#tex2html_wrap_inline1668#

<#227#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp;The symbol #tex2html_wrap_inline1670#
represents the last expression obtained by Jacal. It can ;SPMamp;be used in formulas like any other constant or variable or expression. ;SPMamp;<#227#> e21: 5

e22 : 2

e22: 5

e23 : 3

e23: 25 #math10#;SPMlt; expr₁ ;SPMgt; * ;SPMlt; expr₂ ;SPMgt;

<#4#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;Multiplication of scalar expressions such as numbers, polynomials, rational ;SPMamp;functions and algebraic functions is denoted by the infix operator *. For ;SPMamp;example, ;SPMamp;<#4#> e1 : (2+3*a)*4*a*b^2;

2 2 e1: (8 a + 12 a ) b <#285#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;One can also use * as an infix operator on bunches. In that case, it ;SPMamp;operates componentwise, in an appropriate sense. In case the bunches are ;SPMamp;square matrices, the operator * multiplies corresponding entries of the ;SPMamp;two factors. It does not perform matrix multiplication. To multiply matrices ;SPMamp;one instead uses the operator <#5#>.<#5#> (i.e. a period). More generally, any binary ;SPMamp;scalar operator other than <#228#>ˆ<#228#> can be used on bunches and acts componentwise. ;SPMamp;<#285#> #math11#;SPMlt; matrix₁ ;SPMgt; #tex2html_wrap_inline1676#
;SPMlt; matrix₂ ;SPMgt;

<#8#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp;Matrix multiplication. <#8#> e1 : a:[[1,2,3],[5,2,7]];

[1 2 3] e1: [ ] [5 2 7]

e2 : b:[[3,2],[6,4]];

[3 2] e2: [ ] [6 4]

e3 : b.a;

[13 10 23] e3: [ ] [26 20 46]

#math12#;SPMlt; expr₁ ;SPMgt; + ;SPMlt; expr₂ ;SPMgt;

<#9#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;Addition of scalar quantities or componentwise addition of bunches is accomplished ;SPMamp;by means of the infix operator +. For example, ;SPMamp;<#9#> e2 : a:[[1,3,5],[2,4,7]];

[1 3 5] e2: [ ] [2 4 7]

e3 : b:[2,4];

e3: [2 , 4]

e4 : a+b;

[3 5 7 ] e4: [ ] [6 8 11]

e5 : 3+2;

e5: 5

e6 : c+b;

e6: [2 + c , 4 + c]

e7 : e1+e5;

2 2 e7: 5 + (8 a + 12 a ) b <#10#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp;<#10#> #math13#;SPMlt; expr₁ ;SPMgt; - ;SPMlt; expr₂ ;SPMgt;

<#11#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;The symbol - is used to denote either the binary infix operator subtraction ;SPMamp;or the unary minus. ;SPMamp;<#11#> e1 : -[1,2,3];

e1: [-1 , -2 , -3]

e2 : 3-7;

e2: -4 <#12#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp;<#12#> #math14#;SPMlt; expr₁ ;SPMgt; ± ;SPMlt; expr₂ ;SPMgt;

#math15#;SPMlt; expr₁ ;SPMgt; #tex2html_wrap_inline1683#
;SPMlt; expr₂ ;SPMgt;

<#229#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp;Jacal allows the use of #tex2html_wrap_inline1685#
and #tex2html_wrap_inline1687#
as ambiguous signs ;SPMamp;(unary plus-or-minus, unary minus-or-plus) and as ambiguous infix ;SPMamp;operators (binary plus-or-minus, binary minus-or-plus). The value ±1 ;SPMamp;is also represented by the constant <#13#>sqrt1<#13#>, while #tex2html_wrap_inline1690#
1 is represented by ;SPMamp;#math16##tex2html_wrap_inline1692#
. ;SPMamp;<#229#> e7 : u:+/-3;

e7: 3 4

e8 : u^2;

e8: 9

e9 : +/-(u);

e9: 3

e10 : u-/+3;

e10: b-/+(3 5

#math17#;SPMlt; expr₁ ;SPMgt; / ;SPMlt; expr₂ ;SPMgt;

<#14#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;The symbol for division in Jacal is /. For example, the value returned by ;SPMamp;6/2; is 3.<#14#> e3 : (x^2-y^2)/(x-y);

e3: x + y <#15#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp;<#15#> #math18#;SPMlt; expr₁ ;SPMgt;   =   ;SPMlt; expr₂ ;SPMgt;

<#230#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;In Jacal, the equals sign #tex2html_wrap_inline1698#
is <#16#>not<#16#> used for conditionals and it ;SPMamp;is <#17#>not<#17#> used for assignments. To assign one value to another, use ;SPMamp;either : or : =. The operator = merely returns a value of the form ;SPMamp;#math19#0 = ;SPMlt; expression ;SPMgt;. The value returned by a = b for example is 0 = a - b. ;SPMamp;<#230#> e6 : 1=2;

e6: 0 = -1

;SPMlt; expr₁ ;SPMgt;<#231#> ˆ <#231#>;SPMlt; expr₂ ;SPMgt;

<#286#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;The infix operator <#232#>ˆ<#232#> is used for exponentiation of scalar quantitites or ;SPMamp;for componentwise exponentiation of bunches. For example, 2⁵ returns 32. ;SPMamp;Unlike the other scalar infix operators, one cannot use <#233#>ˆ<#233#> for component- ;SPMamp;wise operations on bunches. Furthermore, one should not try to use <#234#>ˆ<#234#> to ;SPMamp;raise a square matrix to a power. Instead, one should use <#235#>ˆˆ<#235#> ;SPMamp;<#286#> e7 : (1+x)^4;

2 3 4 e7: 1 + 4 x + 6 x + 4 x + x

;SPMlt; matrix ;SPMgt;<#236#> ˆˆ <#236#>;SPMlt; int ;SPMgt;

<#287#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;The infix operator <#237#>ˆˆ<#237#> is used for raising a square matrix ;SPMamp;to an integral power.It can also be used on nonsquare matrices, but the results ;SPMamp;are not documented. ;SPMamp;<#287#> e8 : a:[[1,0],[-1,1]];

[1 0] e8: [ ] [-1 1]

e9 : a^^3;

[1 0] e9: [ ] [-3 1]

<#28#>batch(#math20#;SPMlt; filename ;SPMgt;)<#28#>

<#238#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;The command <#29#>batch<#29#> is used to read in a file containing programs written ;SPMamp;in Jacal. Here, #math21#;SPMlt; filename ;SPMgt; is a string in double quotes. The precise way in which ;SPMamp;one refers to a file is of course system dependent. ;SPMamp;<#238#> e18 : system(;SPMquot;ex pluses;SPMquot;); e18 : system(;SPMquot;ex pluses;SPMquot;); ;SPMquot;pluses;SPMquot; 3 lines, 19 characters :1,#math22#d : aw : 2;x : 5;w^x - x^w;. : wq;SPMquot;pluses;SPMquot;3lines, 19characters

e18 : 0

e19 : batch(;SPMquot;pluses;SPMquot;);

e20 : e20 : 2

e21 : e21 : 5

e22 : e22 : 7

e23 : e19 :

#tex2html_wrap_inline1713#

#tex2html_wrap_inline1714#

[;SPMlt;elt_1;SPMgt;, ;SPMlt;elt_2;SPMgt;,..., ;SPMlt;elt_n;SPMgt;]#math23#

#tex2html_wrap_inline1716#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1717#
;SPMamp; TocollectanynumberofJacalobjectsintoabunch, simplyenclosethemin#tex2html_wrap_inline1718#
;SPMamp; squarebrackets.Forexample, tomakethebunchwhoseelementsare1, 2, 4, type#tex2html_wrap_inline1719#
;SPMamp; $[1, 2, 4]$.Onecanalsonestbunches, forexample,$[1,[[1, 3],[2, 5]],[1, 4]]$.#tex2html_wrap_inline1720#
;SPMamp; Notehoweverthatthebunchwhoseonlyelementis$#tex2html_wrap_inline1721#
#tex2html_wrap_inline1722#
e3 : a : bunch(1, 2, 3);

e3 : [1, 2, 3]

e4 : b : [a];

e4 : [123]

e5 : c : [b];

e5 : [[1, 2, 3]]

e6 : [[[1, 2, 3]]];

e6 : [[1, 2, 3]]

#tex2html_wrap_inline1723#
#tex2html_wrap_inline1724#

#tex2html_wrap_inline1725#

#tex2html_wrap_inline1726#

#tex2html_wrap_inline1727#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1728#
;SPMamp; #tex2html_wrap_inline1729#
;SPMamp; Thecommand#tex2html_wrap_inline1730#
isusedtodeterminethecoefficientofacertainpower#tex2html_wrap_inline1731#
;SPMamp; ofavariableinagivenpolynomial.Here#tex2html_wrap_inline1732#
isapolynomialand#tex2html_wrap_inline1733#
#tex2html_wrap_inline1734#
;SPMamp; isavariable.Iftheoptionalthirdargumentisomitted, thenJacalreturns#tex2html_wrap_inline1735#
;SPMamp; thecoefficientofthevariable#tex2html_wrap_inline1736#
in#tex2html_wrap_inline1737#
.Otherwiseitreturns#tex2html_wrap_inline1738#
;SPMamp; thecoefficientof$#tex2html_wrap_inline1739#
#tex2html_wrap_inline1740#
e14 : coeff ((x + 2)⁴, x, 3);

e14 : 8

e15 : (x + 2)⁴;

234e15 : 16 + 32x + 24x + 8x + x

e16 : coeff ((x + 2)⁴, x);

e16 : 32

e18 : coeffs((x + 2)⁴, x);

e18 : [16, 32, 24, 8, 1]

#tex2html_wrap_inline1741#
#tex2html_wrap_inline1742#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1743#
;SPMamp; Thefunction#tex2html_wrap_inline1744#
providesaninversetothefunction#tex2html_wrap_inline1745#
,#tex2html_wrap_inline1746#
;SPMamp; allowingonetorecoverapolynomialfromitslistofcoefficients.#tex2html_wrap_inline1747#
;SPMamp; #tex2html_wrap_inline1748#
#tex2html_wrap_inline1749#

#tex2html_wrap_inline1750#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1751#
;SPMamp; #tex2html_wrap_inline1752#
;SPMamp; Thecommand#tex2html_wrap_inline1753#
isusedtoextractacolumnofamatrix.Here,#tex2html_wrap_inline1754#
;SPMamp; $ ;SPMlt; matrix ;SPMgt; $ isamatrixand$ ;SPMlt; integer ;SPMgt; $ isapositiveinteger.Ifthat#tex2html_wrap_inline1755#
;SPMamp; integerexceedsthenumberofcolumns, anerrormessagesuchas#tex2html_wrap_inline1756#
;SPMamp; #tex2html_wrap_inline1757#
;SPMamp; #tex2html_wrap_inline1758#
#tex2html_wrap_inline1759#
;SPMamp; #tex2html_wrap_inline1760#
;SPMamp; appears.Hereisanexampleofcorrectuseofthecommand#tex2html_wrap_inline1761#
:#tex2html_wrap_inline1762#
;SPMamp; #tex2html_wrap_inline1763#
#tex2html_wrap_inline1764#
e19 : a : [[1, 2, 4],[2, 5, 6]];

[124]e19 : [][256]

e20 : col (a, 2);

[2]e20 : [][5]#tex2html_wrap_inline1765#
#tex2html_wrap_inline1766#

#tex2html_wrap_inline1767#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1768#
;SPMamp; #tex2html_wrap_inline1769#
;SPMamp; Thecommand#tex2html_wrap_inline1770#
producesalistofallofthecommandavailablein#tex2html_wrap_inline1771#
;SPMamp; Jacal.Itiscalledassfunctionofnoarguments.Explicitly:#tex2html_wrap_inline1772#
;SPMamp; #tex2html_wrap_inline1773#
;SPMamp; #tex2html_wrap_inline1774#
#tex2html_wrap_inline1775#
;SPMamp; #tex2html_wrap_inline1776#
#tex2html_wrap_inline1777#
e21 : commands();

u - / + u + / - transposetranscriptdifferentialtermssystemsylvestershowsetscalarmatrixrowresultantrapplyquitqedpoly_discriminantpolyornumnegatemodminormatrixloadlistofvarsidentgenmatrixgcdfinvfactorexampleeliminatedotproductdividediffdiagmatrixdeterminantdescribedenomcrossproductcontentcommandscolcoeffscoeffbunchbatchb + / - ⁼/ - + *6

#tex2html_wrap_inline1778#
#tex2html_wrap_inline1779#

#tex2html_wrap_inline1780#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1781#
;SPMamp; #tex2html_wrap_inline1782#
;SPMamp; Returnsalistofcontentandprimitivepartofapolynomialwith#tex2html_wrap_inline1783#
;SPMamp; respecttothevariable.ThecontentistheGCDofthecoefficients#tex2html_wrap_inline1784#
;SPMamp; ofthepolynomialinthevariable.Theprimitivepartis$ ;SPMlt; poly ;SPMgt; $ divided#tex2html_wrap_inline1785#
;SPMamp; bythecontent#tex2html_wrap_inline1786#
;SPMamp; #tex2html_wrap_inline1787#
#tex2html_wrap_inline1788#

e24 : content(2*x*y + 4*x²*y², y);

2e24 : [2x, y + 2xy]#tex2html_wrap_inline1789#
#tex2html_wrap_inline1790#

#tex2html_wrap_inline1791#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1792#
;SPMamp; #tex2html_wrap_inline1793#
;SPMamp; TheJacalcommand#tex2html_wrap_inline1794#
computesthecrossproductoftwovectors.#tex2html_wrap_inline1795#
;SPMamp; Bydefinition, thetwovectorsmusteachhavethreecomponents.#tex2html_wrap_inline1796#
;SPMamp; #tex2html_wrap_inline1797#
#tex2html_wrap_inline1798#
e24 : [2x, y + 2xy]

e25 : crossproduct([1, 2, 3],[4, 2, 5]);

e25 : [4, 7, -6]

#tex2html_wrap_inline1799#
#tex2html_wrap_inline1800#

#tex2html_wrap_inline1801#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1802#
;SPMamp; #tex2html_wrap_inline1803#
;SPMamp; TheJacalcommand#tex2html_wrap_inline1804#
isusedtoobtainthedenominatorofarational#tex2html_wrap_inline1805#
;SPMamp; expression.#tex2html_wrap_inline1806#
;SPMamp; #tex2html_wrap_inline1807#
#tex2html_wrap_inline1808#
e26 : denom(4/5);

e26 : 5

#tex2html_wrap_inline1809#
#tex2html_wrap_inline1810#

#tex2html_wrap_inline1811#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1812#
;SPMamp; #tex2html_wrap_inline1813#
;SPMamp; Thecommand#tex2html_wrap_inline1814#
istheheartoftheonlinehelpfacilityofJacal.#tex2html_wrap_inline1815#
;SPMamp; Here,$ ;SPMlt; command ;SPMgt; $ isastringwhichisthenameofacommandand#tex2html_wrap_inline1816#
#tex2html_wrap_inline1817#
;SPMamp; producesabriefdescriptionofthecommandandinmanycasesincludes#tex2html_wrap_inline1818#
;SPMamp; anexampleofitsuse.Togetherwiththecommand#tex2html_wrap_inline1819#
, whichprints#tex2html_wrap_inline1820#
;SPMamp; alistofallavailableJacalcommands, andthecommand#tex2html_wrap_inline1821#
, which#tex2html_wrap_inline1822#
;SPMamp; givesanexampleoftheuseofthecommand, onecaninprincipleuseJacalwithout#tex2html_wrap_inline1823#
;SPMamp; amanualafteronehaslearnedhowtogetstarted.#tex2html_wrap_inline1824#
;SPMamp; #tex2html_wrap_inline1825#
#tex2html_wrap_inline1826#
e27 : describe(col );column.columnofamatrixe27 : describe(resultant);resultant.Theresultofeliminatingavariablebetween2equations(orpolynomials).e27 : describe(+);Addition, plus.a + b

#tex2html_wrap_inline1827#
#tex2html_wrap_inline1828#

#tex2html_wrap_inline1829#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1830#
;SPMamp; #tex2html_wrap_inline1831#
;SPMamp; TheJacalcommand#tex2html_wrap_inline1832#
computesthedeterminantofasquarematrix.#tex2html_wrap_inline1833#
;SPMamp; Attemptingtotakethedeterminantofanon-squarematrixwillproduceanerror#tex2html_wrap_inline1834#
;SPMamp; message.#tex2html_wrap_inline1835#
;SPMamp; #tex2html_wrap_inline1836#
#tex2html_wrap_inline1837#
e1 : a : [[1, 2],[6, 7]];

[12]e1 : [][67]

e2 : determinant(a);

e2 : - 5

#tex2html_wrap_inline1838#
#tex2html_wrap_inline1839#

#tex2html_wrap_inline1840#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1841#
;SPMamp; #tex2html_wrap_inline1842#
;SPMamp; TheJacalcommand#tex2html_wrap_inline1843#
takesasinputalistof#tex2html_wrap_inline1844#
;SPMamp; objectsandreturnsthediagonalmatrixhavingthoseobjects#tex2html_wrap_inline1845#
;SPMamp; asdiagonalentries.Incaseonewantsallofthediagonalentries#tex2html_wrap_inline1846#
;SPMamp; tobeequal, itismoreconvenienttousethecommand#tex2html_wrap_inline1847#
.#tex2html_wrap_inline1848#
;SPMamp; #tex2html_wrap_inline1849#
#tex2html_wrap_inline1850#
e3 : diagmatrix(12, 3, a, s²);

[12000][][0300]e3 : [][00a0][][0002][s]

e4 : diagmatrix([1, 2], 2);

[[1, 2]0]e4 : [][02]

#tex2html_wrap_inline1851#
#tex2html_wrap_inline1852#

#tex2html_wrap_inline1853#

#tex2html_wrap_inline1854#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1855#
;SPMamp; #tex2html_wrap_inline1856#
;SPMamp; Thecommand#tex2html_wrap_inline1857#
treats#tex2html_wrap_inline1858#
and#tex2html_wrap_inline1859#
aspolynomials#tex2html_wrap_inline1860#
;SPMamp; inthevariable#tex2html_wrap_inline1861#
andreturnsapair$#tex2html_wrap_inline1862#
#tex2html_wrap_inline1863#
e5 : divide(x² + y², x - 7*y², x);

224e5 : [x + 7y, y + 49y]

e6 : divide(- 7, 3);

e6 : [- 2, -1]

e11 : divide(x² + y² + z², x + y + z);

22e11 : [- x - y + z, 2x + 2xy + 2y]

e14 : divide(x² + y² + z², x + y + z, y);

22e14 : [- x + y - z, 2x + 2xz + 2z]

e15 : divide(x² + y² + z², x + y + z, z);

22e15 : [- x - y + z, 2x + 2xy + 2y]

#tex2html_wrap_inline1864#
#tex2html_wrap_inline1865#

#tex2html_wrap_inline1866#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1867#
;SPMamp; #tex2html_wrap_inline1868#
;SPMamp; TheJacalfunction#tex2html_wrap_inline1869#
returnsthedotproductoftworowvectors#tex2html_wrap_inline1870#
;SPMamp; ofthesamelength.Itwillalsogivethedotproductoftwomatricesofthe#tex2html_wrap_inline1871#
;SPMamp; samesizebycomputingthesumofthedotproductsofthecorresponding#tex2html_wrap_inline1872#
;SPMamp; rowsor, whatisthesame, thetraceofonematrixtimesthetransposeof#tex2html_wrap_inline1873#
;SPMamp; theotherone.#tex2html_wrap_inline1874#
;SPMamp; #tex2html_wrap_inline1875#
#tex2html_wrap_inline1876#
e28 : a : [1, 2, 3];b : [3, 1, 5];

e28 : [1, 2, 3]

e29 : e29 : [3, 1, 5]

e30 : dotproduct(a, b);

e30 : 20

#tex2html_wrap_inline1877#
#tex2html_wrap_inline1878#

#tex2html_wrap_inline1879#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1880#
;SPMamp; #tex2html_wrap_inline1881#
;SPMamp; Here$ ;SPMlt; eqn_i ;SPMgt; $ isanequationfor$i=1,..., n$ andwhere$ ;SPMlt; var_j ;SPMgt; $ isavariable#tex2html_wrap_inline1882#
;SPMamp; for$j=1,..., m$.#tex2html_wrap_inline1883#
returnsalistofequationsobtainedbyeliminatingthe#tex2html_wrap_inline1884#
;SPMamp; variables$ ;SPMlt; var₁ ;SPMgt; , dots, ;SPMlt; var_m ;SPMgt; $ fromtheequations$ ;SPMlt; eqn₁ ;SPMgt; ,..., ;SPMlt; eqn_n ;SPMgt; $.#tex2html_wrap_inline1885#
;SPMamp; #tex2html_wrap_inline1886#
#tex2html_wrap_inline1887#

e39 : eliminate([x² + y = 0, x³ + y = 0],[x]);

2e39 : 0 = - y - y

e40 : eliminate([x + y + z = 3, x² + y² + z² = 3, x³ + y³ + z³ = 3],[x, y]);

e40 : 0 = 1 - z

#tex2html_wrap_inline1888#
#tex2html_wrap_inline1889#

#tex2html_wrap_inline1890#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1891#
;SPMamp; #tex2html_wrap_inline1892#
;SPMamp; Here,$ ;SPMlt; command ;SPMgt; $ isastringwhichisthenameofaJacalcommand.#tex2html_wrap_inline1893#
#tex2html_wrap_inline1894#
;SPMamp; #tex2html_wrap_inline1895#
givesanexampleoftheuseofthecommand.Seealsothefunction#tex2html_wrap_inline1896#
#tex2html_wrap_inline1897#
;SPMamp; #tex2html_wrap_inline1898#
#tex2html_wrap_inline1899#

e43:example(+);a+b

e43:a+b

#tex2html_wrap_inline1900#
;SPMamp; #tex2html_wrap_inline1901#
#tex2html_wrap_inline1902#

#tex2html_wrap_inline1903#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1904#
;SPMamp; #tex2html_wrap_inline1905#
;SPMamp; TheJacalcommand#tex2html_wrap_inline1906#
takesasinputanintegerandreturns#tex2html_wrap_inline1907#
;SPMamp; alistoftheprimenumbersthatdivideit, eachoccurringwiththeappropriate#tex2html_wrap_inline1908#
;SPMamp; multiplicityinthelist.Ifthenumberisnegative, thelistwillbegin#tex2html_wrap_inline1909#
;SPMamp; with$-1$.#tex2html_wrap_inline1910#
#tex2html_wrap_inline1911#

e45 : factor(120);

e45 : [2, 2, 2, 3, 5]

e46 : factor(- 120);

e46 : [- 1, 2, 2, 2, 3, 5]

#tex2html_wrap_inline1912#
#tex2html_wrap_inline1913#

;SPMlt;function;SPMgt;#math24##tex2html_wrap_inline1915#

#tex2html_wrap_inline1916#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1917#
;SPMamp; #tex2html_wrap_inline1918#
;SPMamp; Thecommand#tex2html_wrap_inline1919#
takesasinputafunctionofonevariable#tex2html_wrap_inline1920#
;SPMamp; andreturnstheinverseofthatfunction.Thefunctionmaybedefined#tex2html_wrap_inline1921#
;SPMamp; inanyofthewayspermittedinJacal, i.e.byanexplicitalgebraic#tex2html_wrap_inline1922#
;SPMamp; definition, byanexplicitlambdaexpressionorbyanimplicitlamba#tex2html_wrap_inline1923#
;SPMamp; expression.If$f$ isthefunction, thentyping$f$#tex2html_wrap_inline1924#
has#tex2html_wrap_inline1925#
;SPMamp; thesameeffectastyping#tex2html_wrap_inline1926#
.#tex2html_wrap_inline1927#
;SPMamp; #tex2html_wrap_inline1928#
#tex2html_wrap_inline1929#
e0 : w(t) : = t + 1;

w(t) : lambda([@1], 1 + @1)

e0 : finv(w);

e0 : lambda([@1], -1 + @1)#tex2html_wrap_inline1930#
#tex2html_wrap_inline1931#

#tex2html_wrap_inline1932#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1933#
;SPMamp; #tex2html_wrap_inline1934#
;SPMamp; TheJacalfunction#tex2html_wrap_inline1935#
takesasargumentstwopolynomialswithinteger#tex2html_wrap_inline1936#
;SPMamp; coefficientsandreturnsagreatestcommondivisorofthetwopolynomials.This#tex2html_wrap_inline1937#
;SPMamp; includesthecasewherethepolynomialsareintegers.#tex2html_wrap_inline1938#
;SPMamp; #tex2html_wrap_inline1939#
#tex2html_wrap_inline1940#
e1 : gcd (x⁴ - y⁴, x⁶ + y⁶);

22e1 : x + y

e2 : gcd (4, 10);

e2 : 2

#tex2html_wrap_inline1941#
#tex2html_wrap_inline1942#

#tex2html_wrap_inline1943#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1944#
;SPMamp; Thefunction#tex2html_wrap_inline1945#
takesasargumentsafunction$ ;SPMlt; function ;SPMgt; $ oftwo#tex2html_wrap_inline1946#
;SPMamp; variablesandtwopositiveintegers,$ ;SPMlt; rows ;SPMgt; $ and$ ;SPMlt; cols ;SPMgt; $.Itreturnsa#tex2html_wrap_inline1947#
;SPMamp; matrixwiththeindicatednumbersofrowsandcolumnsinwhichthe$(i, j)$th#tex2html_wrap_inline1948#
;SPMamp; entryisobtainedbyevaluating$ ;SPMlt; function ;SPMgt; $ at$(i, j)$.Thefunctionmaybe#tex2html_wrap_inline1949#
;SPMamp; definedinanyofthewaysavailableinJacal, i.epreviouslybyanexplicit#tex2html_wrap_inline1950#
;SPMamp; algebraicdefinition, byanexplicitlambdaexpressionorbyanimplicitlambda#tex2html_wrap_inline1951#
;SPMamp; expression.#tex2html_wrap_inline1952#
;SPMamp; #tex2html_wrap_inline1953#
#tex2html_wrap_inline1954#
e4 : @1² + @2²;

22e4 : lambda([@1,@2],@1 + @2)

e5 : genmatrix(e4, 3, 5);

[25101726][]e5 : [58132029][][1013182534]

#tex2html_wrap_inline1955#
#tex2html_wrap_inline1956#

#tex2html_wrap_inline1957#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1958#
;SPMamp; #tex2html_wrap_inline1959#
;SPMamp; Thecommand#tex2html_wrap_inline1960#
takesasargumentapositiveinteger$n$#tex2html_wrap_inline1961#
;SPMamp; andreturnsan$n×n$ identitymatrix.Thisissometimesmoreconvenient#tex2html_wrap_inline1962#
;SPMamp; thanobtainingthissamematrixusingthecommand#tex2html_wrap_inline1963#
.#tex2html_wrap_inline1964#
;SPMamp; #tex2html_wrap_inline1965#
#tex2html_wrap_inline1966#

e6 : ident(4);

[1000][][0100]e6 : [][0010][][0001]

#tex2html_wrap_inline1967#
#tex2html_wrap_inline1968#

#tex2html_wrap_inline1969#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1970#
;SPMamp; Jacalhastheabilitytoworkwithlambdaexpressions, viathecommand#tex2html_wrap_inline1971#
;SPMamp; #tex2html_wrap_inline1972#
.Furthermore, Jacalalwaysconvertsuserdefinitionsoffunctions#tex2html_wrap_inline1973#
;SPMamp; byanymethodintolambdaexpressionsandconvertsthedummyvariablesofthe#tex2html_wrap_inline1974#
;SPMamp; functiondefinitionintosymbolssuchas#tex2html_wrap_inline1975#
,....Jacalcanmanipulate#tex2html_wrap_inline1976#
;SPMamp; lambdaexpressionsbymanipulatingtheirfunctionparts, asin#tex2html_wrap_inline1977#
below.#tex2html_wrap_inline1978#
;SPMamp; Jacalcanalsoinvertafunctionusingthecommand#tex2html_wrap_inline1979#
.#tex2html_wrap_inline1980#
#tex2html_wrap_inline1981#

e12 : lambda([x], x²);

2e12 : lambda([@1],@1)

e13 : lambda([x, y, z], x*y*z);

e13 : lambda([@1,@2,@3],@1@2@3)

e14 : e12 + e13;

2e14 : lambda([@1,@2,@3],@1 + @1@2@3)

#tex2html_wrap_inline1982#
#tex2html_wrap_inline1983#

#tex2html_wrap_inline1984#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1985#
;SPMamp; #tex2html_wrap_inline1986#
;SPMamp; Thecommand#tex2html_wrap_inline1987#
takesasinputarationalexpressionandreturns#tex2html_wrap_inline1988#
;SPMamp; alistofthevariablesthatoccurinthatexpression.#tex2html_wrap_inline1989#
;SPMamp; #tex2html_wrap_inline1990#
#tex2html_wrap_inline1991#

e7 : listofvars(x² + y³);

e7 : [x, y]

e8 : listofvars((x² + y³)/(2*x⁷ + y*x + z));

e8 : [z, x, y]

#tex2html_wrap_inline1992#
#tex2html_wrap_inline1993#

#tex2html_wrap_inline1994#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline1995#
;SPMamp; #tex2html_wrap_inline1996#
;SPMamp; Thejacalcommand#tex2html_wrap_inline1997#
takesasinputastringandreadsina#tex2html_wrap_inline1998#
file#tex2html_wrap_inline1999#
;SPMamp; whosenameisobtainedbyappendingtheextension#tex2html_wrap_inline2000#
tothestring.#tex2html_wrap_inline2001#
;SPMamp; IfyouwanttoreadinafileofJacalcommands, donotuse#tex2html_wrap_inline2002#
.Instead#tex2html_wrap_inline2003#
;SPMamp; usethecommand#tex2html_wrap_inline2004#
.Toloadinthefile#tex2html_wrap_inline2005#
,#tex2html_wrap_inline2006#
;SPMamp; #tex2html_wrap_inline2007#
#tex2html_wrap_inline2008#
e9 : load (;SPMquot;foo;SPMquot;);

e9 : foo

#tex2html_wrap_inline2009#
minor(;SPMlt;matrix;SPMgt;,;SPMlt;row;SPMgt;,;SPMlt;col;SPMgt;);#math25#

#tex2html_wrap_inline2011#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2012#
;SPMamp; #tex2html_wrap_inline2013#
;SPMamp; Thecommand#tex2html_wrap_inline2014#
returnsthesubmatrixof$ ;SPMlt; matrix ;SPMgt; $ obtainedby#tex2html_wrap_inline2015#
;SPMamp; deletingthe$i$throwandthe$j$thcolumn, where$i= ;SPMlt; row ;SPMgt; $ and$j= ;SPMlt; col ;SPMgt; $.#tex2html_wrap_inline2016#
;SPMamp; #tex2html_wrap_inline2017#
#tex2html_wrap_inline2018#

e21 : b : [[1, 2, 3],[3, 1, 5],[5, 2, 7]];

[123][]e21 : [315][][527]

e22 : minor(b, 3, 1);

[23]e22 : [][15]

#tex2html_wrap_inline2019#
#tex2html_wrap_inline2020#

#tex2html_wrap_inline2021#

#tex2html_wrap_inline2022#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2023#
;SPMamp; #tex2html_wrap_inline2024#
;SPMamp; Thefunction#tex2html_wrap_inline2025#
takesasinputstwointegersandreturnsthe#tex2html_wrap_inline2026#
;SPMamp; anintegerwhichiscongruenttothefirstmodulothesecond.Itcan#tex2html_wrap_inline2027#
;SPMamp; alsoreduceapolynomialwithrespecttoagivenequation.#tex2html_wrap_inline2028#
;SPMamp; #tex2html_wrap_inline2029#
#tex2html_wrap_inline2030#

e23 : mod (5, 2);

e23 : - 1

e24 : mod (x⁴ +4, x² = 2);

e24 : 8

#tex2html_wrap_inline2031#
#tex2html_wrap_inline2032#

#tex2html_wrap_inline2033#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2034#
;SPMamp; #tex2html_wrap_inline2035#
;SPMamp; Thefunction#tex2html_wrap_inline2036#
takesarationalexpressionasinputandreturnsa#tex2html_wrap_inline2037#
;SPMamp; numeratoroftheexpression.#tex2html_wrap_inline2038#
;SPMamp; #tex2html_wrap_inline2039#
#tex2html_wrap_inline2040#

e25 : num((x² + y²)/(x² - y²));

22e25 : - x - y

e26 : num(7/4);

e26 : 7

e27 : num(7/(4/3));

e27 : 21

#tex2html_wrap_inline2041#
#tex2html_wrap_inline2042#

#tex2html_wrap_inline2043#

#tex2html_wrap_inline2044#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2045#
;SPMamp; #tex2html_wrap_inline2046#
;SPMamp; Thefunction#tex2html_wrap_inline2047#
takesasinputseithertwoequationsortwovalues.Ifthe#tex2html_wrap_inline2048#
;SPMamp; inputsaretwoequations, then#tex2html_wrap_inline2049#
returnsanequationwhichisequivalent#tex2html_wrap_inline2050#
;SPMamp; totheassertionthatatleastoneofthetwoinputequationsholds.Ifthe#tex2html_wrap_inline2051#
;SPMamp; inputsto#tex2html_wrap_inline2052#
aretwovaluesinsteadoftwoequations, thenthefunction#tex2html_wrap_inline2053#
;SPMamp; #tex2html_wrap_inline2054#
returnsamultiplevalue.Iftheinputsto#tex2html_wrap_inline2055#
consistof#tex2html_wrap_inline2056#
;SPMamp; oneequationandonevalue, then#tex2html_wrap_inline2057#
willreturnthevalue.#tex2html_wrap_inline2058#
;SPMamp; #tex2html_wrap_inline2059#
#tex2html_wrap_inline2060#
e1 : or(x = 2, y = 3);

e1 : 0 = - 6 + 3x + (2 - x)y

e2 : or(2, 3);

2e2 : :@| 0=-6+5:@-:@

e3 : e2²;

2e3 : :@| 0=-36+13:@-:@

e4 : or(x = 2, 17);

e4 : 17

#tex2html_wrap_inline2061#
#tex2html_wrap_inline2062#

#tex2html_wrap_inline2063#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2064#
;SPMamp; #tex2html_wrap_inline2065#
;SPMamp; Here$ ;SPMlt; poly ;SPMgt; $ isapolynomialand$ ;SPMlt; var ;SPMgt; $ isavariable.Thisfunctionreturns#tex2html_wrap_inline2066#
;SPMamp; thesquareoftheproductofthedifferencesoftherootsofthepolynomial$ ;SPMlt; poly ;SPMgt; $#tex2html_wrap_inline2067#
;SPMamp; withrespecttothevariable$ ;SPMlt; var ;SPMgt; $.#tex2html_wrap_inline2068#
;SPMamp; #tex2html_wrap_inline2069#
#tex2html_wrap_inline2070#
e7 : poly_discriminant(x³ -1, x);

e7 : - 27

#tex2html_wrap_inline2071#
#tex2html_wrap_inline2072#

#tex2html_wrap_inline2073#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2074#
;SPMamp; #tex2html_wrap_inline2075#
;SPMamp; Thecommand#tex2html_wrap_inline2076#
isprobablythemostimportantcommandyouneedtoknow#tex2html_wrap_inline2077#
;SPMamp; whenusingJacal, sinceitisthecommandthatlet'syouexitfromJacal.It#tex2html_wrap_inline2078#
;SPMamp; doesnothowever, returnyoutotheoperatingsystem.InsteaditsuspendsJacal#tex2html_wrap_inline2079#
;SPMamp; andreturnsyoutotheunderlyinglisp.YoucanreturntotheJacalsessionwhere#tex2html_wrap_inline2080#
;SPMamp; youleftoffbysimplytyping#tex2html_wrap_inline2081#
.Thisallowsyoutodosomeprogramming#tex2html_wrap_inline2082#
;SPMamp; intheunderlyinglispbeforereturningtoJacal.Ifyoudonotwishtoreturnto#tex2html_wrap_inline2083#
;SPMamp; Jacalbutreallywanttoterminatethesessionandreturntotheoperatingsystem,#tex2html_wrap_inline2084#
;SPMamp; thenaftertyping#tex2html_wrap_inline2085#
, youmustthentypewhatevercommandisusedto#tex2html_wrap_inline2086#
;SPMamp; exitfromtheunderlyinglisptotheoperatingsystem.In#tex2html_wrap_inline2087#
, thecommand#tex2html_wrap_inline2088#
;SPMamp; is#tex2html_wrap_inline2089#
.Iftheunderlyinglispusesthecommand#tex2html_wrap_inline2090#
toreturn#tex2html_wrap_inline2091#
;SPMamp; toheoperatingsystem, thenonecanexitdirectlyfromJacaltotheoperating#tex2html_wrap_inline2092#
;SPMamp; systemwithoutusing#tex2html_wrap_inline2093#
bysimplytyping#tex2html_wrap_inline2094#
.#tex2html_wrap_inline2095#
;SPMamp; #tex2html_wrap_inline2096#
#tex2html_wrap_inline2097#
unix ;SPMgt; jacal

JACALversion1a0, Copyright1989, 1990, 1991, 1992AubreyJafferJACALcomeswithABSOLUTELYNOWARRANTY;fordetailstype`(terms)'.Thisisfreesoftware, andyouarewelcometoredistributeitundercertainconditions;type`(terms)'fordetails.;;;Type(math)tobegin. ;SPMgt; (math)typeqed ()toreturntoschemee1 : qed ();scheme ;SPMgt; (quit)unix ;SPMgt; jacal

JACALversion1a0, Copyright1989, 1990, 1991, 1992AubreyJafferJACALcomeswithABSOLUTELYNOWARRANTY;fordetailstype`(terms)'.Thisisfreesoftware, andyouarewelcometoredistributeitundercertainconditions;type`(terms)'fordetails.;;;Type(math)tobegin. ;SPMgt; (math)typeqed ()toreturntoschemee1 : qed ();scheme ;SPMgt; (math)typeqed ()toreturntoschemee2 : quit();unix ;SPMgt;

#tex2html_wrap_inline2098#
#tex2html_wrap_inline2099#

#tex2html_wrap_inline2100#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2101#
;SPMamp; #tex2html_wrap_inline2102#
;SPMamp; See:#tex2html_wrap_inline2103#
#tex2html_wrap_inline2104#
;SPMamp; #tex2html_wrap_inline2105#
#tex2html_wrap_inline2106#

#tex2html_wrap_inline2107#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2108#
;SPMamp; #tex2html_wrap_inline2109#
;SPMamp; Thefunction#tex2html_wrap_inline2110#
isusedtoaccesselementsofbunches.Itcanalso#tex2html_wrap_inline2111#
;SPMamp; accesselementsnestedatlowerlevelsinabunch.Inparticular, itcanalso#tex2html_wrap_inline2112#
;SPMamp; accessmatrixelements.Intheabovesyntax,$ ;SPMlt; bunch ;SPMgt; $ isthebunchwhoseparts#tex2html_wrap_inline2113#
;SPMamp; onewishestoaccess, and$n, ;SPMlt; int₁ ;SPMgt; , ;SPMlt; int₂ ;SPMgt; ,..., ;SPMlt; int_n ;SPMgt; $ arepositiveintegers.#tex2html_wrap_inline2114#
;SPMamp; Itreturnsthe$ ;SPMlt; int_n ;SPMgt; $-thelementofthe$ ;SPMlt; int_n-1 ;SPMgt; $-thelementof...ofthe#tex2html_wrap_inline2115#
;SPMamp; $ ;SPMlt; int₂ ;SPMgt; $-thelementofthe$ ;SPMlt; int₁ ;SPMgt; $-thelementof$ ;SPMlt; bunch ;SPMgt; $.Onecanhave$n=0$.#tex2html_wrap_inline2116#
;SPMamp; Inthatcase,#tex2html_wrap_inline2117#
simplyreturnsthebunch.#tex2html_wrap_inline2118#
;SPMamp; #tex2html_wrap_inline2119#
#tex2html_wrap_inline2120#
e2 : rapply([[1, 2, 3],[1, 4, 6], 3], 2, 3);

e2 : 6

e6 : rapply([a, b], 2);

e6 : b

e7 : rapply([a, b]);

e7 : [a, b]

#tex2html_wrap_inline2121#
#tex2html_wrap_inline2122#

#tex2html_wrap_inline2123#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2124#
;SPMamp; #tex2html_wrap_inline2125#
;SPMamp; Thefunction#tex2html_wrap_inline2126#
returnstheresultantofthepolynomials#tex2html_wrap_inline2127#
;SPMamp; $ ;SPMlt; poly₁ ;SPMgt; $ and$ ;SPMlt; poly₂ ;SPMgt; $ withrespecttothevariable$ ;SPMlt; var ;SPMgt; $.#tex2html_wrap_inline2128#
;SPMamp; #tex2html_wrap_inline2129#
#tex2html_wrap_inline2130#
e2 : resultant(x² + a, x³ + a, x);

23e2 : a + a

#tex2html_wrap_inline2131#
#tex2html_wrap_inline2132#

#tex2html_wrap_inline2133#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2134#
;SPMamp; #tex2html_wrap_inline2135#
;SPMamp; Thecommand#tex2html_wrap_inline2136#
returnsthe$i$throwofthematrix$ ;SPMlt; matrix ;SPMgt; $, where$i= ;SPMlt; int ;SPMgt; $.#tex2html_wrap_inline2137#
;SPMamp; If$ ;SPMlt; int ;SPMgt; $ islargerthanthenumberofrowsof$ ;SPMlt; matrix ;SPMgt; $, thenJacalprints#tex2html_wrap_inline2138#
;SPMamp; anerrormessage.Thecorrespondingcommandforcolumnsofamatrixis#tex2html_wrap_inline2139#
.#tex2html_wrap_inline2140#
;SPMamp; #tex2html_wrap_inline2141#
#tex2html_wrap_inline2142#
e3 : u : [[1, 2, 3],[1, 5, 3]];

[123]e3 : [][153]

e4 : row(u, 2);

e4 : [1, 5, 3]

#tex2html_wrap_inline2143#
#tex2html_wrap_inline2144#

#tex2html_wrap_inline2145#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2146#
;SPMamp; Thecommand#tex2html_wrap_inline2147#
takesasinputsapositiveinteger#tex2html_wrap_inline2148#
;SPMamp; $ ;SPMlt; size ;SPMgt; $ andanalgebraicexpression$ ;SPMlt; entry ;SPMgt; $ andreturnsan$n×n$#tex2html_wrap_inline2149#
;SPMamp; diagonalmatrixwhosediagonalentriesareallequalto$ ;SPMlt; entry ;SPMgt; $, where#tex2html_wrap_inline2150#
;SPMamp; $n=  ;SPMlt; size ;SPMgt; $.#tex2html_wrap_inline2151#
#tex2html_wrap_inline2152#

e1 : scalarmatrix(3, 6);

[600][]e1 : [060][][006]

#tex2html_wrap_inline2153#
#tex2html_wrap_inline2154#

#tex2html_wrap_inline2155#

#tex2html_wrap_inline2156#

#tex2html_wrap_inline2157#

#tex2html_wrap_inline2158#

#tex2html_wrap_inline2159#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2160#
;SPMamp; TherearevariousflagsthattheJacalusercancontrol, namelytheJacal#tex2html_wrap_inline2161#
;SPMamp; commandlineprompt, thepriorityforprintingtermsinJacaloutput, theinput#tex2html_wrap_inline2162#
;SPMamp; grammarandtheoutputgrammar.Foradiscussion#tex2html_wrap_inline2163#
;SPMamp; ofthevariousgrammarspleaseseesection???ofthismanual.Thecommand#tex2html_wrap_inline2164#
#tex2html_wrap_inline2165#
;SPMamp; iscloselyrelated, allowingonetoseewhatthecurrentsettingsare.#tex2html_wrap_inline2166#
;SPMamp; Ifonechangestheprompt,$ ;SPMlt; string ;SPMgt; $ isastringofalphanumericcharacters#tex2html_wrap_inline2167#
;SPMamp; withoutquotes.Afterthiscommandisexecuted, subsequentcommandswillcuase#tex2html_wrap_inline2168#
;SPMamp; newpromptstobeobtainedfrom$ ;SPMlt; string ;SPMgt; $ byincrementingit.Iftheprompt#tex2html_wrap_inline2169#
;SPMamp; endsinaletter, itwillbetreatedasadigitinbase26andincremented.#tex2html_wrap_inline2170#
;SPMamp; Ifitendsinastringofdigits, thatstringwillbetreatedasanumberinbase#tex2html_wrap_inline2171#
;SPMamp; 10andincremented.#tex2html_wrap_inline2172#
;SPMamp; Theremainingcharactersinthestringwillplaynoroleinthisincrementation.#tex2html_wrap_inline2173#
;SPMamp; #tex2html_wrap_inline2174#
#tex2html_wrap_inline2175#
e1 : setpromptaz9Z;e1 : a + b;

az9Z : a + b

az9AA : a + b;

az9AA : a + b

az9AB : setpromptok99;az9AB : a + b;

ok99 : a + b

ok100 : a + b;

ok100 : a + b

ok101 :

#tex2html_wrap_inline2176#
#tex2html_wrap_inline2177#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2178#
;SPMamp; Thefollowingexamplesshowhowonechangestheinputgrammarorthe#tex2html_wrap_inline2179#
;SPMamp; outputgrammar.#tex2html_wrap_inline2180#
;SPMamp; #tex2html_wrap_inline2181#
#tex2html_wrap_inline2182#
e1 : a : [[[1, 2, 3]]];

e1 : [[1, 2, 3]]

e2 : setoutgrammarstandard;e2 : a;

e2 : [[[1, 2, 3]]]

e3 : setoutgrammarscheme;e3 : a;

(definee3#(#(#(123))))e4 : (1 + x)⁵;

(definee4(+ 1(*5x)(*10(^x2))(*10(^x3))(*5(^x4))(^x5)))e6 : setingrammarscheme;e6 : (+ e41);

(definee6(+ 2(*5x)(*10(^x2))(*10(^x3))(*5(^x4))(^x5)))e7 : (setingrammardisp2d )e7 : diagmatrix(3, 6);

(definee7#(#(30)#(06)))e8 : setoutgrammardisp2d;e8 : e7;

[30]e8 : [][06]

e9 : setoutgrammarstandard;e9 : e7;

e9 : [[3, 0],[0, 6]]

#tex2html_wrap_inline2183#
#tex2html_wrap_inline2184#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2185#
;SPMamp; Notethatintheaboveexamples, itispossibletoinputandoutputexpressions#tex2html_wrap_inline2186#
;SPMamp; inschemebysettingtheingrammarand /oroutgrammarto#tex2html_wrap_inline2187#
.Doingsoresult#tex2html_wrap_inline2188#
;SPMamp; inlinearoutput(aswith#tex2html_wrap_inline2189#
)asopposedtoatwodimensional#tex2html_wrap_inline2190#
;SPMamp; display(aswith#tex2html_wrap_inline2191#
).Theanalogueof#tex2html_wrap_inline2192#
forschemeoutput#tex2html_wrap_inline2193#
;SPMamp; isschemeprettyprinting.Tohavesuchoutput, settheoutputgrammarto#tex2html_wrap_inline2194#
;SPMamp; #tex2html_wrap_inline2195#
.#tex2html_wrap_inline2196#
;SPMamp; #tex2html_wrap_inline2197#
#tex2html_wrap_inline2198#

e4 : setoutgrammarschemepretty;e4 : (1 + x)⁵;

(definee4(+ 1(*5x)(*10(^x2))(*10(^x3))(*5(^x4))(^x5)))

#tex2html_wrap_inline2199#
#tex2html_wrap_inline2200#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2201#
;SPMamp; JacalalsoallowsforoutputtobeautomaticallytypesetinTEX.Thiscanbe#tex2html_wrap_inline2202#
;SPMamp; quiteusefulifonewantstousetheresultsofone'scomputationsinpublished#tex2html_wrap_inline2203#
;SPMamp; articles.Continuingwiththeexampleof$(1+x)⁵$ above, wehave.#tex2html_wrap_inline2204#
;SPMamp; #tex2html_wrap_inline2205#
#tex2html_wrap_inline2206#

e5 : setoutgrammartex;e5 : e4;

e5 : 1 + 5x + 10x² +10x³ +5x⁴ + x⁵

e6 : (1 + 1/x)³/(1 - 1/y)⁴;

e6 : #tex2html_wrap_inline2207#
1 + 3x + 3x² + x³#tex2html_wrap_inline2208#
y⁴#tex2html_wrap_inline2209#
x³-4x³y+

6x³y²-4x³y³+x³y⁴

#tex2html_wrap_inline2210#
#tex2html_wrap_inline2211#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2212#
;SPMamp; Afterbeingincludedinadocumentinmathdisplaymode, thesetwoexamples#tex2html_wrap_inline2213#
;SPMamp; willappearinthefollowingway.#tex2html_wrap_inline2214#
;SPMamp; #tex2html_wrap_inline2215#
#math26#1 + 5x + 10x² +10x³ +5x⁴ + x⁵#math27##tex2html_wrap_inline2219#
1 + 3x + 3x² + x³#tex2html_wrap_inline2220#
y⁴#tex2html_wrap_inline2221#
x³-4x³y+6x³y²-4x³y³+x³y⁴#math28##tex2html_wrap_inline2223#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2224#
;SPMamp; ThatlooksmuchbetterinanarticlethantheactualJacaloutputindisp2d#tex2html_wrap_inline2225#
;SPMamp; grammar.#tex2html_wrap_inline2226#
;SPMamp; #tex2html_wrap_inline2227#
#tex2html_wrap_inline2228#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2229#
;SPMamp; Thefollowingexamplesshowhowtosetthepriorityofprintingterms.#tex2html_wrap_inline2230#
#tex2html_wrap_inline2231#
e10 : a;

e10 : [[[1, 2, 3]]]

e11 : showprioritya;

;;;notasimplevariable : (((123).()).())

e12 : showpriorityb;

e12 : 128

e13 : showpriorityc;

e13 : 128

e14 : b + c;

e14 : b + c

e15 : c + b;

e15 : b + c

e16 : setpriorityb200;e16 : b + c;

#tex2html_wrap_inline2232#
#tex2html_wrap_inline2233#

#tex2html_wrap_inline2234#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2235#
;SPMamp; Thecommand#tex2html_wrap_inline2236#
enablestheJacalusertoexaminethecurrentsetting#tex2html_wrap_inline2237#
;SPMamp; ofvariousflagsaswellastolisttheflagsthatcanbesetbytheuserand#tex2html_wrap_inline2238#
;SPMamp; todisplayotherinformation.Tochangethesettingsoftheflags, usethecommand#tex2html_wrap_inline2239#
;SPMamp; #tex2html_wrap_inline2240#
.Toseealltheinformationaccessiblethroughthe#tex2html_wrap_inline2241#
command,#tex2html_wrap_inline2242#
;SPMamp; type#tex2html_wrap_inline2243#
.Toseetheavailablegrammars, type#tex2html_wrap_inline2244#
.#tex2html_wrap_inline2245#
;SPMamp; Toseethecurrentinputgrammartype#tex2html_wrap_inline2246#
.Toseethecurrent#tex2html_wrap_inline2247#
;SPMamp; outputgrammar, type#tex2html_wrap_inline2248#
.Toseethecurrentpriorityfor#tex2html_wrap_inline2249#
;SPMamp; printingexpressions, type#tex2html_wrap_inline2250#
.#tex2html_wrap_inline2251#
;SPMamp; #tex2html_wrap_inline2252#
#tex2html_wrap_inline2253#
e1 : showall;

promptpriorityoutgrammaringrammargrammarsalle1 : showprompt;

e1 : e1

e3 : showpriority;

: @(differential : @)@3@2@17differentialtermstsystemsylvestershowpriorityshowsetscalarmatrixrowresultantrapplyquitqedpromptprioritypoly_discriminantpolyornumnegatemodminormatrixloadlistofvarsidentgenmatrixgcdfinvfactorexampleeliminatee1dotproductdividediffdiagmatrixdeterminantdescribedenomcrossproductcontentcommandscolcoeffscoeffcbunchbatchb - / + b + / - balla⁼/ - + *8e3 : showoutgrammar;

e3 : disp2d

e4 : showingrammar;

e4 : standard

e5 : showgrammars;

e5 : [disp2d, standard, schemepretty, scheme]

#tex2html_wrap_inline2254#
#tex2html_wrap_inline2255#

#tex2html_wrap_inline2256#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2257#
;SPMamp; #tex2html_wrap_inline2258#
;SPMamp; Here,$ ;SPMlt; poly₁ ;SPMgt; $ and$ ;SPMlt; poly₂ ;SPMgt; $ arepolynomialsand$ ;SPMlt; var ;SPMgt; $ isavariable.#tex2html_wrap_inline2259#
;SPMamp; Thefunction#tex2html_wrap_inline2260#
returnsthematrixintroducedbySylvester(``A#tex2html_wrap_inline2261#
;SPMamp; MethodofDeterminingByMereInspectiontheDerivativesfromTwoEquations#tex2html_wrap_inline2262#
;SPMamp; ofAnyDegree,''Phil.Mag.#tex2html_wrap_inline2263#
(1840)pp.132-135, MathematicalPapers,#tex2html_wrap_inline2264#
;SPMamp; vol.I, pp.54-57)forcomputingtheresultantofthetwopolynomials#tex2html_wrap_inline2265#
;SPMamp; $ ;SPMlt; poly₁ ;SPMgt; $ and$ ;SPMlt; poly₂ ;SPMgt; $ withrespecttothevariable$ ;SPMlt; var ;SPMgt; $.Ifonewants#tex2html_wrap_inline2266#
;SPMamp; tocomputetheresultantitself, onecansimplyusethecommand#tex2html_wrap_inline2267#
#tex2html_wrap_inline2268#
;SPMamp; withthesamesyntax.#tex2html_wrap_inline2269#
;SPMamp; #tex2html_wrap_inline2270#
#tex2html_wrap_inline2271#
e5 : sylvester(a0 + a1*x + a2*x² + a3*x³, b0 + b1*x + b2*x², x);

[a3a2a1a00][][0a3a2a1a0][]e5 : [b2b1b000][][0b2b1b00][][00b2b1b0]

#tex2html_wrap_inline2272#
#tex2html_wrap_inline2273#

#tex2html_wrap_inline2274#
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein# ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;##tex2html_wrap_inline2275#
;SPMamp; #tex2html_wrap_inline2276#
;SPMamp; OnecanissuecommandstotheoperatingsystemwithoutleavingJacal.Todothis,#tex2html_wrap_inline2277#
;SPMamp; oneusesthecommand#tex2html_wrap_inline2278#
.Forexample, inaUNIXoperatingsystem, the#tex2html_wrap_inline2279#
;SPMamp; command#tex2html_wrap_inline2280#
willprintthedirectory.Onewayinwhichthe#tex2html_wrap_inline2281#
;SPMamp; command#tex2html_wrap_inline2282#
mightbeespeciallyusefulistoeditfilescontaining#tex2html_wrap_inline2283#
;SPMamp; JacalscriptswithoutleavingJacal, particularlyinnon-UNIXmachinesoron#tex2html_wrap_inline2284#
;SPMamp; machineswithoutGNUemacs.#tex2html_wrap_inline2285#
;SPMamp; #tex2html_wrap_inline2286#
#tex2html_wrap_inline2287#
e6 : system(;SPMquot;mailirmentrude;SPMquot;);

HiIrma, howareyou?

Lancelot.

e6 : 0

e7 : system(;SPMquot;exgleep;SPMquot;);;SPMquot;gleep;SPMquot;26lines, 239characters : a. : 1, COPYING ChangeLog MANUAL Makefile OLD README TAGS builtin.scm code.doc compilem.lisp ext.scm func.scm hist.scm math.lisp math.scm norm.scm parse.scm poly.scm scl.lisp scl.scm sexp.scm stdgrm.scm toploads.scm types.scm unparse.scm vect.scm :a

test.scm . :wq ;SPMquot;gleep;SPMquot; 28 lines, 249 characters

e7: 0

<#215#>terms()<#215#>

<#216#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp;Prints a copy of the GNU General Public License ;SPMamp;<#216#> e6 : terms();

e6 : terms();

GNU GENERAL PUBLIC LICENSE ************************** Version 1, February 1989

Copyright (C) 1989 Free Software Foundation, Inc. 675 Mass Ave, Cambridge, MA 02139, USA

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

Preamble ========

The license agreements of most software companies try to keep users at the mercy of those companies. By contrast, our General Public License is intended to guarantee your freedom to share and change free software---to make sure the software is free for all its users. The General Public License applies to the Free Software Foundation's software and to any other program whose authors commit to using it. You can use it for your programs, too. [ rest deleted for brevity] <#217#>differential(;SPMlt; expr ;SPMgt;)<#217#>

;SPMlt; expr ;SPMgt; '

<#282#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp;The Jacal command <#218#>differential<#218#> computes the derivative of the expression ;SPMamp;;SPMlt; expr ;SPMgt; with respect to a generic derivation. It is generic in the sense that ;SPMamp;nothing is assumed about its effect on the individual variables. The derivation ;SPMamp;is denoted by a right quote. ;SPMamp;<#282#> e6 : differential(x^2+y^3);

2 e6: 2 x x' + 3 y y'

e7 : (x^2+y^3)';

2 e7: 2 x x' + 3 y y'

<#219#>transcript(;SPMlt; string ;SPMgt;)<#219#>

<#283#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;The command <#220#>transcript<#220#> allows one to record a Jacal session. It is called ;SPMamp;with the syntax <#221#>transcript(;SPMlt; string ;SPMgt;);<#221#>, where ;SPMlt; string ;SPMgt; is the name of ;SPMamp;the file in which one wants to keep the transcript of the session. When one wishes ;SPMamp;to stop recording, one types <#222#>transcript();<#222#>. One is then free to use <#223#>transcript<#223#> ;SPMamp;again later in the session on another file. One can use it on the same file, but the ;SPMamp;file is overwritten. Presently, the command <#224#>transcript<#224#> does not echo commands ;SPMamp;to a file. file. ;SPMamp;<#283#> e9 : a:[1,2,3];

e9: [1, 2, 3]

e10 : transcript(;SPMquot;foo;SPMquot;);

e10: foo

e11 : a;

e11: [1, 2, 3]

e12 : transcript(); e12 : system(;SPMquot;cat foo;SPMquot;);

e10: foo

e11 : a;

e11: [1, 2, 3]

e12 : transcript();

e12: 0

<#225#>transpose(;SPMlt; matrix ;SPMgt;)<#225#>

<#284#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;truein#;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;# ;SPMamp; ;SPMamp;The command <#226#>transpose<#226#> takes a matrix as argument and returns the transpose ;SPMamp;of the matrix. ;SPMamp; <#284#> e13 : q:[[1,1,1,1,1],[1,2,3,4,5]];

[1 1 1 1 1] e13: [ ] [1 2 3 4 5]

e14 : transpose(e13);

[1 1] [ ] [1 2] [ ] e14: [1 3] [ ] [1 4] [ ] [1 5]