<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>a first Macaulay2 session</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_reading_spthe_spdocumentation.html">next</a> | <a href="_using_sp__Macaulay2_spwith_sp__Te__Xmacs.html">previous</a> | <a href="_reading_spthe_spdocumentation.html">forward</a> | <a href="_using_sp__Macaulay2_spwith_sp__Te__Xmacs.html">backward</a> | <a href="_getting_spstarted.html">up</a> | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_getting_spstarted.html" title="">getting started</a> > <a href="_a_spfirst_sp__Macaulay2_spsession.html" title="">a first Macaulay2 session</a></div>
<hr/>
<div><h1>a first Macaulay2 session</h1>
<div><p>Your first input prompt will be <tt>i1 : </tt>. In response to the prompt, type <tt>2+2</tt> and press return. The expression you entered will be evaluated - no punctuation is required at the end of the line.</p>
<table class="examples"><tr><td><pre>i1 : 2+2
o1 = 4</pre>
</td></tr>
</table>
The answer is displayed to the right of the output label <tt>o1 =</tt>.<p/>
Here is some arithmetic with fractions.<table class="examples"><tr><td><pre>i2 : 3/5 + 7/11
68
o2 = --
55
o2 : QQ</pre>
</td></tr>
</table>
<p>Notice the additional line of output labeled with <tt>o2 :</tt>. Output lines labeled with colons provide information about the type of output. In this case, the symbol <a href="___Q__Q.html" title="the class of all rational numbers">QQ</a> is our notation for the class of all rational numbers, and indicates that the answer on the previous line is a rational number.</p>
<p/>
Multiplication is indicated with <a href="__st.html" title="a binary operator, usually used for multiplication">*</a>.<table class="examples"><tr><td><pre>i3 : 1*2*3*4
o3 = 24</pre>
</td></tr>
</table>
Powers are obtained with <a href="_^.html" title="a binary operator, usually used for powers">^</a>.<table class="examples"><tr><td><pre>i4 : 2^200
o4 = 1606938044258990275541962092341162602522202993782792835301376</pre>
</td></tr>
</table>
Factorials are obtained with <a href="_!.html" title="factorial">!</a>.<table class="examples"><tr><td><pre>i5 : 40!
o5 = 815915283247897734345611269596115894272000000000</pre>
</td></tr>
</table>
Because some answers can be very long, it is a good idea to run the program in a window that does not wrap output lines, and allows the user to scroll left horizontally to see the rest of the output. (See <a href="_using_sp__Macaulay2_spwith_spemacs.html" title="">using Macaulay2 with emacs</a>.)<table class="examples"><tr><td><pre>i6 : 100!
o6 = 933262154439441526816992388562667004907159682643816214685929638952175999
932299156089414639761565182862536979208272237582511852109168640000000000
00000000000000</pre>
</td></tr>
</table>
Multiple expressions may be separated by semicolons.<table class="examples"><tr><td><pre>i7 : 1;2;3*4
o9 = 12</pre>
</td></tr>
</table>
A semicolon at the end of the line suppresses the printing of the value.<table class="examples"><tr><td><pre>i10 : 4*5;</pre>
</td></tr>
</table>
The output from the previous line can be obtained with <a href="_oo.html" title="the last output value">oo</a>, even if a semicolon prevented it from being printed.<table class="examples"><tr><td><pre>i11 : oo
o11 = 20</pre>
</td></tr>
</table>
Lines before that can be obtained with <a href="_ooo.html" title="the next to the last output value">ooo</a> and <a href="_oooo.html" title="the third to the last output value">oooo</a>. Alternatively, the symbol labeling an output line can be used to retrieve the value, as in the following example.<table class="examples"><tr><td><pre>i12 : o5 + 1
o12 = 815915283247897734345611269596115894272000000001</pre>
</td></tr>
</table>
To enter a string, use quotation marks.<table class="examples"><tr><td><pre>i13 : "hi there"
o13 = hi there</pre>
</td></tr>
</table>
A value can be assigned to a variable with <a href="__eq.html" title="assignment">=</a>.<table class="examples"><tr><td><pre>i14 : s = "hi there"
o14 = hi there</pre>
</td></tr>
</table>
Strings may be concatenated horizontally with <tt>|</tt>, (see <a href="___Net_sp_vb_sp__Net.html" title="join strings or nets">String | String</a>).<table class="examples"><tr><td><pre>i15 : s | " - " | s
o15 = hi there - hi there</pre>
</td></tr>
</table>
or vertically with <tt>||</tt>, (see <a href="___Net_sp_vb_vb_sp__Net.html" title="join nets or strings vertically">Net || Net</a>).<table class="examples"><tr><td><pre>i16 : s || " - " || s
o16 = hi there
-
hi there</pre>
</td></tr>
</table>
A list of expressions can be formed with braces.<table class="examples"><tr><td><pre>i17 : {1, 2, s}
o17 = {1, 2, hi there}
o17 : List</pre>
</td></tr>
</table>
Lists behave like vectors.<table class="examples"><tr><td><pre>i18 : 10*{1,2,3} + {1,1,1}
o18 = {11, 21, 31}
o18 : List</pre>
</td></tr>
</table>
A function can be created with the arrow operator, <a href="_-_gt.html" title="make a function">-></a> .<table class="examples"><tr><td><pre>i19 : f = i -> i^3
o19 = f
o19 : FunctionClosure</pre>
</td></tr>
</table>
To evaluate a function, place its argument to the right of the function.<table class="examples"><tr><td><pre>i20 : f 5
o20 = 125</pre>
</td></tr>
</table>
Functions of more than one variable take a parenthesized sequence of arguments.<table class="examples"><tr><td><pre>i21 : g = (x,y) -> x * y
o21 = g
o21 : FunctionClosure</pre>
</td></tr>
<tr><td><pre>i22 : g(6,9)
o22 = 54</pre>
</td></tr>
</table>
The function <a href="_apply.html" title="apply a function to each element">apply</a> can be used to apply a function to each element of a list.<table class="examples"><tr><td><pre>i23 : apply({1,2,3,4}, i -> i^2)
o23 = {1, 4, 9, 16}
o23 : List</pre>
</td></tr>
<tr><td><pre>i24 : apply({1,2,3,4}, f)
o24 = {1, 8, 27, 64}
o24 : List</pre>
</td></tr>
</table>
The operator <a href="_...html" title="a binary operator, used for sequences of consecutive items">..</a> may be used to generate sequences of consecutive numbers.<table class="examples"><tr><td><pre>i25 : apply(1 .. 4, f)
o25 = (1, 8, 27, 64)
o25 : Sequence</pre>
</td></tr>
</table>
If the first argument to <tt>apply</tt> is an integer <tt>n</tt> then it stands for the list <tt>{0, 1, ..., n-1}</tt>.<table class="examples"><tr><td><pre>i26 : apply(5, f)
o26 = {0, 1, 8, 27, 64}
o26 : List</pre>
</td></tr>
</table>
The function <a href="_scan.html" title="apply a function to each element">scan</a> is analogous to <a href="_apply.html" title="apply a function to each element">apply</a> except that no value is returned. It may be used to implement loops in programs.<table class="examples"><tr><td><pre>i27 : scan(5, i -> print (i, i^3))
(0, 0)
(1, 1)
(2, 8)
(3, 27)
(4, 64)</pre>
</td></tr>
<tr><td><pre>i28 : j=1; scan(10, i -> j = 2*j); j
o30 = 1024</pre>
</td></tr>
</table>
Most computations with polynomials take place in rings that may be specified in usual mathematical notation.<table class="examples"><tr><td><pre>i31 : R = ZZ/5[x,y,z];</pre>
</td></tr>
</table>
(We reserve single letter symbols such as <tt>Z</tt> for use as variables in rings, hence we must use something like <a href="___Z__Z.html" title="the class of all integers">ZZ</a> to stand for the ring of integers. It may remind you of the "blackboard bold" font of AMSTeX. If you prefer <tt>Z</tt> to <a href="___Z__Z.html" title="the class of all integers">ZZ</a>, you may put <tt>Z=ZZ</tt> in your <a href="_initialization_spfile.html" title="">initialization file</a>. The symbols <tt>ZZ/5</tt> represent the quotient ring <tt>Z/5Z</tt>, and then <tt>ZZ/5[x,y,z]</tt> represents the ring of polynomials in the variables x,y, and z with coefficients in the ring <tt>Z/5Z</tt>.)<table class="examples"><tr><td><pre>i32 : (x+y)^5
5 5
o32 = x + y
o32 : R</pre>
</td></tr>
</table>
Rings and certain other types of things acquire the name of the global variable they are assigned to.<table class="examples"><tr><td><pre>i33 : R
o33 = R
o33 : PolynomialRing</pre>
</td></tr>
</table>
To see the original description of a ring, use <a href="_describe.html" title="real description">describe</a>.<table class="examples"><tr><td><pre>i34 : describe R
ZZ
o34 = --[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1]
5 {GRevLex => {3:1} }
{Position => Up }</pre>
</td></tr>
</table>
A free module can be created as follows.<table class="examples"><tr><td><pre>i35 : F = R^3
3
o35 = R
o35 : R-module, free</pre>
</td></tr>
</table>
The i-th basis element of <tt>F</tt> can be obtained as <tt>F_i</tt>. In this example, the valid values for <tt>i</tt> are 0, 1, and 2.<table class="examples"><tr><td><pre>i36 : F_1
o36 = | 0 |
| 1 |
| 0 |
3
o36 : R</pre>
</td></tr>
</table>
Using a list of indices instead will produce the homomorphism corresponding to the basis vectors indicated.<table class="examples"><tr><td><pre>i37 : F_{1,2}
o37 = | 0 0 |
| 1 0 |
| 0 1 |
3 2
o37 : Matrix R <--- R</pre>
</td></tr>
</table>
Repetitions are allowed.<table class="examples"><tr><td><pre>i38 : F_{2,1,1}
o38 = | 0 0 0 |
| 0 1 1 |
| 1 0 0 |
3 3
o38 : Matrix R <--- R</pre>
</td></tr>
</table>
We can create a homomorphism between free modules with <a href="_matrix.html" title="make a matrix">matrix</a> by providing the list of rows of the matrix, each of which is in turn a list of ring elements.<table class="examples"><tr><td><pre>i39 : f = matrix {{x,y,z}}
--warning: function f redefined
o39 = | x y z |
1 3
o39 : Matrix R <--- R</pre>
</td></tr>
</table>
Use <a href="_image.html" title="image of a map">image</a> to get the image of f.<table class="examples"><tr><td><pre>i40 : image f
o40 = image | x y z |
1
o40 : R-module, submodule of R</pre>
</td></tr>
</table>
We may use <a href="_ideal.html" title="make an ideal">ideal</a> to produce the corresponding ideal.<table class="examples"><tr><td><pre>i41 : ideal (x,y,z)
o41 = ideal (x, y, z)
o41 : Ideal of R</pre>
</td></tr>
</table>
We may use <a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a> to compute the kernel of f.<table class="examples"><tr><td><pre>i42 : kernel f
o42 = image {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
3
o42 : R-module, submodule of R</pre>
</td></tr>
</table>
The answer comes out as a module that is expressed as the image of a homomorphism whose matrix is displayed. Integers inside braces to the left of the matrix give the degrees of the basis elements of the target of the matrix; they are omitted if the degrees are all zero. In case the matrix itself is desired, it can be obtained with <a href="_generators.html" title="provide matrix or list of generators">generators</a>, as follows.<table class="examples"><tr><td><pre>i43 : generators oo
o43 = {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
3 3
o43 : Matrix R <--- R</pre>
</td></tr>
</table>
We may use <a href="_poincare.html" title="assemble degrees into polynomial">poincare</a> to compute the Poincare polynomial.<table class="examples"><tr><td><pre>i44 : poincare kernel f
2 3
o44 = 3T - T
o44 : ZZ[T]</pre>
</td></tr>
</table>
We may use <a href="_rank.html" title="compute the rank">rank</a> to compute the rank.<table class="examples"><tr><td><pre>i45 : rank kernel f
o45 = 2</pre>
</td></tr>
</table>
A presentation for the kernel can be obtained with <a href="_presentation.html" title="presentation of a module or ring">presentation</a>.<table class="examples"><tr><td><pre>i46 : presentation kernel f
o46 = {2} | z |
{2} | x |
{2} | -y |
3 1
o46 : Matrix R <--- R</pre>
</td></tr>
</table>
We can produce the cokernel with <a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel</a>; no computation is performed.<table class="examples"><tr><td><pre>i47 : cokernel f
o47 = cokernel | x y z |
1
o47 : R-module, quotient of R</pre>
</td></tr>
</table>
The direct sum is formed with <a href="___Module_sp_pl_pl_sp__Module.html" title="direct sum of modules">Module ++ Module</a>.<table class="examples"><tr><td><pre>i48 : N = kernel f ++ cokernel f
o48 = subquotient ({1} | -y 0 -z 0 |, {1} | 0 0 0 |)
{1} | x -z 0 0 | {1} | 0 0 0 |
{1} | 0 y x 0 | {1} | 0 0 0 |
{0} | 0 0 0 1 | {0} | x y z |
4
o48 : R-module, subquotient of R</pre>
</td></tr>
</table>
The answer is expressed in terms of the <a href="_subquotient.html" title="make a subquotient module">subquotient</a> function, which produces subquotient modules. Each subquotient module is accompanied by its matrix of generators and its matrix of relations. These matrices can be recovered with <a href="_generators.html" title="provide matrix or list of generators">generators</a> and <a href="_relations.html" title="the defining relations">relations</a>.<table class="examples"><tr><td><pre>i49 : generators N
o49 = {1} | -y 0 -z 0 |
{1} | x -z 0 0 |
{1} | 0 y x 0 |
{0} | 0 0 0 1 |
4 4
o49 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i50 : relations N
o50 = {1} | 0 0 0 |
{1} | 0 0 0 |
{1} | 0 0 0 |
{0} | x y z |
4 3
o50 : Matrix R <--- R</pre>
</td></tr>
</table>
The function <a href="_prune.html" title="prune, e.g., compute a minimal presentation">prune</a> can be used to convert a subquotient module to a quotient module.<table class="examples"><tr><td><pre>i51 : prune N
o51 = cokernel {2} | 0 0 0 z |
{2} | 0 0 0 x |
{2} | 0 0 0 -y |
{0} | z y x 0 |
4
o51 : R-module, quotient of R</pre>
</td></tr>
</table>
We can use <a href="_resolution.html" title="projective resolution">resolution</a> to compute a projective resolution of the cokernel of <tt>f</tt>.<table class="examples"><tr><td><pre>i52 : C = resolution cokernel f
1 3 3 1
o52 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o52 : ChainComplex</pre>
</td></tr>
</table>
To see the differentials we examine 'C.dd'.<table class="examples"><tr><td><pre>i53 : C.dd
1 3
o53 = 0 : R <------------- R : 1
| x y z |
3 3
1 : R <-------------------- R : 2
{1} | -y -z 0 |
{1} | x 0 -z |
{1} | 0 x y |
3 1
2 : R <-------------- R : 3
{2} | z |
{2} | -y |
{2} | x |
1
3 : R <----- 0 : 4
0
o53 : ChainComplexMap</pre>
</td></tr>
</table>
We can verify that <tt>C</tt> is a complex by squaring the differential map.<table class="examples"><tr><td><pre>i54 : C.dd^2 == 0
o54 = true</pre>
</td></tr>
</table>
We can use <a href="_betti.html" title="display degrees">betti</a> to see the degrees of the components of C.<table class="examples"><tr><td><pre>i55 : betti C
0 1 2 3
o55 = total: 1 3 3 1
0: 1 3 3 1
o55 : BettiTally</pre>
</td></tr>
</table>
Let's try a harder example. We can use <a href="_vars.html" title="variables">vars</a> to create a sequence of variables.<table class="examples"><tr><td><pre>i56 : R = ZZ/101[a .. r];</pre>
</td></tr>
</table>
We use <a href="_generic__Matrix.html" title="make a generic matrix of variables">genericMatrix</a> to make a 3 by 6 generic matrix whose entries are drawn from the variables of the ring <tt>R</tt>.<table class="examples"><tr><td><pre>i57 : g = genericMatrix(R,a,3,6)
o57 = | a d g j m p |
| b e h k n q |
| c f i l o r |
3 6
o57 : Matrix R <--- R</pre>
</td></tr>
</table>
Then we construct its cokernel with <a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel</a>.<table class="examples"><tr><td><pre>i58 : M = cokernel g
o58 = cokernel | a d g j m p |
| b e h k n q |
| c f i l o r |
3
o58 : R-module, quotient of R</pre>
</td></tr>
</table>
We may use <a href="_resolution.html" title="projective resolution">resolution</a> to produce a projective resolution of it, and <a href="_time.html" title="time a computation">time</a> to report the time required.<table class="examples"><tr><td><pre>i59 : time C = resolution M
-- used 0.001 seconds
3 6 15 18 6
o59 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o59 : ChainComplex</pre>
</td></tr>
</table>
As before, we may examine the degrees of its components, or display it.<table class="examples"><tr><td><pre>i60 : betti C
0 1 2 3 4
o60 = total: 3 6 15 18 6
0: 3 6 . . .
1: . . . . .
2: . . 15 18 6
o60 : BettiTally</pre>
</td></tr>
</table>
We can make a polynomial ring with 18 <a href="___Indexed__Variable.html" title="the class of all indexed variables">IndexedVariable</a>s.<table class="examples"><tr><td><pre>i61 : S = ZZ/101[t_1 .. t_9, u_1 .. u_9];</pre>
</td></tr>
</table>
We can use <a href="_generic__Matrix.html" title="make a generic matrix of variables">genericMatrix</a> to pack the variables into 3-by-3 matrices.<table class="examples"><tr><td><pre>i62 : m = genericMatrix(S, t_1, 3, 3)
o62 = | t_1 t_4 t_7 |
| t_2 t_5 t_8 |
| t_3 t_6 t_9 |
3 3
o62 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i63 : n = genericMatrix(S, u_1, 3, 3)
o63 = | u_1 u_4 u_7 |
| u_2 u_5 u_8 |
| u_3 u_6 u_9 |
3 3
o63 : Matrix S <--- S</pre>
</td></tr>
</table>
We may look at the matrix product.<table class="examples"><tr><td><pre>i64 : m*n
o64 = | t_1u_1+t_4u_2+t_7u_3 t_1u_4+t_4u_5+t_7u_6 t_1u_7+t_4u_8+t_7u_9 |
| t_2u_1+t_5u_2+t_8u_3 t_2u_4+t_5u_5+t_8u_6 t_2u_7+t_5u_8+t_8u_9 |
| t_3u_1+t_6u_2+t_9u_3 t_3u_4+t_6u_5+t_9u_6 t_3u_7+t_6u_8+t_9u_9 |
3 3
o64 : Matrix S <--- S</pre>
</td></tr>
</table>
Let's produce the equations generated by the equations that assert that m and n commute with each other. (See <a href="_flatten.html" title="flatten a list by unnesting lists">flatten</a>.)<table class="examples"><tr><td><pre>i65 : j = flatten(m*n - n*m)
o65 = | t_4u_2+t_7u_3-t_2u_4-t_3u_7 t_2u_1-t_1u_2+t_5u_2+t_8u_3-t_2u_5-t_3u_8
-----------------------------------------------------------------------
t_3u_1+t_6u_2-t_1u_3+t_9u_3-t_2u_6-t_3u_9
-----------------------------------------------------------------------
-t_4u_1+t_1u_4-t_5u_4+t_4u_5+t_7u_6-t_6u_7 -t_4u_2+t_2u_4+t_8u_6-t_6u_8
-----------------------------------------------------------------------
-t_4u_3+t_3u_4+t_6u_5-t_5u_6+t_9u_6-t_6u_9
-----------------------------------------------------------------------
-t_7u_1-t_8u_4+t_1u_7-t_9u_7+t_4u_8+t_7u_9
-----------------------------------------------------------------------
-t_7u_2-t_8u_5+t_2u_7+t_5u_8-t_9u_8+t_8u_9 -t_7u_3-t_8u_6+t_3u_7+t_6u_8
-----------------------------------------------------------------------
|
1 9
o65 : Matrix S <--- S</pre>
</td></tr>
</table>
Let's compute a Gröbner basis for the image of <tt>j</tt> with <a href="_gb.html" title="compute a Gröbner basis">gb</a>.<table class="examples"><tr><td><pre>i66 : gb j
o66 = GroebnerBasis[status: done; S-pairs encountered up to degree 5]
o66 : GroebnerBasis</pre>
</td></tr>
</table>
The resulting Gröbner basis contains a lot of information. We can get the generators of the basis, and even though we call upon <a href="_gb.html" title="compute a Gröbner basis">gb</a> again, the computation will not be repeated.<table class="examples"><tr><td><pre>i67 : generators gb j;
1 26
o67 : Matrix S <--- S</pre>
</td></tr>
</table>
The semicolon prevents the matrix of generators from appearing on the screen, but the class of the matrix appears -- we see that there are 26 generators.<p/>
We can use <a href="_betti.html" title="display degrees">betti</a> to see the degrees involved in the Gröbner basis.<table class="examples"><tr><td><pre>i68 : betti gb j
0 1
o68 = total: 1 26
0: 1 .
1: . 8
2: . 12
3: . 5
4: . 1
o68 : BettiTally</pre>
</td></tr>
</table>
</div>
</div>
</body>
</html>
