and 18 unexposed library operations named ** +--R having 2 argument(s) but none was determined to be applicable. +--R Use HyperDoc Browse, or issue +--R )display op ** +--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 +--R Cannot find a definition or applicable library operation named ** +--R with argument type(s) +--R Variable(y) +--R Polynomial(PrimeField(3)) +--R +--R Perhaps you should use "@" to indicate the required return type, +--R or "$" to specify which version of the function you need. +--E 52 + +\end{chunk} +$=> 1/4 (x + y) (2 x + y [-1 + i sqrt(3)]) (2 x + y [-1 - i sqrt(3)])$ +\begin{chunk}{*} +--S 53 of 63 +factor(x**3 + y**3, [rootOf(isqrt3**2 + 3)]) +--R +--R +--R - isqrt3 - 1 isqrt3 - 1 +--R (52) (y + ------------ x)(y + x)(y + ---------- x) +--R 2 2 +--R Type: Factored(Polynomial(AlgebraicNumber)) +--E 53 + +\end{chunk} +Partial fraction decomposition $=> 3/(x + 2) - 2/(x + 1) + 2/(x + 1)^2$ +\begin{chunk}{*} +--S 54 of 63 +(x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2) +--R +--R +--R 2 +--R x + 2x + 3 +--R (53) ----------------- +--R 3 2 +--R x + 4x + 5x + 2 +--R Type: Fraction(Polynomial(Integer)) +--E 54 + +--S 55 of 63 +fullPartialFraction( _ + % :: Fraction UnivariatePolynomial(x, Fraction Integer)) +--R +--R +--R 2 2 3 +--R (54) - ----- + -------- + ----- +--R x + 1 2 x + 2 +--R (x + 1) +--RType: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer))) +--E 55 + +\end{chunk} +Noncommutative algebra: note that $(A B C)^{(-1)} = C^{(-1)} B^{(-1)} A^{(-1)}$ + +$=> A B C A C B - C^{(-1)} B^{(-1)} C B$ +\begin{chunk}{*} +--S 56 of 63 +A : SquareMatrix(2, Integer) +--R +--R Type: Void +--E 56 + +--S 57 of 63 +B : SquareMatrix(2, Integer) +--R +--R Type: Void +--E 57 + +--S 58 of 63 +C : SquareMatrix(2, Integer) +--R +--R Type: Void +--E 58 + +--S 59 of 63 +(A*B*C - (A*B*C)**(-1)) * A*C*B +--R +--R +--R A is declared as being in SquareMatrix(2,Integer) but has not been +--R given a value. +--E 59 + +\end{chunk} +Jacobi's identity: $[A, B, C] + [B, C, A] + [C, A, B] = 0$ where +$[A, B, C] = [A, [B, C]]$ and $[A, B] = A B - B A$ +is the commutator of $A$ and $B$ +\begin{chunk}{*} +--S 60 of 63 +comm2(A, B) == A * B - B * A +--R +--R Type: Void +--E 60 + +--S 61 of 63 +comm3(A, B, C) == comm2(A, comm2(B, C)) +--R +--R Type: Void +--E 61 + +--S 62 of 63 +comm2(A, B) +--R +--R +--R A is declared as being in SquareMatrix(2,Integer) but has not been +--R given a value. +--E 62 + +--S 63 of 63 +comm3(A, B, C) + comm3(B, C, A) + comm3(C, A, B) +--R +--R +--R A is declared as being in SquareMatrix(2,Integer) but has not been +--R given a value. +--E 63 + +)spool + + +)lisp (bye) +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document}